Euler’s dilogarithm is a complex valued function Li 2Li_2 given by

Li 2(x)= n=1 x nn 2 Li_2(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}

The dilogarithm is a special case of the polylogarithm Li nLi_n. The Bloch–Wigner dilogarithm is defined by

D(z):=Im(Li 2(z))+arg(1z)log|z| D(z) := Im(Li_2(z)) + arg(1-z) log |z|

The dilogarithm has remarkable relations to many areas of mathematics and mathematical physics including scissors congruence, Reidemeister’s torsion, regulators in higher algebraic K-theory, the Bloch group, CFT, Liouville's gravity?, hyperbolic geometry and cluster transformations.

See also the references at mathworld and P.P. Cook’s blog and the related entry quantum dilogarithm.

  • Don Zagier, The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry II, pp. 3–35 (2007) MR2290758 doi:10.1007/978-3-540-30308-4 preprint pdf; Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields, in Arithmetic Algebraic Geometry, Progr. Math. 89, Birkhäuser, Boston, 1990, 391–430 MR1085270

  • A. N. Kirillov, Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61–142 hep-th/9408113 MR1356515 doi; Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Soviet Math. 47 (1989), 2450–2458.

  • B. Richmond, G. Szekeres, Some formulas related to dilogarithm, the zeta function and the Andrews-Gordon identities, J. Aust. Math. Soc. 31 (1981), 362–373 MR633444 doi

  • S. Bloch, Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, in: Proc. Int. Symp. on Alg. Geometry, Kinokuniya, Tokyo 1978.

  • Tomoki Nakanishi, Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case, Nagoya Math. J. 202 (2011), 23-43, MR2804544 doi

  • W. Nahm, Conformal field theory and torsion elements of the Bloch group, in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 67–132 MR2290759 doi

category: analysis, physics

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