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

Abstract : We show that the dual partition function of the pure $\mathcal N=2$ $SU(2)$ gauge theory in the self-dual $\Omega$-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 Painlev\'e III equation of type $D_8$ (radial sine-Gordon equation). In particular, the principal minor expansion of the Fredholm determinant yields Nekrasov combinatorial sums over pairs of Young diagrams.
Document type :
Conference papers

https://hal.archives-ouvertes.fr/hal-01823332
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Submitted on : Tuesday, June 26, 2018 - 1:12:37 AM
Last modification on : Wednesday, February 12, 2020 - 5:37:10 PM

### Citation

P. Gavrylenko, O. Lisovyy. Pure $SU(2)$ gauge theory partition function and generalized Bessel kernel. String Math 2016, Jun 2016, Paris, France. pp.181-208. ⟨hal-01823332⟩

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