Yangian associated with 2D $\mathcal{N}=1$ SCFT

Recently, Maulik and Okounkov proposed an integrable lattice model where the degree of freedom at each site is identical to the Hilbert space of free boson in two dimensions. We give a brief review of their construction and explain the relation with $\mathcal{W}_n$ algebra and Calogero-Sutherland model. As a generalization, we examine the Yangian associated with $\mathcal{N}=1$ superconformal algebra which describes a supersymmetric extension of Calogero-Sutherland model and compare it with the literature.


Introduction
A few years ago, Alday, Gaiotto and Tachikawa proposed a conjecture [1] that the partition function of 4D N = 2 supersymmetric gauge theories is identical to the correlation function of Liouville (Toda) system in 2D. It triggered rapid development of the technology of 2D CFT and in a few years, there appeared some proofs of the claim from the physical [2,3] and mathematical [4,5] sides. Some features of [4,5] are the use of techniques of the integrable models such as the Calogero-Sutherland system and the symmetry behind it. In [4], they used a nonlinear symmetry, degenerate double affine Hecke algebra which is a natural symmetry of Calogero-Sutherland. One may regard three objects, the cohomology ring of Hilbert scheme of points on surfaces (U(1) instanton), the Calogero-Sutherland model and free boson in 2D as the same objects. All of them are characterized by a Hilbert space whose basis is labeled by a Young diagram Y , Jack polynomial in case of the Calogero-Sutherland, the Fock basis for the free boson, and the fixed points of torus action for the equivariant cohomology of Hilbert scheme.
For the description of SU(N) gauge theories, one has to extend the framework to include the basis with N Young tables. We have to compose the states labeled by single Y to give those with N Y s. For the free boson it is defined by the quantum Miura transformation which defines W nalgebra [6]. For the Calogero system, it is given by a coproduct in spherical degenerate DAHA (refered as SH c ) [4] which defines the algebraic structure among the Jack polynomials. A more explicit construction of the coproduct was presented in [5] where they introduced the structure of integrable lattice model to analyze the problem.
A new feature of the last approach is to use an infinite dimensional Fock space F generated by Virasoro operator on each of the lattice site as the internal degree of freedom. 1 It seems to give a new inspiration in the definition of 2D CFT. In this paper, as the simplest application of their idea, we define an integrable lattice model by N = 1 superconformal field theory. It gives an infinite number of commuting charges by the combination of bosonic and fermionic oscillators. It naturally defines a supersymmetric generalization of Calogero-Sutherland model which may be compared with the known results in the literature [10]. Furthermore, the higher generators may define a supersymmetric W algebra [11].
We organize the paper as follows. In section 2, we briefly review the integrable lattice model proposed in [5]. We explain the definition of R-matrix and the construction of the commuting charges. In particular, for the simplest case where the lattice consists of a point, we obtain the Hamiltonian of the Calogero-Sutherland system as one of the conserved charges. For the lattice with N sites, the monodromy matrix gives the definition of associated Yangian. It is defined in terms of N bosons and is equivalent to a combined system with W N -algebra + U(1) boson. In section 3, we consider a generalization to N = 1 SUSY case. We define the R-matrix in terms of the commutation relation of superconformal generators as a reflection operator. The construction guarantees the existence of infinite number of commuting charges. It defines a supersymmetric generalization of Calogero-Sutherland system when the number of lattice is one. For the general cases, the Yangian is defined in terms of N pairs of boson and fermion fields. We compare the second commuting charge with the Hamiltonian of super Calogero-Sutherland model discussed in [10] and find some discrepancies. It may suggest that the supersymmetric extension of Calogero-Sutherland system is not unique.

Yangian for 2D CFT
In this section, we review the construction of the integrable lattice model with a Fock space on each lattice site, proposed by Maulik and Okounkov (MO) [5]. The Fock space is generated by a standard free boson in 2D: The vacuum state |η is defined by α n |η = 0 (n > 0) and α 0 |η = η|η and we denote the Fock space generated from it as F (η). We are going to define a lattice model where we have an independent Fock space F (η i ) on each site i (i = 1, · · · N). The boson field which acts on F (η i ) will have extra index as φ (i) , α (i) n when the distinction is necessary.

