Quantum dilogarithms over local fields and invariants of 3-manifolds

Stavros Garoufalidis; Rinat Kashaev

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arXiv · 2306.01331

Quantum dilogarithms over local fields and invariants of 3-manifolds

Stavros Garoufalidis·Rinat Kashaev

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Abstract

To each local field (including the real or complex numbers) we associate a quantum dilogarithm and show that it satisfies a pentagon identity and some symmetries. Using an angled version of these quantum dilogarithms, we construct three generalized TQFTs in 2+1 dimensions, one given by a face state-integral and two given by edge state-integrals. Their partition functions rise to distributional invariants of 3-manifolds with torus boundary, conjecturally related to point counting of the $A$-polynomial curve. The partition function of one of these face generalized TQFTs for the case of the real numbers can be expressed either as a multidimensional Barnes-Mellin integral or as a period on a curve which is conjecturally the $A$-polynomial curve.

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Stavros Garoufalidis, Rinat Kashaev. 2023-06-02. Quantum dilogarithms over local fields and invariants of 3-manifolds. https://arxiv.org/abs/2306.01331

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