Quantum Spin Systems and Supersymmetric Gauge Theories, 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 ${\mathfrak{sl}}_2$ $XXX$ spin chain (for infinite dimensional complex spin representations), as well as the $S{L}_N$ 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.

arXiv comment76 pages, 3 figures, fix typo

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Cited by in corpus (16)

• Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations
• Surface defects in gauge theory and KZ equation
• Exact WKB and the quantum Seiberg-Witten curve for 4d $N=2$ pure $SU(3)$ Yang-Mills, Part I: Abelianization
• Tetrahedron instantons
• Correlators on the wall and ${\mathfrak{sl}}_n$ spin chain
• Seiberg-Witten Geometry of Four-Dimensional $\mathcal{N}=2$ Quiver Gauge Theories
• Parallel surface defects, Hecke operators, and quantum Hitchin system
• di-Langlands correspondence and extended observables
• R-matrices and Miura operators in 5d Chern-Simons theory
• Bispectral duality and separation of variables from surface defect transition
• Bethe/Gauge Correspondence for ABCDEFG-type 3d Gauge Theories
• Commuting integral and differential operators and the master symmetries of the Korteweg-de Vries equation
• Defects and type D relativistic Toda lattice for some 5d gauge theories
• Dimers for Type D Relativistic Toda Model
• New dimer integrable systems and defects in five dimensional gauge theory
• Elliptic Stable Envelopes for Certain Non-Symplectic Varieties and Dynamical $R$-Matrices for Superspin Chains from the Bethe/Gauge Correspondence