Multiple Bernoulli series, an Euler-MacLaurin formula, and wall crossings

Abstract

We study multiple Bernoulli series associated to a sequence ofvectors generating a lattice in a vector space. The associated multiple Bernoulliseries is a periodic and locally polynomial function, and we give an explicit formula(called wall crossing formula) comparing the polynomial densities in two adjacentdomains of polynomiality separated by a hyperplane. We also present a formula inthe spirit of Euler-MacLaurin formula. Finally, we give a decomposition formula forthe Bernoulli series describing it as a superposition of convolution products of lowerdimensional Bernoulli series and multisplines. The study of these series is motivatedby the work of E. Witten, computing the symplectic volume of the moduli spaceof flat G-connections on a Riemann surface with one boundary component.