Definition of R-matrix
In the integrable models, the R-matrix is defined as operator acting on the direct product of linear space, say V : Here we use the Fock space F (η i ) as such vector space. We write two Fock spaces as F (η i ) (i = 1, 2) in the following. R-matrix is an endmorphism . The rapidity parameter u is related to the difference of η, u : MO's R-matrix is defined to depend only on φ − but not on φ + . We introduce the Virasoro operator for φ − as, ′ denotes the summation without terms involving zero mode and L (0) .We note that we have replaced α − 0 by its eigenvalue on |η 1 1 ⊗ |η 2 2 . L n (u, ρ) satisfies Virasoro algebra with central charge c = 1 − 12ρ 2 except for a constant shift of L 0 . 2 We introduce four operators R ǫ 1 ,ǫ 2 (ǫ i = ±) which act on the Fock space F − (u), the Hilbert space for φ − : where |u − is the Fock vacuum for φ − with α − 0 |u − = u|u − . Obviously the operator R ++ = 1 is trivial. The operator R −− is characterized by relations α n R −− = −R −− α n which can be easily solved. R −+ is a nontrivial operator which reverses the momentum direction and is known as the reflection operator in Liouville theory [12]. R +− is a novel operator which is the focus of this paper. We will see later that the conditions (4,5) uniquely determines R ǫ 1 ,ǫ 2 . We also note that in [13] the reflection operator plays essential role in the construction of AFLT states [2,3].
The R-matrix is defined as R(u) := R +− (u, ρ). The conditions (4,5) are replaced by: In [5], it was shown that this R-matrix satisfies the Yang-Baxter equation (YBE). Instead of following their argument, we give a shortcut of the proof which depends only on the definition in terms of Virasoro generators. For the discussion of Yang-Baxter relation, we introduce a new operatorŘ(u) := PR(u), where P is the operator exchanging two Fock spaces F 1 , F 2 . On the φ − subspace,Ř(u) = R −+ , while on the φ + subspace, R(u) acts trivially. The commuting chart (the YBE) forŘ(u) is The relation u 3 = u 2 − u 1 with the chart above shows the YBE. We can see that the background charge of the φ − subspace serves exactly as the spectral parameter (rapidity parameter).
For later reference, let us write down the YBE for R(u).

Explicit Computation of R(u)
We note that when u → ∞, all terms involving ρ can be neglected, and therefore R(u) behaves as the identity operator under this limit. The R-matrix, therefore, can be expanded as follows The relation between R (n) and r (n) is expressed for the lower orders as, . . .

The defining recursion equation for R(u) can then be expressed as
In general (9) may be solved from the lower orders. It may not be so obvious if such system of equations is consistent. The Jacobi identity serves as such consistency condition, The confirmation of such identity is straightforward but a bit lengthy. We give some explicit computation in appendix A.
Let us demonstrate the explicit form of R-matrix for the lower orders. The recursion formula can be written more explicitly in terms of r (i) as, . . .
These equations leave a degree of freedom of adding constants to r (n) 's, which will be fixed by condition (7). Eq. (12) can be solved to Substituting (12) into (13), we get It is solved as, The higher coefficients r (3) , r (4) will be given in Appendix B.

