Riemann-Hilbert correspondence and blown up surface defects

TL;DR

This paper explores the connection between surface defects in 4D supersymmetric gauge theories and classical integrability, by analyzing monodromy, tau-functions, and the blowup formula in the context of isomonodromic deformations.

Contribution

It establishes a novel link between surface defects in gauge theories and isomonodromic systems, introducing the GIL formula via blowup techniques.

Findings

Surface defect VEVs form a Fuchsian system on a sphere with five singularities.

Calculated monodromy and defined the isomonodromic tau-function.

Derived the GIL formula connecting gauge theory and classical integrability.

Abstract

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional $\mathcal{N}=2$ gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in $SU(2)$ theory with $N_f=4$ fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with $5$ regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.