Orbifold Schur Index and IR formula

We discuss orbifold version of the Schur index defined as the supersymmetric partition function in S^3/Z_n x S^1. We first give a general formula for Lagrangian theories obtained by localization technique, and then suggest a generalization of the Cordova and Shao's IR formula. We confirm the generalized IR formula gives the correct answer for systems with free hypermultiplets if we tune the background fields so that they are invariant under the orbifold action. Unfortunately, we find disagreement for theories with dynamical vector multiplets.


Introduction
Exactly calculable quantities in supersymmetric field theories play important roles in recent progress in quantum field theories. The Schur index of N = 2 superconformal field theories is one of such quantities. There are different ways to calculate the index.
The Schur index is a specialization of the superconformal index of N = 1 superconformal field theories [1,2]. (See [3] for various limits of the superconformal index.) The superconformal index can be regarded as the supersymmetric partition function in S 3 × S 1 . For Lagrangian theories it is defined as the path integral in the background, and we can reduce it as a finitedimensional integral by using localization method. This is also available for a non-Lagrangian theory if we know a UV Lagrangian theory that flows to the theory.
Cordova and Shao [4] proposed an interesting formula for the Schur index which gives the index as the trace of so-called a quantum monodromy operator. With this formula we can calculate the index from the information of the BPS spectrum of the theory in the Coulomb branch. Although the BPS spectrum depends on the Coulomb moduli parameters and jumps on walls of marginal stability the quantum monodromy operator is wall-crossing invariant, and so is the index. The formula is generalized to decorated indices by introducing defect operators [5,6,7].
For class S theories, which are realized on M5-branes wrapped on Riemann surfaces, the Schur index is expressed as a correlation function of a two-dimensional topological field theory on the associated Riemann surface [8,9,10,11,12,13,14].
The Schur index receives contributions of a special class of gauge invariant operators, which are called Schur operators. Beem et. al. [15] shows that the set of Schur operators form a chiral algebra of two-dimensional CFT. Once we identify the chiral algebra associated with a four dimensional theory the Schur index is obtained as the vacuum character of the chiral algebra. The characters of other modules are also important and are related to line and surface operator insertions [16,7]. For theories of class S there is a prescription to obtain the corresponding chiral algebra [17].
The purpose of this paper is investigate a generalization of the Schur index defined by replacing the background manifold S 3 × S 1 by its orbifold S 3 /Z n × S 1 . Such an orbifold generalization for the superconformal index of N = 1 and N = 2 theories has already been investigated in [18,19,20,21,22]. The index is defined in [18] by using Z n which preserves a supercharge of a specific chirality. Because we need only one supercharge (and its hermitian conjugate) for the definition of the superconformal index this orbifolding is consistent with the definition of the index. However, in the case of Schur index, we need to preserve two supercharges with opposite chirality (and their hermitian conjugate). Therefore, we need to modify the definition of the orbifold index by introducing extra SU (2) R × U (1) r twist. The application of the localization technique for the orbifold Schur index is straightforward and will be shown in Section 2. Then, we discuss a generalization of the IR formula to the orbifold case. We suggest a natural generalization of the Cordova and Shao's formula based on a physical interpretation of the formula, and apply it to some simple examples. For systems consisting of free hypermultiplets we find agreement of the UV and IR formulae. Unfortunately, however, for more general systems including dynamical vector multiplets the suggested formula does not give the desired results.

UV formula
Let us consider an N = 2 superconformal field theory defined in S 3 × R t . Let H, J, J, R, and r be the Hamiltonian, the third component of the lefthanded spin, that of the right-handed spin, the SU (2) R Cartan generator, and the U (1) r charge, respectively. The superconformal index is defined by where the trace is taken over all gauge invariant states in S 3 . We denote the set of Cartan generators of the flavor group F by T F = (T F,1 , . . . , T F,r F ). Other Cartan generators are defined by where Q and Q are supercharges with the following quantum numbers.
