Integrable Structure of Superconformal Field Theory and Quantum super-KdV Theory

The integrable structure of the two dimensional superconformal field theory is considered. The classical counterpart of our constructions is based on the $\hat{osp}(1|2)$ super-KdV hierarchy. The quantum version of the monodromy matrix associated with the linear problem for the corresponding L-operator is introduced. Using the explicit form of the irreducible representations of $\hat{osp}_q(1|2)$, the so-called"fusion relations"for the transfer matrices considered in different representations of $\hat{osp}_q(1|2)$ are obtained. The possible integrable perturbations of the model (primary operators, commuting with integrals of motion) are classified and the relation with the supersymmetric $\hat{osp}(1|2)$ Toda field theory is discussed.


Introduction
Conformal field theory (CFT) provides effective tools to classify the fields in the theory we study and to compute their correlation functions. Perturbartion leads the system out of the critical point and breaks the conformal invariance. But special perturbations, called "integrable" still preserve an infinitedimensional abelian algebra of conserved charges, thus leading to an integrable theory.
The authors of [1], [2] showed that in this case the problem could be studied from a point of view of continuous field theory version of the quantum inverse scattering method (QISM) [3], [4]. The proposition is the following: at first, to use CFT symmetries for construction of QISM structures at the scale invariant fixed point and then to study the integrable perturbed model by obtained QISM tools. Our object of study is a model based on the superconformal symmetry (see e.g. [5], [6]). The superconformal field theory has been applied to the physics of 2D disordered systems, to the study of lattice models (the tricritical Ising model) and to the superstring physics. We build basic structures of QISM: quantum monodromy matrices, RTT-relation and fusion relations for the transfer matrices, using underlying superconformal symmetry.
The osp(1|2) supersymmetric Korteweg -de Vries theory (super-KdV) [8]- [11] is used as a classical limit of our quantum system. Due to the Drinfeld-Sokolov reduction of the osp(1|2) affine superalgebra, Miura transformation and Poisson brackets are introduced. Then the monodromy matrix is constructed. The associated auxiliary L-matrix satisfies the r-matrix quadratic Poisson bracket relation and plays a crucial role in the following.
After this necessary preparation, we move to the quantum theory (Sec. 3). The quantum Miura transformation is realized by the free field representation of the superconformal algebra [5], [6]. Then the super-Virasoro module is considered, where the quantum versions of integrals of motion (IM) act. Vertex operators are also introduced, to build quantum monodromy matrix.
The algebraic structure corresponding to the quantum case, coincides with the quantum superalgebra osp q (1|2). We construct the finite dimensional irreducible representations of this quantum Lie superalgebra (Sec. 4). It appears, that in quantum monodromy matrix (in comparison with the classical case) one term is missing in the P-exponent. The quantum L-matrix satisfies the so-called RTT-relation, giving the integrability condition in the quantum case. Considering monodromy matrices in the obtained osp q (1|2) representations, we get the functional relations ("fusion relations") for their traces -"transfer matrices". When the deformation parameter is rational (the case of CFT minimal models), these fusion relations become the closed system of equations, which, due to the conjecture of [1] can be used to find the full set of eigenvalues of transfer matrices. Also, we suppose that they could be transformed to the Thermodynamic Bethe Ansatz equations [7].
Finally, we discuss integrable perturbations of the model and the relation with the supersymmetric Toda field theory.

