# Painlevé Kernels and Surface Defects at Strong Coupling

## TL;DR

This paper connects Painlevé equations to surface defects in a type of quantum field theory, using an operator approach and matrix models.

## Contribution

A new interpretation of Painlevé kernels as Fermi gas density matrices and a strong coupling expression for surface defects in SU(2) super Yang–Mills.

## Key findings

- An explicit expression for eigenfunctions of a Painlevé kernel via an O(2) matrix model.
- Surface defects in SU(2) super Yang–Mills compute these eigenfunctions in the self-dual Ω-background.
- A strong coupling resummation of the instanton expansion for these defects is derived.

## Abstract

It is well established that the spectral analysis of canonically quantized four-dimensional Seiberg–Witten curves can be systematically studied via the Nekrasov–Shatashvili functions. In this paper, we explore another aspect of the relation between \documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {N}}=2$$\end{document}N=2 supersymmetric gauge theories in four dimensions and operator theory. Specifically, we study an example of an integral operator associated with Painlevé equations and whose spectral traces are related to correlation functions of the 2d Ising model. This operator does not correspond to a canonically quantized Seiberg–Witten curve, but its kernel can nevertheless be interpreted as the density matrix of an ideal Fermi gas. Adopting the approach of Tracy and Widom, we provide an explicit expression for its eigenfunctions via an \documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{O(2)}\,}}$$\end{document}O(2) matrix model. We then show that these eigenfunctions are computed by surface defects in \documentclass[12pt]{minimal}
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\begin{document}$${{\,\mathrm{SU(2)}\,}}$$\end{document}SU(2) super Yang–Mills in the self-dual phase of the \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}Ω-background. Our result also yields a strong coupling expression for such defects which resums the instanton expansion. Even though we focus on one concrete example, we expect these results to hold for a larger class of operators arising in the context of isomonodromic deformation equations.