Supersymmetric Vacua and Bethe Ansatz

Abstract

This note is a short announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N = 4 gauge theories with matter, softly broken by the twisted masses down to N = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg SU (2) XXX spin chain is mapped to the two dimensional U (N ) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XY Z spin chain and eight-vertex model are related to the four dimensional theory compactified on T 2 .The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models as well as the dynamical spin chains such as the Hubbard model. Compactifications of four dimensional N = 2 theories on a two-sphere lead to the instanton-corrected Bethe equations. We propose a completely novel way for the Yangian, quantum affine, and elliptic algebras to act as a symmetry of a union of quantum field theories.a On leave of absence from ITEP, Moscow, Russia for discussions. The results of this note, as well as those in [14], were presented at various conferences and workshops 3 and we thank the organizers for the opportunity to present our results. We thank various agencies and institutions 4 for supporting this research. The gauge theoryHere we give a brief review of the relevant gauge theories. 3 The IHES seminars and the theoretical physics conference dedicated to the 50th anniversary of IHES (Bures-sur-Yvette,

Bethe/gauge correspondence on curved spaces

“…In [4][5][6] the correspondence, called the Bethe/gauge correspondence, between the two dimensional N = 2 supersymmetric gauge theories and quantum integrable systems was formulated in full generality. The novelty of this correspondence is the identification of the Planck constant of the quantum integrable system with a twisted mass parameter on the gauge theory side.…”

“…In [4][5][6] explicit formulas for W eff for a very large class of theories were given, including the properly two dimensional theories, three dimensional theories compactified on a circle, and four dimensional theories compactified on the two-torus or subject to the two dimensional Ω-deformation [14]; the interested reader can consult these papers for details. Our notations follow mostly the conventions from these papers.…”

“…Before we turn to the topologically twisted theories, we give a brief review of the relevant physical gauge theories, following the notations of [4,5].…”

“…In [4][5][6] one can find the expressions for the effective twisted superpotentials of the four dimensional N = 1 theory, compactified on a torus T 2 or the four dimensional N = 2 theory subject to a special Ω-background. These formulas are much more complicated and we shall not write them here.…”

“…The correspondence between supersymmetric gauge theory and quantum integrable system was discovered, in an example, in [1], further explored in [2,3], where it was speculated that such a correspondence should be a general property of a larger class of gauge theories in various space-time dimensions, and finally formulated and explained in [4][5][6] . Prior to [1] a connection of two dimensional pure Yang-Mills theory with massive matter to spin and many-body systems systems with long-range interaction appeared in [7][8][9][10] (the embedding of pure two dimensional Yang-Mills theory into the supersymmetric theory was discussed in [11]).…”