Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings
Annales de l'Institut Fourier, Volume 62 (2012) no. 2, p. 821-858

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

Nous étudions les séries de Bernoulli multiples associées à une suite de vecteurs engendrant un réseau dans un espace vectoriel. Elles déterminent une fonction localement polynomiale et périodique. Nous donnons une formule explicite (saut à travers le mur) qui compare les densités polynomiales dans deux domaines adjacents séparés par un hyperplan. Nous utilisons aussi ces polynômes de Bernoulli périodiques pour donner une formule dans l’esprit de la formule d’Euler-MacLaurin. Finalement nous donnons une formule pour la série de Bernoulli multiple comme une superposition de produits de convolutions de mesures polynomiales supportées sur des sous-espaces et de multisplines. L’étude de ces séries est motivée par la formule de Witten calculant le volume symplectique de l’espace des modules des fibrés plats sur une surface de Riemann avec un point marqué.

DOI : https://doi.org/10.5802/aif.2696
Classification:  14H60,  53D30
Keywords: Multiple Bernoulli series, wall crossing formulae, moduli spaces of flat connections, multiple zeta series, splines.
@article{AIF_2012__62_2_821_0,
     author = {Boysal, Arzu and Vergne, Mich\`ele},
     title = {Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {2},
     year = {2012},
     pages = {821-858},
     doi = {10.5802/aif.2696},
     mrnumber = {2985518},
     zbl = {1251.14023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2012__62_2_821_0}
}
Boysal, Arzu; Vergne, Michèle. Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings. Annales de l'Institut Fourier, Volume 62 (2012) no. 2, pp. 821-858. doi : 10.5802/aif.2696. http://www.numdam.org/item/AIF_2012__62_2_821_0/

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