Theta series, wall-crossing and quantum dilogarithm identities

TL;DR

This paper explores the mathematical structures of theta series related to string theory and gauge theories, revealing new quantum dilogarithm identities and their implications for five-brane physics.

Contribution

It derives a quantum version of Kontsevich-Soibelman symplectomorphisms and proves a new five-term relation for Faddeev's quantum dilogarithm at specific parameters.

Findings

Proves a new five-term relation for Faddeev's quantum dilogarithm at b=1.

Derives a quantum transformation of wave-functions induced by symplectomorphisms.

Generalizes the five-term relation for arbitrary parameters, relevant for five-brane physics.

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 $\Phi_b$ 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.