The definition (1) respects Q. Namely, all Cartan generators used in (1) commute with Q. Because µ x is Q-exact the index (1) is independent of x. 1 The superconformal index can be regarded as the supersymmetric partition function in S 3 × S 1 . If the theory has a Lagrangian description with gauge group G we can define the index as the path integral. By using localization technique we can reduce the path integral to the finite dimensional integral where Pexp is the plethystic exponential defined by i is the one-particle index defined by i(p, q, t, z) = tr e 2πi(J+J) x µx p µp q µq t µt z T , where the trace is taken over all one-particle states including gauge noninvariant states. T includes both flavor and gauge Cartan generators, and z is the set of the corresponding fugacities. Namely, z T = z T F F z T G G . dµ is the integral over the gauge fugacities where |W G | is the size of the Weyl group of G. The Schur index is obtained from the superconformal index by the specialization t = 1. Then µ t disappears from (5), and all the remaining Cartan charges commute with not only Q but also Q. Therefore, the Schur index receives contributions of operators that carry µ x = µ p = 0. Such operators are called Schur operators. As a result, the Schur index is a function of the superconformal fugacity q and the flavor fugacities z F .
There is another index that receives from the same set of operators as the Schur index. It is the Macdonald index defined from the superconformal index by taking the limit p → 0. We do not consider the Macdonald index here because its definition does not respect the supercharges Q and Q which are important when we discuss BPS configuration associated with the IR formula. Even so, the Macdonald index is closely related to the Schur index, and can be reproduced from the chiral algebra up to some ambiguity that can be fixed by using information of 4d SCFT [23]. In the recent paper [24] an orbifold version of the Macdonald index is studied, and a close relation to the chiral algebra is observed. It would be important to study relation between the orbifold Schur index defined below and the orbifold Macdonald index in [24].
The one-particle Schur index is given by where i U (1) , i V , and i H represent the contributions of a single U (1) vector multiplet, charged vector multiplets, and half-hypermultiplets, respectively. They are explicitly given as where χ adj ≡ χ adj −r G is the character of the adjoint representation of G with the Cartan contribution subtracted, and χ R H is the character of the G × F representation of the half-hypermultiplets. 2 Because i U (1) does not depend on the gauge fugacities z G we can factor out the Cartan contribution in (7) as λ r G , with λ defined by where (z; q) ∞ is the q-Pochhammer symbol defined by A generalization of the superconformal index to the orbifold background S 3 /Z n × S 1 was first investigated in [18]. They defined the orbifold with the discrete group Z n generated by This is consistent to the definition (1) of the superconformal index in the sense that g n commutes with the supercharge Q. If the theory has gauge and/or flavor symmetry we can turn on holonomies h by using g n ω h T n instead of g n as the Z n generator. Once we choose the Z n generator, it is straightforward to generalize the formula (7) to the orbifold case. What we have to do first is projecting away the contribution of Z n non-invariant states from the oneparticle index (9). This projection is carried out by inserting the projection into the trace in (9). Because g n can be expressed in terms of the Cartan generators appearing in (1) as g n = ω 2J n = ω µx−µp+µq−µt n , insertion of (g n ω h T n ) k is equivalent to the replacement of the fugacities and the one-particle index for the Z n orbifold is By using this orbifold one-particle index, the orbifold index is given by where the factor e ε( h) is the contribution of zero-point energy. For ordinary index this factor just gives an overall factor and usually neglected. However, in the orbifold case this factor is important because it depends on the gauge holonomy h G . The zero-point factor e ε( h) is easily obtained by taking the product of all contributions of one-particle states. For example, if the oneparticle index is expanded as i h n = i c i p a i q b i (· · · ) (We explicitly show p and q dependence for simplicity, and the dots include other fugacities.) then the corresponding zero-point factor is given by Although this infinite product is usually divergent, we can obtain finite result by using an appropriate regularization such as ζ-function regularization. Now, let us apply the same prescription of the orbifolding to the Schur index. We cannot obtain the orbifold Schur index by the specialization t → 1 from the orbifold superconformal index above because this contradicts the orbifold action (17). For the consistency to t = 1 we use instead of g n . Namely, we combine g n with the additional SU (2) R × U (1) twist to keep both Q and Q invariant. Again, we can turn on holonomies h, and the insertion of (g n ω h T n ) k is equivalent to the replacement, (x, p, q, z) → (ω k n x, ω −k n p, ω k n q, ω k h n z).
Therefore, the orbifold one-particle index is With this one-particle index we can calculate the orbifold Schur index of Lagrangian theories by using (19), which we call UV formula in the following. Before discussing the IR formula for the orbifold Schur index, let us apply the UV formula to some simple systems. In the following examples the zeropoint factors give overall factors independent of the holonomies and we omit them.
