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

The relation among arithmetic and physics has an extended historical past, during which the function of quantity idea and of alternative extra summary elements 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 all in favour of the wider interface of quantity concept, geometry, and physics.

This ebook is the results of that intriguing assembly, and collects, in 2 volumes, prolonged models of the lecture classes, by means of shorter texts on certain subject matters, of eminent mathematicians and physicists.

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

The spouse quantity is subtitled: On Random Matrices, Zeta services and Dynamical structures (Springer, 3-540-23189-7).

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

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

**Example text**

R2 be any r2 linearly independent elements of it, (i) and form the matrix with entries D(ξj ), (i, j = 1, . . , r2 ). Then the determinant of this matrix is a non-zero rational multiple of |d|1/2 ζF (2)/π 2r1 +2r2 . 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.

20 Notes on Chapter I . . . . . . . . . . . . . . . . . . . . . . 21 II. 1. 2. 3. 4. Further aspects of the dilogarithm . . . . . . . . . . . . 22 Variants of the dilogarithm function . . . . . . . . . . . . . . . Dilogarithm identities . . . . . . . . . . . . . . . . . . . . . Dilogarithms and modular functions: the Nahm equation . . . . . . Higher polylogarithms . . . . . . . .

We now deﬁne D on X by has derivative given by F (v) = D(ˆ z ) = F (v) + uv 2 for zˆ = (u, v) ∈ X . This is a holomorphic function from X to C/(2πi)2 Z whose behavior under the covering transformations of X → X is given by D((u + 2πir, v + 2πis)) = D((u, v)) + πi(rv − su) + 2π 2 rs (r, s ∈ Z) and whose relation to the Bloch-Wigner function D(z) is given by 1 2 (D(ˆ z )) = D(z) + (¯ uv) (ˆ z = (u, v), π(ˆ z ) = z). (1) (For more details, see [47], pp. ) Now let ξ = nj [zj ] be an element of the Bloch group BC .