Theta Series, Wall-Crossing and Quantum Dilogarithm Identities
Abstract
Motivated by mathematical structures which arise in string vacua and gauge theories with 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 {Φ_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.
- Publication:
-
Letters in Mathematical Physics
- Pub Date:
- August 2016
- DOI:
- 10.1007/s11005-016-0857-3
- arXiv:
- arXiv:1511.02892
- Bibcode:
- 2016LMaPh.106.1037A
- Keywords:
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- High Energy Physics - Theory;
- Mathematical Physics;
- Mathematics - Quantum Algebra
- E-Print:
- 26 pages