U (1) vector multiplet The U (1) contribution in S 3 is given by λ in (13). The Z n action is simple phase rotation of variable q, and we obtain the orbifold one-particle index i n = −2q n /(1 − q n ). The orbifold Schur index is Free hypermultiplet Let us consider the system of a single free hypermultiplet (q, q). This has SU (2) F flavor symmetry. Let z be the fugacity for U (1) F ⊂ SU (2) F and F be the corresponding generator such that q and q carry F = +1 and −1, respectively. The one-particle index before Z n projection is The Z n generator acts on the variables as g n ω hF n : (q, z) → (ω n q, ω h n z).
By definition the generator g n ω hF n must satisfy (g n ω hF n ) n = 1. (Otherwise the fiber bundle associated with the fields is ill-defined.) Due to the fractional SU (2) R charge R = −1/2 of q and q, g n n acts on the hypermultiplet as −1. To compensate this, we need to take fractional holonomy h ∈ Z + 1 2 . Then, after the projection we obtain where [x] n for x ∈ Z is the minimum non-negative integer satisfying [x] n ≡ x mod n. In the Z 2 case, the one-particle indices for two holonomies h = ±1/2 are QED Let us consider U (1) gauge theory with N f hypermultiplets. This theory is not conformal, but we can obtain the index by applying the localization formula. When we consider orbifold index, we should be careful about the fact that U (1) r is broken to the subgroup Z N f ⊂ U (1) r by anomaly. The orbifold action must be consistent to this unbroken symmetry. For example, in the case of Z 2 orbifold, N f must be even. For N f = 2 the Z 2 orbifold index is given by where i h 2 is the one-particle index (27) for a single hypermultiplet. We did not turn on the SU (N F ) flavor symmetry for simplicity, and we omitted the zero-point factor which does not depend on h = ±1/2.

IR formula
The IR formula proposed by Cordova and Shao [4] gives the Schur index by using the information of BPS spectrum in the Coulomb branch.
Let Γ be the charge lattice of flavor and gauge charges, and γ, γ be the associated Dirac pairing. The central charge of a particle is determined by its charge γ ∈ Γ, and we denote it by Z γ . The flavor sublattice Γ F ⊂ Γ is defined as the set of charges γ which have vanishing pairing γ, γ = 0 with arbitrary γ ∈ Γ.
Let L be the set of primitive charges such that the charge of an arbitrary BPS particle is given by nγ with γ ∈ L and n ∈ Z + . The BPS spectrum at a point of Coulomb branch is encoded in a set of functions K γ (z) defined for each γ ∈ L. These functions are called quantum Kontsevich-Soibelman factors. The functional form of K γ (z) depends on the helicity of the BPS particle γ. (By an abuse of notation we use γ as a label of sorts of particles.) In the following we deal with only BPS particles belonging to a half-hyper multiplet, and the function K γ for such a particle is given by where (q) k is the q-factorial defined by (q) k = k i=1 (1 − q i ). The Cordova and Shao's IR formula [4] is where γ∈L is the phase ordered product according to arg Z γ . r is the rank of the theory, which is the number of massless U (1) gauge fields at a generic point of the Coulomb branch. X γ are operators satisfying the quantum torus algebra 3 The trace tr X γ is defined as an isomorphic map from Γ F to C * . If γ / ∈ Γ F tr X γ = 0.
To physically interpret this formula, we should understand the structure of BPS configurations in the Coulomb branch [5]. In general, a BPS configuration contains massive BPS particles with different charges. For a configuration in S 3 × R t to preserve the supercharges Q and Q, which are necessary to define the Schur index, the particles must be aligned along a large circle in S 3 , and the position θ γ on the circle is determined by the central charge Z γ by θ γ = arg Z γ . Thus we can specify the particle distribution in a BPS configuration by giving a set of occupation numbers n γ for γ ∈ L. We denote this set by {n γ } γ∈L .
If we expand the function K γ as then the index is given as the summation over the occupation numbers: The summand gives the contribution of a specific set of occupation numbers, and consists of the following three factors.
• The factor λ r is the 1-loop contribution of the massless vector multiplets.
• The factor C is the contribution associated with internal degrees of freedom of BPS particles.
• The trace factor can be identified with the classical contribution of massless vector multiplets. The angular momentum induced by the Poynting vector due to the existence of mutually nonlocal charges contributes to the index by the factor e 2πi(J+J) q J+J , and q-dependence of the trace factor gives this factor. For example, let us consider two adjucent charges γ and γ on the large circle. If they carry mutually non-local charges, the induced massless gauge fields contribute to the angular momentum J + J by ± 1 2 γ, γ . The angular momentum is independent of the distance between charges while the signature depends on the order of the charges along the circle. This order dependence is correctly reproduced by the algebra (31). This factor also provides the dependence on the flavor fugacities. Now we suggest an IR formula for the orbifold as a natural generalization of the original one. The Z n acts on the large circle as a shift by 2π/n and the orbifolding makes it a circle with circumference 2π/n. In other words, the large circle in the covering space consists of n fundamental regions, and only BPS particles in one of the fundamental regions are independent. Let L n be the set of primitive charges associated with a specific fundamental region. The charge distribution of a BPS configuration in the orbifold is specified by {n γ } γ∈Ln . Therefore, the index should be given as the summation over {n γ } γ∈Ln .
