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Quantum curves and q-deformed Painlevé equations

  • Giulio Bonelli
  • Alba GrassiEmail author
  • Alessandro Tanzini
Article

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.

Keywords

Painlevé equations Supersymmetric gauge theory Topological string theory Spectral theory 

Mathematics Subject Classification

34M55 81T60 14H70 51P05 81Q10 15B52 81T13 

Notes

Acknowledgements

We would like to thank Mikhail Bershtein, Jie Gu, Oleg Lisovyy, Marcos Mariño, Andrei Mironov, Alexei Morozov and especially Yasuhiko Yamada for many useful discussions and correspondence. We are also thankful to Jie Gu for a careful reading of the manuscript. We thank Bernard Julia and the participants of the workshop “Exceptional and ubiquitous Painlevé equations for Physics” for useful discussions and the stimulating atmosphere. The work of GB is supported by the PRIN project “Non-perturbative Aspects Of Gauge Theories And Strings”. GB and AG acknowledge support by INFN Iniziativa Specifica ST&FI. AT acknowledges support by INFN Iniziativa Specifica GAST.

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© Springer Nature B.V. 2019

Authors and Affiliations

  • Giulio Bonelli
    • 1
    • 2
  • Alba Grassi
    • 3
    • 4
    Email author
  • Alessandro Tanzini
    • 1
    • 2
  1. 1.International School of Advanced Studies (SISSA)TriesteItaly
  2. 2.INFN, Sezione di TriesteTriesteItaly
  3. 3.International Center for Theoretical Physics, ICTPTriesteItaly
  4. 4.Simons Center for Geometry and PhysicsSUNYStony BrookUSA

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