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Theta Series, Wall-Crossing and Quantum Dilogarithm Identities

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Abstract

Motivated by mathematical structures which arise in string vacua and gauge theories with \({{\mathcal{N}=2}}\) supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi–Yau string vacua, such theta series encode instanton corrections from k Neveu–Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich–Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge k. Consistency with wall-crossing implies a new five-term relation for Faddeev’s quantum dilogarithm \({\Phi_b}\) at b = 1, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary b and k, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum.

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Authors and Affiliations

  1. Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, 34095, Montpellier, France

    Sergei Alexandrov

  2. CERN PH-TH, Case C01600, CERN, 1211, Geneva 23, Switzerland

    Boris Pioline

  3. Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Boris Pioline

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  1. Sergei Alexandrov
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  2. Boris Pioline
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Alexandrov, S., Pioline, B. Theta Series, Wall-Crossing and Quantum Dilogarithm Identities. Lett Math Phys 106, 1037–1066 (2016). https://doi.org/10.1007/s11005-016-0857-3

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  • DOI: https://doi.org/10.1007/s11005-016-0857-3

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