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Letters in Mathematical Physics

, Volume 106, Issue 8, pp 1037–1066 | Cite as

Theta Series, Wall-Crossing and Quantum Dilogarithm Identities

  • Sergei Alexandrov
  • Boris PiolineEmail author
Article
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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.

Keywords

contact geometry quantization fivebranes cluster algebras 

Mathematics Subject Classification

13F60 33B30 53C26 81R05 81T30 81T60 
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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Laboratoire Charles Coulomb (L2C)UMR 5221 CNRS-Université de MontpellierMontpellierFrance
  2. 2.CERN PH-THCase C01600, CERNGeneva 23Switzerland
  3. 3.Laboratoire de Physique Théorique et Hautes EnergiesCNRS UMR 7589, Université Pierre et Marie CurieParis Cedex 05France

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