Gapped Boundary Phases of Topological Insulators via Weak Coupling

The standard boundary state of a topological insulator in 3+1 dimensions has gapless charged fermions. We present model systems that reproduce this standard gapless boundary state in one phase, but also have gapped phases with topological order. Our models are weakly coupled and all the dynamics is explicit. We rederive some known boundary states of topological insulators and construct new ones. Consistency with the standard spin/charge relation of condensed matter physics places a nontrivial constraint on models.


Anomalies And Boundary States Of A Topological Insulator
The topological insulator [1]- [10] is a fascinating and experimentally accessible example of a symmetry-protected topological (SPT) [11]- [16] phase of matter. In general, an SPT phase seems almost trivial in bulk at long distances. It is gapped and lacks long range topological order. Yet subtle features of the bulk physics force nontrivial behavior on the boundary of a material that is in an SPT phase. The boundary in general may spontaneously break some of the symmetries, have gapless excitations, or have topological order.
A topological insulator for our purposes is a 3 + 1-dimensional 1 time-reversal invariant system of electrons coupled to background electromagnetism with gauge field A. (We write T for time-reversal.) We will take A to be a classical background field, which does not fluctuate. We model the system by relativistic Dirac fermions with an electron mass that we assume may be position-dependent. Time-reversal symmetry forces the electron mass parameter m to be everywhere real. (The electron mass term is thus mΨΨ with real m, while a T-violating contribution proportional to Ψγ 5 Ψ is absent.) A topological insulator can be modeled as a system in which m is real and positive outside the material and negative inside. Thus the mass parameter changes sign near the surface of the material. As in [18], such a sign change leads to the appearance of massless fermions on the interface between the two regions. These gapless modes propagate along the boundary as a 2 + 1-dimensional massless Dirac fermion, and are characteristic of the standard boundary state of the topological insulator.
One could make a chiral rotation of the electron field inside the material by an angle ±π to reduce to the case that m is positive inside as well as outside the material. The chiral anomaly means that this rotation induces the interaction where X is the worldvolume of the material, F = dA is the field strength of electromagnetism, and A(R) is a certain bilinear function of the Riemann tensor R, defined in eqn. (A.1). (The overall sign of I ind depends upon whether one makes a chiral rotation by an angle +π or −π to reverse the sign of the mass parameter.) The second term in eqn. (1.1) represents a contribution to the theta-angle of the electromagnetic field, which [19,20] is θ = π inside a topological insulator assuming that it vanishes outside. Under T, θ = π is mapped to θ = −π.
The physics of a bulk system is invariant under θ → θ + 2π, so θ = π is equivalent in bulk to θ = −π. The same consideration appears to show that the gravitational term π A(R) is irrelevant, 2 as it naively represents a gravitational theta-term with a coefficient θ grav = 2π, which one might expect to be equivalent to θ grav = 0. However, we will see that this term plays a role when considered together with the spin/charge relation of condensed matter physics. The subtlety of the topological insulator comes from the fact that although the interaction (1.1) is T-invariant on a spacetime manifold without boundary, it is not T-invariant by itself if the integral in (1.1) runs only over the interior X of a topological insulator. For one thing, the proof of T-invariance of a theory with the interaction I ind depends on integrality of 1 2 X F ∧ F/(2π) 2 (which ensures that I ind , which is odd under T, is T-invariant mod 2πZ). This integrality holds if X is a compact spin manifold without boundary, but not if it is the worldvolume of a topological insulator, which in the real world always has a nonempty boundary. On the contrary, if X has boundary W , then π X F ∧ F/(2π) 2 can serve as the definition of the Chern-Simons function of a gauge field A on W : Another explanation of why merely including a term I ind in the effective action violates Tinvariance is that although a global chiral rotation by an angle π everywhere in spacetime is indeed equivalent to a global chiral rotation by an angle −π, a chiral rotation by a spatiallydependent angle that varies from 0 outside the material to +π inside is not equivalent to one that varies from 0 to −π.
In a T-invariant system that has the interaction I ind in its interior, something interesting must happen on the boundary to maintain T-invariance. The standard boundary state of the topological insulator achieves this via the presence of gapless charged fermions on the boundary, whose appearance was briefly explained above. A purely 2 + 1-dimensional system of massless charged electrons propagating on a three-manifold W suffers from what is commonly called a parity anomaly [21,22,23,17]. Classically, this system of fermions has time-reversal symmetry and a global U (1) A symmetry to which A couples. In the quantum theory one has to give up at least one of these symmetries. But if W is the boundary of a four-manifold X and the bulk physics on X includes the interaction I ind , then T symmetry and U (1) A can be simultaneously preserved. This can be understood as an example of anomaly inflow [24], in fact a rather subtle example as the relevant anomaly is a mod 2 effect.
This interplay between the physics in the bulk and on the surface, which is controlled by symmetries and anomalies, makes the system that we have just described robust. Small 2 One may also question more generally whether considerations involving coupling to gravity, and thus to a general spacetime manifold, are meaningful in condensed matter physics, which lacks a microscopic relativistic symmetry. Experience seems to show that such considerations do give valid results, possibly because considerations that can be seen by considering the behavior on a general spacetime manifold could instead be extracted from entanglement properties of many-body quantum wavefunctions. symmetry-preserving deformations cannot change the essential properties: the existence of the interaction I ind in bulk, and the presence of gapless charged fermions on the boundary. What about large symmetry-preserving deformations?
The tight structure based on anomalies makes it challenging to gap the boundary while preserving the symmetries. Clearly, if the massless fermions of the standard gapless boundary state are absent, something else must contribute the same anomaly. The boundary hence cannot be a trivial gapped system. But the boundary can support a 2 + 1 dimensional topological quantum field theory (TQFT) that has the same anomalous realization of time-reversal symmetry T and global U (1) A symmetry as the free fermions of the standard gapless boundary state. Examples have been described in [25]- [31], following work on gapped boundary states of certain bosonic systems [32]. The goal of this paper is to further understand the constraints on these systems, to give a description in which all of the dynamics is completely explicit, and to find additional symmetry-preserving gapped boundary states.
The models we construct are similar to models constructed in [25] and [28] for boundary states of a topological insulator and in [33] and [34] for boundary states of a topological superconductor. In these papers, starting with the standard gapless boundary state of a topological insulator, the first step is to put the boundary in a superconducting state, for example by proximity to a slab of s-wave superconductor. The boundary then supports quasiparticlesvortices -that carry ordinary magnetic flux and obey exotic statistics. The key second step is to assume a process of "vortex condensation" after which the electromagnetic gauge symmetry is restored and the vorticity is only conserved modulo some integer n. But in the real world, the ordinary magnetic flux through a two-dimensional surface (such as the surface of a topological insulator) is an exactly conserved quantity, which cannot be reduced by any boundary dynamics to a mod n conservation law. Related to this, the dynamics postulated in these papers is not very explicit. To avoid such issues, we reinterpret their construction so that the flux carried by the vortices is the flux of an emergent gauge field. With suitable hypotheses on symmetry breaking, this flux can perfectly well be conserved mod n for some n. All of the required dynamics is explicit and elementary, rather as in the "composite fermion" derivation of the Moore-Read state [35,36]. Similarly, [25] exhibits a second and simpler gapped boundary state obtained from the first using the process of "anyon condensation." Again, the assumed dynamics must be strongly coupled. Instead, we construct a weakly coupled system leading to the same topological quantum field theory. Analogous remarks apply concerning the relation of our treatment to the treatment of a topological superconductor in [33] and [34].

Coupling To Background Fields
In condensed matter physics, one commonly discusses three distinct descriptions of a given system (see fig. 1). First, we have a microscopic Hamiltonian describing the electrons of the material. This system is quite complicated and typically cannot be analyzed explicitly. Second, we consider a simplified model, which captures some collective excitations of the microscopic system. There is no claim that this model gives a complete and accurate description of the system. The idea is only that it is a simpler model that captures the essential properties of the original microscopic theory; they are in the same universality class. We will refer to this model as a phenomenological model. Finally, we have the long-distance, macroscopic description of the physics. description. Instead, we replace it with a simplified model that captures some degrees of freedom. This model is not supposed to be exact. This model is weakly coupled and allows us to find its long distance behavior easily. The hope is that it is in the same universality class as the original microscopic theory.
Since it is often nontrivial to derive one of these descriptions from the others, it is useful to constrain them by finding appropriate consistency conditions. We have already mentioned that we impose T and U (1) A symmetry. Specifically, we impose that the 2 + 1-dimensional boundary theory combined with the bulk term I ind respects these symmetries.
A refinement of these constraints follow from what in particle physics is known as 't Hooft anomaly matching [37]. Instead of considering these three systems (the microscopic system, the simplified mode, and the long distance macroscopic description) as they are, we couple them to a nontrivial electromagnetic and gravitational background. In the real original system of interest, the electromagnetic vector potential A is topologically trivial and the spacetime metric is flat. But the microscopic Schrodinger equation is consistent in more generality.
For example, even though in nature, magnetic monopoles are not available to us, we can entertain introducing a magnetic monopole into our material. (In the current context this was done in [19,38,39].) This is consistent at the level of the microscopic Schrodinger equation so it must also be consistent in the other two descriptions. Similarly, we can place the system on a curved spatial manifold, or, if we are optimistic (see footnote 2), on a fairly general curved spacetime. Some conditions involving spin are needed, as discussed momentarily. Time-reversal symmetry means that we can define the system even on a spacetime that is not necessarily orientable. When we do this, some properties must match between the microscopic and macroscopic description. As with the magnetic monopole, the real system of interest exists in flat spacetime, but placing it in a hypothetical curved spacetime turns out to be a powerful tool to learn about the flat spacetime theory. Finally, any system made from electrons only (coupled to a background of nuclei that is inert except for lattice vibrations, and possibly supplemented with photons) obeys what we will call the spin/charge relation of condensed matter physics. This asserts that -in a system of finite volume, which is ultimately constructed from a finite number of electrons and nuclei -states of odd electric charge have half-integral spin and states of even electric charge have integral spin. 3 If the spin/charge relation is satisfied microscopically, then it must be satisfied in any phenomenological description.
A useful way to incorporate the spin/charge relation, described in detail in section 2.3, is as follows. Any system of fermions can be defined on a spin manifold. But a system that obeys the spin/charge relation can be formulated on a more general class of manifolds known as spin c manifolds. (For recent application of spin c manifolds in a condensed matter context, see [40].) If we start with a microscopic system that obeys the spin/charge relation, then any more phenomenological description of the same system should make sense on a spin c manifold. This gives constraints on phenomenological models that are interesting and useful, even if we are really not interested in the physics on spin c manifolds.

Outline
The outline of this paper is as follows.
In section 2, we review the standard gapless boundary state of a topological insulator. We describe the action of free 2 + 1-dimensional fermions and its symmetries. We give a first description of the anomaly and we explain the spin/charge constraint and its relation to spin c .
In section 3, we present a simple one-parameter family of boundary models for the topological insulator. In one phase, these models describe the standard gapless boundary state of the topological insulator. In another phase, a gap develops and a nontrivial boundary TQFT is found. The models are weakly coupled and the analysis is explicit and elementary. The study of these models uncovers a new anomaly associated with the spin/charge relation. We describe the collective excitations of these models and show that they exhibit non-Abelian statistics. (This should not be a surprise given previous literature [25]- [31].) We also compute the partition functions of these models and clarify a number of subtle issues. Section 4 is devoted to the long distance macroscopic description of the system. We identify the low energy TQFT and explore its properties. This motivates us to find additional models generalizing those in section 3. We recognize many of these TQFTs as those found in [25]. More detail on these models, and an "abelian" model that almost works as a boundary state of a topological insulator, can be found in section 5.
Our main focus in this paper is on boundary states of a topological insulator, but the constructions are also applicable to topological superconductors, as we discuss in section 6.
A number of appendices present additional background information. In Appendix A, we summarize the proper normalization of some of the topological terms we use in three and four dimensions. In Appendix B, we review the Callias index theorem, which is useful in the analysis of vortices in these models. In Appendix C, we summarize some useful information about Chern-Simons gauge theories, clarify some possibly confusing points in the literature, and present an explicit gauge theory Lagrangian for the TQFT of the Moore-Read state [35] and its generalizations.

Action And Symmetries
Let X be the 3 + 1-dimensional worldvolume of a topological insulator and W its 2 + 1dimensional boundary. In general, we assume that X and W are possibly curved Riemannian manifolds with a spin structure. When gauge fields are present, we write / D 0 for the Dirac operator for a fermion coupled to the geometry only (so on a flat manifold, / D 0 is simply the flat space Dirac operator / ∂) and / D for the full Dirac operator including the gauge fields. A topological insulator is gapped in bulk. Its standard boundary state has a massless Dirac fermion Ψ that propagates on W . Its action is where A is the U (1) gauge field of electromagnetism. To describe in a simple way the action of the discrete symmetries T and C (time-reversal and charge conjugation), it is convenient to use, in Lorentz signature, a basis of 2 × 2 real gamma matrices γ µ , µ = 0, 1, 2, that obey {γ µ , γ ν } = 2η µν , η µν = diag(−1, 1, 1).

(2.2)
With real gamma matrices, a Majorana fermion λ is just a real 2-component fermion field, which we think of as a column vector. Then λ is defined simply to be λ tr γ 0 , where tr denotes transpose. A Dirac fermion is a complex 2-component fermion field Ψ; for a Dirac fermion Ψ, with hermitian adjoint Ψ † (both of which are viewed as column vectors), one defines Ψ = (Ψ † ) tr γ 0 . T is conventionally defined so that electric charge is even under T and magnetic vorticity is odd. Its action on A is then T(A 0 (t, x)) = A 0 (−t, x) T(A i (t, x)) = −A i (−t, x), i = 1, 2. (2. 3) It is clumsy to always write such a pair of equations, and we will usually abbreviate such a pair by writing T(A(t, x)) = −A(−t, x), (2.4) where we write the transformation of the spatial components and it is understood that the time component transforms oppositely. 4 C acts by C(A(t, x)) = −A(t, x). Often we write even more briefly T(A) = −A or CT(A) = A.
To describe the action of the discrete symmetries on Ψ, it is most convenient to expand Ψ in terms of a pair of Majorana fermions λ 1 , λ 2 : Ψ = (λ 1 + iλ 2 )/ √ 2. T acts by T(λ 1 (t, x)) = γ 0 λ 1 (−t, x), T(λ 2 (t, x)) = −γ 0 λ 2 (−t, x). (2.8) Allowing for the fact that T is antiunitary so that i is odd under T, this definition implies that T(Ψ(t, x)) = γ 0 Ψ(−t, x), (2.9) which somewhat deceptively makes it appear that Ψ transforms linearly under T. C acts by C(λ 1 ) = λ 1 , C(λ 2 ) = −λ 2 , C(Ψ) = Ψ † . (2.10) The action of CT is accordingly CT(λ i (t, x)) = γ 0 λ i (−t, x), i = 1, 2, CT(Ψ(t, x)) = γ 0 Ψ † (−t, x). (2.11) In condensed matter physics, T can be a good microscopic symmetry, while C and CT (which map electrons to positrons) are not. 5 Nevertheless, in this paper, we will keep track of all of these symmetries. One reason for this is that all boundary states we construct do have C and CT as well as T symmetry. Also, many of the results have applications to topological superconductors, with CT playing the role of T, and accordingly CT will be important in section 6.

Anomalies
In constructing an alternative boundary state for the topological insulator, we want to match certain anomalies of the standard boundary state that we have just described.
The most basic and familiar anomaly is that the standard boundary state has anomalous T symmetry. This means that if the action I Ψ of eqn. (2.1) is understood as the action of an abstract 2 + 1-dimensional system, then it cannot be quantized in a way that maintains T symmetry (and also the U (1) A symmetry corresponding to conservation of electric charge). But it can be quantized in a T-invariant fashion if the three-manifold W is the boundary of a four-manifold X and the electromagnetic theta-angle θ differs by π between the vacuum and X. Usually it is assumed that θ = 0 in vacuum and then anomalous T symmetry means that θ = π inside the bulk of a topological insulator.
The T anomaly just described is commonly called a "parity" anomaly, but this terminology is a little misleading because in 2 + 1 dimensions, parity, which acts on all spatial coordinates by x i → −x i , i = 1, 2, is an element of the connected part of the rotation group. We prefer 5 In particular, there is no particular constraint on the fermi energy for the massless boundary fermions of a 3 + 1-dimensional topological insulator except that it is within the bulk energy gap. So in relativistic terms, the boundary fermions are similar to massless Dirac fermions with a nonzero chemical potential. This chemical potential violates C and CT. Of course, C and CT are also generically violated by all sorts of higher order terms in the Hamiltonian of the massless fermions.
to call this an anomaly in T and R symmetry, where R is a spatial reflection, for example The other anomalies to consider involve coupling to gravity, that is, they involve the question of what happens if W and X are curved. It is not completely obvious that such anomalies are relevant in condensed matter physics, where it is natural to consider a general curved spatial manifold, but not a general curved spacetime. However, experience -notably [41,17] with the 3 + 1-dimensional topological superconductor -appears to show that such anomalies are relevant. Probably this means that the results that can be deduced from anomalies involving coupling to gravity can alternatively be deduced by more subtle considerations of locality. This has not yet been demonstrated.
Here is one example of an anomaly involving gravity. Consider any 2 + 1-dimensional T-invariant state that is continuously connected to a system of free bose and fermi fields. 6 Assume that T 2 = (−1) F , where (−1) F is the operator that equals −1 for fermions and +1 for bosons. Then the action of T on fermions can be diagonalized so that each Majorana fermion λ transforms as T(λ(t, x)) = ±γ 0 λ(−t, x), for some choice of sign. Let n + be the number of gapless Majorana fermions transforming under T with a + sign and n − the number transforming with a − sign. A bare mass term iλλ is T-invariant if and only if λ and λ transform under T with opposite signs, so only the difference ν T = n + − n − is an invariant when T-conserving bare masses are turned on and off. When interactions are taken into account [42,33,34,43], ν T is no longer a Z-valued invariant, but is an invariant mod 16. In fact, T symmetry can be used to place any given T-invariant theory on an unorientable manifold. When this is done, one obtains a theory whose partition function is in general not well-defined; it has an anomaly that depends on the value of ν T mod 16. This was illustrated in an example in [41] and explained more systematically in [17]. For the standard boundary state of the topological insulator, n + = n − = 1 so ν T = 0.
Similarly, in a CT-invariant model, assuming that (CT) 2 = (−1) F , let n + be the number of gapless Majorana fermions transforming under CT as CT(λ(t, x)) = +γ 0 λ(t, x) and n − the number transforming as CT(λ(t, x)) = −γ 0 λ(t, x). Only the difference ν CT = n + −n − is invariant under turning CT-conserving bare masses on and off. When interactions are considered, ν CT is an invariant mod 16. This can again be understood in terms of the anomaly that arises when CT symmetry is used to place a theory on an unorientable 3-manifold. For the standard boundary state of a topological insulator, ν CT = 2.
In condensed matter physics, as long as electric charge is conserved, ν T and ν CT are highly constrained by the spin/charge relation. In Appendix A.4, we show that in a system that obeys the spin/charge relation, ν CT is even and ν T does not give any information beyond what one can learn on an orientable manifold. (In the absence of the spin/charge relation, the only general restriction is that ν T and ν CT are congruent mod 2.) In condensed matter physics, CT is not really a symmetry so different boundary states of a topological insulator can in general have different values of ν CT . Hence in constructing boundary states of the topological insulator, we will allow states with different values of ν CT . The distinction between states with different values of ν CT , though not relevant for a topological insulator, is relevant for topological superconductors, as we discuss in section 6.
The reader may have noticed that we have not discussed what might appear to be the most obvious anomaly involving coupling of a 2 + 1-dimensional system to gravity: the gravitational contribution to the anomaly in T symmetry, or alternatively the jump in the gravitational theta-angle in crossing from vacuum to the bulk of a topological insulator. What should be meant by the gravitational theta-angle depends on whether the spin/charge relation is assumed. For a theory that obeys the spin/charge relation, the appropriate gravitational analog of θ = π is θ 2 = π, in the notation of Appendix A.4. The standard topological insulator has θ 2 = 0, but in Appendix A.4 and also in section 4, we describe symmetry-preserving boundary states (gapless or gapped) for a hypothetical material of θ 2 = π.

