A quantum generalization of Rogers' five term, or "pentagon" dilogarithm identity is suggested. It is shown that the classical limit gives usual Rogers' identity. The case where the quantum identity is realized in finite dimensional space is also considered and the quantum dilogarithm is constructed as a function on Fermat curve, while the identity itself is equivalent to the restricted star-triangle relation introduced

Abstract. The quantum discrete Liouville model in the strongly coupled regime, 1 < c < 25, is formulated as a well defined quantum mechanical problem with unitary evolution operator. The theory is selfdual: there are two exponential fields related by Hermitean conjugation, satisfying two discrete quantum Liouville equations, and living in mutually commuting subalgebras of the quantum algebra of observables.

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 È 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.

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