Dilogarithm identities. (English) Zbl 0894.11052

Kirillov, Anatol N.

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 \(n\leq 2\) only) functional equations is given. For odd levels the \(\widehat{sl}_2\) 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 \(\widehat{sl}_n\) are obtained. Connection between dilogarithm identities and algebraic \(K\)-theory (torsion in \(K_3(\mathbb R))\) is discussed. Relations between crystal basis, branching functions \(b_\lambda^{k\Lambda_0} (q)\) 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 \(b_\lambda^{k\Lambda_0} (q)\) 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.
For the entire collection see [Zbl 0839.00030].

Reviewer: O.Ninnemann (Berlin)

MSC:

11Z05	Miscellaneous applications of number theory
11-02	Research exposition (monographs, survey articles) pertaining to number theory
33-02	Research exposition (monographs, survey articles) pertaining to 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. in quantum mechanics
17B68	Virasoro and related algebras
11P81	Elementary theory of partitions
05E15	Combinatorial aspects of groups and algebras (MSC2010)
17B37	Quantum groups (quantized enveloping algebras) and related deformations
19M05	Miscellaneous applications of \(K\)-theory

Keywords:

thermodynamic Bethe Ansatz limit; Virasoro algebras; Kac-Moody algebras; conformal field theory; survey; dilogarithm identities; functional equations; Kuniba-Nakanishi’s dilogarithm; partition identities; string functions; affine Lie algebra; algebraic \(K\)-theory; crystal basis; branching functions; Kostka-Foulkes polynomials; Melzer and Milne conjectures; Robinson-Schensted-Knuth correspondence