Abstract
The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the N site mathfrak{sl} 2X X X spin chain (for infinite dimensional complex spin representations), as well as the SLN Gaudin system, which reduces, in a limiting case, to that of the N-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.
Highlights
Introduction and summaryThe BPS/CFT correspondence [1] relates the algebra and geometry of two dimensional conformal field theories, and their q-deformations, to the algebra and geometry of the moduli space of vacua of four dimensional N = 2 supersymmetric gauge theories, and their various deformations, such as Ω-deformation, lift to higher dimensions, inclusion of extended objects and so on
The analogue of curve counting is the enumeration of instantons in four dimensional gauge theory, while the role of the mirror complex geometry is played by the two dimensional conformal field theory
In this paper we explore a specific corner of the BPS/CFT correspondence, where the techniques developed in the four dimensional instanton counting are applied to a seemingly very distant problem: calculating a quantum mechanical wave-function of a many-body system, or a spin chain
Summary
In this limit our Mellin-Barnes-type integrals can be evaluated by the saddle point approximation. We find the the saddle point equations of the surface defect partition function look like the nested Bethe equations, which can be solved in terms of the holonomy matrix of the classical limit of the XXXsl2-spin chain In this way we recover Sklyanin’s separated variables [42, 43]. Using the nonperturbative Dyson-Schwinger equations we are able to generate infinitely many bulk gauge invariant chiral ring observables, whose vacuum expectation values are the eigenvalues of the mutually commuting differential operators (Hamiltonians) acting on the surface defect partition functions, which are the higher quantum integrals of motion of the XXXsl spin chain. In this way we obtain a generalization of the results of [53], which can be recovered for special values of masses and Coulomb parameters
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