nLab quantum dilogarithm

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

8-6)

• S. Alexandrov, B. Pioline, Theta series, wall-crossing and quantum dilogarithm identities, Lett. Math. Phys. 106:1037 (2016) doi

• S. L. Woronowicz, Quantum exponential function, Rev. Math. Phys. 12 (2000) 873–920

• Bernhard Keller, On cluster theory and quantum dilogarithm identities, pp. 85–116 in A. Skowronski, K. Yamagata K. (eds.), Representations of algebras and related topics, Eur. Math. Soc. 2011 doi arXiv/1102.4148

Appearance 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)