The Spin/Charge Relation And Spin c Manifolds
The gravitational anomalies that were mentioned in section 2.2 are conveniently studied by formulating a theory on a possibly curved four-manifold X, thus placing the boundary state on a possibly curved three-manifold W . But what class of manifolds should be considered?
Since the microscopic systems of interest involve fermions, one might want to restrict X to be a spin manifold. But in a certain sense this is too restrictive. In condensed matter physics, as long as we consider systems in which electric charge is conserved, there is a fundamental spin/charge relation that was already mentioned in section 1.2. A system that satisfies the spin/charge relation can be formulated not just on a spin manifold but on a more general type of manifold known as a spin c manifold. On a possibly curved spin manifold, one has a covariant derivative D (0) µ that acts on neutral spin 1/2 fermions coupled to gravity only. In the presence of electromagnetism, one additionally has a U (1) gauge field A and, for any integer n, a covariant derivative D µ is not well-defined by itself and A µ is not well-defined globally as an Abelian gauge field. Rather, A µ is what we will call a spin c connection. Though the notion of a spin c manifold can be defined purely topologically, for our purposes a spin c manifold always has a spin c connection A µ .
Concretely, A µ is not well-defined globally as an Abelian gauge field because the field strength F = dA does not obey standard Dirac quantization of flux. The standard quantization would say that if C ⊂ X is an oriented two-dimensional cycle, then C F/2π should be an integer. The spin c condition says instead that Here w 2 is the second Stieffel-Whitney class of X. All that matters for our purposes is that in general C w 2 is equal to 0 or 1 mod 2. Eqn. (2.12) shows that in general F does not obey standard Dirac quantization, but 2F does, since if we multiply by 2, the right hand side of (2.12) becomes integer-valued. Accordingly, although a spin c connection A µ is not well-defined as an Abelian gauge field, 2A µ is an ordinary Abelian gauge field. If we start with a microscopic system that can be placed on a general spin c manifold, then any effective field theory that describes its low energy excitations and likewise any topological field theory that describes a possible gapped phase must make sense on a general spin c manifold. This will ensure that the states of the effective field theory or topological field theory (in any sample of finite volume) will satisfy the spin/charge relation.
In sections 3.2 and 3.6, we will see that even if a classical theory of fermions satisfies the spin/charge relation, there may be a quantum anomaly in this relation; equivalently, even if a classical theory can be formulated on a spin c manifold, it may be that quantum mechanically a spin rather than spin c structure is required. Models in which this happens cannot arise as effective field theories describing boundary states of a topological insulator. This will give a nontrivial constraint on our constructions.
The ability to formulate a theory on a spin c manifold likewise gives a nontrivial constraint on the topological field theories that can be used to describe gapped boundary states at low energies. We will describe this in the context of Chern-Simons gauge theories with Abelian gauge group. Here we will discuss purely 2 + 1-dimensional couplings (as opposed to couplings that are possible on the boundary of a four-manifold). For example, A could be the electromagnetic gauge field interacting with the 2 + 1-dimensional world-volume W of a quantum Hall system. The Chern-Simons coupling CS(A) is supposed to depend mod 2πZ only on the restriction of A to W (and possibly on the spin structure of W ).
We need a few preliminary facts, which we just state here, with explanations in Appendix A.3. It is relatively well-known that if A is an Abelian gauge field, then on a compact threedimensional spin manifold W , the Chern-Simons functional is well-defined mod 2πZ, ensuring that it makes sense as a contribution to a quantum effective action of a purely 2 + 1-dimensional system. (In the absence of a chosen spin structure on W , CS(A) is only well-defined mod πZ and must appear in the effective action with an even integer coefficient.) What if A is a spin c connection rather than an ordinary U (1) gauge field? Then a modified version of CS(A) with a certain gravitational correction is well-defined mod 2πZ: 14) The ellipses refer to a purely gravitational contribution (Ω(g) in eqn. (A.21); here g is the metric tensor of W ). Now if A is a spin c connection, and a is an ordinary U (1) gauge field, then A + a is a spin c connection. So another expression that is well-defined mod 2πZ is CS(A + a, g) = 1 4π W (ada + 2adA + AdA) + . . . , (2.15) with the same gravitational correction. Subtracting the last two formulas, the purely gravitational term cancels out and we learn that is well-defined mod 2πZ. On a spin manifold, with a and A being ordinary U (1) gauge fields, the two terms on the right hand side of eqn. (2.16) would be separately well-defined mod 2πZ. In the more general case of a spin c manifold, with spin c connection A, the two terms are separately well-defined only mod πZ but their sum is well-defined mod 2πZ. Given these facts, let us consider a 3d theory with emergent U (1) gauge fields a 1 , a 2 , . . . , a n , as well as the electromagnetic potential A. Here a i are dynamical 3d fields and A is a classical background field. (Later, A will actually be the restriction of a 4d field to 3d.) We consider a theory with Chern-Simons action If W is a spin manifold, then the condition for this action to be well-defined mod 2πZ is simply that the coefficients k ij and q i should all be integers. (If W is a purely bosonic manifold with no spin structure, then in addition the diagonal elements k ii must be even.) However, suppose that I CS is supposed to be part of the microscopic description of a theory that satisfies the usual spin/charge relation. In this case, I CS must be well-defined mod 2πZ on any spin c manifold W with spin c connection A. In view of our discussion of eqn. (2.16), the condition for this (beyond integrality of the k ij and q i ) is As a check, note that the condition (2.18) is invariant under field redefinitions a i → a i +N i A provided the coefficients N i are even, but not otherwise. This reflects the fact that 2A is an ordinary U (1) gauge field but A is not, so if a is a U (1) gauge field, then a redefinition a → a+2A makes sense but a → a + A does not. More generally, let us arrange a i as a column vector  , and consider an allowed 7 change of variables with M ∈ GL(n, Z) and N a column vector with even integer entries. The reader can verify that this transforms k ij and q i to another set of integers, still satisfying (2.18), and also adds to the action an integer multiple of 8CS(A). As explained above, on a spin c manifold, the basic Chern-Simons invariant CS(A), normalized as in eqn. (2.13), requires a gravitational correction to make it well-defined mod 2πZ. Is some multiple of CS(A) well-defined mod 2πZ without a gravitational correction? The answer is that the basic multiple of CS(A) with this property is That a factor of 8 is necessary here is explained in Appendix A.3. That a factor of 8 is sufficient may be seen as follows. On any three-manifold W (not necessarily spin), for any A check is that the change of variable (2.19) shifts the action by a multiple of 8CS(A) and maintains its consistency.
The restriction (2.18) has been previously discussed [44] more along the following lines. Consider the Wilson line operators exp in i k ij a j , (2.21) with integers n i . These particular line operators are "transparent"; they have trivial braidings and describe the worldlines of excitations that can move to the bulk of the system. Using standard formulas for Abelian Chern-Simons theory, these operators have spins n i k ij n j /2. Their coupling to A shows that their charges are q i n i . Requiring that these line operators should satisfy the usual spin/charge relation gives eqn. (2.18).

Some Models
In this section, we will describe a simple class of boundary states for the topological insulator that are gapped but nonetheless have the same symmetries and anomalies as the standard boundary state that we reviewed in section 2.1. Boundary states with this property inevitably have nontrivial topological order, and in our examples this will arise in a simple way.
In constructing models, we want to maintain the spin/charge relation of condensed matter physics: states of odd charge have half-integral spin and states of even charge have integral spin. To do so, we simply will take all fermi fields to have odd electric charge and all bose fields to have even electric charge.
In our construction, we assume that on the boundary W of the topological insulator, there propagates an emergent U (1) gauge field a as well as the U (1) gauge field A of electromagnetism. We call the combined gauge group U (1) A × U (1) a . T-invariance implies that the action has no Chern-Simons terms, so for a we assume a Maxwell-like action proportional to f µν f µν , where f µν = ∂ µ a ν − ∂ ν a µ . We normalize a µ so that f µν has the standard Dirac flux quantum 8 of 2π.
We also add a boundary Dirac fermion χ that couples to U (1) A × U (1) a with charges (1, 2s), for some integer s. Assigning to χ odd charge under U (1) A is motivated by the spin/charge relation, as just explained. In addition, the fact that χ has odd charge under A and even charge under a means that χ generates the same T-anomaly as the fermion field Ψ of the standard boundary state of eqn. (2.1). (This T-anomaly is a mod 2 effect, as it involves the question of whether the effective theta-angle is an odd or even multiple of π. So it only depends on the values of the U (1) A × U (1) a charges mod 2; the χ field of charges (1, 2s) therefore produces the same anomaly as the Ψ field of charges (1, 0). See section 3.6 for a fuller explanation.) We assume that a and A transform in the same way under T and CT, as described in eqns. (2.3) and (2.6). We likewise assume that χ transforms like Ψ under T and CT (eqns. (2.9), (2.11)). These assumptions ensure that the kinetic energy is Tand CT-invariant. They also ensure that the model has ν CT = 2, just like the standard boundary state of a topological insulator. Let us write q and k for the ordinary electric charge and the conserved charge of the emergent gauge field. T and CT are conventionally defined so that q is even under T and odd under CT. Since we take a to transform like A under the discrete symmetries, the same is true of k: In addition to a and χ, we add two complex scalar fields w and φ of charges (0, 1) and (2, 4s) under U (1) A × U (1) a . The purpose of adding these two fields is to make it possible to exhibit two different phases. In one phase, w has an expectation value, while φ has zero expectation value and a positive mass squared. In this phase, U (1) A × U (1) a is spontaneously broken to U (1) A . Expanding around this vacuum, the only gapless field is the Dirac fermion χ, which has charge 1 under the unbroken symmetry U (1) A . At low energies, this phase is indistinguishable from the standard boundary state of the topological insulator.
We also want a phase in which the χ field undergoes s-wave pairing, with ab χ a χ b = 0 (here a, b = 1, 2 is a spinor index). We could introduce some sort of attractive interactions that would cause such pairing to occur dynamically, but instead to get a simple explicit model we introduce the scalar field φ, whose vacuum expectation value describes pairing.
To make possible a Yukawa coupling of φ to χ, we take φ to transform under the discrete symmetries as By contrast, for w we take 9 If we expand w in terms of real scalar fields as w = (w 1 + iw 2 )/ √ 2, then the above formulas imply These formulas have obvious analogs for φ. The field χw 2s has the quantum numbers of a bulk electron, so we can include in the action or the Hamiltonian a coupling of the bulk electron field e (restricted to the boundary) to χw 2s : The theory admits a gauge-invariant, Lorentz-invariant, and discrete symmetry preserving Yukawa coupling of φ to χ, (Here a, b = 1, 2 are spinor indices, ab is the corresponding antisymmetric tensor, and h is a real coupling constant.) Bearing in mind that for Majorana fermions λ and λ, i ab λ a λ b is hermitian, it is straightforward to verify that I Yuk is hermitian. To verify T-invariance, which requires the minus sign in eqn. (3.3), one uses the fact that i is odd under T but (because the matrix γ 0 in the transformation law T(χ(t, x)) = γ 0 χ(−t, x) has determinant +1) ab χ a χ b is even. Finally, C invariance is clear. The phase with φ = 0 preserves the discrete symmetries if we accompany them by suitable gauge transformations. If we take φ to be real (and, say, positive), then C needs no accompanying gauge transformation, but T and CT must be accompanied by a U (1) a gauge transformation, which we can take to have gauge parameter exp(2πi/8s). In the phase with φ = 0, U (1) a is broken to a subgroup Z 4s , generated by a discrete gauge symmetry K = exp(2πik/4s); we also set K 1/2 = exp(2πik/8s). The unbroken time-reversal symmetry of the system is T = TK 1/2 . Eqn. (3.2) implies that TK 1/2 = K −1/2 T, so the usual relation T 2 = (−1) F for free electrons is unchanged: Likewise the effective CT transformation is But since CT commutes with K 1/2 , the usual (CT) 2 = (−1) F becomes Individual quasiparticles can have K = 1 -for example, w corresponds to a quasiparticle with K = exp(2πi/4s). But as K is a gauge transformation in the emergent gauge group, it leaves invariant any globally-defined state of a sample of finite size. So for any state of a finite sample, all quasiparticles together always combine to a state of K = 1, and therefore to a state on which T = T, CT = CT, and all the standard relations are obeyed. The unbroken electric charge in the phase with φ = 0 is a linear combination of q and k under which φ is neutral. This combination is In the phase with φ = 0, all bosons and fermions have bare masses at tree level. So potentially, this is a gapped, symmetry-preserving boundary state for the topological insulator. However, a restriction on the value of s is needed, as we will see in section 3.2.
The model that we have described is analogous to the composite fermion model [36] of the Moore-Read state [35] of a fractional quantum Hall system. In that context, the composite fermion is coupled to U (1) A × U (1) a where a is an emergent Abelian gauge field. And pairing of the composite fermion is a key ingredient. There are a few differences. The Moore-Read state has no T symmetry, so the gauge fields used in constructing it can and do have Chern-Simons couplings. Also, the composite fermion of the Moore-Read state is spin-polarized (it has only one spin state of spin +1/2), and its pairing is p-wave pairing. Finally, in the context of the Moore-Read state, there is no analog of Higgsing to a standard state, and hence there is no analog of w.
The models that we have constructed turn out to be related to those of [25]. The main difference is our use of the emergent gauge field a. As we have already mentioned in the introduction, this allows us to analyze the models in a reliable way using standard weak coupling techniques. This is what we will do below.

Monopole Operators And The Anomaly
In a theory constructed from electrons only, all local gauge-invariant operators must obey the usual spin/charge relation. Here "gauge-invariant" operators are those that are invariant under any emergent gauge symmetries, in our case U (1) a . The spin/charge relation says that operators of half-integral spin carry odd electric charge, and those of integer spin carry even electric charge.
We have constructed our models so that the obvious U (1) a -invariant local operators -the ones constructed as polynomials in the fields and their derivatives -obey the spin/charge relation. But we must also consider monopole operators.
A monopole operator (also known as an 't Hooft operator) is defined by postulating a Dirac monopole singularity at a specified point p ∈ R 3 and performing a path integral in the presence of this singularity. Very near p, all fermi and bose fields that we have introduced can be treated as free fields. Moreover, the terms in the action that violate conformal invariance are all irrelevant at short distances. As in [45], which we follow in the analysis below, the asymptotic conformal invariance near p enables one to determine the quantum numbers of monopole operators in radial quantization. This amounts to making a conformal mapping from R 3 \p (that is, R 3 with the point p omitted) to R × S 2 . We view the R direction as (Euclidean) "time." To construct the basic monopole operators with minimum charge, we place on S 2 one unit of flux of a. The quantum states obtained by quantization in this situation correspond to the monopole operators.
In general, Chern-Simons couplings of gauge fields can generate nontrivial quantum numbers for monopole operators. However, because of T symmetry, our models have no such couplings. That being so, we get nontrivial quantum numbers only from quantization of the fermion zeromodes. The relevant zero-modes are time-independent modes that are zero-modes of the Dirac operator on S 2 .
The χ field, because it has U (1) a charge 2s, has 2s zero-modes χ i , i = 1, . . . , 2s. These modes all have the same 2d chirality and they transform with spin J = (2s − 1)/2 under rotations of S 2 . The complex conjugates of the χ i are zero-modes χ * i of the adjoint field χ † . These zero-modes can be normalized to give, after quantization, a system of standard creation and annihilation operators, Because of the existence of these zero-modes, the lowest energy level in the monopole sector is degenerate. It contains a state | ↓ that is annihilated by the χ i , Acting repeatedly with the χ † i gives a state that they annihilate: Each time we act with one of the χ † i , we shift the U (1) A × U (1) a charges of a state by (−1, −2s). Since we do this 2s times to map | ↓ to | ↑ , the charges of | ↑ exceed those of | ↓ by 2s(−1, −2s). On the other hand CT exchanges | ↓ with | ↑ and ensures that the charges of those two states are equal and opposite. Hence | ↓ has charges (s, 2s 2 ) and | ↑ has charges (−s, −2s 2 ).
Let us construct monopole operators that are U (1) A × U (1) a -invariant. To construct such an operator from the Hilbert space that we have just described, one has to act on the state | ↓ with precisely s creation operators. The resulting states are of the form For any integer s, the spin (2s−1)/2 of the operators χ † i is half-integral. The gauge-invariant states in eqn. (3.17) are obtained by acting on the spinless state | ↓ with s of these "creation operators," so they have half-integer or integer spin depending on whether s is odd or even. If O is the monopole operator corresponding to a linear combination of the states (3.17), then O is U (1) A × U (1) a -invariant and its spin is s/2 mod Z.
Thus, such an O satisfies the spin/charge relation if and only if s is even. Henceforth, therefore, in investigating this class of models, we restrict to even s.
All monopole operators of unit charge appear in the operator product expansion of the O constructed above with an ordinary local operator (a gauge-invariant polynomial in the fields and their derivatives). So for even s, all charge 1 monopole operators satisfy the spin/charge relation. Taking products of these operators, we learn that for even s, this relation is satisfied by monopole operators of any charge.
For odd s, with O(p) a monopole operator of charge 1, if |Ψ is any state that satisfies the spin/charge relation, then O(p)|Ψ is a state that violates it. Since the operator O(p) is a source of one flux quantum of f = da, the states that violate the spin/charge relation are simply those that have an odd number of such flux quanta.

Breaking A Symmetry
In condensed matter physics, the only exact symmetries are the ones that can be defined microscopically (such as electric charge conservation and time-reversal symmetry). There might be additional emergent symmetries in the infrared, but in a more precise description, they should all be explicitly broken, possibly by operators that are irrelevant in the renormalization group sense. For example, the Lorentz invariance and the C and CT symmetry of the models introduced in section 3.1 are all not natural in condensed matter physics and should be explicitly broken, possibly by irrelevant interactions. There is one more symmetry of these models as presented so far that does not correspond to any microscopic symmetry in condensed matter physics and so must be broken explicitly in a more precise treatment. This is the symmetry generated by the conserved current We have normalized j µ so that -with a µ understood to obey standard Dirac quantization of flux -the corresponding charge u = d 2 x j 0 = d 2 x f 12 /2π has integer eigenvalues. u is simply the number of flux quanta. The operators that violate conservation of u are simply the monopole operators that we analyzed in section 3.2. We can eliminate an unphysical global conservation law from the model by adding to the action a charge 1 monopole operator with a small coefficient ε. For even s, a suitable linear combination of the states (3.17) carries zero spin. So if O is the corresponding monopole operator, with hermitian conjugate O † of opposite monopole charge, we can eliminate the conservation of u, preserving all other symmetries, by adding ε(O + O † ) to the action, or equivalently to the Hamiltonian.
In section 3.2, we have constructed the monopole operators at short distances without worrying about what sort of states they act on. This of course depends on the physics at long distances. In the phase that corresponds to the standard boundary state of a topological insulator, with w = 0, U (1) a is completely broken. This leads to the existence of vortices with any integer value of the vorticity u. The monopole operators create and annihilate these vortices, so adding a u = 1 monopole operator to the action means vortices can annihilate to quasiparticles that do not carry vorticity.
In the gapped phase with φ = 0, matters are more interesting. A discrete subgroup of U (1) a is unbroken, so there are stable quasiparticles with fractional u. They are discussed in section 3.4. However, in a sample of finite size, the total u of all quasiparticles is always an integer, and in the presence of a perturbation ε(O +O † ), there is no conservation law associated to this integer.
The phase with φ = 0 is gapped at ε = 0, so adding ε(O + O † ) to the action with small ε does not produce any nontrivial dynamics. The only purpose of this step is to explicitly break a symmetry, reducing the number of quasiparticle types. In treatments of similar models that have appeared in the literature, the analogous step has involved "vortex proliferation" or "condensation of vortices," usually understood to involve some not completely explicit dynamics.

