Quantum Spin Systems and Supersymmetric Gauge Theories, I

TL;DR

This paper explores the deep connections between supersymmetric gauge theories and quantum spin systems, deriving explicit formulas and establishing relations that shed light on integrable models and the AGT conjecture.

Contribution

It provides an explicit formula for the Jost function of the $ ext{XXX}$ spin chain and relates spin chain Hamiltonians to the gauge theory's twisted chiral ring, advancing understanding of their interplay.

Findings

Derived explicit Jost function formula for the $ ext{XXX}$ spin chain.

Established relations between spin chain Hamiltonians and gauge theory rings.

Provided new evidence supporting the AGT conjecture.

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 $\mathfrak{sl}_{2}$ $XXX$ spin chain (for infinite dimensional complex spin representations), as well as the $SL_N$ 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.