[Séries de Bernoulli multiples, Formule d’Euler-MacLaurin et Formules de saut]
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.
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.
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.
@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},
zbl = {1251.14023},
mrnumber = {2985518},
language = {en},
url = {http://www.numdam.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 - http://www.numdam.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 http://www.numdam.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, Tome 62 (2012) no. 2, pp. 821-858. doi : 10.5802/aif.2696. http://www.numdam.org/articles/10.5802/aif.2696/
[1] Paradan’s wall crossing formula for partitions functions and Khovanski-Pukhlikov differential operator, Annales de l’Institut Fourier, Volume 59 (2009), pp. 1715-1752 | DOI | Numdam | MR
[2] Arrangement of hyperplanes I: Rational functions and Jeffrey-Kirwan residue, Ann. scient. Éc. Norm. Sup., Volume 32 (1999), pp. 715-741 | Numdam | MR | Zbl
[3] Arrangement of hyperplanes II: The Szenes formula and Eisenstein series, Duke Math. J., Volume 103 (2000), pp. 279-302 | DOI | MR
[4] Translates of multivariate splines, Linear Algebra Appl., Volume 52 (1983), pp. 217-234 | MR | Zbl
[5] Topics in hyperplane arrangements, polytopes and box splines to appear (available on the personal web page of C. Procesi)
[6] Partition functions and generalized Dahmen-Micchelli spaces, arXiv : math/0805.2907. Transformation Groups, Volume 15 (2010) no. 3, pp. 751-773 | DOI | MR
[7] Symplectic fibrations and multiplicity diagrams, Cambridge University Press, 1996 | MR | Zbl
[8] Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. (2), Volume 148 (1998) no. 1, pp. 109-196 | DOI | MR | Zbl
[9] Wall-crossing formulas in Hamiltonian geometry (arXiv:math/0411306)
[10] Localization of the Riemann-Roch character, J. Funct. Anal., Volume 187 (2001), pp. 442-509 | DOI | MR
[11] The moment map and equivariant cohomology with generalized coefficients, Topology, Volume 39 (2001), pp. 401-444 | DOI | MR
[12] Iterated Residues and Multiple Bernoulli Polynomials, International Mathematics Research Notices, Volume 18 (1998), pp. 937-956 | DOI | MR | Zbl
[13] Residue theorem for rational trigonometric sums and Verlinde’s formula, Duke Math. J., Volume 118 (2003), pp. 189-227 | DOI | MR
[14] and Kostant partition functions (arXiv:math/1006.4149)
[15] Residue formulae for vector partitions and Euler-MacLaurin sums, Advances in Applied Mathematics, Volume 30 (2003), pp. 295-342 | DOI | MR
[16] A Remark on the Convolution with Box Splines arXiv: math/1003.1574 (to appear in Annals of Mathematics)
[17] On quantum gauge theories in two dimensions, Commun. Math. Phys., Volume 141 (1991), pp. 153-209 | DOI | MR | Zbl
[18] Two dimensional gauge theories revisited, J. Geom. Phys., Volume 9 (1992), pp. 303-368 | DOI | MR | Zbl
Cité par Sources :
