  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 $\mathcal{N}=2$ supersymmetric gauge theories in four
dimensions and operator theory. Specifically, we study an example of an
integral operator associated with Painlev\'e 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 $\mathrm{O}(2)$ matrix model.
We then show that these eigenfunctions are computed by surface defects in
$\mathrm{SU}(2)$ super Yang-Mills in the self-dual phase of the
$\Omega$-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.
