Pure SU(2) gauge theory partition function and generalized Bessel kernel

Source abstract

We show that the dual partition function of the pure N = 2 \mathcal N=2 S U ( 2 ) \mathrm {SU}\left (2\right ) gauge theory in the self-dual Ω \Omega -background (a) is given by Fredholm determinant of a generalized Bessel kernel and (b) coincides with the tau function associated to the general solution of the Painlevé III equation of type D 8 D_8 (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.