The dilogarithm function.

*(English)*Zbl 1176.11026
Cartier, Pierre (ed.) et al., Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9–21, 2003. Berlin: Springer (ISBN 978-3-540-30307-7/hbk). 3-65 (2007).

From the introduction: The dilogarithm function, defined in the first sentence of Chapter I, is a function which has been known for more than 250 years, but which for a long time was familiar only to a few enthusiasts. In recent years it has become much better known, due to its appearance in hyperbolic geometry and in algebraic \(K\)-theory on the one hand and in mathematical physics (in particular, in conformal field theory) on the other. I was therefore asked to give two lectures at the Les Houches meeting introducing this function and explaining some of its most important properties and applications, and to write up these lectures for the Proceedings.

The first task was relatively straightforward, but the second posed a problem since I had already written and published an expository article on the dilogarithm some 15 years earlier. (In fact, that paper, originally written as a lecture in honor of Friedrich Hirzebruch’s 60th birthday, had appeared in two different Indian publications during the Ramanujan centennial year – see J. Math. Phys. Sci. 22, No. 1, 131–145 (1988; Zbl 0669.33001)). It seemed to make little sense to try to repeat in different words the contents of that earlier article. On the other hand, just reprinting the original article would mean omitting several topics which were either developed since it was written or which were omitted then but are of more interest now in the context of the appearances of the dilogarithm in mathematical physics.

The solution I finally decided on was to write a text consisting of two chapters of different natures. The first is simply an unchanged copy of the 1988 article, with its original title, footnotes, and bibliography, reprinted by permission from the book “Number theory and related topics”, Pap. Ramanujan Colloq., Bombay/India 1988, Stud. Math., Tata Inst. Fundam. Res. 12, 231–243 (1989; Zbl 0744.33011)).

In this chapter we define the dilogarithm function and describe some of its more striking properties: its known special values which can be expressed in terms of ordinary logarithms, its many functional equations, its connection with the volumes of ideal tetrahedra in hyperbolic 3-space and with the special values at \(s=2\) of the Dedekind zeta functions of algebraic number fields, and its appearance in algebraic \(K\)-theory; the higher polylogarithms are also treated briefly. The second, new, chapter gives further information as well as some more recent developments of the theory. Four main topics are discussed here. Three of them – functional equations, modifications of the dilogarithm function, and higher polylogarithms – are continuations of themes which were already begun in Chapter I. The fourth topic, Nahm’s conjectural connection between (torsion in) the Bloch group and modular functions, is new and especially fascinating.

We discuss only some elementary aspects concerning the asymptotic properties of Nahm’s \(q\)-expansions, referring the reader for the deeper parts of the theory, concerning the (in general conjectural) interpretation of these \(q\)-series as characters of rational conformal field theories, to the beautiful article by Nahm on pp. 67–132 in this volume [Zbl 1193.81092]. ...

Each of the two chapters has its own bibliography, that of Chapter I being a reprint of the original one and that of Chapter II giving some references to more recent literature. I apologize to the reader for this somewhat artificial construction, but hope that the two parts of the paper can still be read without too much confusion and perhaps even with some enjoyment. My own enthusiasm for this marvelous function as expressed in the 1988 paper has certainly not lessened in the intervening years, and I hope that the reader will be able to share at least some of it.

The reader interested in knowing more about dilogarithms should also consult the long article [Prog. Theor. Phys., Suppl. 118, 61–142 (1995; Zbl 0894.11052)] of A. N. Kirillov, which is both a survey paper treating most or all of the topics discussed here and also contains many new results of interest from the point of view of both mathematics and physics.

For the entire collection see [Zbl 1104.11003].

The first task was relatively straightforward, but the second posed a problem since I had already written and published an expository article on the dilogarithm some 15 years earlier. (In fact, that paper, originally written as a lecture in honor of Friedrich Hirzebruch’s 60th birthday, had appeared in two different Indian publications during the Ramanujan centennial year – see J. Math. Phys. Sci. 22, No. 1, 131–145 (1988; Zbl 0669.33001)). It seemed to make little sense to try to repeat in different words the contents of that earlier article. On the other hand, just reprinting the original article would mean omitting several topics which were either developed since it was written or which were omitted then but are of more interest now in the context of the appearances of the dilogarithm in mathematical physics.

The solution I finally decided on was to write a text consisting of two chapters of different natures. The first is simply an unchanged copy of the 1988 article, with its original title, footnotes, and bibliography, reprinted by permission from the book “Number theory and related topics”, Pap. Ramanujan Colloq., Bombay/India 1988, Stud. Math., Tata Inst. Fundam. Res. 12, 231–243 (1989; Zbl 0744.33011)).

In this chapter we define the dilogarithm function and describe some of its more striking properties: its known special values which can be expressed in terms of ordinary logarithms, its many functional equations, its connection with the volumes of ideal tetrahedra in hyperbolic 3-space and with the special values at \(s=2\) of the Dedekind zeta functions of algebraic number fields, and its appearance in algebraic \(K\)-theory; the higher polylogarithms are also treated briefly. The second, new, chapter gives further information as well as some more recent developments of the theory. Four main topics are discussed here. Three of them – functional equations, modifications of the dilogarithm function, and higher polylogarithms – are continuations of themes which were already begun in Chapter I. The fourth topic, Nahm’s conjectural connection between (torsion in) the Bloch group and modular functions, is new and especially fascinating.

We discuss only some elementary aspects concerning the asymptotic properties of Nahm’s \(q\)-expansions, referring the reader for the deeper parts of the theory, concerning the (in general conjectural) interpretation of these \(q\)-series as characters of rational conformal field theories, to the beautiful article by Nahm on pp. 67–132 in this volume [Zbl 1193.81092]. ...

Each of the two chapters has its own bibliography, that of Chapter I being a reprint of the original one and that of Chapter II giving some references to more recent literature. I apologize to the reader for this somewhat artificial construction, but hope that the two parts of the paper can still be read without too much confusion and perhaps even with some enjoyment. My own enthusiasm for this marvelous function as expressed in the 1988 paper has certainly not lessened in the intervening years, and I hope that the reader will be able to share at least some of it.

The reader interested in knowing more about dilogarithms should also consult the long article [Prog. Theor. Phys., Suppl. 118, 61–142 (1995; Zbl 0894.11052)] of A. N. Kirillov, which is both a survey paper treating most or all of the topics discussed here and also contains many new results of interest from the point of view of both mathematics and physics.

For the entire collection see [Zbl 1104.11003].