The Vortex
The gapped phase with φ = 0 has quasiparticles with fractional vorticity. In fact, since φ has charge k φ = 4s under U (1) a , a vortex in the φ field, around which the phase of φ changes by 2π, has flux R 2 f = 2π/k φ = 2π/4s and thus has u = 1/4s.
We prefer to define the integer-valued charge v = 4su. The purpose of this is to describe the vorticity in a language analogous to what we use for the unbroken Z 4s subgroup of U (1) a . This subgroup is generated by K = exp(2πik/4s), where k is an integer-valued invariant that is conserved mod 4s. So we likewise define an integer-valued vorticity v that (once monopole operators are included, as in section 3.3) is conserved mod 4s.
Vorticity transforms under the discrete symmetries oppositely to the electric charge. Thus the analog of eqn. (3.2) for the vorticity is (3.20) We have written these formulas in terms of the discrete symmetries T and CT that are unbroken in the phase under study.
For the basic case of v = 1 (or −1), there is a well-known classical solution that describes a vortex at rest. This solution is rotation-invariant up to a gauge transformation, and (after suitable gauge-fixing) it is unique up to a spatial translation. In particular, this standard vortex solution is CT -invariant. We will see that this fact is very useful.
In a purely bosonic theory with U (1) a broken to Z 4s , and no other pertinent degrees of freedom, quasiparticles would be labeled by the pair (k, v) and would satisfy Abelian statistics. For example, in the simplest version of Z 4s gauge theory, a quasiparticle of charges (k, v), when winding around another quasiparticle of charges (k , v ), would acquire a phase exp(2πi(kv − k v)/4s).
The main difference between the model under study here and a simple Z 4s gauge theory results from the fact that in a vortex field with v = 1, the χ field has a single zero-mode. The phase of this zero-mode can be chosen to make it CT -invariant, and this fixes the phase of the zero-mode up to sign. But actually, even without CT symmetry, hermiticity of the Hamiltonian and fermi statistics imply that the space of fermion zero-modes always has a natural real structure, 10 which in the case of a single zero-mode fixes the mode (once we normalize it) up to sign. Such an unpaired real fermion zero-mode is usually called a Majorana zero-mode.
The zero-mode that arises in this problem in a vortex field is somewhat analogous to fermion zero-modes that appear, for example, in the field of a 1 + 1-dimensional kink or a 3 + 1dimensional monopole [18]. In all these examples, when a fermion mass arises from symmetry breaking, the fermion acquires a zero-mode in a field of suitable topology. In a 2+1-dimensional model with the same relevant property (the context was a topological insulator with pairing induced on its surface by proximity to an s-wave superconductor), this mode was found in [46]. It was later interpreted [47] in terms of the Callias index theorem [48], following earlier work [49,50]; we review this interpretation in Appendix B. More recently it was also investigated in the context of topological superconductors [34]. This zero-mode is very robust. Even if we perturb away from the idealized models that were introduced in section 3.1, adding irrelevant interactions (for example, breaking the symmetries C and CT that are not natural in condensed matter physics), a vortex field of this system with v = 1 still has an exact Majorana zero-mode. Local interactions can never lift a single Majorana zero-mode.
As in the composite fermion approach to the Moore-Read state (where an analogous zeromode appears [36,51] for somewhat different reasons), the occurrence of a single Majorana zero-mode in the field of a vortex leads to non-Abelian statistics. Let us consider a system of k widely separated vortices in R 2 . To avoid extraneous subtleties involving properties of the Clifford algebra, 11 we take k to be divisible by 8. (In any event, a globally defined state in a sample of finite volume would have k a multiple of 4s, which is divisible by 8 since s is even.) Simply because an isolated vortex would have an exact fermion zero-mode and the theory 10 In a real time Hamiltonian framework, any fermion field can always be expanded in real components. Upon doing so, the general form of a quadratic fermion Hamiltonian allowed by hermiticity and fermi statistics is H = i kl h kl Ψ k Ψ l , where h is a real antisymmetric matrix and for clarity we write the formula in terms of a discrete sum over modes. As h is real, the space of its zero-modes is a real vector space. 11 For example, if k is odd, there is no completely natural quantization of k fermion zero-modes; if k is congruent to 2 mod 4, the operator (−1) F = γ 1 γ 2 . . . γ k that plays an important role below squares to −1 rather than +1. Such issues are not relevant for our present purposes and to avoid them, we take k to be divisible by 8, which ensures that k fermion zero-modes can be represented by real matrices of rank 2 k/2 . is gapped, it follows that in the limit that the vortices are widely separated, the field χ has k modes that are exponentially close to being zero-modes. Whether these modes are exact zero-modes or are only exponentially close to being zero-modes will not be important in what follows. 12 Let γ i , i = 1, . . . , k, be the Majorana zero-mode associated to the i th vortex. The γ i can be normalized so that after quantization they obey a Clifford algebra: An irreducible representation R of this algebra has 2 k/2 states. So this k-vortex system has, on the average, √ 2 quantum states per vortex. The nonintegrality of this number is a symptom of non-Abelian statistics.
Suppose that we adiabatically move the vortices around, keeping them widely separated. The quantum state varies according to the Berry connection [52]. The monodromies around closed orbits 13 in the configuration space of k vortices give a representation of the braid group B k on the Hilbert space R.
It turns out that this representation can be easily determined using two elementary facts: (1) Each vortex has its own zero-mode, which is uniquely determined up to sign. So the braid group can only act by permutations and sign changes of the k objects γ 1 , γ 2 , . . . , γ k .
(2) The action of the braid group commutes with the operator (−1) F , which counts the number of fermions mod 2. This operator anticommutes with all of the fermion zero-modes γ 1 , γ 2 , . . . , γ k . Up to an irrelevant sign, the operator on the Hilbert space R that has this property (and squares to 1) is the chirality operator, the product of all the gamma matrices: Now let us consider a basic operation: the exchange of two vortices, say labeled 1 and 2, that move half way around each other in, say, the counterclockwise direction ( fig. 2). How does this elementary move act on the gamma matrices? One might expect that the answer would involve a simple transposition (γ 1 , γ 2 ) → (γ 2 , γ 1 ), but this is not possible as it would reverse the sign of the chirality operator (−1) F . Rather, we must use the fact that the γ i were naturally defined only up to sign. The basic operation acts on the gamma matrices with a sign change: Actually, we could equally well take (γ 1 , γ 2 ) → (−γ 2 , γ 1 ). The two operations differ by conjugation by (γ 1 , γ 2 ) → (γ 1 , −γ 2 ) (or (γ 1 , γ 2 ) → (−γ 1 , γ 2 )). The sign choice in eqn. (3.22) fixes the relative signs of the γ i , which have been arbitrary until this point. The crucial minus sign in eqn. (3.22) means that B 12 does not coincide with its inverse, and so we get different results depending on whether particle 1 moves around particle 2 in a clockwise or counterclockwise direction. In fact, B 2 12 = 1 but B 4 12 = 1. The fact that B 2 12 = 1 means that braiding of vortices gives a representation of the braid group that does not merely come from a representation of the permutation group. The fact that the operation (3.22), extended to a k particle system, does give a representation of the braid group can be verified as follows.
Start with an initial configuration in which k vortices, labeled 1, 2, . . . , k, are arranged on the real axis in ascending order. The braid group B k on k letters is generated by elements B i,i+1 , i = 1, . . . , k − 1 that exchange the i th and i + 1 th objects in a counterclockwise direction. As above, we take To get a representation of B k , the only relation that we have to check is commutes with the other gamma matrices? It is where CT -invariance implies that the overall phase must be a sign ±1, and to satisfy eqn. (3.23), this sign must be independent of i. The reason for this sign ambiguity is the following. There is a 1-dimensional representation of the braid group -the representation associated to fermionsin which each B i,i+1 is equal to −1. Tensoring with this representation changes the sign in eqn. (3.26). The two choices give representations that are definitely inequivalent; the eigenvalues of B i,i+1 are either i ±1/2 or i ±3/2 . We will see in section 3.5 that the basic vortex corresponds to the + sign in (3.26). The opposite sign arises for another CT -invariant quasiparticle of the model (a state of vorticity 3).
The representation of the braid group that we have just described is very close to one that arises in Ising conformal field theory in two dimensions, or in the corresponding Chern-Simons topological field theory in three dimensions. The conformal blocks of the spin fields of the Ising model are characterized by a representation of the braid group in which B i,i+1 acts as described above, up to an Abelian factor (that is, a phase factor) that violates CT symmetry. (This representation was first found in the RCFT context and appeared explicitly in [53]. It was later used in the context of non-Abelian anyons in [54,55].) Clearly, since the models that we are studying are CT -conserving, they do not generate this CT -violating phase. We will see instead that the models considered here can be constructed by combining in a certain sense an Ising model with a theory (a version of Z 4s gauge theory) that cancels the phase factor. By itself, this latter theory would generate Abelian statistics.
So far we have considered only vortices with v = 1. What happens for v > 1? Assuming CT symmetry, a system of v widely separated vortices of vorticity 1 has v zero-modes, all transforming the same way under CT , since the individual vortices are identical (or since this is predicted by the Callias index theorem). We choose the sign of the operator CT so that the approximate zero-modes all transform as +1 under CT . This remains so when we bring the individual vortices together to make a quasiparticle of vorticity v.
We write γ j , j = 1, . . . , v for the fermion zero-modes in a quasiparticle of vorticity v. If we add to the underlying Lagrangian or Hamiltonian irrelevant interactions that explicitly break CT symmetry, with a small coefficient, this will induce an effective Hamiltonian for the vortex where m kl is a real, antisymmetric matrix. Generically, this completely lifts all of the fermion zero-modes if v is even, and leaves 1 zero-mode if v is odd. In a gapped system, adding an irrelevant operator with a small coefficient cannot affect the topological field theory that describes the braiding of quasiparticles. So this braiding can be computed assuming that a vortex of vorticity v has 1 real fermion zero-mode if v is odd and none if v is odd. Accordingly, vortices of vorticity v have non-Abelian statistics as described above if v is odd, and have only Abelian statistics if v is even. 14 If instead we maintain CT symmetry and take it into account, then the vortices have a richer structure. This will be more fully explored in section 3.5, but as a preliminary, we note that only the value of v mod 8 is important. This is true for essentially the same reason that there is a mod 8 periodicity in the theory of the Majorana chain [56]. CT symmetry does not allow a quadratic Hamiltonian as in eqn. (3.27), but it does allow a quartic Hamiltonian where p ijkl is real and completely antisymmetric. A Hamiltonian of this form can remove zeromodes in groups of 8. The precise meaning of "removing zero-modes in groups of 8" is the following. Quantization of 8 fermion zero-modes gives 2 8/2 = 16 states. A quartic fermion Hamiltonian generically selects a single CT -invariant ground state in this 16-dimensional space (moreover, as explained in Appendix B, this can be done in such a way that the state in question is bosonic and has spin zero). Thus in the presence of a generic CT -conserving Hamiltonian, real fermion zero-modes can be omitted in groups of 8. As a check on the claim that the statistics of a vortex of vorticity v only depend on the value of v mod 2, we will consider the braiding of a vortex of vorticity 1 with one of some arbitrary vorticity p. The v = 1 vortex supports a single gamma matrix γ 1 and the v = p vortex supports p additional gamma matrices γ 2 , γ 3 , . . . , γ p+1 . Braiding of the v = 1 vortex counterclockwise around the composite v = p vortex is simply described by a product B p,p+1 B p−1,p . . . B 1,2 of the basic moves described above. The result is (γ 1 , γ 2 , . . . , γ s+1 ) → (γ 2 , γ 3 , . . . , γ p+1 , (−1) p γ 1 ). For even p, there is no nontrivial sign, and the braiding involves a simple permutation of the gamma matrices. For odd p, we get the same sign change that we found earlier for p = 1, leading to the same non-Abelian statistics as before. This reasoning can be straightforwardly extended to consider braiding of vortices with any two given vorticities p and p .
An alternative explanation of why nonabelian statistics arise precisely for odd vorticity is as follows. Consider a collection of n widely separated quasiparticles of vorticity and gives an irreducible representation of the whole set of v gamma matrices. Now suppose instead that the v i are all odd numbers v i = 2k i + 1, and assume that n is even so that v = i v i is even. 16 An irreducible representation of 2k i + 1 gamma matrices has dimension 2 k i , so this is the dimension of the Hilbert space H i that we might associate to the i th quasiparticle. The tensor product ⊗ i H i then has dimension 2 i k i , while the Hilbert space of the whole system, which has to represent the whole set of i v i = 2 i k i + n/2 gamma matrices, will have dimension 2 i k i +n/2 . So the Hilbert space of the whose system is not simply ⊗ i H i ; it is H = Q ⊗ i H i , where Q, known in RCFT as the space of conformal blocks, has dimension 2 n/2 . The existence of a space of conformal blocks of dimension greater than 1 is the sign of nonabelian statistics. More generally, suppose that there are n e quasiparticles of even vorticity and n o of odd vorticity. A counting as above shows that the dimension of the space of conformal blocks is 2 no/2 .

Quasiparticles
In this section, we describe the quasiparticles of the theory more systematically.

Enumerating the Quasiparticles
First we consider the "elementary" quasiparticles, the ones that do not carry vorticity. These are the fermion field χ and the boson w. (In the phase with φ = 0, there is no quasiparticle corresponding to φ.) χ is electrically neutral and has spin 1/2. In enumerating quasiparticles, we only keep track of the spin mod 1, since integer spin can be converted to orbital angular momentum. So we do not distinguish the two components of χ, which have spin ±1/2. w has spin 0 and electric charge q = −1/2s. By fusion of χ with w or w, we can make additional quasiparticles w n and w n χ, with the relations which reflect the following. Since χ has a Majorana mass, a pair of χ excitations can be created or annihilated. And the bulk coupling I e of eqn. (3.7), where e is the electron field, means that a combination of quasiparticles χw 2s can be converted to an electron e and disappear into the bulk; similarly χw 2s is equivalent to a hole. In describing the low energy physics, it is possible to consistently keep track of the electric charge only modulo 2, basically because a pair of electrons can carry integer angular momentum, which is irrelevant in the low energy topological field theory. We can thus consider e 2 and e 2 to be trivial, and thus we can set w 4s = w 4s = 1 and identify w n with w 4s−n . In fact, we have implicitly done that in enumerating the quasiparticles in the last paragraph. If we want to keep track of electric charge as a real number (and not just its reduction mod 2), then we have to distinguish w n from w 4s−n . In this case, the full set of quasiparticles of zero vorticity would be w n , w n , χw n , χw n , with the relations (3.29), along with χw 2s = e and ww = 1. For brevity, in the following we use the smaller set of quasiparticles.
The quasiparticles w n and χw n are mapped to themselves by the effective time-reversal symmetry T of the low energy theory. Since T 2 = (−1) F , the states w n are Kramers singlets and χw n are Kramers doublets. C maps χ to itself and w n to w n . From this, the action of CT follows. Since (CT ) 2 = (−1) F K (eqn. (3.11)) and χ is invariant under (−1) F K, χ transforms as a Kramers singlet under CT . CT exchanges w n with w n with phases that are determined by the relation (CT ) 2 = (−1) F K.
If CT symmetry is weakly broken by a perturbation such as that of eqn. (3.27), then it is straightforward to enumerate the quasiparticles for any v. For odd v, there is a single real fermion zero-mode γ 0 . It obeys γ 2 0 = 1 and can be represented as 1 (or −1) in the vortex state (that is, in the Fock vacuum obtained by quantizing the nonzero-modes of χ in the field of the vortex). In effect, γ 0 has an expectation value in the vortex state. Moreover, γ 0 is a mode of χ. So acting with χ on the vortex does not produce a new quasiparticle type. We can still act with w n , n = 0, . . . , 4s − 1, producing 4s quasiparticle types for odd v.
For even v, with CT symmetry weakly broken, there are no fermion zero-modes and, just as for v = 0, we can make 2 × 4s = 8s quasiparticle types by acting with χ and/or w n , n = 0, . . . , 4s − 1. For both even and odd v, it is a little tricky to determine the spins of these quasiparticles. We will return to this shortly.

Consequences of CT Symmetry
CT symmetry leads to a richer structure, though it does not change the list of quasiparticles. As a preliminary, the gauge charges q and k and the spin or angular momentum j are all odd under CT . So CT symmetry ensures that the filled Dirac sea of negative energy modes of χ does not contribute to any of those quantum numbers. There is similarly no contribution from the bosons. The quasiparticle quantum numbers are therefore determined entirely by quantization of the fermion zero-modes. What we are really interested in here are the fractional parts of the quasiparticle quantum numbers, which are observables of the topological field theory that governs quasiparticle interactions at low energies. This topological field theory is not affected by weak CT -violating perturbations, so the determination below of the fractional parts of the quantum numbers is also valid in the absence of CT symmetry.
Even in the presence of CT symmetry, the quasiparticle structure in a sector of vorticity v only depends on v mod 8, as explained in the discussion of eqn. (3.28). So we can limit ourselves to 0 ≤ v ≤ 7. But actually, even in this range, there is a symmetry v → 8 − v. In fact, T maps v to −v, which, since v is only defined mod 4s, is equivalent to 4s − v. Because of the anomaly described in section 3.2, we take s to be even, so 4s is a multiple of 8 and hence 4s − v is congruent mod 8 to 8 − v. Combining these statements, we see that everything is determined by what happens for 0 ≤ v ≤ 4. (Some subtlety is needed in applying this statement, because T , which was used in relating v to 8 − v, does not commute with CT (eqn. (3.12).) For v = 1, the vortex has just one χ zero-mode. As explained in section 3.5.1, this mode has an expectation value in the vortex state, as a result of which acting with χ does not produce a new quasiparticle. But we obtain 4s quasiparticle types by acting with powers of w. These are mostly exchanged in pairs by CT , since CT exchanges w and w.
To determine the spins of these quasiparticles, we need to take into account that in a vortex sector, the conserved angular momentum j is not just the conventional angular momentum j, but is a linear combination of j with the emergent gauge charge k: Since w has j = 0, k = 1, it carries j = v/4s, so it has spin 1/4s in a field of v = 1. Thus if Ψ 1 is the basic vortex of v = 1, the quasiparticles w n Ψ 1 have charge q = −n/2s and spin j = n/4s.
For v = 2, there are two χ zero-modes. As explained in Appendix B, these modes have j = ±1/2. Quantizing those modes gives two states of j = ±1/4. These two states are electrically neutral, of course, and are exchanged by the action of χ.
The fact that the spins of the states obtained by quantizing the fermion zero-modes are ±1/4 enables us to answer a question that was left open in section 3.4. Starting with a rotation-symmetric classical state of v = 2, separate the two vortices in a way that preserves the symmetry under a π rotation that exchanges the two vortices (but not, of course, the full rotation symmetry). In this process, j is not conserved, but it is conserved mod 2, since the π rotation symmetry is preserved. So the possible angular momentum states of a system of two basic vortices of v = 1 are of the form 2p ± 1/4, p ∈ Z. This means that the eigenvalues of a π rotation are exp (iπ(2p ± 1/4)) = exp(±iπ/4). This π rotation is a way to describe the braiding of two vortices. The eigenvalues in the braiding of two vortices are thus exp(±iπ/4), which means that we must use the + sign in eqn. (3.26).
The remaining quasiparticles of v = 2 are obtained by acting with w n on these two states of j = ±1/4. For v = 2, w has j = 1/2s. So the states obtained by acting with w n have spins j = ±1/4 + n/2s and electric charge q = −n/2s.

Some More Delicate Questions
There remain some more delicate questions concerning the transformation of the quasiparticles under symmetries. In particular, according to eqn. (3.11), the theory should admit symmetry operators CT , (−1) F , and K that should be be linked by (and more trivial relations: K 4s = 1 and CT , (−1) F , and K all commute). We discuss below the action of T . For quasiparticles of vorticity 0, one can use classical field theory to define symmetry operators that obey all expected relations. This is essentially how eqn. (3.31) was deduced in section 3.1. Even when one includes vortices, it remains true that for any state of the system that can be defined in a compact sample, the symmetry operators can be defined and obey the expected relations. Concretely, this is because, in a compact sample, the total vorticity is always a multiple of 17 4s and therefore (since s is even) is divisible by 8. So the fermion zero-modes of the whole system, taken together, generate a Clifford algebra whose rank is a multiple of 8. Such a Clifford algebra has a real representation in which all symmetries are realized in the expected way.
By contrast, if v is not a multiple of 8, then the fermion zero-modes of a quasiparticle of vorticity v generate a Clifford algebra whose rank is not a multiple of 8, and in this case, in general the symmetries are not realized "locally" (i.e. on just one quasiparticle) in the expected way. An analog is the theory of the Majorana chain [56]. A state of the whole system realizes the symmetries (in that case, T and (−1) F ) as expected, but if one attempts to factorize the Hilbert space as a tensor product of a space associated to one end of the chain and a space associated to the other, then the symmetries cannot be defined on the individual factors or do not act in the expected way.
In the present problem, we find that the symmetries act in a fairly normal way if v is even, but not if v is odd. There are several ways to explain this. Quasiparticles of even vorticity obey abelian statistics, so that, as noted at the end of section 3.4, if one considers states with only such quasiparticles, the Hilbert space of the whole system is the tensor product of subspaces associated to individual quasiparticles. This is enough to ensure that one can define all of the expected operators for each quasiparticle, but allows the possibility that in doing so, one will run into a central extension of the expected symmetry group. In a sense, the occurrence in eqn. (3.31) of the operator K, which is trivial for a global state of the system but nontrivial for an individual quasiparticle, represents such a central extension. Alternatively, as we explain at the end of section 4.2, by considering a topological insulator interacting with magnetic monopoles, one can create a situation in which a compact sample can have a state of any even vorticity (not necessarily a multiple of 4s), 18 so it must be possible to define the symmetries for any even value of v.
Let us begin by explaining the difficulty in realizing the symmetries when v is odd. The problem is particularly obvious for v = 1. There is only one fermion zero-mode γ 1 and it has an expectation value in the vortex state. In eqn. (3.31), the operator (−1) F is supposed to anticommute with all fermion operators (and commute with bosons), and likewise χ is odd under 19 K. So both (−1) F and K should anticommute with γ 1 . Clearly, an expectation value of γ 1 is not compatible with the existence of a symmetry (−1) F or K under which γ 1 is odd. 20 However, there is no obstruction to realizing the products (−1) F K or K 2 , which commute with γ 1 .
More generally, for any odd v, there are v zero-modes γ 1 , . . . , γ v , and the operator γ = γ 1 γ 2 . . . γ v commutes with them and so is a c-number in an irreducible representation of the algebra. But γ is odd under (−1) F and under K, so those operators cannot act in an irreducible representation of the algebra. Again, there is no problem with (−1) F K or K 2 .
Since (−1) F K commutes with all the zero-modes, it acts as a c-number in an irreducible representation of the Clifford algebra. To decide what this c-number should be, we return to eqn. (3.31), in which CT is supposed to be an antiunitary symmetry. We now run into further subtleties, which depend on the properties of the rank v Clifford algebra generated by the zero-modes. For v = 1, the Clifford algebra has a 1-dimensional representation, with the only gamma matrix being γ 1 = 1. On this space, we can define CT = K (complex conjugation), obeying (CT ) 2 = 1. So we must define (−1) F K = 1 on the vortex ground state for v = 1. K 2 should commute with CT , so it should act on this state as 21 ±1. For v = 3, there are three zero-modes, which we can represent as 2 × 2 matrices γ i = σ i (the Pauli matrices). Now we run into the fact that on this two-dimensional space, it is not possible to define CT , because there does not exist any antiunitary operator that commutes with the γ i , as CT is supposed to do. There does exist the antiunitary operator Kσ 2 , which anticommutes with the γ i , and obeys (Kσ 2 ) 2 = −1. We cannot interpret this operator as CT , but we can interpret it as CT K, which is supposed to anticommute with the γ i . As (CT K) 2 = (−1) F K 3 , we learn that we must take (−1) F K 3 = −1 for v = 3. As for K 2 , having fixed its sign for v = 1, we can determine its sign for v = 3 by fusing three vortices of v = 1.
More generally, for any odd value of v, since there is no way to define K, it is clear that we at most may be able to define CT or CT K, but not both. Continuing in the above vein, one finds that for v = 5, one can define CT (and it obeys (CT ) 2 = −1), while for v = 7, one can 18 In section 4.2, we show that with unit magnetic charge inside the sample, one gets on the surface a quasiparticle of vorticity ±2 together with s quasiparticles of type w or w. Since the symmetries can certainly be defined for the classical fields w and w, and they can also be defined for a global state of the whole system, it must be possible to define the symmetries for a quasiparticle of vorticity ±2. 19 (−1) F and K differ because they act differently on w. 20 The problem in defining K in the sector with odd v is reminiscent of the problem in defining the color of magnetic monopoles in grand unified theories [57]. In both cases we have an unbroken gauge theory, but the "global gauge charge" cannot be defined in a topological nontrivial sector. 21 There seems to be no natural way to fix this sign. In the language of section 4.2, the signs depends on whether we associate K 2 with the one-form global symmetry e 2i c or W ψ e 2i c . These are exchanged by CT so neither is preferred. define CT K (and it obeys (CT K) 2 = 1).
Now we consider the case of even v, which is more straightforward as explained above (but which still has some subtleties that we will not fully explore). For v = 2, there are two gamma matrices, which can be given a real 2 × 2 representation. So the antilinear operator CT that commutes with them is simply, up to an irrelevant phase, K. In particular, it obeys (CT ) 2 = 1, so the vortex states are Kramers singlets of CT . Accordingly, we expect (−1) F K = 1. Because (−1) F and K should commute with CT and anticommute with the γ i , they must be, up to sign, (−1) F = −K = γ = γ 1 γ 2 . As a check, the eigenvalues of K = exp(2πik/4s) are ±i, corresponding to k = ±s. These are the expected values as the γ i (being modes of χ) carry k = 2s or K = −1, so that by the usual logic [18] of charge fractionation, the states that the γ i act on must have k = ±s. We should note, however, that the square of (−1) F is −1, not the expected +1. We can regard this as representing a (further) central extension of the algebra, a possibility allowed by the general remarks above.
For v = 4, the gamma matrices γ 1 , . . . , γ 4 can be represented in a four-dimensional Hilbert space H. It is not possible to take them to be all real. One can have three of the gamma matrices real and the fourth imaginary, for example With this choice, the CT transformation that commutes with the γ i is uniquely determined, up to an irrelevant phase, to be CT = Kγ 123 , where γ i 1 i 2 ...i k is an abbreviation for γ i 1 γ i 2 . . . γ i k . This operator obeys (CT ) 2 = −1, so the states are all in Kramers doublets, and we want (−1) F K = −1. The operators (−1) F and K must anticommute with the γ i and commute with CT . Up to sign, this forces (−1) The fact that K = −(−1) F for these states is a kind of anomaly. We have found the vortex states by quantizing the theory of the χ field coupled to φ and a (but with w playing no role). In a restricted theory without the field w, at the classical level K = +(−1) F for all fields, but we have found that in the v = 4 sector, the sign in this relation is reversed. This is closely related to the anomaly that was described in section 3.2. To understand the relation, set s = 1, so that a state of v = 4 can be created by a monopole operator. The anomaly says that the relation between spin and charge for monopole operators is the opposite of the classical one, and this corresponds to the minus sign in the relation K = −(−1) F .
The action of T is largely determined by the relation T 2 = (−1) F and the fact that T exchanges states of vorticity v with those of vorticity −v. The nontrivial cases are the Tinvariant values v = 0 and v = 2s. For v = 0, the action of T is determined from the classical considerations of section 3.1. As for v = 2s, it is equivalent to v = 0 if s is divisible by 4. If instead s is congruent to 2 mod 4, then the rank of the v = 2s Clifford algebra is congruent to 4 mod 8, and the analysis of the action of T in this case is similar to what we said above concerning the action of CT for v = 4.
In section 4 we will rederive this spectrum of quasiparticles and the action of the various symmetries on them using the low-energy TQFT description of this system.

