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Communications in Mathematical Physics

, Volume 346, Issue 3, pp 967–994 | Cite as

Operators from Mirror Curves and the Quantum Dilogarithm

  • Rinat Kashaev
  • Marcos MariñoEmail author
Article
First Online:

Abstract

Mirror manifolds to toric Calabi–Yau threefolds are encoded in algebraic curves. The quantization of these curves leads naturally to quantum-mechanical operators on the real line. We show that, for a large number of local del Pezzo Calabi–Yau threefolds, these operators are of trace class. In some simple geometries, like local \({\mathbb{P}^2}\), we calculate the integral kernel of the corresponding operators in terms of Faddeev's quantum dilogarithm. Their spectral traces are expressed in terms of multi-dimensional integrals, similar to the state-integrals appearing in three-manifold topology, and we show that they can be evaluated explicitly in some cases. Our results provide further verifications of a recent conjecture which gives an explicit expression for the Fredholm determinant of these operators, in terms of enumerative invariants of the underlying Calabi–Yau threefolds.

Keywords

Topological String Formal Power Series Integral Kernel Trace Class ABJM Theory 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Section de MathématiquesUniversité de GenèveGeneva 4Switzerland
  2. 2.Section de Mathématiques et Département de Physique ThéoriqueUniversité de GenèveGeneva 4Switzerland

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