How instanton combinatorics solves Painlevé VI, V and IIIs (Q2847995)

The authors conjecture explicit combinatorial series representations for the tau functions \(\tau_{VI}(t)\), \(\tau_{V}(t)\) and \(\tau_{III}(t)\), associated with the classical Painlevé VI, V and III equations, around the critical point \(t=0\). In all the cases, the expansions have the form NEWLINE\[NEWLINE\tau(t)=t^{\theta}\sum_{n\in{\mathbb Z}}C (\sigma+n) s^n t^{(\sigma+n)^2}{\mathcal B}(\sigma+n,t),NEWLINE\]NEWLINE where \(\sigma\) and \(s\) are constants of integration, \(C(\sigma+n)\) is the coefficient explicitly expressed in terms of the Barnes functions, and NEWLINE\[NEWLINE{\mathcal B}(\sigma,t)=g(t)\sum_{\lambda,\mu\in{\mathbb Y}}{\mathcal B}_{\lambda,\mu}(\sigma)t^{|\lambda|+|\mu|}NEWLINE\]NEWLINE is the double sum over all Young diagrams \(\lambda\) and \(\mu\) with the explicitly given coefficients \({\mathcal B}_{\lambda,\mu}(\sigma)\).NEWLINENEWLINEFor the PVI case, the above combinatorial expansion follows from the identification of the tau-function \(\tau_{VI}(t)\) with a correlation function of primary fields in 2D conformal field theory with a central charge \(c=1\) using the relation of the latter correlation function to the instanton partition functions in super symmetric Yang-Mills theories which, in its turn, is expressible in terms of sums over Young diagrams. Expansions for \(\tau_V(t)\) and \(\tau_{III}(t)\) follow using the known scaling limit transitions. The elaborated formulae are confirmed by some numeric tests. The authors discuss also the cases of the classical solutions and the cases related to the integrable kernels. The latter cases are characterized by the special choice of the monodromy data of the associated linear equations.