The Partition Function
Here we will examine in detail the partition function of the Dirac fermion χ, coupled to B = A + 2sa. One goal is to show carefully that the models we have studied have precisely the same anomalies as the standard gapless boundary state of a topological insulator, and thus are fit as boundary states. A second goal is to describe the partition function in a way that will motivate a topological field theory description that we will give in section 4. We will consider only orientable spacetimes. 22

Review Of The Parity Anomaly
Naively the fermion partition function is given by the determinant det D of the hermitian operator D = i / D. Because D has real eigenvalues, this determinant is naturally real. But infinitely many eigenvalues are negative, and this leads to a problem in defining the sign of det D.
Naively, det D = (−1) n − | det D| where n − is the number of negative eigenvalues of D. As n − is infinite, some regularization is needed. Formally let n + be the number of positive eigenvalues of D and N = n + + n − the total number of eigenvalues. In trying to regularize n − , it is better to first subtract the infinite constant N/2, thus replacing n − by n − − N/2 = (n − − n + )/2. The difference n + − n − is still ill-defined, but any reasonable regularization of it gives the same result, the η-invariant of Atiyah, Patodi, and Singer [59]. For example, one may define where the sum runs over all eigenvalues λ i of the operator D, and we define (A different but equivalent regularization was used originally [59].) Thus we regularize (n − − n + )/2 by replacing it with −η/2, and so we replace (−1) n − = exp(±iπn − ) with exp(∓iπη/2). Finally, the regularized version of the path integral of the χ field is [23] Z χ = |det D| exp(∓iπη/2).
The parity anomaly is often described in terms of the Chern-Simons function rather than the η-invariant, but this is not quite correct. The Atiyah-Patodi-Singer theorem [59] would let us replace exp(iπη) with exp(iCS), where CS is a Chern-Simons function that is well-defined mod 2π. Here CS is CS(A, g) of eqn. (2.14), which includes the gravitational contribution Ω(g) of eqn. (A.21). We cannot make this sort of replacement in eqn. (3.34). Naively, we would want to replace exp(iπη/2) with exp(iCS/2), but as CS is only well-defined mod 2π, exp(iCS/2) has an ill-defined sign. The parity anomaly revolves around the overall sign of the partition function, so we cannot understand it properly via a formula with an ill-defined sign.
Because η is not an integer, the sign ∓ in the exponent in eqn. (3.34) matters. In an approach based on Pauli-Villars regularization, this sign comes from the sign of the regulator mass [23,17]. This sign violates T and R symmetry, which say that on an orientable manifold the path integral should be real. This violation is often loosely called the parity anomaly, though it is more precise to call it an anomaly in T and R symmetry. This anomaly is a mod 2 effect because the original problem only involved the sign of the path integral. If we had two χ fields, the path integral would have been formally det 2 D, which is naturally positive. In terms of the above derivation, we could define the path integral for one χ field using a + sign in eqn. (3.34), and that of the second using a − sign. This amounts to using a Tand R-preserving regularization. The phase factors would cancel, leaving us with a real, positive partition function. (Alternatively, we could use a Tand R-violating regularization, such that the two χ fields contribute with the same sign, say +. Then the partition function is proportional to exp(+iπη), which violates T and R. But now we can add to the bare action a properly normalized Chern-Simons term making the phase exp(+iπη − iCS) = 1, where the last step uses the Atiyah-Patodi-Singer theorem. The combination of the regularization and this bare term respects T and R.) To avoid a possible confusion, we would like to add a clarifying comment. The free fermion theory without the gauge field A has both a global U (1) symmetry and a time-reversal symmetry T. It has a conserved U (1) current j µ and all the flat space correlation functions of operators at separated points are U (1)-and time-reversal invariant. The problem of the anomaly can be seen as a possible failure of these symmetries at coincident points; specifically, the product Such contact terms can be probed when the theory is coupled to a classical background gauge field A. Even then, if A is infinitesimal, we can add to the Lagrangian a counterterm proportional to AdA to cancel the contact term and thus preserve T. The anomaly is the statement that this cannot be done when A is not infinitesimal and the gauge group is compact. Then, the allowed AdA counterterms must have properly quantized coefficients and cannot cancel arbitrary fractional contact terms in j µ (x)j ν (0). The situation here is similar to the 't Hooft anomaly. There a system has some global symmetries, but the currents of these symmetries suffer from contact terms, which prevent us from coupling them to background gauge fields.
Once the U (1) symmetry is gauged, there is no way to eliminate the anomaly in T and R symmetry in a purely three-dimensional context. But those symmetries can potentially be restored, as we discuss next, if the three-manifold W on which the χ field is defined is the boundary of an oriented four-manifold X over which the spin c structure of W extends. Physically, X will be the worldvolume of a topological insulator.

The APS Index Theorem And T-Invariance
The path integral of the topological insulator can be analyzed [60,17] by using the APS theorem for the index 23 I of the Dirac operator on X: Here is the curvature of B = A + 2sa, and A(R) is a certain quadratic expression in the Riemann tensor R (eqn. (A.1)). If X has no boundary, then there is no η/2 term on the right hand side of eqn. (3.35), and this formula reduces to the standard Atiyah-Singer formula for the index of the Dirac operator on a compact four-dimensional spin manifold without boundary.
To restore T and R symmetry, we assume the existence on X of the bulk "topological" couplings suggested by the index formula: 24 The combined boundary and bulk path integral, including a factor of exp(−I top ) from the bulk, is then In the last step, we used the APS index formula (3.35).
Since the combined partition function |det D|(−1) I is real, it respects T and R symmetry. C (and therefore CT) symmetry is also manifest, since both factors |det D| and (−1) I in eqn. (3.38) are invariant under a sign change of B = A + 2sa. Now let us discuss the physical meaning of the bulk couplings that we have had to add. We first review the standard boundary state of a topological insulator, so we set s = 0, B = A, and G = F . Also, first we consider the case that X is a spin manifold and A is a U (1) gauge field, and then the slightly more subtle case that X is a spin c manifold with spin c connection A.
If X is spin, the bulk couplings π A(R) and (π/2)F ∧ F/(2π) 2 are each T-conserving. The F 2 coupling corresponds to an electromagnetic θ-angle of π. This assertion depends on the fact that 1 2 F ∧ F/(2π) 2 is an integer on a spin manifold, but has no further divisibility properties. (See Appendix A for this and some further remarks in this paragraph.) The two Tand Rconserving values of the electromagnetic theta-angle are 0 and π, and the F 2 coupling in I top means that the topological insulator is the case θ = π. However, on a spin manifold, the bulk coupling π A(R) is trivial. This is because X A(R) is an even integer on a compact spin manifold X without boundary, so π X A(R) always vanishes mod 2π. Thus on a spin manifold, the gravitational coupling π X A(R) does not influence the bulk physics, and accordingly, the effect of I top in bulk can be summarized by saying that the electromagnetic theta-angle equals π. In the spin c case, the combined gauge and gravitational coupling I top is T-invariant in bulk, but separately the gauge and gravitational couplings in I top are not T-invariant. These matters are discussed in Appendix A.

Extension To s = 0
Now let us consider the more general case with s = 0 and G = F + 2sf . There is something non-trivial that we have to check in order to decide if the formula (3.38) is physically sensible. The emergent gauge field a and its curvature f = da are defined only on the boundary W , Figure 3: Two manifolds X and X with the same boundary W are glued together -after reversing the orientation of X so that the orientations are compatible -to make a compact oriented manifold X * without boundary. (With a view to later generalizations, X and X are depicted as being topologically different.) not on the four-manifold X. But the formula (3.38) involves G = F + 2sf , so it presumes an extension of f over X. Such an extension may not exist 25 and if it exists it is not unique.
To begin with, let us assume that a and therefore f = da can be extended over X and show that the choice of this extension does not matter. The dependence of Z com on the choice of extension comes from the part of the topological couplings I top that depend on f . This part leads to a factor We need to know whether U X depends on how a and f were extended over X.
There is a standard way to answer such a question. Let X be another copy of X (or in more general applications that we come to presently, some other four-manifold with the same boundary as X) with some possibly different extension of a over X . By gluing together X and X along their common boundary, with a reversal of orientation of X so that their orientations agree along W , we make an oriented four-manifold X * without boundary ( fig. 3). The choices of extension of a over X and over X glue together to an extension over X. We want to know whether U X as defined in eqn. (3.39) is equal to U X , defined by the same formula evaluated on X . The ratio is We want to know whether this always equals 1. First suppose that X * is a spin manifold and A is a U (1) gauge field. In this case, the cohomology classes F/2π and f /2π are integral, so the integrals X * F ∧ f /(2π) 2 and X f ∧ f /(2π) 2 are both integers. In the exponent in eqn. (3.40), these integers multiply 2πis or 2πis 2 , where s is an integer, so the ratio U X /U X is indeed always equal to 1.
If instead X * is a spin c manifold and A is a spin c connection, then matters are different, and we will rediscover the anomaly that was found in section 3.2. Indeed, in the spin c case, in general F/2π is a half-integral class, so p = X F ∧ f /(2π) 2 takes values in Z/2. To ensure that U X /U X is always 1, we now need exp(2πisp) = 1 for all p ∈ Z/2; in other words, we need s to be even. This is the condition found in section 3.2 for the model to make sense in the spin c context.
The analysis of the case that a and f cannot be extended over the worldvolume X of the topological insulator is a little technical, and the following paragraph could be omitted on first reading. Even if a and f do not extend over X, cobordism theory ensures that there is always some four-manifold X over which both A and a, and therefore F and f , as well as the spin structure of W , can be extended. If we were studying a topological insulator with worldvolume X rather than X, then as in eqn. (3.38) the appropriate partition function would be |det D|(−1) I , where now I is the Dirac index on X (with APS boundary conditions) coupled to A + 2sa. But of course, we really want to do physics on X, not X . What is wrong with the formula |det D|(−1) I is that although, as above, it does not depend on how a is defined on X , it does depend on the unphysical choice of how A is defined on X . To cancel this dependence, we need another factor. An appropriate general formula is Here X * is the four-manifold without boundary that is built as in fig. 3 by gluing X to X, and I In all of this, we have treated χ as a massless Dirac fermion coupled to the gauge field or spin c connection A + 2sa. What happens in the gapped phase in which χ acquires a mass by coupling to the Higgs field φ? Since φ has charges (2, 4s), when it has an expectation value, A + 2sa is reduced at low energies to a Z 2 gauge field (more precisely, the spin c structure that it defines reduces to a spin structure). χ then splits up as a pair of Majorana fermions coupled to the same spin structure or Z 2 gauge field. They get equal and opposite masses ±m from the coupling to φ, and this replaces |det D| in the formulas for the partition function with the strictly positive quantity Pf(D + im)Pf(D − im) = |Pf(D + im)| 2 (here Pf denotes the Pfaffian). The sign of the partition function is still as given in the formulas (3.38) or (3.41).
If B = A + 2sa is generic, then the eigenvalues of the Dirac operator are generically nondegenerate, and the partition function should change sign when an eigenvalue passes through zero. In terms of the formula (3.38) for the path integral of the system, 26 there is of course no sign change of |det D| when an eigenvalue of D passes through zero, but when this happens η jumps by ±2, causing a sign change in exp(∓iπη/2). Likewise, I jumps by ±1, causing a sign change in (−1) I . In the gapped phase, with the gauge group reduced at low energies to Z 2 , the eigenvalues of D have even multiplicity because of a version of Kramers doubling. Since there are no gapless fermions, no sign change in the path integral is expected when a pair of eigenvalues passes through zero. Indeed, when that happens, η jumps by ±4, so that exp(∓iπη/2) is a smoothly varying function, and I jumps by ±2, so that (−1) I is unchanged and is a topological invariant. The topological invariance of the overall sign (−1) I of the path integral is essential for the existence of the topological field theory interpretation that we construct in section 4, and the fact that exp(−iπη/2) is smoothly varying in the gapped phase is necessary for the relation to the Ising model that we discuss in section 3.6.4.
The gapped phase also has vortices. In the core of a vortex, the gauge symmetry is restored. The vortex represents a singularity in the low energy Z 2 gauge field or spin c connection A+2sa. The singularity is associated to a monodromy of −1, familiar from the spin field of the Ising model, and from the Ramond sector in string theory. Analogs of eqns. (3.38) and (3.41) and of our statements above and also in section 3.6.4 below hold in the presence of vortices. But one has to take into account the existence of a Majorana zero-mode in the vortex core, and a detailed explanation is somewhat more technical.

Relation To The Ising Model
We make here a few remarks as preparation for section 4, in which we will describe a topological field theory interpretation of the physics in the gapped phase.
The combined partition function Z com of eqn. (3.38) is the product of two phase factors, namely (where now we make a choice of sign in the exponent) and (3.43) Φ 1 comes from integrating out the boundary fermions, and Φ 2 from a bulk coupling. 27 To interpret Φ 2 in topological field theory, we will just write an Abelian Chern-Simons theory that reproduces this factor. But what is the topological field theory interpretation of Φ 1 ?
One of the simplest rational conformal field theories (RCFT's) in two dimensions is the Ising model. From a chiral point of view, this is the theory of a Majorana-Weyl fermion ψ of dimension 1/2. What is the chiral algebra of the Ising model? There are two possible answers. From one standard point of view, one considers the chiral algebra to consist of holomorphic fields of integer dimension. From this point of view, the chiral algebra of the Ising model is simply generated by a stress tensor of central charge c = 1/2. This chiral algebra has three representations, usually denoted by 1, σ, and ψ, whose dimensions are 0, 1 16 , and 1 2 respectively. Alternatively, we can define a Z 2 -graded chiral algebra in which we allow holomorphic fields of half-integer spin. With this definition, the chiral algebra of the Ising model is simply the Z 2 -graded algebra generated by the ψ field itself. We will say that the two points of view correspond respectively to RCFT's and to spin-RCFT's.
RCFT's in two dimensions are associated to three-dimensional topological quantum field theories (TQFT's), and likewise spin-RCFT's are associated to three-dimensional spin topological field theories (spin-TQFT's). A TQFT is defined on an oriented three-manifold, and similarly a spin-TQFT is defined on an oriented three-manifold with a choice of spin structure. (In both cases, in general there is a framing anomaly.) The Ising model viewed as an RCFT or as a spin-RCFT is associated to a 3d theory that we will call the Ising TQFT or spin-TQFT.
Representations of the chiral algebra of an RCFT correspond to line operators -or more physically to quasiparticles -in the associated TQFT. So the Ising TQFT has three line operators corresponding to 1, σ, ψ. The analogous statement in the case of a spin-RCFT has a subtlety that is explained in Appendix C. In this case, each representation of the Z 2 -graded chiral algebra leads to two distinct lines in the 2 + 1-dimensional TQFT. The spins of these two lines differ by 1 2 mod 1 and each of them is associated with a different representation of the subalgebra of the chiral algebra consisting of fields of integer spin. In the case of the Ising spin-TQFT, since there is only one representation, there are just two distinct lines, the trivial line with spin 0 and a line W ψ with spin 1 2 . The Ising spin-TQFT will have a partition function on an oriented three-manifold W with a choice of spin structure. As for any TQFT or spin-TQFT, this partition function can be computed, starting with simple special cases, by applying a sequence of braiding and surgery operations. However, in the case of the Ising spin-TQFT, these braiding and surgery operations all take place in a 1-dimensional space, since the Z 2 -graded chiral algebra generated by ψ has only one representation. Accordingly, the partition function of the Ising spin-TQFT will always be of modulus 1.
To illustrate some of these ideas, let us consider the simple example of the S 2 × S 1 partition function of the Ising spin-TQFT. It needs a spin structure around the S 1 factor, which we denote in an obvious way by s = ±1. For s = +1 this partition function is Tr H S 2 (−1) F and for s = −1 it is Tr H S 2 1, where H S 2 is the Hilbert space associated with S 2 (in the absence of any quasiparticles) and (−1) F is the usual operator that distinguishes bosons and fermions. Since H S 2 consists of a single bosonic state, the S 2 × S 1 partition function is equal to one for every s. Now, the theory has a nontrivial line associated with the fermion ψ. We can let this line pierce the S 2 at a point and wrap the S 1 factor. The only change in the S 2 × S 1 partition function calculation is that now H S 2 should be replaced by H S 2 ψ , the Hilbert space of the sphere pierced by the line. This Hilbert space consists of a single fermionic state. Therefore, the path integral in the presence of this line is −s. As expected, the path integral is a phase and it depends on the spin structure.
In general, we claim that the partition function of the Ising spin-TQFT (with no line operator insertions) is simply exp(−iπη/2), where η is the eta-invariant of a Majorana fermion on W , coupled to the relevant spin structure. This statement can be understood as a consequence of the Dai-Freed theorem [58]. Suppose that W has a non-empty boundary Σ. Then exp(−iπη/2) is not well-defined by itself (its definition fails when the Dirac operator on Σ acquires a zero-mode), but the Dai-Freed theorem says that the product of exp(−iπη/2) with the path integral of a chiral fermion ψ on Σ is smooth and well-defined as a function of the metric on W . This mirrors the standard relation between a TQFT or spin-TQFT in three dimensions and the corresponding RCFT or spin-RCFT in two dimensions: exp(−iπη/2) is the partition function of the bulk theory associated to the boundary theory of ψ. So exp(−iπη/2) is the partition function of the Ising spin-TQFT.
As a check, one can consider the dependence on the metric of W . If η is the eta-invariant of a Majorana fermion coupled to a spin structure on W , then the dependence of exp(−iπη/2) on the metric on W is determined by the APS index theorem and corresponds to a framing anomaly with c = ±1/2 (where the sign depends on a choice of orientation), as expected for the Ising spin-TQFT. In fact, the APS index theorem implies that the metric dependence of exp(−iπη/2) is the same as that of (3.44) (Like exp(−iπη/2), Θ really depends only on the metric of W , not X.) We note that Θ is the inverse of the gravitational factor in Φ 2 (eqn. (3.43)), so Θ cancels the framing anomaly of the Ising spin-TQFT. This cancellation is an inevitable consequence of the fact that the overall partition function is real, as is manifest in eqns. (3.38) and (3.41), so it cannot have a net framing anomaly. So in section 4, we will interpret the phase factor Φ 2 in terms of an Abelian Chern-Simons TQFT, and Φ 1 in terms of the Ising spin-TQFT. This does not mean that the spin-TQFT that governs the gapped phase of our models is simply the product of those two theories. The two theories are coupled because the spin structure seen by the Ising spin-TQFT and entering in the definition of Φ 1 = exp(−iπη/2) is the one determined by A + 2sa, which also appears in Φ 2 .
To better understand this coupling, we need to understand how the Ising TQFT is related to the Ising spin-TQFT. 28 The partition function of the Ising TQFT is obtained from that of the Ising spin-TQFT by simply summing over spin structures. In other words, the Ising spin-TQFT assigns a partition function to an oriented three-manifold W with a choice of spin structure; if we sum this partition function over the choice of spin structure, we get the partition function of the Ising TQFT. (This statement is equivalent to the statement that if the Ising TQFT is quantized on a two-manifold Σ, its physical states have a basis corresponding to spin structures on Σ. But the general relation between 3d TQFT and 2d RCFT says that the physical states of the Ising TQFT are the conformal blocks of the Ising RCFT on the same surface. And the conformal blocks of the Ising TQFT have a basis corresponding to spin structures on Σ.) So far a and A have been classical background fields, but we still need to sum over the quantum field a. Since A + 2sa couples to χ, which is a fermion, it might seem like the sum over a will amount to the sum over spin structures, making the spin-TQFT that results from the sum an ordinary (non-spin) TQFT that no longer depends on a particular choice of spin structure. However, we should not forget the w quanta (and correspondingly the Wilson lines e i a in the low energy TQFT). These mean that the sum over a still leaves dependence on the spin structure. One way to see that is to recall that the fundamental electron has the quantum numbers of χw 2s . It is U (1) a invariant, but has spin 1 2 . Therefore, it is sensitive to the spin structure even after the sum over a. We conclude that the low energy theory must be a spin-TQFT.
In section 4, when we represent the two factors Φ 1 and Φ 2 of the partition function by the Ising spin-TQFT and by an Abelian theory, we will have to take into account the fact that these two factors couple to the same Z 2 gauge field A + 2sa. One way to describe how the two factors are coupled is the following. We start with two decoupled theories and then gauge a common Z 2 one-form symmetry [61,62]. We will describe that procedure in a concrete way in section 4.2. Alternatively, when the TQFT is presented as a Chern-Simons gauge theory, this one-form gauging is implemented by modding the gauge group by Z 2 . We will do that in section 4.4. In our case, both the non-Abelian Ising TQFT and the Abelian TQFT are not spin. But their Z 2 quotient is a spin-TQFT. In Appendix C we will review this modding of the gauge group and will demonstrate it in examples. In particular, we will show how it can turn an ordinary TQFT to a spin-TQFT.
The product Φ 1 Φ 2 = (−1) I is T-invariant, but the individual factors Φ 1 and Φ 2 are not. An alternative and T-conjugate factorization can be obtained by complex conjugating both Φ 1 and Φ 2 (or equivalently by reversing the sign in eqn. (3.34)). The construction that we will make based on factoring (−1) I as Φ 1 Φ 2 has the advantage of giving a concise description of the low energy physics, but it has the drawback of somewhat hiding the T and CT symmetry.

A Topological Field Theory For The Low Energy Physics
In this section we study the low energy behavior of our model of section 3.1 and then we generalize it. Clearly, since the system is gapped, the low energy description is a topological field theory. This theory should exhibit all the symmetries of the system. These include timereversal and charge conjugation as well as U (1) A . Also, the topological theory should exhibit the same anomalies as the microscopic theory. Finally, since the microscopic theory can be placed on a spin c manifold, the low energy theory should have the same property.
As we will see, it will be convenient to describe the topological field theory as having two sectors, coupled in a certain way. This separation into two sectors makes it easy to analyze the model, though it obscures some of the global symmetries, in particular T and CT .
One way to understand the two sectors is to look at the partition function (3.38) and its discussion in section 3.6.4. One sector, which we will discuss in section 4.1, leads to The other sector, which we will add in section 4.2, leads to Φ 1 = exp(−iπη/2).
In this section we will describe the TQFT of the system. We will suppress the gravitational interactions and will focus on the gauge theory.

The Abelian Sector
In the models of section 3.1, the expectation value of φ breaks the U (1) a gauge symmetry to Z 4s . Therefore, our low energy theory should include a Z 4s gauge theory. One way to describe Z 4s gauge theory is to replace the Higgs mechanism of φ by a Lagrange multiplier U (1) gauge field c with the coupling [63,64] 1 2π cd(4sa + 2A) . The equation of motion of c states that 4sa + 2A is a trivial gauge field. Its field strength d(4sa + 2A) vanishes and its periods are trivial. This does not mean that a is a trivial gauge field. Instead, we can shift a by a Z 4s gauge field and still satisfy these constraints. Therefore, the dynamical part of a is a Z 4s gauge field. 29 Next, we integrate out χ. As we said, we are going to separate the expressions in section 3.6 into Φ 1 = exp(−iπη/2) (which we will discuss in section 4.2) and . Ignoring the gravitational term A(R), Φ 2 corresponds to Abelian Chern-Simons couplings: As expected, the AdA term is not properly normalized to be a purely three-dimensional term. This reflects the need to attach the system to the 3+1-dimensional bulk. The ada term deforms the Z 4s gauge theory in eqn. (4.1). It makes it a Dijkgraaf-Witten theory [65] with k = 2s 2 (see Appendix C.3). The adA term is properly normalized. But since k is always even, the model satisfies the spin/charge relation only for even s. 30 Let us analyze the theory based on the sum of equations (4.1) and (4.2):  where we used the fact that s is even. This means that we can restrict to n a , n c = 0, ±1, . . . , ±(2s− 1), 2s. It is also straightforward to find the spins of these lines S na,nc = n a n c 4s − n 2 c 16 mod 1 .
(4.7) 29 The Lagrangian of eqn. (4.1) can also be understood by starting with the microscopic field φ = |φ|e iϕ and considering the effective Lagrangian of the gauge mode ϕ. The field c is dual to the compact scalar ϕ. 30 Here we see another perspective on the spin/charge anomaly we discussed in section 3.2. When s is odd we can restore the spin/charge relation in the low energy theory by adding to it an odd multiple of 1 2π adA. One way to do it is to add this term to the microscopic theory with χ. This would restore the relation and would allow A to be a spin c connection. But since this term violates T , we do not do that. This is the hallmark of an anomaly. The classical theory preserves T and satisfies the spin/charge relation, but the quantum theory cannot have both. It either preserves T , or satisfies the spin/charge relation.

