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.
Keywords: Multiple Bernoulli series, wall crossing formulae, moduli spaces of flat connections, multiple zeta series, splines.
Mot clés : séries de Bernoulli, fonctions zeta multiples, espaces de modules de fibrés plats, splines.
Boysal, Arzu 1; Vergne, Michèle 2
@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}, pages = {821--858}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {62}, number = {2}, year = {2012}, doi = {10.5802/aif.2696}, mrnumber = {2985518}, zbl = {1251.14023}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2696/} }
TY - JOUR AU - Boysal, Arzu AU - Vergne, Michèle TI - Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings JO - Annales de l'Institut Fourier PY - 2012 SP - 821 EP - 858 VL - 62 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2696/ DO - 10.5802/aif.2696 LA - en ID - AIF_2012__62_2_821_0 ER -
%0 Journal Article %A Boysal, Arzu %A Vergne, Michèle %T Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings %J Annales de l'Institut Fourier %D 2012 %P 821-858 %V 62 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2696/ %R 10.5802/aif.2696 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2696/
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