The first factor in the summand should be replaced by the U (1) factor λ r n with λ n given in (23). The second factor associated with the internal degrees of freedom of BPS particles should be the product of C γ (n γ ) for γ ∈ L n . The third factor, as we mentioned above, can be regarded as the classical contribution to the angular momentum and the flavor charges. The angular momentum can be obtained by integrating the contribution of the Poynting vector over the orbifold. The same result is obtained by first calculating the angular momentum for the covering space S 3 , and dividing the result by n. This is also the case for the flavor charges. Correspondingly, the trace factor should be replaced by the n-th root of the trace for the configuration in the covering space. Combining these, we obtain This is the IR formula we want to discuss in the next section. An additional comment for the trace factor would be in order. As we mentioned above, only charge distribution in a specific fundamental region is independent, and the distribution in other regions should be determined by the Z n symmetry. The phase ordered trance in (34) must be taken over all charges in the covering space. It is important that the Z n acts on charges non-trivially, and the product is not simple n-th power of the product in the fundamental region. In the n = 2 case, for example, the Z 2 action flips the signs of charges as we will explicitly see in the next section, and the trace factor takes the form tr(X n 1 γ 1 X n 2 γ 2 · · · X n 1 −γ 1 X n 2 −γ 2 · · · ) 1/2 .

Comparison
Let us apply the IR formula (34) to a few simple examples and compare the results with what are obtained by the UV formula. We first consider the system of a free hypermultiplet q = (q, q). Before computing the index by the UV and IR formulae it is important to check the consistency between BPS configurations in the Coulomb branch and the orbifold action.
In the system of a single hypermultiplet, we have only two sorts of particles with flavor charge ±γ. In a BPS configuration there are particles with charge +γ at θ γ = arg Z γ and anti-particles with charge −γ at θ γ + π. This cannot be invariant under Z n action with n ≥ 3. Even for n = 2 the Z 2 exchanges particles and anti-particles, and the orbifolding seems to contradict the BPS configuration in the Coulomb branch. When n = 2, actually, we can take it back to the original configuration by performing additional charge conjugation. This additional transformation is also necessary to keep the mass term in the Lagrangian Z 2 invariant. To make the hypermultiplet massive we need to turn on the vev φ = m of the scalar component φ of the non-dynamical background vector multiplet coupling to the U (1) F current. Because φ carries U (1) r charge +2, the g 2 action changes its sign. This is compensated by the additional charge conjugation, which flip the sign of the background vector multiplet.
The charge conjugation exchanging q and q is realized by the SU (2) F transformation q → U q with This anti-commutes with the U (1) F generator iσ z , and works as the charge conjugation. For consistency, we should set the U (1) F Wilson line to be z = ±1. (A generic Wilson line is not allowed because the operator z F does not commute with iσ x , and incompatible with the orbifold action.) Now we have chosen the Z 2 generator g 2 U consistent to the Coulomb branch vev. Let us calculate the index by using UV and IR formulae. When z = ±1, the orbifold with the charge conjugation twist is in fact equivalent to the Z 2 orbifold with fractional holonomies studied in the previous section up to SU (2) F rotation, because the fractional holonomy ω hF 2 = ±iσ z is SU (2) F conjugate to the charge conjugation U = iσ x . Therefore, the index is still given by (27) with z = ±1. By setting z = 1, the one-particle index becomes and the Schur index is On the IR side, we have only one charge γ in L 2 . The suggested formula (34) gives and this agrees with (37). By introducing multiple hypermultiplets we can realize more general orbifolds. Let us consider the system of k free hypermultiplets. There are 2k sorts of particles. For Z 2k invariance, we need to tune the Coulomb branch vev φ and introduce an appropriate twist U k . Let φ = diag(a 1 , . . . , a k ) ⊗ σ z ∈ sp(k) be the Coulomb branch vev. (We use the basis in which the Sp(k) invariant tensor is given by J = 1 k ⊗ .) We tune a i so that the central charges are Z 2k symmetric and given by The action of g 2k shifts a i to a i+1 for i = 1 ∼ k − 1 and a k to −a 1 . To keep the background unchanged, we need a twist U k such that the transformation φ = U k φU † k acts on a i as The following U k realize this transformation: (This is a generalization of the charge