On the integrability of N=2 Landau-Ginzburg models: A graph generalization of the Yang-Baxter equation

The study of the integrability properties of the N=2 Landau-Ginzburg models leads naturally to a graph generalization of the Yang-Baxter equation which synthetizes the well known vertex and RSOS Yang-Baxter equations.A non trivial solution of this equation is found for the t2 perturbation of the A-models, which turns out to be intimately related to the Boltzmann weights of a Chiral-Potts model. CERN-TH.6963/93 UGVA 07/622/93 August 1993 ∗Permanent address: Instituto de Matemáticas y F́ısica Fundamental, CSIC; Serrano 123, E–28006 Spain. 0 Introduction N=2 Landau-Ginzburg models have been extensively used to describe relevant and SUSY preserving perturbations of N=2 superconformal field theories [13]. The resulting massive field theories are in some cases integrable and the associated scattering S-matrices satisfy the bootstrap and factorization equations [2, 3] . Those Landau-Ginzburg superpotentials which are Morse functions contain in their spectrum all the soliton configurations which interpolate between the different critical points. Evenmore in some special cases a closed scattering theory ,in the sense of bootstrap [4], can be defined choosing as fundamental particles a set of Bogomolnyi solitons. Two main questions arise in the study of N=2 Landau-Ginzburg models: a) when the massive theory defined by a given non degenerated superpotential is integrable and b) which is the associated scattering S-matrix which satisfy the requirements of bootstrap and factorization. During the last few years these two questions have been considered, which much or less succes, under different points of view: Toda theories [5], quantum affine algebras [6], holomorphic field theories [7], graph rings [8]. In this letter we shall present an algebraic framework which we hope will capture the integrability of the N=2 Landau-Ginzburg models. The cornerstone of our approach is a graph generalization of the Yang-Baxter equation, whose non triviality is illustrated with the t2 perturbation of the A-models. N=2 Landau-Ginzburg models: Graph Quantum Groups The Landau-Ginzburg models that we shall consider have a N=2 supersymmetry algebra generated by the SUSY chargesQ, Q̄ together with the topological charges T, T̄ , fermion number operator F and momentum operators P, P̄ satisfying the relations: (Q) = (Q̄) = {Q+, Q̄−} = {Q−, Q̄+} = 0 {Q, Q} = P, {Q̄+, Q̄−} = P̄ (1) {Q, Q̄} = T, {Q−, Q̄−} = T̄ [ F , Q ] = ±Q, [ F , Q̄ ] = ∓Q̄± Each LG model is characterized by a superpotential whose critical points define the vertex of a graph. The links of this graph are associated to the fundamental Bogomolnyi solitons. 1 The graph we are refering to is the same used in [7] for the classification of N=2 theories and it is 1 Let us denote by a, b, · · · the vertices of the LG-graph, then a Bogomolnyi soliton associated to the link (a, b) and rapidity θ give rise to a two dimensional irrep πa,b(θ) of the N=2 algebra ( 1 ) given by: πa,b(θ)(Q ) =   0 0 √ ma,be θ/2 0 