A review of classical super-KdV theory
The classical limit of constructions of papers [1], [2] leads to the Drinfeld-Sokolov KdV hierarchies related to the corresponding affine Lie algebras A (1) 1 and A (2) 2 . Our quantum model gives in classical limit the super-KdV hierarchy [8]- [11] related to B(0, 1) (1) (or osp(1|2)) affine Lie superalgebra. The supermatrix L-operator, corresponding to super-KdV theory is the following one: where D u,θ = ∂ θ + θ∂ u is a superderivative, variable u lies on a cylinder of circumference 2π, θ is a Grassmann variable, Ψ(u, θ) = φ(u) − iθξ(u)/ √ 2 is a bosonic superfield; h, v + , v − , X − , X + are generators of osp(1|2) (for more information see [12], [13]): Here [,] means supercommutator: [a, b] = ab − (−1) p(a)p(b) ba and the parity p is defined as follows: p(v ± ) = 1, p(X ± ) = 0, p(h) = 0. The "fermionic" operator L F considered together with a linear problem L F χ(u, θ) = 0 is equivalent to the "bosonic" one: The fields φ, ξ satisfy the following boundary conditions: where "+" corresponds to the so-called Ramond (R) sector of the model and "-" to the Neveu-Schwarz (NS) one. The Poisson brackets, given by the Drinfeld-Sokolov construction are the following: The L-operators (1), (3) correspond to the super-modified KdV, they are written in the Miura form. Making a gauge transformation to proceed to the super-KdV L-operator one obtains two fields: which generate the superconformal algebra under the Poisson brackets: These brackets describe the second Hamiltonian structure of the super-KdV hierarchy. One can obtain an evolution equation by taking one of the corresponding infinite set of local IM (they could be obtained by expanding log(t 1 (λ)), where t 1 (λ) is the supertrace of the monodromy matrix, see below): . .

Free field representation of Superconformal algebra and Vertex operators
To quantize the introduced classical quantities, we start from a quantum version of the Miura transformation (6), the so-called free field representation of the superconformal algebra [5]: [ Recall that there are two types of boundary conditions on ξ: ξ(u+2π) = ±ξ(u). The sign "+" corresponds to the R sector,the case when ξ is integer modded, the "-" sign corresponds to the NS sector and ξ is half-integer modded. The variable ǫ in (13) is equal to zero in the R case and equal to 1 in the NS case. One can expand T (u) and G(u) by modes in such a way: T (u) = n L −n e inu − c 16 , G(u) = n G −n e inu , whereĉ = 5 − 2( β 2 2 + 2 β 2 ) and L n , G m generate the superconformal algebra: In the classical limit c → −∞ (the same is β 2 → 0) the following substitution: iĉ {, } reduce the above algebra to the Poisson bracket algebra of super-KdV theory.
Let now F p be the highest weight module over the oscillator algebra of a n , ξ m with the highest weight vector (ground state) |p determined by the eigenvalue of P and nilpotency condition of the action of the positive modes: P |p = p|p , a n |p = 0, ξ m |p = 0 n, m > 0. (16) In the case of the R sector the highest weight becomes doubly degenerate due to the presence of zero mode ξ 0 . So, there are two ground states |p, + and |p, − : |p, + = ξ 0 |p, − . Using the above free field representation of the superconformal algebra one can obtain that for genericĉ and p, F p is isomorphic to the super-Virasoro module with the highest weight vector |p : in the NS sector and module with two highest weight vectors in the Ramond case: The space F p , now considered as super-Virasoro module, splits in the sum of finite-dimensional subspaces, determined by the value of L 0 : p . The quantum versions of local integrals of motion should act invariantly on the subspaces F (k) p . Thus, the diagonalization of IM reduces (in a given subspace F (k) p ) to the finite purely algebraic problem, which however rapidly become very complex for large k. It should be noted also that in the case of the Ramond sector G 0 does not commute with IM (even classically), so IM mix |p, + and |p, − . At the end of this section we introduce another useful notion -vertex operator. We need two types of them: V

