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 N $$ \mathcal{N} $$ = 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 sl ̂ N $$ {\hat{\mathfrak{sl}}}_N $$ current algebra with one of the vertex operators corresponding to the N-dimensional sl N $$ {\mathfrak{sl}}_N $$ representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the XXX sl 2 $$ {XXX}_{{\mathfrak{sl}}_2} $$ spin chain of N Heisenberg-Weyl modules over Y ( sl 2 $$ {\mathfrak{sl}}_2 $$ ). We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.