Adding The Non-Abelian Sector
The Abelian model of eqn. (4.3) cannot be the whole story. It ignores the factor Φ 1 = exp(−iπη/2) in the partition function and it does not capture the results about the vortices and the non-Abelian statistics of the quasiparticles that we discussed in sections 3.4 and 3.5.
Here we add another sector to account for these issues. As explained in section 3.6.4, this should be a topological theory, which we can refer to as the "Ising system." It has three line observables W 1 , W σ , and W ψ , whose spins are 0, 1 16 , and 1 2 respectively. These lines have the standard non-Abelian statistics of the Ising model.
As a first attempt in coupling the two systems, we do the following. The Abelian theory (4.3) is a Z 4s gauge theory. This means that 2sa + A is a Z 2 gauge field. (More precisely, it can be a spin c connection.) It is subject to the Dijkgraaf-Witten term and the coupling to A (4.2). The same Z 2 gauge field appears in the argument of η. So our action is the sum of the Abelian Z 4s gauge theory and − π 2 η(2sa + A). Let us first assume that the two factors are decoupled and consider the theory of each of them separately. Above we describes the Abelian sector in terms of two U (1) gauge fields a and c (4.3). We could try to do it for also for − π 2 η(2sa + A), introducing two U (1) gauge fields x, y with "action" This is not really a sensible action in the usual sense, because π 2 η(x) is not continuous mod 2π when its argument is a U (1) rather than Z 2 gauge field. However, this "action" does lead to a sensible path integral which moreover reproduces the Ising TQFT, because after performing the integral over y, x is constrained to be a Z 2 gauge field, and then the integral over x reduces to the sum over spin structures of exp(−iπη/2).
Next, we consider the sum of equations (4.3) and (4.8) In order to identify x = 2sa + A, we mod out the U (1) c × U (1) y gauge group by a Z 2 that acts on the two factors. The quotient can be chosen so that y and c are no longer good gauge fields but c = 2c and y = y + c are good U (1) gauge fields. In terms of them, the action becomes with the same gauge field 2sa + A in both the Ising and Abelian sectors. In section 4.4 we will present another action for this system in terms of a Chern-Simons theory of a continuous group without the problem of the first term in (4.11) not being a good action off-shell. But for now, we take a simple lesson from this exercise. The model that we are trying to understand can be constructed by coupling the Abelian sector to an Ising system and dividing by a Z 2 which acts simultaneously on c and on the Ising sector. The operation of taking a Z 2 quotient of the product of the Abelian sector and the Ising system can be described in many different ways. From the RCFT point of view, taking the quotient amounts [66] to extending the chiral algebra by the operator associated with E = W 2s,0 W ψ . From the TQFT point of view, this can be described as gauging a Z 2 global oneform symmetry that acts on the two sectors [61,62]. From the Chern-Simons gauge theory perspective the quotient is obtained [67] by dividing the gauge group by Z 2 , as we have done in the above derivation.
Gauging the one-form symmetry amounts to projecting on lines that have trivial braiding with E = W 2s,0 W ψ , thus making it a "transparent line." (The line operators with trivial braiding with E are the ones that in the above derivation can be written in terms of c and y rather than c and y.) The line observables of the system are restricted to be W na,nc n c even W na,nc W ψ n c even W na,nc W σ n c odd,  respectively. Every one of these lines represents a quasiparticle of our system. Let us compare this description of the quasiparticles to the discussion in section 3.5. First, it is clear that the line W 1,0 represents the elementary w quanta. More generally, the integer n a represents the U (1) a charge denoted by k. The relation W 4s 1,0 = e −2i A of eqn. (4.6) means that k is conserved modulo 4s. Also, the classical object e −2i A in the right hand side has a simple physical interpretation. The field φ has U (1) A × U (1) a charges (2, 4s). Its condensation means that the corresponding line W 4s 1,0 e 2i A has trivial correlation functions and can be replaced by the unit operator. The fundamental fermion χ is represented by W ψ . The Ising relation W 2 ψ = 1 corresponds to χ 2 = 1 in eqn. (3.29). It is consistent with the fact that χ 2 has the same charges as φ, which condensed. The transparent line E = W 2s,0 W ψ , which we used in the Z 2 quotient, is related to the underlying electron (compare with the relation w 2s χ = e in eqn. (3.29)). Note that this line cannot be set to 1. As it represents the electron, it has half-integer spin and as such it is not trivial. Yet, its trivial braiding reflects the fact that the electron can leave the 2+1-dimensional boundary and move into the 3 + 1-dimensional bulk.
Next, we identify the vorticity as v = n c . A justification of this is that in the theory (4.3), an insertion of the line operator exp iv c causes a monodromy of a that would be expected for vorticity v. For nonzero values of v = n c we recover the spectrum of quasiparticles above. For example, the basic vortex corresponds to W 0,1 W σ , whose spin vanishes (see eqn. (4.13)).
Two such vortices can fuse in two possible channels, namely W 0,2 and W 0,2 W ψ , and therefore we have non-Abelian statistics. These two lines represent v = 2 vortices. Since the spins of these two lines are ± 1 4 , the eigenvalues of the braiding matrix are e iπ(2·0± 1 4 ) = e ±iπ/4 , as we found in section 3.5. Similarly, the vorticity v = 3 line W 0,3 W σ has spin 1 2 mod 1. Two such vortices can fuse to the two quasiparticles W 0,6 and W 0,6 W ψ , whose spins are ± 1 4 . Therefore, the eigenvalues of the braiding matrix of two v = 3 vortices are e iπ(2· 1 2 ± 1 4 ) = e ±3iπ/4 . We can also add elementary quanta to the vortices. Adding χ is represented by attaching W ψ to the lines. For odd v this is trivial, because of the Ising fusion rule ψ × σ = σ. This is consistent with the statement above and in section 3.5 that for odd v there is only a single state. For even v, this maps W 0,v ↔ W 0,v W ψ , i.e. it maps between the two states of the vortex. We can also add k w quanta to the vortex by changing n a → n a + k. This shifts the spin of the line by kv/4s mod 1 (see eqn. (4.13)), exactly as expected.
Next, we would like to identify the action of the unbroken Z 4s gauge theory on the lines. Naively, we might expect the Z 4s generator K to be a one-form global symmetry and we can attempt to identify it with e i c . This cannot be right, because the line e i c was projected out by the quotient and the line W σ e i c cannot be used as a one-form generator, because its fusion is non-Abelian. Although we can identify K 2 = e 2i c or K 2 = W ψ e 2i c , it seems that we cannot identify K in the TQFT. This should not be too surprising, since we have seen in section 3.5 that K can be defined in the even vorticity sectors, but only even powers of K can be defined in the odd vorticity sectors.
Finally, let us discuss the action of the discrete symmetries. The charge conjugation symmetry C acts as and it is manifestly a symmetry of our system. Time-reversal is more subtle. From the discussion in section 3, it is clear that time-reversal changes the direction of time but also acts on the gauge fields as 31 This is a symmetry of the c da term in the action, and it is also a symmetry of the remaining Abelian part of the action together with the eta-invariant (the product of those terms, or rather of their exponentials, is the time-reversal invariant expression (−1) I ).
Because of the choice of sign that went into factoring the model in terms of an Abelian sector times a non-Abelian one, time-reversal symmetry is not manifest in our presentation. But the spectrum of quasiparticles that we have found is clearly T -invariant.
Given that s should be even, the simplest TQFT that arises from this construction has s = 2. Precisely this model was constructed 32 in [25], where it was called T 96 . Using a procedure known as "anyon condensation," these authors also constructed another model from it, with fewer quasiparticles. This procedure does not result from any weakly coupled dynamics. In section 4.3, we will present a weakly coupled theory that leads at long distances to that model. But for the time being we would like to describe their procedure using another language. 31 We use an abbreviated notation for the action time-reversal that was described in section 2.1. 32 If one sets b = c + a, then the abelian action (4.3) for s = 2 becomes 1 4π (8bdb − 8cdc) + 1 2π (2Adb), which is U (1) 8 × U (1) −8 with a specific coupling of A, as in [25]. The "anyon condensation" procedure amounts to modding out the gauge group by a discrete subgroup. In our case, the Abelian sector has a U (1) a × U (1) c gauge symmetry and we turn it into U (1) a × U (1) c /Z 2 . We implement the quotient by expressing the Lagrangian in terms of the fields a = 2a and c = c + a and view them as good U (1) gauge fields, while a and c are not. This turns equation . This is a Z 4 gauge theory with k = −2 and it satisfies the spin/charge relation. This is precisely the minimal model of [25] and [28]. 33 We will end this subsection by describing an interesting phenomenon. As in [19,38,39], let us consider taking a magnetic monopole of unit magnetic charge from outside the material through the boundary into the material. Outside the material, the monopole has vanishing electric charge; inside the material, because of the Witten effect, it has electric charge ± 1 2 . It is 33 One way of thinking about this quotient is by identifying a one-form global symmetry of the original theory and then gauging it. What we did here is to gauge the global symmetry that shifts a → a + ζ simultaneously with c → c + ζ with the same Z 2 connection ζ. The theory also has a Z 2 one-form global symmetry that shifts only a → a + ζ. Gauging this symmetry leads to the theory with s = 1 in equation (4.3), but as we remarked there, this would violate the spin/charge relation. The conflict between gauging this Z 2 one-form symmetry (as opposed to the one that acts both on a and on c) and the spin/charge relation can be interpreted as a mixed 't Hooft anomaly between them. expected that the charge is compensated by leaving charge ∓ 1 2 on the surface of the material. In a gapless phase, such electric charge can be deposited on the surface using the charged massless fermions there (this depends on zero-modes that the fermions develop in the presence of a unit of magnetic flux). But how does it happen in the gapped phase? The world-line of the magnetic monopole is an 't Hooft line. It is characterized by having nonzero flux through an S 2 that surrounds it: S 2 F = 2π. This fact has interesting consequences at the point p where the 't Hooft line penetrates the material (the red dot in figure 4). Momentarily ignoring possible production of quasiparticles, the equation of motion of c sets on the boundary 2sda + dA = 0, forcing da to be nonzero near p. Consider a Euclidean S 2 in the surface that surrounds the point p. Assuming that 2sda + dA = 0 on the S 2 , we have S 2 da = − 1 2s S 2 dA = − 2π 2s , which is not properly quantized. This means that there must be a quasiparticle worldline intersecting the surface and contributing a singularity da = 2π 2s δ, with δ a delta function. With this delta function, the integral S 2 da vanishes. The interpretation is that a quasiparticle is emitted from p. Its world-line is described by a line operator that ends at p. More explicitly, this singularity is produced by a line operator whose dependence on c is a factor of exp(−2i c); i.e. this line operator describes a quasiparticle with vorticity v = −2. We would also like the flux integral S 2 dc to vanish. Using the equations of motion, this means that the line W should include a factor of exp(−si a), i.e. the vortex has s quanta of w. Since w has electric charge 1/2s, this accounts for the expected deposition in the surface of electric charge 1/2. We conclude that W = W −s,−2 in the notation of eqn. (4.12); the red dot in figure 4 represents an operator in the TQFT on which such a line can end. We can also discuss the CT image of this picture. Here the emitted line is W s,−2 W ψ . It differs from the previous case by an electron line E = W 2s,0 W ψ . This electron line continues in the bulk along the monopole line (the dashed line) and changes the sign of its electric charge.

More General Models
More general models can be studied in a similar way. For example, let us generalize the model introduced in section 3.1 by adding pairs of fermions χ i and χ i with U (1) A × U (1) a charges (1, 2s + n i ) and (1, 2s − n i ) with n i ∈ Z. Without loss of generality, we take all n i > 0. These charges have been chosen to be compatible with the spin/charge relation and to allow T -and CT -preserving couplings of the form φχ i χ i + w n i χχ i + w n i χχ i + h.c. On the left hand side, the first term describes the unbroken Z 4s gauge theory, the second term represents the contribution of χ, and the terms with the sum represent the contribution of the other fermions. (We explain in section 5.2 that the minus sign in the sum is needed if one wants a formalism without additional eta-invariants.) The right hand side depends only on n = i n i . This Abelian topological field theory is characterized by a k-matrix and charge vector k = 0 4s 4s 2s(s + 2n) , q = 2 s + n . (4.19) We again find a spin/charge anomaly. The classical system satisfies this relation, but the quantum theory satisfies it only when s + n is even.
As in the original model with χ only, we should add the non-Abelian sector associated with η, which can be represented by the Ising topological field theory and a Z 2 quotient.
As above, C acts as in eqn. (4.14) and it is a manifest symmetry of our system. As usual, time-reversal is more subtle. We still take T to reverse the time and to act as a → −a, A → −A, but the action on c should be T : c → c + na . In general, different microscopic theories can lead to the same macroscopic topological theory. We have already seen that a macroscopic description of this class of models depends only on n and not on the individual n i . Additional identifications can be found using the freedom to shift c. Since we need n + s to be even, we can shift c → c − n+s 2 a and write equations (4.18) and (4.19) as  Note that a microscopic theory with even n (including the original model without χ i and χ i ) needs even s and a microscopic theory with odd n needs odd s. Yet the resulting macroscopic theories can be described uniformly (eqn. 4.21).
Can there be other models that lead to additional low energy topological field theories? We can start with U (1) A and some other emergent gauge symmetry with some fermions and scalars. For simplicity, let us limit the search to models associated with a U (1) A × U (1) a gauge symmetry (i.e., with only a single emergent gauge field U (1) a ) and φ with charges (2l, m), where the U (1) A charge of φ has to be even in order to preserve the spin/charge relation. We also allow arbitrary fermions with various charges such there is no anomaly associated with U (1) a and the spin/charge relation of U (1) A is satisfied. The most general Abelian sector resulting from such a theory can be written as a sum of 1 2π cd(ma + 2lA) and a linear combination of terms of the form ada, Ada, and AdA.
Next we impose time-reversal. For that we need to know which Z 2 gauge field the Ising sector couples to. So far, it has been A + 2sa. Here we take it to be more generally lA + m 2 a (and hence m must be even). The motivation for this choice is that an expectation value of φ turns this combination into a Z 2 gauge field. Clearly, other options are also possible. We generalize the action of T to include c → c + ra + 2pA with r, p ∈ Z. Using the fact that the Ising system plus equation (4.3) is T -invariant with c → c, we learn that m = 4s with s ∈ Z, and that the Lagrangian should be 1 8π (8c + 4(ra + 2pA) + 2sa + lA)d(2sa + lA) . If we want anomalous time-reversal symmetry (which means that the coefficient of AdA/4π should be a half-integer), then l should be odd. Using this fact, the spin/charge relation sets r = s mod 2. This allows us to redefine c → c − r+s 2 a and to turn eqn. (4.23) to  The congruence (4.27) can actually be understood based on matters that are explained in section 2.3 and Appendices A.2 and A.3. Let us first consider a conventional topological insulator, whose bulk action, in the notation of Appendix A.2, has θ 1 = π and θ 2 = 0. Thus the bulk action is πI 1 = π X A(R) + 1 2 F ∧F (2π) 2 . In addition to this bulk action, there may be boundary contributions in the effective action. These boundary terms must be properly normalized three-dimensional Chern-Simons couplings. The coupling πI 1 has a metric-dependence because of the A(R) term. However, this metric dependence is just right to cancel the metric dependence of the term −πη/2 in the effective action; this is ensured by the APS index theorem. This cancellation is a consequence of time-reversal symmetry (without which the partition function of a gapped boundary state can depend on the boundary metric, via the framing anomaly of 3d TQFT). Accordingly, any additional boundary contributions to the effective action must be independent of the metric and can only depend on A and a. In particular, as explained in section 2.3 and Appendix A.3, for a spin c connection A, a properly normalized Chern-Simons coupling of the form AdA must be an integer multiple of (8/4π) AdA; in other words it must contribute to k c an integer multiple of 8.
Based on this, it might seem that k c should be congruent to 1/2 mod 8. However, we should also consider the possibility described in Appendix A.2 of a material with θ 1 = θ 2 = π. Such a material is not a standard topological insulator, but something more exotic. The coupling πI 2 contributes 4 to the effective value of k c . The boundary Chern-Simons couplings will still contribute a multiple of 8. So a material with θ 1 = θ 2 = π will have a boundary state with k c ∼ = 9/2 mod 8.
A microscopic construction of the models with (p, l) = (0, 1) was given above. Other values of (p, l) can also be found. For example, rescaling all the U (1) A charges in the models above by l, we can construct the low energy theory with p = 0 and arbitrary l. Another option is to start with φ and χ with U (1) A × U (1) a charges (2l, 4s), (3l, 6s) with coupling φ 3 χχ. For s even this leads to the effective theory (4.24) with p = 1 and arbitrary l. We conclude that a large class of microscopic models leads at long distances to the same theory (4.24), which is labeled by s ∈ Z and the integers (p, l) as in eqn. (4.25). As we said above, by enlarging the emergent gauge symmetry U (1) a and by adding several φ-like fields, many additional models can be constructed.
Finally, we would like to show that (as remarked at the end of section 4.2) our simplest topological field theory parameterized by s = l = 1, p = 0 describes the minimal model of [25] and [28]. To see that, substitute s = l = 1, p = 0, and a = b + 2c in eqn. (4.24) to find 34 This is a U (1) 8 × U (1) −2 theory with only the U (1) 8 field c coupled to A, as in [25] and [28]. Surprisingly, our simplest microscopic models with a single χ field lead to more complicated low-energy theories (or at least low-energy theories with more quasiparticles), because they need even s. And we derive the simplest topological theory (4.28) using a more complicated microscopic model with at least three fermions. For s = 1, the model just described is the only one in the family (4.24) that obeys the appropriate congruence conditions for a standard topological insulator. For s = 2, there are two possibilities: l = 1, p = 0, or l = 3, p = 1. These models both have abelian sectors U (1) 8 × U (1) −8 (with different couplings to A) and as noted in section 4.2, the l = 1 model is the model T 96 of [25].

An Explicit Chern-Simons Lagrangian for the Whole System
Here we give an explicit local Lagrangian that combines the Abelian and non-Abelian sectors in the above analysis and describes the whole system. The Abelian Z 4s sector is represented by the U (1) a × U (1) c gauge theory (4.24). The Ising system can be viewed as a coset theory [68]. Such a coset can be described [67] by an SU (2)  where u is a gauge field of U (2) = (SU (2) × U (1))/Z 2 . (See Appendix C for more details.) The overall sign of these terms is related to the orientation of the Ising system and was set to agree with the discussion of our microscopic models. Note that even though eqn.   The field e describes a decoupled trivial U (1) −1 theory and we included A in its coupling to make the spin/charge relation manifest. It is easy to check that the first two terms in eqn. (4.31) depend only on the traceless part of u; i.e. the field Tr u does not have a quadratic kinetic term. This part of the theory can be viewed as U (2) 2,0 . (Note that this theory is spin, but the coupling to A is such that A can be a spin c connection.) Tr u is a Lagrange multiplier implementing d(2sa + lA) = 0. It leaves an unbroken Z 4s ⊂ U (1), whose gauge field is a. This Z 4s gauge theory has k = 2s 2 . Since e decouples, the whole system is effectively SU (2) 2 × (Z 4s ) k=2s 2 /Z 2 . It is straightforward to check that this simpler theory leads to the correct spectrum of lines.

Some Additional Topics
In this section, we describe a very simple T-invariant topological field theory and provide some more detail on models were described in section 4.
We consider a U (1) 2 × U (1) −1 Chern-Simons gauge theory with the Lagrangian This theory gives an explicit realization of the semion-fermion theory of [69], denoted SF. It has four non-trivial lines generated by W 1,0 = exp(i a) and W 0,1 = exp(i b) (see Appendix C), whose spins modulo 1 are 1 4 and 1 2 , respectively. In the language of [69], W 1,0 is the semion s and W 0,1 is the fermion f . The lines satisfy W 2 1,0 = W 2 0,1 = 1. It is easy to check that this The action on the lines is T : We combine this with the usual T(A) = −A. Under this time-reversal transformation, the theory based on (5.4) is invariant, provided we add to the Lagrangian 1 8π AdA. As in our previous examples, we see that this theory is T-invariant with the same anomaly as a topological insulator; it is T-invariant when coupled to a 3 + 1-dimensional bulk with θ = π.
Both the coupling of A in the Lagrangian (5.4) and the transformation laws (5.5) are incompatible with the spin/charge relation. Therefore, the theory needs a spin structure. A must be a U (1) gauge field rather than a spin c connection. More generally, the original theory (5.1) needs a spin structure. It can be coupled to a spin c connection A by adding to it 1 2π Adb (rather than 1 2π Ada, as in eqn. (5.4)), but then the T transformation (5.2) cannot be extended to nonzero A.
To explore this theory further, let us set A = 0 and place it on a spatial torus T 2 and choose a basis of one-cycles labeled by a, b and a spin structure labeled in an obvious way by s a,b = ±1. The Hilbert space H of our theory is two-dimensional, since U (1) 2 has two states on T 2 and U (1) −1 (for any spin structure) has only one. A line operator wrapping a 1-cycle in T 2 gives an operator that acts on H, and in a suitable basis we have This action of W 1,0 is standard and follows (up to a choice of basis) from the fact that W 2 1,0 = 1 and {W a 1,0 , W b 1,0 } = 0. The form of W 0,1 around the two cycles was found by recalling that W 0,1 is the worldline of a spinor and is otherwise trivial.
Time-reversal should act as a 2 × 2 matrix T combined with complex conjugation. Because of eqn. (5.3), we should have We solve these (up to an overall sign) by for s a = 1 . We conclude that the two states are Kramers singlets in the case of an even spin structure (either s a , or s b , or both are −1), but form a Kramers doublet for the odd spin structure (s a = s b = 1). Despite the difficulty with the spin/charge relation, this model has various potential applications. In eqn. (4.28), we have already encountered U (1) 2 (with opposite orientation and without the coupling to A) as a factor in a T-invariant system that satisfies the spin/charge relation. The semion-fermion theory SF has been used in [69] in studying topological superconductors and we will encounter it in that guise in section 6.1. As another possible application, by multiplying the action (5.4) by 2 (or by any even integer), we get a T-invariant theory that no longer needs a spin structure and might be a boundary state for the bosonic topological insulators of [32]. (After rescaling the action, T 2 is still the operation (a, b) → (−a, −b), but this is now a nontrivial global symmetry.)

