# Frontiers in Number Theory, Physics, and Geometry: On - download pdf or read online

By Pierre E. Cartier, Bernard Julia, Pierre Moussa, Pierre Vanhove

The relation among arithmetic and physics has a protracted background, within which the position of quantity thought and of different extra summary components of arithmetic has lately turn into extra prominent.

More than ten years after a primary assembly in 1989 among quantity theorists and physicists on the Centre de body des Houches, a moment 2-week occasion occupied with the wider interface of quantity thought, geometry, and physics.

This e-book is the results of that fascinating assembly, and collects, in 2 volumes, prolonged models of theВ lecture classes, byВ shorter texts on designated issues, of eminent mathematicians and physicists.

The current quantity has 3 components: Conformal box Theories,В Discrete teams, Renomalization.

The significant other quantity is subtitled:В On Random Matrices, Zeta features andВ Dynamical SystemsВ (Springer, 3-540-23189-7).

**Read Online or Download Frontiers in Number Theory, Physics, and Geometry: On Conformal Field Theories, Discrete Groups and Renormalization PDF**

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**Extra resources for Frontiers in Number Theory, Physics, and Geometry: On Conformal Field Theories, Discrete Groups and Renormalization**

**Sample text**

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. The Dilogarithm Function 5 Chapter I. The dilogarithm function in geometry and number theory1 The dilogarithm function is the function deﬁned by the power series Li2 (z) = ∞ n=1 zn n2 for |z| < 1 . The deﬁnition and the name, of course, come from the analogy with the Taylor series of the ordinary logarithm around 1, − log(1 − z) = ∞ n=1 zn n for |z| < 1 , which leads similarly to the deﬁnition of the polylogarithm Lim (z) = ∞ n=1 zn nm for |z| < 1, m = 1, 2, .

The Dilogarithm Function 21 Notes on Chapter I. A The comment about “too little-known” is now no longer applicable, since the dilogarithm has become very popular in both mathematics and mathematical physics, due to its appearance in algebraic K-theory on the one hand and in conformal ﬁeld theory on the other. Today one needs no apology for devoting a paper to this function. B From the point of view of the modern theory, the arguments of the dilogarithm occurring in these eight formulas are easy to recognize: they are the totally real algebraic numbers x (oﬀ the cut) for which x and 1 − x, × if non-zero, belong to the same rank 1 subgroup of Q , or equivalently, for which [x] is a torsion element of the Bloch group.

It turns out, however, that there are other natural dilogarithm functions besides Li2 and D which have interesting properties. In this section we shall discuss six of these: the Rogers dilogarithm L, which is similar to D but is deﬁned on P1 (R) (where D vanishes); the “enhanced” dilogarithm D, which takes values in C/π 2 Q and is in some sense a combination of the Rogers and Bloch-Wigner dilogarithms, but is only deﬁned on the Bloch group of The Dilogarithm Function 23 C rather than for individual complex arguments; the double logarithm Li1,1 , the simplest of the multiple polylogarithms, which has two arguments but can be expressed in terms of ordinary dilogarithms; the quantum dilogarithm of Faddeev and Kashaev, which will play a role in the discussion of Nahm’s conjecture in §3; and, very brieﬂy, the p-adic and the modulo p analogues of the dilogarithm.