Quantum Spin Systems and Supersymmetric Gauge Theories, I

Understand

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.