More On Some Models From Section 4
In section 4.3, we described models that can be constructed by adding additional pairs of Dirac fermion fields to the basic model of section 3.1. Here we will supply some more detail. For brevity we consider the case of adding to the original χ field of U (1) A × U (1) a charges (1, 2s) a single additional pair χ and χ of charges (1, 2s + n) and (1, 2s − n).
The first question that we want to address is the T and CT transformations 35 of χ, χ , and χ and the Yukawa couplings that make possible both Higgsing to the standard gapless boundary state and also to the gapped boundary state described in section 4.3.
In general, if χ is a charged Dirac fermion coupled to A and/or a, then given that A and a transform under T as in eqn. (2.3), the transformation of χ under T is essentially uniquely determined. When χ is expanded as a sum of two Majorana fermions, one will have to transform under T with a factor of +γ 0 and one with a factor of −γ 0 , as in eqn. (2.8); after rotating to the right basis, the transformation of χ under T will then be as in eqn. (2.9). By contrast, the two Majorana components of χ will transform the same way under CT, and there is a nontrivial choice of whether they will both transform as +γ 0 (as in eqn. (2.11)) or as −γ 0 . This choice will determine whether χ contributes +2 or −2 to ν CT . The contribution of χ to ν T is always 0.
We would like to choose the signs in the CT transformations of χ, χ , and χ so that Higgsing to the conventional boundary state of a topological insulator will be possible. This means that the signs must be such that the net contribution to ν CT will be +2. One might think that the "new" fields χ and χ should make canceling contributions to ν CT , and that the +2 comes from the original χ field. It turns out, however, that to make possible the Higgsing that we want, χ should contribute −2 and χ , χ contribute +2 each.
In one phase, the scalar field w of charges (0, 1) has an expectation value, breaking U (1) A × U (1) a to U (1) A . After this breaking, χ, χ , and χ are all simply Dirac fermions of ordinary electric charge 1. To reduce to the standard gapless boundary state of a topological insulator, we want w to couple to these fermions in such a way that two linear combinations become massive, leaving only one massless Dirac fermion. In general, let χ 1 and χ 2 be Dirac fermions of charges (1, p 1 ) and (1, p 2 ), transforming in the standard way under T. For definiteness suppose that p 1 ≥ p 2 . Then in the phase with w = 0, χ 1 and χ 2 can get a mass from a T-invariant coupling This is hermitian, 36 and each term is separately T-invariant. The two terms are exchanged by CT, which is a symmetry if and only if χ 1 and χ 2 transform with opposite signs under CT.
In the phase with φ = 0, we want all fermions to get masses by Yukawa couplings to φ. The only such couplings allowed by U (1) A × U (1) a gauge symmetry are φχχ + h.c. and φχ χ + h.c. We have already discussed the T, C, and therefore also CT invariance of the former coupling in discussing eqn. (3.8). The same reasoning applies to a similar coupling φχ χ + h.c. and shows that it is CT-invariant if and only if χ and χ transform the same way under CT. 35 We study these symmetries only at the classical level and do not analyze quantum anomalies on an unorientable manifold. 36 The difference from the situation described in footnote 38 is that as χ 1 and χ 2 are assumed to transform with opposite signs under CT, this expression is CT-conserving.
So we must choose χ and χ to each contribute +2 to ν CT , and hence χ to contribute −2. The allowed couplings involving w are, for n > 0, w n χχ + h.c. and w n χχ + h.c. Such couplings yield a mass matrix of rank 2, leaving one massless Dirac fermion in the phase with w = 0.
Next, we would like to justify more precisely the treatment of the model that was given in section 4.3. We recall from section 3.6 that the partition function of the original χ field of charges (1, 2s) can be factored as This description was used in section 4.3.
Finally, we will discuss the quantization of the monopole operators, to recover the result found in section 4.3: the model obeys the spin/charge relation if and only if s + n is even. The discussion of eqns. (3.14)-(3.16) is a useful starting point. There we found that in quantization on S 2 in the presence of a single flux quantum of f = da, the original χ field has a state |↑ of spin 0 and charges (−s, −2s 2 ). The fields χ and χ have analogous states |↑ and |↑ of respective charges (−(s + n/2), −2(s + n/2) 2 ) and (−(s − n/2), −2(s − n/2) 2 ). Adding these values, the combined system in the presence of one unit of flux has a state | ↑↑ ↑ of charges (−3s, −6s 2 − n 2 ). Starting from this state, there are many ways to construct a state that is invariant under the emergent gauge symmetry U (1) a , and hence will correspond to a gaugeinvariant monopole operator. One simple choice (not the choice that we made in section 3.2) is to act with 6s 2 +n 2 powers of w, which has charges (0, 1). Here we have to remember that w has half-integral spin in the field of a magnetic monopole of U (1) a . Hence the state w 6s 2 +n 2 | ↑↑ ↑ has electric charge −3s and has spin (6s 2 + n 2 )/2 mod 1, or equivalently n/2 mod 1. This state obeys the spin/charge relation if and only if s + n is even. If this one gauge-invariant state obeys the spin/charge relation, then so do all gauge-invariant states in this sector, since they can be obtained by acting on this state by a product of gauge-invariant products of elementary fields.

Preliminaries
With a few simple twists, the models constructed in this paper can be interpreted as gapped symmetry-preserving boundary states of a topological superconductor.
The main difference, for our purposes, between a superconductor and an insulator is that pairing occurs in the superconductor, reducing the gauge group of electromagnetism from U (1) to Z 2 . This means that in a superconductor, the electromagnetic gauge field A is gaugeequivalent to −A. Accordingly, there is no natural role for a symmetry C that reverses the sign of A, and no natural a priori distinction between the time-reversal symmetries that we have called T and CT.
Apart from the residual Z 2 of electromagnetism, the important symmetry in a topological superconductor is time-reversal. As in our discussion of the parity anomaly in section 3.6.1, here the flat space theory is T-invariant, but there is an anomaly after coupling to a background field. In this case the background field is the metric and the anomaly is visible only when the theory is placed on a non-orientable manifold.
In adapting the models of this paper to a topological superconductor, either T or CT can play the role of time-reversal. However, the more interesting case is that the time-reversal symmetry of the topological superconductor is identified with what until now has been called CT. The reason for this is simply that the models of this paper have ν T = 0, which means that if they are interpreted as boundary states of a superconductor with T as the time-reversal symmetry, then this is a topologically trivial superconductor. However, the same models have ν CT = 2, meaning that if they are interpreted as boundary states of a superconductor with CT as the time-reversal symmetry, then this is a topologically non-trivial superconductor with ν sc = 2.
Hence, in applications to a topological superconductor, we will take CT as the time-reversal transformation. However, the name CT is rather misleading in this context, since there is no natural notion of C in a superconductor, so we will rename CT as T sc . (It might cause too much confusion with the rest of the paper if we rename CT as T in this section only.) In our analysis so far of boundary states of topological insulators, the spin/charge relation of condensed matter physics has provided an important constraint. By contrast, for models of the class considered in this paper, there is no such constraint for a topological superconductor. We will give two explanations of this fact.
First of all, in any of our models of topological insulator boundary states that violate the spin/charge relation, that relation could be restored (as discussed in footnote 30) by adding an additional coupling This coupling shifts by 1 the electric charge of any state with a unit flux of da/2π, so it restores the spin/charge relation, if that relation is not otherwise satisfied. But the coupling (6.1) is T-violating, so it is forbidden in the context of a topological insulator. However, in a superconductor, since A is gauge-equivalent to −A, the coupling of eqn. (6.1) is actually T sc -conserving. By adding this coupling, if necessary, one can eliminate any problem in the spin/charge relation.
For a second explanation, consider a spin manifold with some chosen spin structure and with a corresponding spin connection D 0 . Let A be a Z 2 gauge field. Then D 0 + A is the spin connection of some other spin structure, which we will call the effective spin structure. So any boundary theory that makes sense on a general spin manifold can be interpreted as a boundary state of a superconductor, simply by coupling it to the effective spin structure. The spin/charge relation is manifestly obeyed, since the boundary degrees of freedom couple only to the effective spin structure.
Thus, in the original model with a single χ field, for applications to a topological superconductor, we can consider odd as well as even values of s. For a minimal example, we return to eqn. (4.3), where now we set s = 1 and omit the coupling to A (since A is now absorbed in the effective spin structure that is used in defining the Chern-Simons action and the η-invariant). Setting b = a + 2c, we can rewrite (4.3) as

Varying ν sc
In short, all models so far studied in this paper, including models that violate the spin/charge relation in the context of a topological insulator, can be interpreted as boundary states for a topological superconductor with ν sc = 2.
We can easily make boundary states with any even value of ν sc just by taking tensor products of a number of ν sc = 2 models. (The methods of this paper do not suffice to make boundary states of a topological superconductor with odd ν sc .) We instead will consider the following simple variant of that idea. We generalize the models of section 3.1 with a single χ field simply by introducing r identical fermion fields χ i , i = 1, . . . , r, with exactly the same charges (1, 2s) under U (1) A ×U (1) a and exactly the same transformation under T and CT as in section 3.1. This will give a model with ν sc = 2r. We assume that the χ i all have the same coupling to φ as in the familiar model with r = 1: Then in the phase with φ = 0, the model is gapped. (This model is not the same as the tensor product of r copies of the original model, because the various χ i all couple to the same a, w, and φ.) 37 In section 5.1, we described SF as U (1) 2 × U (1) −1 . The role of U (1) −1 was to convert U (1) 2 into a spin theory (thus providing the "transparent" fermion of the semion-fermion theory SF) while also canceling the framing anomaly of U (1) 2 . In the present context, T-Pfaffian is already a spin theory, so we do not need a U (1) −1 factor to provide a transparent fermion, and moreover the microscopic construction with χ makes manifest that the model has no framing anomaly. However, the fact that we are getting the transparent fermion of SF from T-Pfaffian means that it may be slightly oversimplified to identify our theory (U (1) −8 × Ising)/Z 2 × U (1) 2 as T-Pfaffian×SF.
Our main interest is to apply this model to a topological superconductor (in which case we care about only T sc = CT and not T), but if r is odd, the model also provides a possible boundary state for a topological insulator with θ = π. So we will start the analysis keeping track of all symmetries, and only later concentrate on the application to the topological superconductor.
For any r, the partition function of this model, on an orientable manifold, for a given choice of the Z 2 gauge field 2sa + A, can be computed without further ado. It is simply the r th power of what we computed in eqn. (3.38) (or its generalization (3.41)). In the gapped phase, the partition function with a single χ field is simply (−1) I , so if the number of χ fields is r, the partition function is (−1) rI . This depends only on the value of r mod 2, strongly suggesting that the topological field theory describing the model depends on r only mod 2 at least if we restrict to orientable manifolds. For odd r, this topological field theory should thus be the one that we studied in section 4; for even r, since the partition function is identically 1, the topological field theory is simply (on orientable manifolds) a trivial version of Z 4s gauge theory with no Dijkgraaf-Witten term.
There is actually a simple way to prove that the topological field theory derived from this model only depends on the value of r mod 2. While maintaining T-invariance, it is possible to add a bare mass term for two of the χ i that preserves all symmetries except CT. For example, we can take with m real. 38 Once the system is gapped for φ = 0 using the Yukawa couplings (6.3), it remains gapped if the bare mass term I bare is turned on. This is true for all m. For m → ∞, χ r−1 and χ r decouple and we get the same model again but with r replaced by r − 2. But in general, the topological field theory that describes a gapped system at low energies is invariant under deformation of parameters as long as the system remains gapped. Hence, as suggested by the discussion of the partition function in the last paragraph, the low energy topological field theory is the same for any odd number r of χ fields. Since the trajectory in theory space by which we interpolated between r and r − 2 is CTviolating, it is natural to suspect that the theories with different values of r are in different universality classes and can actually be distinguished by some observables that are sensitive to CT symmetry. Such an observable is actually the number of fermion zero-modes mod 8 in a vortex field, which determines subtle properties of the quasiparticles. In a basic v = 1 vortex, each of the χ i has a single Majorana zero-mode, making a total of r zero-modes, all transforming the same way under CT. In the absence of CT symmetry, generic perturbations can add to the Hamiltonian a term bilinear in the zero-modes, lifting the zero-modes in pairs. So without CT symmetry, a universality class is only characterized by the value of r mod 2. (T relates v = 1 to v = −1, and so places no constraint on what happens for v = 1.) But in the CT-conserving case, just as in the theory of the Majorana chain [56], a perturbation to the Hamiltonian can only lift zero-modes in groups of 8. Therefore the number mod 8 of zero-modes in a vortex field is an invariant of a CT-conserving universality class. It is equal to the value of r mod 8.
The anomaly that arises if CT is used to place a theory on an unorientable manifold depends on ν CT mod 16. Since ν CT = 2r, it follows that on an unorientable manifold, the partition function (and even its anomaly) depends on r mod 8. Presumably, if we understood how to generalize the low energy topological field theory to an unorientable manifold, it would capture this information. The arguments given above claiming to show that the low energy theory depends only on r mod 2 involve either computing the partition function on an orientable manifold or making a deformation that violates CT. So these arguments do not tell us about the low energy description when CT is used to go to an unorientable manifold. Now let us apply all this to a topological superconductor, meaning that we set T sc = CT, and take A to be of order 2. For a basic example, we also set s = 1, leading (if r = 1) to T-Pfaffian×SF, as described in section 6.1. If r is even, we get a model with abelian statistics that can be a boundary state of a topological insulator with ν sc = 2r divisible by 4. Here we will focus on the case of odd r, corresponding to ν sc congruent to 2 mod 4.
The associated topological field theory is always T-Pfaffian×SF, but with four possible actions of T sc , depending on whether we take r = 1, 3, 5, or 7, and always with ν sc = 2r. Indeed, it has been claimed in [69] that there are two possible actions of time-reversal on T-Pfaffian and two on SF, combining to four possible actions on T-Pfaffian×SF. The four possibilities have been denoted T-Pfaffian ± ×SF ± . Moreover, these four cases have been claimed to correspond to ν sc = 2, 6, 10, and 14 -in other words, to ν sc = 2r with r = 1, 3, 5, 7.
Let us compare our construction in more detail to the assertions in [69]. First of all, the theories T-Pfaffian + and T-Pfaffian − are supposed to correspond respectively to the case that the basic vortex of vorticity v = 1 is a Kramers singlet or a Kramers doublet under timereversal. In our approach, this vortex has r fermion zero-modes (all of angular momentum zero). Quantizing those zero-modes gives the spinor representation of Spin(r). This representation is real for r = 1, 7 and pseudoreal if r = 3, 5. The real and pseudoreal cases allow respectively an antiunitary symmetry of square 1 or −1. So the first factor of T-Pfaffian ± ×SF ± is T-Pfaffian + if r = 1, 7 and T-Pfaffian − if r = 3, 5. This seems to agree with what is claimed in [69]. However, we should note the following subtlety. Similarly to what is explained in section 3.5.3, the antiunitary symmetry that can be defined in a sector of v = 1 actually depends on the value of r mod 4; it is really T sc if r = 1, 5, and T sc K if r = 3, 7. (In each of these statements, T sc is the effective time-reversal symmetry of the low energy phase and obeys T sc 2 = (−1) F K.) On the other hand, the distinction between SF + and SF − is supposed to be as follows. Let s be the "semion," the unique nontrivial quasiparticle of U (1) 2 , and f the transparent fermion (which corresponds to a line operator of U (1) −1 in the approach of section 5.1, and to χw 2 in the microscopic construction based on the χ field). The two quasiparticles s and sf have spins ±1/4 and are exchanged by T sc . SF + and SF − are distinguished by the sign of T 2 sc in acting on these two states. To analyze this problem in our framework, we first must identify s and sf in our approach. Here s is supposed to correspond to the non-trivial line exp(i b) = exp(i (a + 2c)) of U (1) 2 (we recall that in eqn. (6.2), the U (1) 2 gauge field is b = a + 2c). The coefficient of c in the exponent on the right hand side is the vorticity v = 2. Likewise, sf has vorticity 2, since f = χw 2 has vorticity 0.
For r = 1, the vortex of v = 2 was analyzed in section 3.5. It has two fermion zero-modes, whose quantization leads to two states, denoted Λ ± , with spins ±1/4 and k = ±1. They are exchanged by χ. (These two states can be further dressed with powers of w, and this will be important presently. In general, these two states have k = ±s; we set s = 1.) More generally, for any r, the v = 2 vortex has 2r fermion zero-modes. Quantization of these zero-modes leads again to two quasiparticles Λ ± , which now correspond to the two spinor representations of Spin(2r). They have 39 spin ±r/4 and k = ±r. However, in topological field theory, we only care about the value of the spin mod 1, and similarly (for s = 1) we only care about k mod 4. So in fact, for odd r, the two quasiparticles obtained by quantizing the fermion zero-modes have spin ±1/4 and k = ±1, just as for r = 1. Moreover, for odd r, the two spinor representations of Spin(2r) are complex conjugates, so Λ ± are exchanged by any antiunitary symmetry.
The states Λ + and Λ − are actually part of the T-Pfaffian theory; they correspond to I 2 and ψ 2 in the terminology of [69]. By contrast, the states s and sf of SF have k = 0, and correspond to wΛ + and f wΛ + = χwΛ + = wΛ − . (We use the fact that the transparent fermion is f = χw 2 and that χΛ ± = Λ ∓ , since the chirality operator that distinguishes Λ + and Λ − anticommutes with χ. We also use ww = 1.) Because of time-reversal symmetry, there is an arbitrary choice of which of s and f s corresponds to wΛ + and which to wΛ − . We will set s = wΛ − , sf = wΛ + . We will determine the action of T sc 2 on both pairs Λ + , Λ − and wΛ − , wΛ + . To determine how T sc 2 acts on Λ + and Λ − , we need to know whether the representation of a Clifford algebra of rank 2r (which we get by quantizing the 2r fermion zero-modes) is real or pseudoreal. Because of the mod 8 periodicity of the Clifford algebra, the answer depends only on the value of r mod 4. For r ∼ = 1 mod 4, the representation is real (as we have noted above for r = 1), leading to T sc 2 = 1, and for r ∼ = 3 mod 4, it is pseudoreal, 40 corresponding to T sc 2 = −1. (In [69], it is asserted that the action of T sc 2 on I 2 and Ψ 2 is ill-defined, but in our framework this seems to be well-defined.) So in general, on these states, T sc 2 = (−1) (r−1)/2 . Given this, to understand how T sc 2 acts on s = wΛ − and sf = wΛ + , we just need to know how it acts on w and w. This is determined by T sc 2 = (−1) F K. Since w and w are invariant under (−1) F but K(w) = iw, K(w) = −iw, we get T sc 2 (s) = i(−1) (r−1)/2 s, T sc 2 (f s) = −i(−1) (r−1)/2 f s. This agrees 41 with [69] if we assume that (as suggested by the values of ν sc claimed in that paper), SF + corresponds to r = 1, 5 and SF − corresponds to r = 3, 7.
An interesting detail is that the sign with which T sc 2 acts on s, sf is correlated with the sign with which it acts on I 2 = Λ + , ψ 2 = Λ − , even though one pair of states is in SF and one is in T-Pfaffian. Hence, although the model under consideration admits four possible actions of T sc , corresponding to r = 1, 3, 5, 7, it is not clear that this can be understood in terms of two possible actions on SF and two on T-Pfaffian. This may be related to the following. It may be oversimplified to interpret the theory under consideration here as T-Pfaffian×SF, because as observed in footnote 37, in the present context the transparent fermion of SF is actually a quasiparticle χw 2 of T-Pfaffian.
Unfortunately, our methods do not give a natural way to study T-Pfaffian ± by itself so we have no good way to directly compute ν sc for TPfaffian ± . Moreover, though SF can be 39 The zero-modes of each of the fermion fields χ 1 , . . . , χ r contribute equally, so the spins and the values of k of Λ ± are r times what they are for r = 1. 40 It suffices to take r = 3, corresponding to a Clifford algebra of rank 6. We can represent 6 gamma matrices by An antiunitary symmetry T sc that commutes with these matrices is, up to an irrelevant phase, T sc = Kγ 1 γ 2 γ 3 (where K is complex conjugation), satisfying T sc 2 = −1. 41 We should note that the overall sign of T sc 2 acting on the pair s, f s depends on some arbitrary choices. We made one arbitrary choice of whether to identify s with Λ + or Λ − . There is also an arbitrary choice of whether to identify the time-reversal symmetry of [69] with what in our language is T sc or the equally good antiunitary symmetry T sc K −1 , which obeys (T sc K −1 ) 2 = (−1) F K −1 .) usefully described as U (1) 2 × U (1) −1 as in section 5.1, the fact that off-shell T 2 = 1 in this model prevents us from being able to straightforwardly formulate the model on an unorientable manifold; and this in turn leaves us with no straightforward way to directly compute ν sc for SF ± .

