Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also certain multiple zeta values.

[1]  K. Ueno,et al.  Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries , 1989 .

[2]  C. Woodward,et al.  The index formula for the moduli of G-bundles on a curve , 2009 .

[3]  R. Carter.  Lie Groups , 1970, Nature.

[4]  ON WITTEN MULTIPLE ZETA-FUNCTIONS ASSOCIATED WITH SEMI-SIMPLE LIE ALGEBRAS IV , 2009, Glasgow Mathematical Journal.

[5]  E. Witten.  On quantum gauge theories in two dimensions , 1991 .

[6]  Michele Vergne,et al.  Residues formulae for volumes and Ehrhart polynomials of convex polytopes. , 2001, math/0103097.

[7]  M. Vergne,et al.  Multiple Bernoulli series, an Euler-MacLaurin formula, and wall crossings , 2010, 1008.0263.

[8]  Andras Szenes,et al.  Iterated Residues and Multiple Bernoulli Polynomials , 1997 .

[9]  Y. Komori,et al.  A study on multiple zeta values from the viewpoint of zeta-functions of root systems , 2012, 1205.0182.

[10]  P. Orlik,et al.  Unitary reflection groups and cohomology , 1980 .

[11]  Kohji Matsumoto,et al.  Mean value theorems for the double zeta-function , 2012, 1203.2242.

[12]  Andras Szenes.  Residue theorem for rational trigonometric sums and Verlinde's formula , 2001 .

[13]  Multiple zeta values and zeta-functions of root systems , 2011 .

[14]  M. Brion,et al.  ARRANGEMENT OF HYPERPLANES, II: THE SZENES FORMULA AND EISENSTEIN SERIES , 2000 .

[15]  D. Goldfeld,et al.  Multiple Dirichlet series, L-functions and automorphic forms , 2012 .

[16]  Michele Vergne,et al.  Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue , 1999 .

[17]  P. Gunnells,et al.  Evaluation of Dedekind sums, Eisenstein cocycles, and special values of L-functions , 1999, math/9909141.

[18]  Nicolas Bourbaki,et al.  Groupes et algèbres de Lie , 1971 .

[19]  Raoul Bott,et al.  The Yang-Mills equations over Riemann surfaces , 1983, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.