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\begin{document}

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\section*{dilogarithm}

Euler's \textbf{dilogarithm} is a complex valued function $Li_2$ given by

\begin{displaymath}
Li_2(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}
\end{displaymath}
The dilogarithm is a special case of the [[polylogarithm]] $Li_n$. The \textbf{Bloch--Wigner dilogarithm} is defined by

\begin{displaymath}
D(z) := Im(Li_2(z)) + arg(1-z) log |z|
\end{displaymath}
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 transformation]]s.

See also the references at \href{http://mathworld.wolfram.com/Dilogarithm.html}{mathworld} and P.P. Cook's \href{http://ppcook.blogspot.com/2005/12/dilogarithms.html}{blog} and the related entry [[quantum dilogarithm]].

\begin{itemize}%
\item Don Zagier, \emph{The dilogarithm function}, in Frontiers in Number Theory, Physics, and Geometry II, pp. 3--35 (2007) \href{http://www.ams.org/mathscinet-getitem?mr=2290758}{MR2290758} \href{http://dx.doi.org/10.1007/978-3-540-30308-4}{doi:10.1007/978-3-540-30308-4} \href{http://maths.dur.ac.uk/~dma0hg/dilog.pdf}{preprint pdf}; \emph{Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields}, in Arithmetic Algebraic Geometry, Progr. Math. \textbf{89}, Birkh\"a{}user, Boston, 1990, 391--430 \href{http://www.ams.org/mathscinet-getitem?mr=1085270}{MR1085270}


\item A. N. Kirillov, \emph{Dilogarithm identities}, Progr. Theoret. Phys. Suppl. 118 (1995), 61--142 \href{http://arxiv.org/abs/hep-th/9408113}{hep-th/9408113} \href{http://www.ams.org/mathscinet-getitem?mr=1356515}{MR1356515} \href{http://dx.doi.org/10.1143/PTPS.118.61}{doi}; \emph{Identities for the Rogers dilogarithm function connected with simple Lie algebras}, J. Soviet Math. 47 (1989), 2450--2458.


\item R.M. Kashaev, \emph{Heisenberg double and the pentagon relation}, St. Petersburg Math. J. 8 (1997) 585-- 592 \href{http://arxiv.org/abs/q-alg/9503005}{q-alg/9503005}


\item Rinat M. Kashaev, Tomoki Nakanishi, \emph{Classical and quantum dilogarithm identities}, \href{https://arxiv.org/abs/1104.4630}{arxiv/1104.4630} \href{https://doi.org/10.3842/SIGMA.2011.102}{doi} (treatment in terms of [[cluster algebra]]s)


\item Tomoki Nakanishi, \emph{Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case}, Nagoya Math. J. \textbf{202} (2011), 23-43, \href{http://www.ams.org/mathscinet-getitem?mr=2804544}{MR2804544} \href{http://dx.doi.org/10.1215/00277630-1260432}{doi}


\item B. Richmond, G. Szekeres, \emph{Some formulas related to dilogarithm, the zeta function and the Andrews-Gordon identities}, J. Aust. Math. Soc. \textbf{31} (1981), 362--373 \href{http://www.ams.org/mathscinet-getitem?mr=633444}{MR633444} \href{http://dx.doi.org/10.1017/S1446788700019492}{doi}


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


\item W. Nahm, \emph{Conformal field theory and torsion elements of the Bloch group}, in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 67--132 \href{http://www.ams.org/mathscinet-getitem?mr=2290759}{MR2290759} \href{http://dx.doi.org/10.1007/978-3-540-30308-4_2}{doi}


\item Marco Aldi, Reimundo Heluani, \emph{Dilogarithms, OPE, and twisted T-duality}, International Mathematics Research Notices 2014:6, 1528–1575, \href{https://doi.org/10.1093/imrn/rns258}{doi}


\item Andreas Deser, \emph{Lie algebroids, non-associative structures and non-geometric fluxes}, \href{}{arXiv}


\item Fock, Vladimir V.; Goncharov, Alexander B., \emph{Cluster ensembles, quantization and the dilogarithm}, Annales scientifiques de l'École Normale Supérieure, Serie 4, \textbf{42}:6 (2009) 865-930 \href{https://doi.org/10.24033/asens.2112}{doi}


\item E. Aldrovandi, \emph{On hermitian-holomorphic classes related to uniformization, the dilogarithm, and the Liouville Aaction}, Commun. Math. Phys. (2004) 251: 27. \href{https://doi.org/10.1007/s00220-004-1168-6}{doi}


\item S. Alexandrov, B. Pioline, \emph{Theta series, wall-crossing and quantum dilogarithm identities}, Lett. Math. Phys. \textbf{106}:1037 (2016) \href{https://doi.org/10.1007/s11005-016-0857-3}{doi}


\item S. L. Woronowicz, \emph{Quantum exponential function}, Rev. Math. Phys. \textbf{12} (2000) 873–920.



\end{itemize}
category: analysis, physics



\end{document}