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Dilogarithm identities

Anatol N. Kirillov
(
- Tokyo U., Komaba and
- Steklov Math. Inst., St. Petersburg
)

Oct 28, 1994

87 pages

Part of Quantum field theory, integrable models and beyond. Proceedings, YITP Workshop, Kyoto, Japan, February 14-17, 1994

Published in:

Prog.Theor.Phys.Suppl. 118 (1995) 61-142

Contribution to:

Workshop on Quantum Field Theory, Integrable Models and Beyond

e-Print:

hep-th/9408113 [hep-th]

DOI:

10.1143/PTPS.118.61

View in:

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Abstract:

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

\hat{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

\hat{sl_n}

are obtained. Connection between dilogarithm identities and algebraic

K

-theory (torsion in

K_3({\bf 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). Connection between "finite-dimensional part of crystal base" and Robinson-Schensted-Knuth correspondence is considered.

lectures: Kyoto 1994/02/14
field theory: conformal
algebra: Kac-Moody
analytic properties
mathematical methods: dilogarithm
bibliography

References(72)

Figures(0)

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