Yangian
As a standard strategy in the integrable models, one can construct the monodromy operator (Tmatrix) through the R-matrix. We use the R-matrix as the L-operator and treat the Fock space F (say F 0 ) as the auxiliary space. The T -matrix is defined as, which defines an endmorphism on It can be formally expanded as, where the summation is over the Hilbert space F 0 . Since it is the Fock space of free boson, one may parametrize it by a Young diagram λ = (λ 1 , λ 2 , · · · ) with λ 1 ≥ λ 2 ≥ · · · ≥ λ r > λ r+1 = · · · = 0. The operators α λp for the Young diagram λ described above. The coefficient O λν gives the endmorphism on ⊗ N i=1 F i . The set of endmorphism on F defines a Lie algebra and we will denote it as g. The T -matrix gives its Yangian Y (g) and satisfies the fundamental algebraic relation, From the monodromy matrix, one can define mutually commuting operators, the transfer matrix T (u). In the standard construction, it is given as the trace of T(u) over the auxiliary space. Here such construction does not work, the trace over the Hilbert space F diverges in general. Instead, one may define the transfer matrix by using the vacuum expectation value of T(u): Here we take u = (η a − η)/ √ 2 with a = 1, · · · , N. As we show below, such transfer matrices commute with each other from the condition (7). Here is a short computation to prove this: , and the stability conditions for the vacua, (we note that the above equation becomes The computation from here is straightforward, coefficients of the 1/u-expansion of the transfer matrix give infinitely many conserved charges. The existence of such operators implies the integrable structure. For the one-site case (N = 1), the first two charges are 3 Extracting the independent part from the second charge and rescale it, H := 2 with Q := − √ 2ρ. This can be identified with the Calogero-Sutherland (CS) Hamiltonian written in terms of free boson with the identification Q = √ β − 1/ √ β where β is the coupling constant of CS [14,15]. We note that CS is a system of M interacting particles on a circle with a long range interaction. The description in terms of free boson is exact only in the large M limit.
When N > 1, these conserved charges are generally the sum of the one-site charge for each site, c (n) i for the i-th site, with additional cross-terms. The first two charges are The higher-rank charges can be computed in the same way. 3 Since there are no α − 0 in r (n) 's, we will adopt the notation 0 0| . . . |0 0 for VEV, unless it can be confusing. We also drop the suffix ( (1) ) in α n for simplicity.

U(1)⊗W N Structure in the Integrable System
We will investigate the algebraic structure of the above integrable system in this subsection. To get prepared, we need the commutator results for the one-site case n here is the stress tensor without zero modes for α n . We will denote the above result as [H, α n ] = −2nL (0) n ± Qn 2 α n for short.
In the N = 2 case, the full Hamiltonian is n It coincides with the Hamiltonian for the generalized Calogero-Sutherland system [13,[16][17][18]. It is easy to show that The φ − part in those formula, L − n := L n gives the Virasoro operator which is included in the W N algebra (plus the contribution of U(1) factor) with the central charge, (see for example, chapter 5 of [19] for the detail).
We have to mention that the relation between the algebra SH c [4] (a short name for spherical degenerate double affine Hecke algebra) and Yangian which we consider here. The algebra SH c has infinite number of generators D r,s with r, s ∈ Z with r ≥ 0. The algebra contains a parameter β and it is reduced to W 1+∞ algebra when β → 1. While the algebra is complicated, it can be generated by the algebra by three operators D ±1,0 , D 0,2 . The expression for them coincides with a α (a) ±1 and H for the representation by N bosons. Since they coincide with the Yangian generators, SH c and Yangian are the same symmetry. It is known that the representation of SH c by N bosons is identical with W N algebra with U(1) factor (some detail of the correspondence was shown in [4,20].) So the Yangian construction gives an alternative and a systematic definition of W N algebra. W ∞ [µ] symmetry [21] is known to unify W N -algebra. In this sense, it should be the same symmetry as SH c or the Yangian while the latter contains extra U(1) factor.
There is another interesting link between W N algebra and MO's R-matrix. The definition of generators of W N algebra (with diagonal U(1) factor) is given in terms of N bosons through the quantum Miura transformation [6], (28) On the left hand side, there is an ambiguity of ordering N bosons. Different ordering gives different but equivalent set of W generators. MO's R-matrix plays a role of changing the order, The proof is straightforward, by using the notation φ ± := 1 f is some function independent of φ − and T − is the Liouville stress tensor on the φ − subspace, thus we can see that eq. (29) holds.

Extension to N = 1 Superconformal Field Theory
In this section, we will introduce an R-matrix respecting the superconformal algebra and build the corresponding integrable system. The superconformal algebra is given by the following OPEs between the stress tensor and an h = 3/2 primary field T F (z), whereĉ is the central charge of super Virasoro algebra which is related to the bosonic case as c = 3ĉ 2 . For notational record, the free fermion in 2D can be expanded as For NS sector r ∈ Z + 1 2 and for Ramond r ∈ Z. In the following we focus on the NS sector. We write the Fock space generated by oscillators α n , ψ r from the vacuum |η (α n |η = ψ r |η = 0, (n, r > 0) and α 0 |η = η|η ) asF (η).

