By Don Zagier (auth.), Pierre Cartier, Pierre Moussa, Bernard Julia, Pierre Vanhove (eds.)

The relation among arithmetic and physics has a protracted heritage, during which the position of quantity thought and of different extra summary elements of arithmetic has lately develop 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 all for the wider interface of quantity concept, geometry, and physics.

This e-book is the results of that intriguing assembly, and collects, in 2 volumes, prolonged models of the lecture classes, via shorter texts on precise themes, of eminent mathematicians and physicists.

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

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

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Additional info for Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization

Example text

If instead of taking any r2 linearly independent elements we choose the ξj to 2 It should be mentioned that the definition of B F which we gave for F = C or Q must be modified slightly when F is a number field because F × is no longer divisible; however, this is a minor point, affecting only the torsion in the Bloch group, and will be ignored here. The Dilogarithm Function 19 be a basis of BF /{torsion}, then this rational multiple (chosen positively) is an invariant of F , independent of the choice of ξj .

8 The Bloch-Wigner function D(z) and its generalizations . . . . . . 10 Volumes of hyperbolic 3-manifolds . . . . . . . . . . . . . . . 13 . . and values of Dedekind zeta functions . . . . . . . . . . . . . 16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Notes on Chapter I . . . . . . . . . . . . . . . . . . . . . . 21 II. 1. 2. 3. 4. Further aspects of the dilogarithm . . . . .

It is given by Vol(∆) = D(z0 , z1 , z2 , z3 ), (7) where z0 , . . , z3 ∈ C are the vertices of ∆ and D is the function defined in (5). In the special case that three of the vertices of ∆ are ∞, 0, and 1, equation (7) reduces to the formula (due essentially to Lobachevsky) Vol(∆) = D(z). (8) 14 Don Zagier In fact, equations (7) and (8) are equivalent, since any 4-tuple of points z0 , . . , z3 can be brought into the form {∞, 0, 1, z} by the action of some element of SL2 (C) on P1 (C), and the group SL2 (C) acts on H3 by isometries.

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