Classical and Quantum Dilogarithm Identities

Using the quantum cluster algebra formalism of Fock and Goncharov, we present several forms of quantum dilogarithm identities associated with periodicities in quantum cluster algebras, namely, the tropical, universal, and local forms. We then demonstrate how classical dilogarithm identities naturally emerge from quantum dilogarithm identities in local form in the semiclassical limit by applying the saddle point method

Spectral equations for the modular oscillator

Motivated by applications for non-perturbative topological strings in toric Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local $\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We illustrate our analytical results by numerical calculations.Comment: 23 pages, 9 figures, references added and interpretation of the numerical results of Section 6 correcte

On the Spectrum of the Local P2 Mirror Curve

We address the spectral problem of the formally normal quantum mechanical operator associated with the quantised mirror curve of the toric (almost) del Pezzo Calabi-Yau threefold called local P2 in the case of complex values of Planck's constant. We show that the problem can be approached in terms of the Bethe ansatz-type highly transcendental equations.Open access funding provided by University of Geneva.The work is partially supported by the Australian Research Council and Swiss National Science Foundation

Centrally extended mapping class groups from quantum Teichmüller Theory

The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmüller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in [9] for the Thompson groups

Spectral Equations for the Modular Oscillator

Motivated by applications for non-perturbative topological strings in toric Calabi– Yau manifolds, we discuss the spectral problem for a pair of commuting modular conjugate (in the sense of Faddeev) Harper type operators, corresponding to a special case of the quantized mirror curve of local P 1 × P 1 and complex values of Planck’s constant. We illustrate our analytical results by numerical calculations