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. © 2019, Springer Nature B.V.

Quantum curves and q-deformed Painlevé equations / Bonelli, G.; Grassi, A.; Tanzini, A.. - In: LETTERS IN MATHEMATICAL PHYSICS. - ISSN 0377-9017. - 109:9(2019), pp. 1961-2001. [10.1007/s11005-019-01174-y]

Quantum curves and q-deformed Painlevé equations

Bonelli G.;Tanzini A.
2019

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 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. © 2019, Springer Nature B.V.
109
9
1961
2001
https://doi.org/10.1007/s11005-019-01174-y
https://arxiv.org/abs/1710.11603
Bonelli, G.; Grassi, A.; Tanzini, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/103734
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