More On T-Pfaffian
T-Pfaffian has also been studied from another point of view, motivated by applications to topological insulators. We would like to analyze this in the context of the present paper.
For our starting point, we simply copy a model from eqn.
(2) of [30] (the same model has also been considered elsewhere, e.g. [70]). The model contains an emergent U (1) gauge field a and a composite Dirac fermion Ψ cf . The Lagrangian density is A µ ∂ ν a λ . Similarly, we have to reverse the roles of T and CT in (2.9) and (2.11): By contrast, so far in this paper, all gauge fields and fermions have had the standard T and CT transformations. The transformations in eqn. (6.7) correspond to ν T = 2, ν CT = 0, and these values will be unaffected by the redefinitions below. The value of ν CT is not important for a topological insulator, but ν T mod 16 is a meaningful invariant of a T-invariant condensed matter system, and we observe that it differs from the value ν T = 0 appropriate to the usual topological insulator. (This obstruction to interpreting the model (6.5) as a boundary state of a topological insulator will not be affected by modifications of the model that we consider shortly.) If we assume that a obeys standard Dirac quantization, the model has several difficulties. Ψ cf couples to a with odd charge, which induces the usual T and CT anomaly. Since a is supposed to be an emergent gauge field that only lives on the boundary of a 3 + 1-dimensional system, we cannot compensate for this with the help of a bulk coupling. Likewise the Ada coupling is not properly quantized. We can resolve both of these issues if we assume that the flux quantum of a is 4π rather than the usual 2π. With this in mind, we write a = 2a where a obeys standard Dirac quantization. Dropping the prime, we thus rewrite the Lagrangian as A µ ∂ ν a λ . (6.8) This version of the model has also been considered previously (see eqn. (36) in [71]).
This theory is well-defined, but it is not a boundary state for a topological insulator since it lacks the usual AdA anomaly in time-reversal symmetry. We can get a theory that does have the appropriate anomaly if we simply replace 2a everywhere with 2a + A. This gives the version of the model that we will actually study: In perturbation theory, the two models (6.8) and (6.9) are equivalent, since we can go from the first to the second by the change of variables a = a + 1 2 A (where again we drop the prime from a ). But the models are really not equivalent since a = a + 1 2 A is not a valid change of variables. On the contrary, in going from (6.8) to (6.9) we have changed the Dirac condition on quantization of magnetic flux. (In fact, if A is a spin c connection, the coefficient of A in such a change of variables should be an even integer.) Actually, because Ψ cf now has electric charge 1 with respect to A, we now have a model that can be a boundary state for a T-conserving system in which the electromagnetic thetaangle in bulk is θ = π. We now want to ask two questions about the model: (1) Is it Tand CT-invariant? (2) Is it consistent with the usual spin/charge relation of condensed matter physics?
Concerning the first question, if A is understood as a U (1) gauge field, then the model is Tand CT-invariant. Indeed, the starting point (6.5) was invariant classically under the T and CT transformations (6.6). By following through the various changes of variable, one can find the classical T and CT symmetries of (6.9). They are obtained by just replacing a by a + A on the right hand side of (6.6): (6.10) Although this transformation is satisfactory if A is a U (1) gauge field, it is not if A is a spin c connection. In that case the transformation (6.10) is not well-defined: T or CT would map a gauge field a to a spin c connection ±(a + A). Therefore, as it stands, this model cannot, while also maintaining T-invariance, satisfy the spin/charge relation in the strong sense that A can be understood as a spin c connection. An interesting manifestation of the difficulty arises if we add a scalar field w of charges (0, 1) under U (1) A × U (1) a , to make possible Higgsing to the standard boundary state of a topological insulator. Then T will relate w to another scalar field of charges (1,1), violating the spin/charge relation. (If both w and its T conjugate have expectation values, maintaining T invariance, then electromagnetic gauge invariance is spontaneously broken.) As in section 4, the low energy topological field theory description of this model includes an Ising sector that involves an eta-invariant of the Dirac operator coupled to A + 2a. It is coupled by a Z 2 quotient to an Abelian sector that can be constructed as in section 4.1. In fact, the only difference from the s = 1 case of eqn. (4.3) is that in the action of the Abelian sector, we have to include the Chern-Simons terms that are already present in L. The action of the Abelian sector is thus If we set b = a + 2c, this becomes 1 4π (−8cdc + 2bdb) + 1 2π 2Adc − 1 8π AdA. (6.12) After including the Ising sector, this differs from the familiar T-Pfaffian model by a decoupled U (1) 2 of b.
The coupling to A is not relevant for applications to a topological superconductor, but it is certainly relevant for the topological insulator. As we have seen, with this coupling, the theory cannot simultaneously be T-invariant and consistent with the spin/charge relation. Its low energy description differs from the T-Pfaffian theory by a U (1) 2 factor that suffers from a similar problem, and by the value of ν T . It may well be that, as proposed in the literature, T-Pfaffian by itself is a satisfactory boundary state for a topological insulator. Unfortunately, in the context of the present paper, we do not have a natural procedure to generate T-Pfaffian without the U (1) 2 factor. gravity). Its index is The right hand side of this formula is therefore again an integer (not necessarily even, in this case). Subtracting these two formulas to cancel the A(R) term, we learn that on a spin manifold X, if F is a closed two-form that obeys Dirac quantization (so that it is the field strength of some U (1) gauge field A), then What we have gained by assuming that X is spin is the factor of 1/2 in front of this formula. If X is not necessarily spin, then X F ∧ F/(2π) 2 is an integer but not necessarily even. Simple examples show that on a spin manifold, X F ∧ F/(2π) 2 has no divisibility beyond what is claimed in eqn. (A.4). Next, suppose X is not necessarily a spin manifold, but a more general spin c manifold with spin c connection A. In this case, X A(R) is no longer an integer, but σ is still an integer, so we deduce from eqn.
The Dirac operator / D for a spin 1/2 particle coupled to A with charge 1 is still defined; its index I( / D) is still an integer and is still given by eqn. (A.3). So also This can also be proved by writing the left hand side as 1 8 X (2F/2π) ∧ (2F/2π), and observing that as 2F/2π is an integral class, the integral X (2F/2π) ∧ (2F/2π) is an integer.
A canonical example of a four-manifold that is not a spin manifold is X = CP 2 . The second Betti number of X is 1, so X has essentially only one interesting two-cycle, which is a copy of CP 1 ⊂ CP 2 . Let G be a closed two-form on X normalized to have unit Dirac flux on CP 1 : Then a standard topological argument shows that CP 2 is not a spin manifold, but it has a spin c structure with a spin c connection A whose field strength F = dA satisfies F = G/2. From this and eqn. (A.7), we learn that We see that the quantization in eqn. (A.6) cannot be strengthened. We can learn the same from the fact that CP 2 has σ = 1, which implies that the denominator in eqn. (A.5) cannot be reduced.
To summarize, a four-dimensional manifold X has For an ordinary U (1) gauge field A, the instanton number is If A is a spin c connection, then

A.2 Theta-Angles
It follows from what we have said that a four-dimensional field theory on a spin c manifold X has two θ-like parameters, corresponding to a topological interaction θ 1 I 1 + θ 2 I 2 , with We have normalized I 1 and I 2 to be integer-valued, so θ 1 and θ 2 both have 2π periodicity. Tand CT-invariance hold when both θ 1 and θ 2 equal 0 or π.
In equation (A.13), we have written the topological interactions in one useful basis. Another useful basis is given by I 1 and I 2 = 8I 1 − I 2 : (A.14) This has the advantage that I 2 is a purely gravitational coupling. A coupling πσ is the appropriate gravitational analog of θ = π for a system that obeys the usual spin/charge relation. We note that in the Tand CT-conserving case, with both θ 1 and θ 2 being 0 or π, θ 1 I 1 + θ 2 I 2 is equivalent to θ 1 I 1 + θ 2 I 2 . (This is certainly not true for generic values of the θ i .) The interior of a conventional topological insulator has θ 1 = π, θ 2 = 0. It is also possible to conceive a material with θ 1 = θ 2 = π. Such a material has a bulk electromagnetic θ-angle of π, like a conventional topological insulator, but the fact that θ 2 = 0 means that its allowed boundary states are different.
The topological interaction at θ 1 = θ 2 = π is This is precisely I( / D 3 ), the index of the Dirac operator / D 3 for a spin 1/2 particle of electric charge 3. This observation leads to an easy construction of a gapless boundary state for a material with θ 1 = θ 2 = π. We simply assume existence on the boundary of a massless Dirac fermion of electric charge 3. This leads to a T-anomaly by the same logic as applies for a boundary fermion of charge 1. Following the reasoning of section 3.6.2, to restore T-invariance we need in the path integral a bulk factor (−1) I( / D 3 ) = exp(iπI( / D 3 )). But this coincides with exp(iπI 1 + iπI 2 ), so the charge 3 fermion is a possible boundary state for a T-invariant system with (θ 1 , θ 2 ) = (π, π). Similarly, for any integer k, a massless Dirac fermion of charge 8k ± 1 is a possible boundary state at (θ 1 , θ 2 ) = (π, 0), and a massless Dirac fermion of charge 8k ± 3 is a possible boundary state at (θ 1 , θ 2 ) = (π, π). Symmetry-preserving gapped boundary states appropriate for (θ 1 , θ 2 ) = (π, π) are described in section 4.3.
What happens if we specialize to spin manifolds? Then I 2 is divisible by 8, so πI 2 is a multiple of 2π and θ 2 = π cannot be distinguished from θ 2 = 0. Moreover, X A is an even integer and that term can be dropped in πI 1 . So in the context of a spin manifold, θ 1 = π just means that the ordinary electromagnetic theta-angle equals π.
On the other hand, on a spin-manifold there is an integer-valued coupling Accordingly, it is possible to have on a spin manifold a T-invariant gapped system with partition function exp(iπI 2 ) = (−1) I( / D 0 ) . Such a system cannot be defined on a spin c manifold in a Tinvariant fashion, since I 2 is in general not integer-valued on a spin c manifold. Thus, in a T-conserving condensed matter system that conserves electric charge and satisfies the usual spin/charge relation, one will not encounter the interaction πI 2 .

A.3 2 + 1-Dimensional Chern-Simons Terms
Naively, the Chern-Simons functional of a U (1) gauge field A on an oriented three-manifold W is defined by the familiar integral: This formula is satisfactory in a topologically trivial situation, but in general it is difficult to interpret as A may have Dirac string singularities. A procedure that avoids this problem is as follows. There always exists an oriented four-manifold X of boundary W such that A extends over X. Moreover, if W has a chosen spin structure, 42 then X can be chosen so that the spin structure of W extends over X. Picking such an X and A, and defining F = dA as before, we attempt the definition This is manifestly gauge-invariant, but in general it depends on the choice of X and of the extension of A over X. Had we used in this procedure another spin manifold X , again with some chosen extension of A, then eqn. (A.18) would be replaced by The difference between the two definitions is where X * is a compact spin-manifold without boundary made by gluing X to X after reversing the orientation of X ( fig. 3 of section 3.6.3). According to eqn. (A.4), the right hand side of this formula is an integer multiple of 2π. In other words, our procedure for defining CS(A) has given a result that is independent of the choices mod 2πZ. Now let us repeat this in the spin c case. The operator / D 0 acting on a neutral spin 1/2 fermion is not available, but the operator / D acting on a spin 1/2 fermion of charge 1 still exists. Its index is still given by the formula (A.3), so the integral on the right hand side of that equation is still Z-valued. This formula contains the gravitational term A(R) as well as the gauge theory term F ∧ F/8π 2 , and there is no longer a convenient way to remove the gravitational term by subtraction. So if we use integrality of the index to define a Chern-Simons-like functional, this functional will have to have a gravitational contribution. Picking again a spin c manifold X with boundary W and an extension of A over X, we define Here Ω(g) is a sort of gravitational Chern-Simons term. The same argument as before shows that CS(A)+Ω(g) is independent of the choices mod 2π. Moreover, the normalization of CS(A) is the same as it was in the spin case (for example, in a topologically trivial situation, CS(A) is still given by the naive formula (A.17)). However, only the sum CS(A) + Ω(g), and not either of the two terms separately, is independent of the choice of X mod 2πZ. In section 2.3, a simple argument was given to show that CS(A) is well-defined mod 2π/8, or equivalently that 8CS(A) is well-defined mod 2π. The discussion leading to eqn. (A.11) shows that this is the sharpest result of its type: we cannot replace 8 by any smaller integer. We see that in the spin c case, CS(A), defined by the usual formula, is well-defined only modulo 2π/8.

A.4 Unorientable Spacetimes
Throughout this paper, we have considered orientable spacetimes only. Without aiming to explain the generalization of all statements in this paper to unorientable spacetimes, we will here sketch the generalization of the bulk couplings corresponding to θ 1 = π and θ 2 = π.
The unorientable analog of a spin c manifold is called a pin c manifold, and (in a theory that satisfies T 2 = (−1) F , so that fermions are in Kramers doublets) the unorientable analog of a spin manifold is a pin + manifold. In a theory that satisfies the spin/charge relation, the more general pin c concept is natural, but we will also run into a variant of pin c below (footnote 44).
An important preliminary is that either T or CT can be used to define a theory on an unorientable manifold X. In one case, fields undergo a T or CR transformation in going around an unorientable loop, and in the other case they undergo a CT or R transformation. 43 Concretely, if we use CT symmetry to define a theory on X, then the electromagnetic field strength F is an ordinary two-form on X, while if we use T, then it is a twisted two-form that changes sign in going around an orientation-reversing loop.
It is more straightforward to generalize the interaction πI 2 to a pin c manifold, so we begin with this. On a spin c or pin c manifold of any dimension, the integral class 2F/2π is congruent mod 2 to w 2 , the second Stieffel-Whitney class. (This is true whether F is an ordinary two-form or a twisted one.) So exp(iπI 2 ) is equivalent to (−1) X w 2 2 , and this generalizes θ 2 = π to a pin c manifold. On a pin + manifold, w 2 = 0 and this interaction is trivial. Now let us consider the opposite case of a gapped theory with θ 1 = π, θ 2 = 0. On an orientable manifold, its partition function is (−1) I( / D) . But the index of the Dirac operator is only defined on an orientable manifold. How can we generalize (−1) I( / D) to an unorientable manifold? The answer to this question is that we should use η, the eta-invariant of the Dirac operator, rather than the index. On an orientable manifold of even dimension, the spectrum of the Dirac operator is symmetric under λ ↔ −λ. Recalling the definition (3.32) of η, this means that modes with λ = 0 do not contribute to η, which just receives a contribution of +1 from each zero-mode. The index I likewise receives no net contribution from non-zero eigenvalues; on the other hand, a zero-mode contributes ±1 to I, depending on its chirality. So on an orientable manifold of even dimension, I and η are congruent mod 2, and therefore (−1) I = (−1) η .
This can be used as the starting point to generalize θ 1 = π to an unorientable manifold, but some subtleties arise. First, let us assume that T or equivalently CR has been used to define a theory on an unorientable manifold X. This means that in going around an orientationreversing loop in X, a fermion field χ of charge 1 is exchanged with its adjoint χ of charge −1. Given this, one cannot define a Dirac operator that acts solely on χ; only a Dirac operator that acts on the pair χ χ can be defined. 44 In eqn. Unlike η, η is still defined when we use T symmetry to go to an unorientable manifold. When we do that, η is always an even integer (as it is in the orientable case) and is a topological invariant mod 4. 45 Therefore, in this situation, a suitable generalization of (−1) I is (−1) η/2 . This gives a generalization of θ 1 = π when we use T to go to an unorientable manifold. 43 We here use the CRT theorem of relativistic field theory (more commonly called the CPT theorem), via which T is related to CR and CT to R. In Euclidean signature, it is more natural to think about CR and R rather than T and CT. 44 The fact that A is odd under CR also means that it is not what is usually called a pin c connection but a variant of this with the group O(2) replacing SO(2) = U (1). CR takes values in the disconnected component of O (2). 45 An unorientable manifold X has a canonical oriented double cover X. Instead of computing η in terms of all fermion modes of either charge on X, we can compute it in terms of modes of charge 1 on X. When we do However, using T to go to an unorientable manifold in a theory that satisfies the spin/charge relation has not really given us a new probe of the physics. The mod 2 invariant (−1) I has been generalized to a mod 2 invariant (−1) η/2 , but since η is always an even integer in this situation and is only a topological invariant mod 4, this mod 2 invariant is all that we get.
If instead CT or equivalently R symmetry is used to define a theory on an unorientable spacetime X, matters are different. There is no mixing between modes of opposite charge, so η is naturally defined. But in this situation, η is not necessarily an integer. So before generalizing to an unorientable spacetime, we should first rewrite (−1) I as exp(−iπI) (the sign in the exponent is an arbitrary choice here), and then we can generalize this to an unorientable spacetime as exp(−iπη). On a pin c manifold, η is a topological invariant mod 46 2, so exp(−iπη) is a topological invariant. But what are the possible values of η mod 2 on a four-dimensional pin c manifold? The answer is that η is always an integer multiple 47 of 1/4. Hence, there are actually 8 classes of CT-conserving theory that can be defined on a general pin c manifold, with the partition function being exp(−iπρη), ρ = 0, . . . , 7. If we specialize these 8 classes to a spin c manifold, we can only detect the value of ρ mod 2, since on an orientable manifold, exp(−iπρη) = (−1) ρI . On an orientable manifold, these 8 classes of theory have θ 1 = πρ mod 2π.
If we consider a charge-conserving theory that lacks the conventional spin/charge relationship, then a few things are different. We should work only on a pin + manifold, not a more general pin c manifold. The basic eta-invariant to consider is that of a Majorana fermion coupled to gravity only. We will denote it as η. It is a topological invariant mod 48 4. So exp(−iπνη/2) is a topological invariant for any integer ν. The values of η are always integer multiples of 1/4 (the minimum value again occurs for RP 4 ), so ν is an invariant mod 16.
In a theory that conserves both T and CT and does not have the standard spin charge relation, we should distinguish two mod 16 invariants ν T and ν CT , depending on which symmetry is used to define the theory on an unorientable manifold. Thus we consider a theory whose partition function on a pin + manifold X is exp(−iπν T η/2) or exp(−iπν CT η/2), depending on which symmetry is used when the fields traverse an orientation-reversing cycle in X. If X happens to be orientable (and thus spin), these factors must be equal since in that case there this, because X is orientable, only zero-modes contribute. The number of those zero-modes is even because, as X has an orientation-reversing symmetry, the index of the charge 1 Dirac operator on X is 0. To show that η is a topological invariant mod 4, we may reason as follows on the original unorientable manifold X. (For background, see [17], especially the discussion of eqns. (2.47) and (B.16).) In general, in even dimensions, because there is no Chern-Simons function, η/2 is the index of the Dirac operator with APS boundary conditions and hence is a topological invariant except for jumps that occur when eigenvalues pass through 0. In four dimensions, even on an unorientable manifold, the eigenvalues of the hermitian Dirac operator with values in a real representation (such as we have here because we include both signs of the charge) all have even multiplicity because of a version of Kramers doubling. When a pair of eigenvalues passes through 0, η jumps by ±4, so it is a topological invariant mod 4. 46 As in footnote 45, η is a topological invariant except for jumping that occurs when an eigenvalue passes through 0. When this happens, η jumps by ±2. So it is a topological invariant mod 2. There is no further restriction in general. The difference from footnote 45 is that we are dealing with fermions in a complex representation, so there is no Kramers doubling. 47 This minimum value is assumed for the pin c manifold RP 4 . See Appendix C of [17]. The argument there is presented for pin + manifolds but the same reasoning applies for pin c . 48 The reasoning is the same as in footnote 46, except that on a pin + manifold, the eigenvalues have a two-fold Kramers degeneracy, and hence the jumps in η are by ±4. was no need to make a choice of T or CT. Since on an orientable manifold, η/2 is an arbitrary integer (η receives contributions only from zero-modes, and there are an even number of them because of Kramers doubling), it follows that ν T and ν CT are congruent to each other mod 2. This is the only general relationship between them, as one can see by considering examples constructed from massless free fermions with suitable transformations under T and CT. Each massless Majorana fermion can independently contribute ±1 to ν T and to ν CT . Now let us consider a gapped, charge-conserving theory that does obey the usual spin/charge relation. Suppose that, when placed on an unorientable manifold using CT, this theory has partition function exp(−iπρη). If we specialize to a pin + manifold, then η reduces to η and the partition function is supposed to be exp(−iπν CT η/2). Evidently, the relation between ρ and ν CT is In particular, ν CT is even for theories that obey the usual spin/charge relation. This is consistent with the fact that ρ is a mod 8 invariant and ν CT is a mod 16 invariant.

