Quantum spin systems and supersymmetric gauge theories. Part I

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 $$ \mathfrak{sl} $$ 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.

Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations

“…The separation of variables of the quantum integrable system is deeply involved with our study. It was indeed shown in [42] that, in the limit ε 2 → 0, the vacuum expectation value Ψ(q, z) (4.11) of the regular surface defect admits a Mellin-Barnes integral representation. This integral transform led to the expression of the eigenfunction in separated variables for the XXX sl 2 spin chain.…”

“…The Ωbackground uplifts these relations to differential equations in coupling constants obeyed by the vacuum expectation value of the defect observable. The non-perturbative Dyson-Schwinger equation can effectively used to exactly derive such differential equations, as shown in [31,35,[37][38][39]42].…”

“…With the same restriction on the spin representations, the IR duality discovered in [27] between four-dimensional N = 2 theories and two-dimensional N = (2, 2) theories gave a four-dimensional account for the quantum spin chains. Finally, it was shown in [42] that the classical XXX sl 2 spin chain arises in the Seiberg-Witten geometry of the four-dimensional N = 2 theory [21,59], as well as a relation between the XXX sl 2 spin chain coordinate systems and the defect gauge theory parameters, with more general sl 2 -representations which are neither highest-weight nor lowest-weight. The extension to the supergroup gauge theories is also discussed in [65][66][67].…”

“…The correlators of the gauge origami system obey Dyson-Schwinger relations which express the compactness of the moduli space of spiked instantons [28,36], implying nontrivial equations on the gauge theory correlation functions [31,35,[37][38][39][40][41][42][43][44], including the chiral ring relations of the gauge theory [40], both with and without the surface defects in the Ω-background. Some of these equations are identified with the Belavin-Polyakov-Zamolodchikov (BPZ) equations [45] and the Knizhnik-Zamolodchikov (KZ) equations [46] satisfied by the CFT correlation functions, leading to a direct proof of various incarnations of the BPS/CFT correspondence [31,38,40,43].…”

“…By concatenating the Lax operators we get the monodromy matrix of the spin chain. We note that the construction generalizes both the setup of [24,25] by incorporating unbounded weight representations, the so-called HW-modules [43], and the setup of [42], by quantization. We also show that the higher-rank qq-characters yield non-perturbative Dyson-Schwinger equations which express the spin chain transfer matrix in gauge theory language.…”