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

TL;DR

This paper develops a new Euler-MacLaurin type formula using multiple Bernoulli series, introduces wall crossing and decomposition formulas, motivated by Witten's volume formulas for moduli spaces of flat bundles.

Contribution

It presents novel formulas involving multiple Bernoulli series, extending classical Euler-MacLaurin and wall crossing concepts in the context of moduli space volume calculations.

Findings

Derived a new Euler-MacLaurin type formula using multiple Bernoulli series

Established a wall crossing formula relevant to moduli space volumes

Provided a decomposition formula for these series

Abstract

Using multiple Bernoulli series, we give a formula in the spirit of Euler MacLaurin formula. We also give a wall crossing formula and a decomposition formula. The study of these series is motivated by formulae of E.Witten for volumes of moduli spaces of flat bundles over a surface.