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arXiv · 1710.11603

Quantum curves and $q$-deformed Painlevé equations

Giulio Bonelli·Alba Grassi·Alessandro Tanzini
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Abstract

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 $\mathbb{P}^1\times \mathbb{P}^1$ case, which is related to q-difference Painlevé with affine $A_1$ symmetry, to $SU(2)$ Super Yang-Mills in five dimensions and to relativistic Toda system.

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Giulio Bonelli, Alba Grassi, Alessandro Tanzini. 2018-01-01. Quantum curves and $q$-deformed Painlevé equations. https://arxiv.org/abs/1710.11603

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