2017/10/31 by Bonelli, Giulio, Grassi, Alba, Tanzini, Alessandro · 2 citations
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1710.11603
We propose that the grand canonical topological string partition functions satisfy finite-difference equations in the closed string moduli. In the case of genus one mirror curve these are conjectured to be the q-difference Painlevé equations as in Sakai's classification. More precisely, we propose that the tau-functions of q-Painlevé equations are related to the grand canonical topological string partition functions on the corresponding geometry. In the toric cases we use topological string/spectral theory duality to give a Fredholm determinant representation for the above tau-functions in terms of the underlying quantum mirror curve. As a consequence, the zeroes of the tau-functions compute the exact spectrum of the associated quantum integrable systems. We provide details of this construction for the local ℙ1× ℙ1 case, which is related to q-difference Painlevé with affine A1 symmetry, to SU(2) Super Yang-Mills in five dimensions and to relativistic Toda system.