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 $\mathcal{N}=2$ 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 $XX{X}_{{\mathfrak{sl}}_2}$ spin chain of $N$ Heisenberg-Weyl modules over $Y({\mathfrak{sl}}_2)$. We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.

arXiv comment85 pages; v2. published version; v3. journal reference number corrected; v4. references added

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Cited by in corpus (6)

• 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
• Branches, quivers, and ideals for knot complements
• Intersecting Defects and Supergroup Gauge Theory
• Elliptic Quantum Curves of 6d SO(N) theories
• Seiberg-Witten Geometry of Four-Dimensional $\mathcal{N}=2$ Quiver Gauge Theories