Quantum Monodromy Matrix and Fusion Relations
In this section we will construct the quantum versions of monodromy matrices, operators L s and t s . The classical monodromy matrix is based on the osp(1|2) affine Lie algebra.
In the quantum case the underlying algebra is quantum osp q (1|2) [14] with q = e iπβ 2 and generators, corresponding to even root α 0 and odd root α: s (λ) of osp q (1|2) can be characterized by integer number s. The triple h α 0 , e ±α 0 forms the sl q (2) subalgebra and the whole osp q (1|2) finite dimensional irreducible representation is decomposed into the direct sum: j are the representations of sl q (2) with spin j (j runs through integer and half integer numbers). The odd generators e ±α acting on µ (q) j mix µ (q) j+1/2 and µ (q) j−1/2 representations. In the classical limit (q → 1) the representation π (q) s (λ) splits into the direct sum of the irreducible representations π r (λ) of osp(1|2): π (1) k=0 π s−2k (λ). In this sum k runs through integer numbers. The structure of irreducible finite dimensional representations of osp q (1|2) is similar to those of (A (2) 2 ) q [2]. This is the consequence of the coincidence of their Cartan matrices.
After these preparations we are ready to introduce the quantum analogue of L s operators: = π (q) s (λ) e −iπP hα 0 P exp One can see that the term e −2φ(u) 2e 2 α is missing in the P-exponent in comparison with the classical case (11). This is the general result for superalgebras and we will return to this in [17]. Analyzing the singularity properties of the integrands in P-exponent of L (q) s (λ) one can find that the integrals are convergent for −∞ <ĉ < 0 and need regularization for a wider region. Now let's prove that in the classical limit L (q) will coincide with L. First let's analyse the products of the operators we have in the P-exponent. The product of the two fermion operators can be written in such a way: where κ is equal to zero in the NS sector and equal to 1 in the R sector Also, for vertex operators we have: : e aφ(u) :: We can rewrite these products, extracting the singular parts: : e aφ(u) :: where c k (u) and d k (u) are operator-valued functions of u. Now the L (q) (λ) operator. can be expressed in the following way: K(u) ≡ : e 2φ(u) : e α 0 + ξ(u) : e −φ(u) : e α .
Here we have divided the interval [0, 2π] into small intervals [x m , x m+1 ] with x m+1 − x m = ∆ = 2π/N. Studying the behaviour of the first two iterations when β 2 → 0: we conclude that the second iteration can give contribution to the first one.
To see this let's consider the expression that comes from the second iteration: Now, using the above operator products and seeking the terms of order ∆ 1+β 2 (only those can give us the first iteration terms in β 2 → 0 limit) one obtains that their contribution is: Considering this in the classical limit we recognize the familiar terms from L: Collecting all τ (1) m one obtains the desired result: L (1) = L. Using information about representations one can also get: k=0 L s−2k (λ), where k runs over integer numbers.

Discussion
Returning to the quantum Miura transformation (13) it should be noted that one can choose another version: The reason why we introduce this one is the following: we have two candidates to perturb the model without spoiling the conservation laws, the so-called "integrable perturbations" [19]: Operator V 1 is screening for the deformation (13), the dimension of other one is h 1,5 (ĉ) − 1/2. But one should be able also to consider V 2 as screening -that is one should use (33). In this case the dimension of other one is 1/2 + h 1,2 (c) where c = 13 − 6(β 2 + 1/β 2 ) is the central charge of Virasoro algebra, generated by (33) without fermion term. When one of these operators is screening, another one is chosen as a perturbation, so one can relate the obtained model with osp(1|2) supersymmetric Toda field theory [18] with the corresponding Lagrangian: L = D u,θ ΦDū ,θ Φ − e −Φ − θθe 2Φ . Really, when β 2 is rational the obtained model can be treated as superconformal minimal model, perturbed by h 1,5 (ĉ) − 1/2 dimensional operator, or as minimal model, perturbed by 1/2 + h 1,2 (c) dimensional operator. We conjecture, that the same one could obtain from the quantum group reduction of the theory with the mentioned Lagrangian.
The solution of the functional system of equations (32) due to the conjecture of [1] determines the whole set of eigenvalues of t (q) s (λ) in the model. In the case when q is root of unity, this system should become a closed system of equations (quantum group truncation). Also, due to the results of [1] and [2] one can suppose that these functional equations can be transformed to the so-called Thermodynamic Bethe Ansatz equations [7], giving the description of the "massless S-matrix" theory associated with minimal superconformal field theories.