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Dilogarithm identities. (English) Zbl 0894.11052
Inami, Takeo (ed.) et al., Quantum field theory, integrable models and beyond. Proceedings of a workshop, Kyoto, Japan, February 14-17, 1994. Kyoto: Kyoto University, Yukawa Inst. for Theoretical Physics, Prog. Theor. Phys., Suppl. 118, 61-142 (1995).

The main purpose of this paper, based on a series of lectures given at Japanese Universities, is a survey of dilogarithm identities and related topics. It contains a wealth of information and an extensive list of references. The contents are conveniently described in the author’s summary.

We study the dilogarithm identities from algebraic, analytic, asymptotic, $K$-theoretic, combinatorial and representation-theoretic points of view. We prove that a lot of dilogarithm identities (hypothetically all!) can be obtained by using the five-term relation only. Among those the Coxeter, Lewin, Loxton and Browkin ones are contained. Accessibility of Lewin’s one variable and Ray’s multivariable (here for $n\le 2$ only) functional equations is given. For odd levels the ${\stackrel{^}{sl}}_{2}$ case of Kuniba-Nakanishi’s dilogarithm conjecture is proven and additional results about remainder term are obtained. The connections between dilogarithm identities and Rogers-Ramanujan-Andrews-Gordon type partition identities via their asymptotic behavior are discussed. Some new results about the string functions for level $k$ vacuum representation of the affine Lie algebra ${\stackrel{^}{sl}}_{n}$ are obtained. Connection between dilogarithm identities and algebraic $K$-theory (torsion in ${K}_{3}\left(ℝ\right)\right)$ is discussed. Relations between crystal basis, branching functions ${b}_{\lambda }^{k{{\Lambda }}_{0}}\left(q\right)$ and Kostka-Foulkes polynomials (Lusztig’s $q$-analog of weight multiplicity) are considered. The Melzer and Milne conjectures are proven. In some special cases we are proving that the branching functions ${b}_{\lambda }^{k{{\Lambda }}_{0}}\left(q\right)$ are equal to an appropriate limit of Kostka polynomials (the so-called Thermodynamic Bethe Ansatz limit). The connection between the “finite-dimensional part of crystal base” and the Robinson-Schensted-Knuth correspondence is considered.

##### MSC:
 11Z05 Miscellaneous applications of number theory 11-02 Research monographs (number theory) 33-02 Research monographs (special functions) 11G55 Polylogarithms and relations with $K$-theory 33B15 Gamma, beta and polygamma functions 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 81T40 Two-dimensional field theories, conformal field theories, etc. 17B68 Virasoro and related algebras 11P81 Elementary theory of partitions 05E15 Combinatorial aspects of groups and algebras 17B37 Quantum groups and related deformations 19M05 Miscellaneous applications of $K$-theory