Definition of R-matrix
The super Virasoro operators are written by free boson and fermion as, The R-matrix is defined as an endomorphism onF ⊗F . To describe the Fock space we introduce two sets of free bosons and fermions φ (a) , ψ (a) and introduce φ ± := 1 ) as before. We write the super Virasoro generator in terms of φ − , ψ − . Their mode expansion contains parameters ρ and zero mode (u) of φ − as (again we use the expansion in w plane): which givesĉ = 1 − 8ρ 2 SCFT. We define the reflection operators R ǫ 1 ,ǫ 2 by requiring (6)  R ǫ 1 ,ǫ 2 (u, ρ)L B n (u, ρ) = L B n (ǫ 1 u, ǫ 2 ρ)R ǫ 1 ,ǫ 2 (u, ρ) , R ǫ 1 ,ǫ 2 (u, ρ)L F n (u, ρ) = L F n (ǫ 1 u, ǫ 2 ρ)R ǫ 1 ,ǫ 2 (u, ρ) . (37) We keep also the normalization condition (7). The R matrix is defined as R +− (u, ρ) as the bosonic case: The YBE can be proved as the bosonic case. We can write down the recursion equation for the coefficients R (n) as The Jacobi identity for these recursion relation gives nontrivial consistency conditions. We give an explicit proof for such relations also in Appendix A. It implies that the R (m) can be determined uniquely to the full order.

Explicit Form of R-matrix and Conserved charges
R-matrix Let us explore the first few terms of the expansion of R-matrix. The first set of equations for r (1) is The solution is Substituting this result into equations for r (2) , we get With an identity Conserved charges Then we can go further to calculate conserved charges with the same definition of transfer matrix (19). The first charge, is just a sum of N copies of that of a one-site system.
It may be better to write the second charge in the form of Hamiltonian, It is divided as a sum over the diagonal and off-diagonal part as, where For the N = 1 case, the diagonal part H (a) gives a supersymmetric generalization of the Calogero-Sutherland Hamiltonian (23). By construction it is one of the infinite commuting charges defined by c (n) (n = 1, 2, 3, · · · ). In this sense, it gives a definition of integrable model -a supersymmetric generalization of Calogero-Sutherland model.

Comparison with Super CS Model
A supersymmetric extension of Calogero-Sutherland model has been developed for some years since [10]. They introduced both bosonic and fermionic coordinates x i , θ i (i = 1, · · · , N) and defined an integrable system with similar long range interaction. Let us quote the definition in their recent article [22] where the relation with the superconformal field theory is well-explained. The operator which plays the role of the Hamiltonian is given as, (α := 1/β), They argued that the eigenstates can not be completely specified by the Hamiltonian since there are some degeneracy. To resolve them, we need another operator ∆, These operators can be rewritten in the language of free boson and fermion oscillators in the N → ∞ limit as in the bosonic case [14,15]. We introduce the analog of power sum for n > 0, We introduce bosonic and fermionic oscillators by the correspondences (again n > 0, r > 0) where the index r for fermion is assumed to be half-integer. Based on such correspondence, they derived the correspondence between super Jack polynomial and the null states of N = 1 superconformal field theory. [22] For the comparison with our model, we derive an oscillator expression for the two conserved charges [23], The Hamiltonian looks similar to the expression (48) if we identify Q ≡ √ α − 1 √ α but there are some discrepancy. For example, ψψα term does not have terms like α −r−s ψ r ψ s (r, s > 0). It implies that the construction of the supersymmetric generalarization of Calogero-Sutherland model may not be unique. Indeed, their construction and ours contain the same bosonic part and have an infinite number of commuting charges. In that sense, both constructions define a solvable model with the same bosonic part.

Conclusion
In this paper, the Yangian for the N = 1 superconformal algebra was constructed along the line of proposal [5]. We proved the consistency conditions for the recursion formula and derived higher conserved charges, the existence of which implies the integrability of the system defined in terms of N pairs of free boson and fermion.
Obviously there are many open questions in the model. In the bosonic case, as we briefly explained, the Yangian is equivalent to W algebra and SH c . We expected that a similar correspondence exists in our model. So far, however, in our preliminary study, the super W algebra [11] is not included in the Yangian nor the connection with the super KP hierarchy [24] could be found. In this sense, some modification will be necessary to find the link with the recent developments of super W ∞ symmetry (see for example, [25,26]).