Solutions to the reflection equation and integrable systems for N = 2 SQCD with classical groups

Integrable systems underlying the Seiberg-Witten solutions for the N = 2 SQCD with gauge groups SO(n) and Sp(n) are proposed. They are described by the inhomogeneous XXX spin chain with specific boundary conditions given by reflection matrices. We attribute reflection matrices to orientifold planes in the brane construction and briefly discuss its possible deformations. solving N = 2 SUSY gauge theories, there have been a lot of attempts to realize the structures behind it, in order to get any kind of understanding and, after all, derivation. In particular, one of the important structures that underlines the Seiberg-Witten (SW) anzatz and reflects its symmetry properties is integrability [2]. Concretely, in [2] it has been shown that the SW solution of the pure gauge N=2 SUSY theories with SU(N c) gauge group can be described in the framework of the periodic Toda chain with N c sites. Since then, there have been a lot of different examples of the correspondence (SW solution ←→ integrable system) considered [3, 4, 5, 6, 7, 8, 9]. The list of examples includes 4d, 5d and 6d theories with matter hypermultiplets in adjoint or fundamental representations included. However, all the examples from this extensive list mainly dealt with the SU(N c) group. Not much has been known of other groups up to the recent time. In fact, the first paper dealt with integrable structures for other classical groups was [3]. The authors of [3] considered the pure gauge theory with gauge group G that is one of the classical groups SO(n) or Sp(n) and demonstrated that the corresponding SW anzatz can be described by the Toda chain associated with the root system of the dual affine algebra G ∨ (one should also specifically match the rank of this algebra, see below). This result has been recently generalized to the theories with adjoint matter, which are described by the elliptic Calogero-Moser model [6]. For these systems, Lax representation with the spectral parameter has been constructed for the classical groups other than SU(N c) in [10] (and the proper brane picture has been suggested in [11, 12]). However, including the fundamental matter for all thee classical groups has remained a problem. Indeed, the theories