B The Callias Index Theorem
In this appendix, we will use the Callias index theorem [48] to study time-independent solutions of the Dirac equation for χ. (This theorem has been elucidated in [72] and is based in part on analytical foundations established in [73]. See also [47,49,50].) It is helpful to write the Dirac equation in a completely real form. To this end, we first introduce a basis of real gamma matrices: We also write χ in terms of Majorana fermions: χ = (χ 1 + iχ 2 )/ √ 2. It is convenient to arrange these Majorana fermions as a column vector χ 1 χ 2 , and to introduce a set of real gamma matrices that act on this column vector: We take the matrices γ i , i = 0, 1, 2 that act on the pair χ 1 χ 2 to commute with the matrices γ µ , µ = 0, 1, 2 that act on the spinor indices carried by χ 1 and χ 2 . The covariant derivative of χ (on a flat spacetime) is usually written D µ = (∂ µ + iA µ )χ. On the column vector χ 1 χ 2 , this is equivalent to According to eqn. (2.11), CT acts on χ 1 χ 2 simply as multiplication by γ 0 (along with a reversal of the time, which will not be important here as we will be considering time-independent solutions). However, according to eqn. (3.10), the appropriate unbroken symmetry in the gapped phase is not CT but CT = CTK 1/2 , where K 1/2 is a gauge transformation that acts on χ as multiplication by i. On the column vector χ 1 χ 2 , K 1/2 acts as multiplication by γ 0 . Hence the action of CT is Setting φ = (φ 1 + iφ 2 )/ √ 2, the Dirac equation obeyed by χ is We drop the time-derivative (since we want to study time-independent solutions) and we multiply by iγ 0 to define a hermitian Dirac operator: D is imaginary and antisymmetric. We observe that the matrix M = γ 0 γ 0 that represents the action of CT obeys M 2 = 1 and anticommutes with D: (The full CT operation, which is the product of M with complex conjugation, commutes with D, since D is imaginary.) This is the situation in which one can define an integer-valued index I, the number of M = 1 zero-modes of D minus the number of M = −1 zero-modes of D. Actually, the ability to define an index depends on the fact that the operator D has a discrete spectrum near zero. This is true as long as φ is bounded away from zero near spatial infinity, giving χ a mass and ensuring that any low-lying modes of χ must be localized near the origin in space. This integer is a topological invariant, in the sense that it is invariant under deformations of the operator D that do not change its behavior at infinity in either real space or momentum space. For example, although A µ is nonzero in the standard vortex solution, it vanishes at infinity, and therefore I would be unchanged if we simply set A µ to 0. The index only depends on the winding at spatial infinity of the field φ that determines the fermion mass.
If explicit CT -violating terms are added to the action, then D will be replaced by an operator that no longer anticommutes with M (or with any similar matrix), and zero-modes of χ will no longer be governed by an integer-valued index. However, fermi statistics imply that fermion zero-modes can only be lifted in pairs. So even if CT is explicitly violated, the number of χ zero-modes will always be equal mod 2 to the index I that one can define in the CT -conserving case. Generically, the number of χ zero-modes will always be as small as possible subject to the constraints implied by the index, so the total number of zero-modes is generically |v| in the CT -conserving case, and 0 or 1 in the CT -violating case.
According to the Callias index theorem, I is given by a sort of winding number that should be computed at infinity in phase space. In other words, we consider a phase space R 4 = R 2 ×R 2 where the first factor is parameterized by position components x 1 , x 2 and the second factor by momentum components p 1 , p 2 . We let D + be the part of D that maps states of M = 1 to states of M = −1. We undo the passage to quantum mechanics, replacing derivatives by momentum components, ∂ µ → ip µ . D + thus becomes a function of the x's and p's. Because χ is gapped near spatial infinity (and the operator D is "elliptic," meaning that it is invertible for large Euclidean momenta), the operator D + is invertible when restricted to a large sphere S 3 at infinity in R 4 . The winding number of this invertible operator, integrated over S 3 , is equal to the index.
To implement this program, we want to write D + explicitly as a map from a space V + of modes with χ = 1 to a space V − of modes with χ = −1. We use a notation in which | ↑↑ represents a joint eigenstate of γ 0 and γ 0 with both having eigenvalue i, | ↑↓ represents a joint eigenstate with respective eigenvalues i and −i, etc. So | ↑↓ and | ↓↑ give a basis of V + , while | ↑↑ and | ↓↓ give a basis of V − . To write D + : V + → V − as a matrix, we represent the basis vectors | ↑↓ and | ↓↑ of V + as 1 0 and 0 1 , respectively, and similarly we represent basis vectors | ↑↑ and | ↓↓ of V − as 1 0 and 0 1 . After setting A µ to zero (since it will not affect the index), a calculation gives In a suitable gauge, the standard vortex solution of vorticity v = 1 is invariant under rotations of R 2 together with constant gauge transformations. The scalar field φ is The function f (|x|) vanishes at infinity as | φ |/| x|. However, it will not affect the index if we just set f = 1 (which means that the fermion mass grows linearly at infinity), since we can interpolate from a realistic f to f = 1 in such a way that the fermions are always gapped near spatial infinity. With f = 1, D + is simply linear in the x i and p j and the winding number at infinity is easy to calculate. After relabeling (x 1 , x 2 , p 1 , p 2 ) as (P 0 , P 1 , P 2 , P 3 ) in a fairly obvious way, we find D + = P 0 + i σ · P , (B.10) where σ are the Pauli sigma matrices. If we restrict D + to the sphere S 3 defined by 3 s=0 P 2 s = R 2 for large R, then up to a factor of R, D + is just the identity map from S 3 to SU (2) ∼ = S 3 . This identity map is the basic example of a map of winding number 1, and that is the value of the Callias index in this situation.
For an example of a field of vorticity v > 0, we can just take φ at infinity to be defined by This just multiplies the winding number by v. To get vorticity v < 0, we take Again the winding number if v. So in all cases, the winding number of the map D + , and therefore the index predicted by the Callias index theorem, is equal to the vorticity v.
For any value of the vorticity v, it is possible to have a classical field that is rotationinvariant, or more precisely, invariant under a combined rotation plus gauge transformation. Indeed, if j is the standard angular momentum generator and k is the gauge charge of the emergent gauge field, then the ansatz (B.11) or (B.12) is invariant under the combination The factor of 1/4s arises because φ has k = 4s. We set k * = k/4s so j = j + vk * . (B.14) χ and χ have k = ±2s and so k * = ±1/2. A stable quasiparticle with |v| > 1 does not necessarily exist in the theory, since the interaction between vortices may be repulsive, as in a Type II superconductor. However, if a stable vortex of given v exists, one may expect that classically it is described by a j -invariant solution.
(Even if this is not the case, the vortex is deformable to a j -invariant situation, so computing in the j -invariant case should suffice for determining the deformation-invariant properties of the spectrum.) To understand the quantization of the vortex states, it is necessary to know not just the number of fermion zero-modes in the vortex field, but their j quantum numbers.
The framework to determine those quantum numbers conceptually -as opposed to attempting to directly solve the Dirac equation -is to compute what is known as the character-valued index of the Dirac operator. Define the positive operator H = D 2 . For an angle α, define The idea behind this definition is that (as in the standard definition of the index of the Dirac operator) nonzero eigenvalues of H appear in pairs with the same j but opposite signs of M , and so cancel out of the trace. Hence F (α) is independent of β, and it can be computed effectively for small β in a high temperature expansion, as we discuss below. On the other hand, since the nonzero modes of H make no net contribution to F (α), that function can be computed just in the space of fermion zero-modes, and therefore captures the desired information about the j eigenvalues of those modes. The result that comes from the high temperature expansion is meaning that the zero-modes have spins j = (v − 1)/2, (v − 3)/2, . . . , −(v − 1)/2. Before explaining how this formula is obtained, we look at the first few cases.
For v = 1, there is only one zero-mode, and it has j = 0. Indeed, if the operator D has only one zero-mode, it must have j = 0, since complex conjugation (or CT symmetry) ensures that the spectrum of zero-modes is symmetric under j ↔ −j .
For v = 2, the two zero-modes have spins ±1/2. Upon quantization of these modes, one gets a pair of states of spins ±1/4, as asserted in section 3.5.
For v = 3, the three zero-modes have spins 1, 0, −1. Quantization of the modes of spins ±1 gives two states of spins ±1/2, and existence of the third zero-mode of spin zero means that the v = 3 vortex satisfies non-Abelian statistics.
Finally, for v = 4, the four zero-modes have spins 3/2, 1/2, −1/2, −3/2. Quantization gives a pair of states of spins ±1 (which in topological field theory is equivalent to 0) and another pair of states of spins ±1/2. Since the fermion zero-modes anticommute with the operator (−1) F that distinguishes bosons and fermions, one pair of states is bosonic and one is fermionic. Which is which is discussed in section 3.5.
The actual computation of the character-valued index F (α) uses a high temperature expansion, valid for small β. For a proof of the character-valued index theorem for the Dirac operator using path integrals, see [74]. Here we will describe an equivalent computation in a Hamiltonian approach.
For small β, when we compute F (α) = Tr M e iαj exp(−βH) via an integral over particle orbits, we have to consider orbits that are periodic up to a rotation by an angle α. Let R α be the operator that rotates the x plane by angle α. It has a unique fixed point at x = 0. An orbit that propagates from x to R α x in imaginary time β has a very large action unless |(1 − R α ) x| β. For fixed nonzero α, as β → 0, the condition is equivalent to | x| β. On such short distances, a particle can be treated as free; the background gauge fields and scalar fields play no role. When we set the background field to zero, H = D 2 becomes simply the is just the space of scalar wavefunctions on R 2 , and H is a four-dimensional Hilbert space obtained by quantizing the four real components of χ. The trace that we want factorizes as the product of a trace in L 2 (R 2 ) and a trace in H.
We start with the first factor. This means that we consider H to act on scalar wavefunctions on R 2 , and j reduces to the standard angular momentum generator −i(x 1 ∂ 2 − x 2 ∂ 2 ). The operator M acts trivially on L 2 (R 2 ) and can be dropped. We compute (B.17) To compute the trace in H, we observe the following. As operators on H, the usual angular momentum is j = i 2 γ 0 and the charge generator is in a way that preserves angular momentum. More specifically, we want to verify that we can do this in a way that reduces the spectrum of a vortex of vorticity v + 8 to that of one of vorticity v. The group theoretic fact that we will use is the following. In the group SO(8) that acts on 8 gamma matrices γ 1 , . . . , γ 8 , pick a Spin (7) subgroup, so that the γ i transform in the spinor representation 8 of Spin (7). In this representation, there is an invariant fourth order antisymmetric tensor. Take p ijkl to be a multiple of this tensor. The group SO(8) has two spinor representations, which we denote as 8 and 8 . One of them, say the 8 , is irreducible under Spin (7) while the other decomposes as 1 ⊕ 7. If the sign of the Hamiltonian is chosen correctly, then the 1 of Spin (7) is its ground state and in particular is nondegenerate. It remains to show that this can be done in such as way that the ground state is an angular momentum eigenstate with j = 0. For this, we first consider the case of a vortex of vorticity 8, which has precisely 8 zero-modes of angular momenta j = ±7/2, ±5/2, ±3/2, ±1/2.
The ground state of ∆H will certainly be j -invariant if j is a generator of Spin (7). For this, we take for j a generator of Spin(7) that in the 7 has eigenvalues 0, ±1, ±2, ±4. This generator has in the 8 of Spin (7) the desired eigenvalues of eqn. (B.22). More generally, we can in a similar fashion reduce a vortex V + of vorticity v + 8 to a vortex V − of vorticity v. V + has 8 extra zero-modes that have no counterpart in V − ; their angular momenta are j = ±(v + 7)/2, ±(v + 5/2), ±(v + 3/2), ±(v + 1/2). To construct a quartic Hamiltonian that will lift these modes in a j -invariant way, we proceed as before, embedding j in Spin (7) so that in the representation 7, it has eigenvalues 0, ±1, ±2, ±(4 + v).

C Some Useful Facts About Chern-Simons Gauge Theories
The purpose of this appendix is to review and clarify some known facts about 2 + 1-dimensional Chern-Simons gauge theories. In the last subsection we will use this information to exhibit an explicit Chern-Simons Lagrangian that describes the non-Abelian statistics of the Read-Rezayi states [76], which generalize the Moore-Read state [35]. The analysis in that subsection is similar to the analysis in section 4.4.

C.1 U (1)
Consider a U (1) Chern-Simons theory with the Lagrangian where a is a U (1) gauge field and A is a classical background U (1) field, which couples to the current j = 1 2π da .

(C.2)
In order not to clutter the notation, we will first take k positive. We denote this theory as U (1) k . For even k the theory is well defined on any manifold. For odd k we need the manifold to be spin and the answers depend on the choice of spin structure. Also, as we discussed in section 2.3, even for odd k we can use a spin c classical gauge field A to define the theory as a non-spin theory. This is consistent with the identification of the lines labeled by n and by n + k. Using that and the expression for the current (C.2) we learn that A couples to the charge Q = − n k . (C.5) Note that because of the identification of the lines, only Q mod 1 is meaningful. If we want to regard Q as a real number, we cannot claim any identifications of lines. Some of the 2 + 1-dimensional Chern-Simons theories have a corresponding 2d rational conformal field theory (RCFT) interpretation. In the case of U (1) k , the relevant theory is that of a rational boson. Its chiral algebra includes the current ∂φ and is extended by the operator e i √ kφ of dimension k 2 . This is indeed an integer for k even. 49 For k odd, we can still use the expressions (C.4) and (C.5) for the spin, the holonomy, and the charge. However, now we can no longer identify the lines labeled by n and by n + k. These two lines induce the same holonomy, but their spins differ by 1 2 mod 1. For example, the line E = e ik a does not induce a nontrivial holonomy, but its spin is half-integer. Therefore, this line is nontrivial and cannot be ignored; its expectation value will depend on the spin structure of the three-manifold. Correspondingly, there are 2k inequivalent lines labeled by n = 0, ±1..., ±(k − 1), k . (C.6) Unlike the case of k even, here Q = −n/k is meaningful mod 2.
In the fractional quantum Hall application, the various Wilson lines represent the worldlines of quasiparticles. The special line E represents the underlying electrons. It has spin 1 2 and charge 1. And since it does not induce any holonomy, it has trivial braiding with all quasiparticles (only Fermi statistics).
Let us consider some examples: k = 1 has two lines, the trivial line with vanishing spin and E with half-integer spin. From the 2d RCFT perspective, this theory corresponds to the rational boson at the free fermion radius with the fermion e iφ associated with the line E. k = 2 has two lines with spins modulo one S = 0, 1 4 . This theory U (1) 2 is the same as SU (2) 1 . k = 3 has six lines with spins modulo one S = 0, 1 6 , 1 6 , 1 2 , 2 3 , 2 3 . From the 2d RCFT perspective this theory is the first member of the N = 2 supersymmetric minimal models. The two supercharges are associated with E, whose 2d dimension is 3 2 . k = 4 has four lines with spins 0, 1 8 , 1 8 , 1 2 . This theory is similar to the k = 1 theory. It is a free fermion theory, but now it includes also two complex conjugate representations of dimension 1 8 . These are spin fields. Finally, we would like to point out that the U (1) m theory can be viewed as a Z 2 quotient of the U (1) 4m theory. Of course, as groups U (1)/Z 2 = U (1). But since the quotient changes the normalization of the gauge fields, the map is nontrivial. We start with U (1) 4m with the Lagrangian 4m 4π ada. The Z 2 quotient is implemented by stating that a is not a good gauge field, but b = 2a is. In terms of it 4m 4π ada = m 4π bdb, which describes U (1) m . This Z 2 quotient can also be described as gauging a one-form global symmetry [61,62]. It can be implemented by identifying a line in the original theory whose braidings will be trivial in the quotient theory [66,67]. In this case, the relevant line is e 2im a , with dimension m 2 . Therefore, the quotient theory is a spin-TQFT for odd m and a non-spin-TQFT for even m. Imposing that this line has trivial braidings projects out all the lines e in a with odd n. When m is even we should also identify the lines e in a ∼ e i(n+2m) a , whose spins differ by an integer [66]. When m is odd the spins of these two lines differ by half an integer and they should not be identified. As a check, the fact that the line with n = 1 is projected out is equivalent to the statement in the last paragraph that a is not a good gauge field in the quotient theory.

C.2 U (1) n
More generally, if the gauge group is U (1) n , the theory is characterized by a K matrix, a q vector, and a Chern-Simons contact term k c . In terms of them, the Lagrangian is 1 4π K ij a i da j + 1 2π q i a i dA + 1 4π k c AdA . (C.7) All the coefficients in the Lagrangian must be integers. If all the diagonal elements of K as well as k c are even, the theory does not need a spin structure. If some of these are odd, the theory is a spin Chern-Simons theory; it can be defined on a spin manifold and the results depend on the spin structure.
In the special case where K ii − q i ∈ 2Z for all i k c ∈ 8Z (C. 8) we can place the theory on an arbitrary manifold with a choice of a spin c structure, letting A be a spin c connection. The first restriction guarantees that all the monopole operators of a satisfy the proper spin/charge relation, which is needed for the spin c structure. And the condition on k c is needed for the Chern-Simons contact term for 2A to be well defined on an arbitrary spin c manifold. See also the discussion in section 2.3. The independent lines are e in i a i whose spin and charge are S = 1 2 K ij n i n j and Q = −K ij q i n j , where K ij is the inverse matrix of K ij .

C.3 Z N
Here we give a continuum description of Z N Chern-Simons gauge theory [63,64].
We consider a special case of the theories in section C.2 based on the gauge group U (1) a × U (1) c and the Lagrangian N 2π adc + k 4π ada. (C.9) We can also couple the model to A. The field c acts as a Lagrange multiplier constraining the holonomy e i a to be an N th root of unity. Therefore, this theory is a Z N gauge theory with e i a the Z N Wilson line. The line e i c creates a holonomy for the Z N Wilson line. The term with k is easily identified as the Dijkgraaf-Witten term [65] of the Z N gauge theory. Here it is given a continuum description in terms of a U (1) a × U (1) c gauge theory. For k ∈ 2Z the theory is not spin. But for odd k it depends on the spin structure. Using c → c + a we can identify k ∼ k + 2N and hence the inequivalent non-spin theories are labeled by k = 0, 2, . . . , 2N − 2. And if we allow dependence on the spin structure we can have k = 0, 1, . . . , 2N − 1.
The theory has line operators Their correlation functions are W na,nc (γ)W n a ,n c (γ ) ∼ exp 2πi N 2 #(γ, γ )(N n a n c + N n c n a − kn c n c ) , (C. 12) where #(γ, γ ) is the linking between the two lines γ and γ . From equations (C.11) and (C.12), we see that the lines W na,nc and W na+N,nc have the same correlation functions and the same spins. They should be identified. The lines W na,nc and W na+k,nc+N have the same correlation functions, but their spins satisfy S(W na+k,nc+N ) = S(W na,nc ) − k 2 . Therefore, for even k they should be identified. For odd k, the theory needs a spin structure. For odd k, the line E = W k,N has half-integer spin and is nontrivial.
We conclude that for k even there are only N 2 lines and for k odd there are 2N 2 lines. In other words the lines satisfy W N 1,0 = 1 W k 1,0 W N 0,1 = E , where for k even we can set E = 1 and for k odd we have E 2 = 1.
(C. 15) Here k 1,2 ∈ Z. In order not to clutter the equations we take k 2 positive, but we allow k 1 to be negative. For k 1 ∈ 2Z the theory is non-spin and the lines are labeled by j = 0, . . . , k 2 2 and n = 0, ±1, . . . , |k 1 | 2 . On a spin manifold with a choice of spin structure (or by coupling to a spin c classical gauge field A) we can also have odd k 1 and then n = 0, ±1, . . . , |k 1 |.
The U (2) = (SU (2) × U (1))/Z 2 theory is characterized by two levels k 1,2 and is denoted by U (2) k 1 ,k 2 . One way to describe it is to combine a and b in eqn. (C.15) to a conventionally normalized U (2) gauge field c = b + a1 with the Lagrangian Tr (cdc + 2 3 ccc) + k 1 − 2k 2 4 · 4π (Tr c)d(Tr c) . (C.16) As a three-dimensional gauge theory it differs from the SU (2) × U (1) theory of equation (C.15) by having additional bundles. Therefore, there might be additional conditions on k 1,2 . One way to find them follows from focusing on a specific U (1) ⊂ U (2) gauge field of the form c 11 0 0 0 , whose "effective Lagrangian" is 2k 2 +k 1 4·4π c 11 dc 11 . Therefore, the theory is consistent only when 2k 2 + k 1 ∈ 4Z (C. 17) and it depends on the spin structure unless 2k 2 + k 1 ∈ 8Z. In particular, k 1 must be even. It can be shown by adapting to U (2) the arguments given for U (1) in Appendix A that there are no further conditions. Let us understand this in more detail. We start with the SU (2) k 2 × U (1) k 1 theory and perform the quotient. From the three-dimensional point of view this means that we gauge a Z 2 one-form global symmetry [61,62]. This is achieved by summing over additional bundles that are not SU (2)×U (1) bundles. One way to think about it to start with the original SU (2)×U (1) theory and to identify in it a Wilson line that induces a nontrivial holonomy in SU (2) × U (1), which is trivial in U (2). This holonomy should be −1 in each of the factors. Then we should impose that this line has trivial correlation functions. In the SU (2) factor this line should have j = k 2 2 . The holonomy around it is in the center Z 2 ⊂ SU (2). Looking at equation (C.4) it is clear that from the U (1) factor we need n = |k 1 | 2 and hence k 1 should be even. The spin of the full line is When this spin is an integer, i.e. when 2k 2 + k 1 = 0 mod 8, the quotient makes sense on every manifold. Alternatively, we can have 2k 2 + k 1 = 4 mod 8 and then the quotient makes sense only on a spin manifold.
In general, when we perform such a quotient the spectrum of lines is modified by three rules [66,67]. First, of the original lines labeled by (j, n) we should keep only those that have trivial braiding with the special line (j = k 2 2 , n = |k 1 | 2 ). This leads to a selection rule j + n 2 ∈ Z . (C. 19) Indeed, this is the condition that (j, n) is a U (2) representation rather than a faithful SU (2) × U (1) representation. Second, we should identify some lines. It is easy to see that (j, n) and k 2 2 − j, ( k 1 2 + n) mod |k 1 | (C.20) induce the same holonomy. Their spins differ by ) . (C.21) For k 1 +2k 2 = 0 mod 8, when the theory is not a spin theory, we use the selection rule (C. 19) to see that these two representations have the same spin modulo one and therefore they should be identified. For k 1 + 2k 2 = 4 mod 8, when the theory is a spin theory these two representations induce the same holonomy, but their spins differ by half an integer and therefore they should not be identified.
The third rule of a quotient [66,67] applies to lines that are fixed points of the identification. Such lines should be treated more carefully. But it is easy to see from (C.20) that no such fixed point exists in this case.
Let us consider the special case k 2 = k, k 1 = −2k. Since 2k 2 +k 1 = 0, this is not a spin theory. Since k 1 is negative, this particular Chern-Simons theory can be coupled on the boundary of a three-manifold to a two-dimensional field theory that has both left movers realizing SU (2) k and right movers realizing U (1) −2k . But in this particular case, there is also another option for the boundary theory [66,67]. Since U (1) 2k is conformally embedded in SU (2) k , this theory describes also the GKO coset SU (2) k U (1) 2k [68]. This theory is known as the parafermion theory. (Recall that our normalization of k 1 differs by a factor of 2 from the standard RCFT normalization.) The special case of k = 2 corresponds to the Ising model. Here the possible Wilson lines are labeled by j = 0, 1 2 , 1 and n = 0, ±1, 2. The selection rule (C.19) and the identification (C.20) show that the distinct lines are the trivial line with (j = 0, n = 0), W σ with (j = 1 2 , n = 1), and W ψ with (j = 1, n = 0). They correspond to the identity operator (spin 0), the spin field (spin 1 16 ), and the free fermion (spin 1 2 ). Note that even though this is a theory of 1 + 1-dimensional fermions, the corresponding 2 + 1-dimensional Chern-Simons theory is not a spin theory. In the language of section 3.6.4, this theory is the Ising TQFT, not the Ising spin-TQFT.
We can quotient the U (2) theory again by Z 2 and turn the gauge group into U (2)/Z 2 = SO(3) × U (1). This is possible only when k 2 is even and k 1 = 0 mod 4. One way to see that is based on identifying the special lines in the SU (2) k 2 × U (1) k 1 theory that have trivial braiding in the quotient. They are (j = k 2 2 , n = 0), whose spin is k 2 4 and (j = 0, n = |k 1 | 2 ), whose spin is |k 1 | 8 . The product of these lines is the special line in the quotient leading to U (2) and should not be considered separately. The theory that results from the additional quotient is SO(3) k 2 × U (1) k 1 /4 . It is non-spin when k 2 = 0 mod 4 or k 1 = 0 mod 8. Otherwise the theory is spin. Note that when the two factors of the gauge group are both spin, i.e. when k 2 = 2 mod 4 or k 1 = 4 mod 8, we do not need to double the number of lines twice. l = 2 is usually denoted by U (1) 4 . l = 3 is usually denoted by SU (2) 2 .
Let us lift the spin-TQFT of SO(m) 1 to a non-spin TQFT. Tensoring it with SO(n) 1 and summing over the spin structures the partition function is Z n (SO(m) 1 ) = s Z s (SO(n) 1 )Z s (SO(m) 1 ) = s Z s (SO(n + m) 1 ) = Z(Spin(n + m) 1 ) .

(C.24)
The resulting theory is Spin(n + m) 1 . The simple case n = 0 lifts SO(m) 1 to the non-spin theory Spin(n) 1 .
A more interesting case corresponds to n = −2. It lifts SO(m) 1 to Spin(m − 2) 1 . For m = 3 this lifts SU (2) 2 /Z 2 to the non-spin Ising theory. We can also write this as where the first term represents Z s , the second is z s Z s and the third is the result of the sum over s. In the third step the numerator is the result of independently summing each factor over s and the Z 2 quotient guarantees that s in the two sectors is the same. The last expression is the Chern-Simons description of the RCFT coset SU (2) 2 U (1) 4 that describes the non-spin Ising theory. As in Appendix C.4, it can also be written as U (2) 2,−4 .

C.6 Chern-Simons Gauge Theories for Non-Abelian FQHE
Here we use the analysis in the previous subsections to find a Chern-Simons description of the series of models of non-Abelian statistics of Read and Rezayi [76], which generalize the famous Moore-Read state [35].
The Read-Rezayi construction starts with the chiral algebra The first factor, the parafermion theory, was analyzed above. It is not a spin theory. The level of the U (1) factor is taken to be a multiple of k so that we can mod out by Z k . More explicitly, this guarantees that the U (1) factor has a global Z k one-form symmetry, which can be gauged. The quotient by Z k is achieved by ensuring that the line (j = 0, n = 2, l = L) has trivial braiding. (Here the labels of the line are the SU (2) k × U (1) −2k × U (1) kL representations.) Its spin is S = L−2 2k mod 1 and therefore we need For M odd it is a fermion and the theory is spin and for M even it is a boson. Let us describe the 2 + 1-dimensional Chern-Simons theory. The parafermion theory is described by a U (2) k,−2k theory (C.16) with gauge field c and the U (1) k(kM +2) theory by a gauge field b. The Lagrangian of the U (2) k,−2k × U (1) k(kM +2) theory is k 4π Tr (cdc + 2 3 ccc) − k 4π (Tr c)d(Tr c) + k(kM + 2) 4π bdb + k 2π Adb .
(C. 28) In this form the gauge group is U (2) k,−2(k+2) × U (1) 1 and the U (1) 1 part decouples. Note that we did not shift b by A in order to preserve the spin/charge relation. An interesting aspect of the theory (C.34) is the role of the decoupled U (1) 1 factor. The theory without it is still spin. So its presence does not add additional lines; the theory without it has all the necessary lines. However, keeping the U (1) 1 factor, the theory has a purely left moving chiral algebra (N = 2 supersymmetry). But if we remove the U (1) 1 factor, the corresponding two-dimensional theory needs both left-moving SU (2) k and right-moving U (1) −2(k+2) modes. Finally, in the form (C.34) the theory is U (2) k,−2(k+2) and it is straightforward use the expressions in section C.4 to work out the spins and charges of the quasiparticles. k = 2, M = −3 Here eqn. (C.26) becomes (Ising × U (1) −8 ) /Z 2 . This is the topological field theory of the T-Pfaffian state [26,27], which is expected to play a role both in topological insulators and topological superconductors [69,34], and which we have discussed in section 6. Substituting these values in eqn. (C.30) we find the Lagrangian