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

TL;DR

This paper explores the deep connections between intersecting surface defects in supersymmetric gauge theories, conformal field theory, and quantum integrable systems, revealing new relations and mathematical structures.

Contribution

It establishes a novel correspondence linking gauge theory defect correlators with conformal blocks and quantum spin chains, demonstrating new difference equations and algebraic relations.

Findings

Correlation functions satisfy fractional quantum T-Q relations.

Fourier transforms relate to conformal blocks of affine current algebra.

Identifies gauge theory correlators with states in quantum spin chains.

Abstract

We propose an interesting BPS/CFT correspondence playground: the correlation function of two intersecting half-BPS surface defects in four-dimensional $\mathcal{N}=2$ 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 $\widehat{\mathfrak{sl}}_N$ current algebra with one of the vertex operators corresponding to the $N$-dimensional $\mathfrak{sl}_N$ representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the $XXX_{\mathfrak{sl}_2}$ spin chain of $N$ Heisenberg-Weyl modules over $Y(\mathfrak{sl}_2)$. We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.