Theta Series, Wall-Crossing and Quantum Dilogarithm Identities
Tóm tắt
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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