Euler’s dilogarithm is a complex valued function $Li_2$ given by
The dilogarithm is a special case of the polylogarithm $Li_n$. The Bloch–Wigner dilogarithm is defined by
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; 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