The quantum dilogarithm was discovered by physicist Anatole Kirillov and by Ludwig Fadeev and Kashaev. The latter also discovered a fundamental pentagon relation for the quantum dilogarithm, which is an analogue of the classical Roger’s identities and makes a connection to the current subject of cluster transformations. Cf. also classical dilogarithm.
Some references:
L.D.Fadeev, R.M.Kashaev, Quantum dilogarithm, Mod. Phys. Lett. A. 9 (1994) p.427–434.
V V Bazhanov, N Yu Reshetikhin, Remarks on quantum dilogarithm, J. Phys. A: Math. Gen. 28 2217–2226 (1995) doi:10.1088/0305-4470/28/8/014
A. N. Kirillov, Dilogarithm identities, hep-th/9408113, in: Quantum field theory, integrable models and beyond (Kyoto, 1994). Progr. Theoret. Phys. Suppl. No. 118 (1995), 61–142, doi; Dilogarithm identities, partitions, and spectra in conformal field theory. Algebra i Analiz 6 (1994), no. 2, 152–175, pdf.
A.B. Goncharov, The pentagon relation for the quantum dilogarithm and quantized $M_{0,5}$, arxiv:0706.4054
V. V. Fock, A. B. Goncharov, The quantum dilogarithm and representations of quantum cluster varieties, arxiv:math/0702397
V. V. Fock, A. B. Goncharov, Cluster ensembles, quantization and the dilogarithm II: The intertwiner, arxiv:math/0702398
Ettore Aldrovandi, Leon A. Takhtajan, Generating functional in CFT and effective action for twodimensional quantum gravity on higher genus Riemann surfaces, Comm. Math. Phys. 188 (1997), no. 1, 29–67; Generating functional in CFT on Riemann surfaces. II. Homological aspects, Comm. Math. Phys. 227 (2002), no. 2, 303–348, arXiv:math.AT/0006147.
R. M. Kashaev, Quantization of Teichmuller spaces and the quantum dilogarithm, Lett. Math. Phys., Vol. 43, No. 2, 1998
R.M.Kashaev, The hyperbolic volume of knots from the quantum dilogarithm, Letters in mathematical physics 39, n3, 1997
Aug 9-13, 2010 – workshop on quantum dilogarithm in Aarhus: web
Bernhard Keller, On cluster theory and quantum dilogarithm identities, arxiv/1102.4148
Apopearance of quantum dilogarithm in computations related to some cases of mirror symmetry and in topological strings are studied in
Rinat Kashaev, Marcos Mariño, Operators from mirror curves and the quantum dilogarithm, Comm. Math. Phys. 2016 (Online First) arxiv/1501.01014 doi
Rinat Kashaev, Marcos Mariño, Szabolcs Zakany, Matrix models from operators and topological strings, 2, Annales Henri Poincaré 2016 (Online First) doi; continuation of M. Mariño, S. Zakany, Matrix models from operators and topological strings, arxiv/1502.02958 (part I has less coverage of quantum dilogarithm aspects)