conjugation (35) in the k = 1 case.) Just by the same reason as the Z 2 case we set the Wilson line to vanish. The diagonalization of U k gives the eigenvalues ±α m with For each pair (α m , −α m ) of eigenvalues this is the same as the action of the fractional holonomy h = m + 1/2 on a single hypermultiplet, and the oneparticle index of the whole system is the sum of (26) over h = 1/2, . . . , k−1/2: The Schur index is I n = Pexp i n = E q (1), and this is the same as what we obtain from the IR formula. Each fundamental region of the large circle contains one sort of particles and the IR formula (34) gives I = E q (1). We can consider more general case in which n is not 2k but its divisor n = 2k/d. Such an orbifold is defined by using (g 2k U k ) d as the Z n generator, and the one-particle index becomes and the Schur index is I n = Pexp i n = E q (1) d . Again, this is reproduced by the IR formula (34) by taking account of the d sorts of particles in a fundamental region. Up to here we have found good agreement between UV and IR results. Let us move on to a more complicated system, QED with N f hypermultiplets. Although this system is not conformal, it is known that the two formulae give the same answer for the ordinary Schur index [4], and it is natural to expect this holds for orbifolds, too. However, disappointingly, we find discrepancy for orbifold index in this case.
For example, let us consider Z 2 orbifold of the 2-flavored QED. The UV formula gives the index (28). When we use the IR formula we need to take account of the existence of two sorts of particles in the fundamental region. Let γ 1 and γ 2 be their charges and n 1 and n 2 be the corresponding occupation numbers. In the covering space the charges of particles in a fundamental region are always canceled by the mirror images in the other fundamental region, and the trace factor in (34) does not impose any constraints on n 1 and n 2 . If we assume the trivial flavor holonomy, the trace factor simply gives 1. This means the index is essentially the same as that for two half-hyper multiplets in S 3 , and given by This is obviously different from the result of the UV formula (28).

Discussions
In this paper we generalized the Schur index to the orbifold S 3 /Z n in such a way that the Z n action preserves the two supersymmetries respected by the definition of the Schur index. We also naively generalized the Cordova and Shao's IR formula for Schur index to the orbifold case. It reproduces the correct index for a system of free hypermultiplets when the fugacities and holonomies are chosen in a Z n invariant way. However, it is far from satisfactory. It has the following deficits.
• Although any SCFT admits Z n orbifold with an arbitrary n = 2, 3, . . . in the UV description, the IR formula works only for special values of n depending on the theory.
• Even for a system of free hypermultiplets it is not possible to turn on generic Wilson lines.
• For systems with dynamical vector multiplets the formula does not reproduce the correct answer.
The second and third deficits may be related to each other. Incompatibility of the generic Wilson lines seems to prevent us from performing path integral over all gauge field configurations. At present, we cannot claim anything definite for this point, and more investigation is desired. It is important to study more general systems to clarify to what extent our formula works and (if possible) how we should improve it to make it applicable to general systems. In particular, it would be interesting and important to study the orbifold Schur index of non-Lagrangian theories such as Argyres-Douglas theories. Lagrangian theories that flow to a class of Argyres-Douglus theories are proposed in [25,26,27,28,29,30,31] and this enable us to apply the UV formula of the orbifold index to the non-Lagrangian theories.
It is also interesting to analyze the relation between the orbifold index and chiral algebra. In the unorbifolded case, the chiral algebra is realized in a complex plane, which is identified by Weyl rescaling with S 1 × R t ⊂ S 3 × R t , where S 1 is the large circle in S 3 on which BPS particles are aligned. The Virasoro generators L 0 and L 0 on the plane are related to the superconformal generators by L 0 = 1 2 (H + J + J) = 1 2 (µ x + µ p ) + µ q , For Schur operators with µ x = µ p = 0 L 0 = 0 and the Z n generator g n = ω µx−µp+µq n is given by Namely, this is Z n orbifold acting on the worldsheet. Therefore, in the context of the chiral algebra, the orbifolding should be regarded as the insertion of a twist operator at the origin. Such a relation is studied for an orbifold version of the Macdonald index in [24]. It would be interesting to do a similar analysis for the orbifold Schur index.