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

Abstract

We propose an interesting BPS/CFT correspondence playground: the correlation function of two intersecting half-BPS surface defects in four-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric SU(N) gauge theory with 2N fundamental hypermultiplets. We show it satisfies a difference equation, the fractional quantum T-Q relation. Its Fourier transform is the 5-point conformal block of the $$ {\hat{\mathfrak{sl}}}_N $$ sl ̂ N current algebra with one of the vertex operators corresponding to the N-dimensional $$ {\mathfrak{sl}}_N $$ sl N representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the $$ {XXX}_{{\mathfrak{sl}}_2} $$ XXX sl 2 spin chain of N Heisenberg-Weyl modules over Y ($$ {\mathfrak{sl}}_2 $$ sl 2 ). We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.

Aspects of Supergroup Gauge Theory

“…Hence, the situation with both n α,1,2 realizes intersecting defects crossing at the origin [DGH10, BTZ12, GLFPP17, PP17, GLFMS17, NPZ18]. See also [JLN21]. Computing the partition function of this configuration, it is given in the form of (deformation of) supermatrix model, which implies that we have an emerging supergroup structure originated from physical (non-supergroup) theory by considering the intersecting defects.…”

A Panorama Of Physical Mathematics c. 2022

“…Presumably [739], this relates the "analytic Langlands correspondence" of [340,342,717] to the bosonic instantonic theory, also known as the curved βγ βγ-system [740][741][742][743] on the complete flag variety G/B, and to the higher rank generalizations of [271,744,745]. Finally, the main ingredient of both analytic and classical GLP, Hecke operators [341] (which commute with quantized Hitchin hamiltonians), show up in the N = 2 description of the theory, as the so-called Q-observables, the specific surface defects, which can be engineered using the folded instanton construction [746,747]. In this way explicit formulae for the eigenvalues of the Hecke operators of [340,341] can be obtained by localization.…”

Defect in Gauge Theory and Quantum Hall States

“…The main statement in [13] proves certain vanishing conditions for the expectation values of the qqobservables, both with or without defects. These vanishing conditions, called non-perturbative Dyson-Schwinger equations, can be used to construct KZ-type equations [14] satisfied by the partition function [15,16]. In the NS-limit, the KZ-equations becomes a Schrödinger-type equation satisfied by the partition function.…”

“…Recent developments in BPS/CFT correspondence [6,7,11] notices differential equations of two dimensional conformal field theories, such as KZ-equation [7,16,20] and KZB-equations can be verified by adding a regular surface defect in the supersymmetric gauge theory. These conformal equations becomes eigenvalue equations of the integrable model in the Nekrasov-Shatashivilli limit (NS-limit for short) ε 2 → 0.…”