# Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings

Arzu Boysal[1]; Michèle Vergne[2]

• [1] Bogaziçi University Faculty of Arts and Science Department of Mathematics 34342, Bebek-Istanbul (Turkey)
• [2] Institut Mathématique de Jussieu 175 rue du Chevaleret 75013 Paris (France)
• Volume: 62, Issue: 2, page 821-858
• ISSN: 0373-0956

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## Abstract

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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.

## How to cite

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Boysal, Arzu, and Vergne, Michèle. "Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings." Annales de l’institut Fourier 62.2 (2012): 821-858. <http://eudml.org/doc/251122>.

@article{Boysal2012,
abstract = {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.},
affiliation = {Bogaziçi University Faculty of Arts and Science Department of Mathematics 34342, Bebek-Istanbul (Turkey); Institut Mathématique de Jussieu 175 rue du Chevaleret 75013 Paris (France)},
author = {Boysal, Arzu, Vergne, Michèle},
journal = {Annales de l’institut Fourier},
keywords = {Multiple Bernoulli series; wall crossing formulae; moduli spaces of flat connections; multiple zeta series; splines; multiple Bernoulli series},
language = {eng},
number = {2},
pages = {821-858},
publisher = {Association des Annales de l’institut Fourier},
title = {Multiple Bernoulli series, an Euler-MacLaurin formula, and Wall crossings},
url = {http://eudml.org/doc/251122},
volume = {62},
year = {2012},
}

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
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 2
SP - 821
EP - 858
AB - 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.
LA - eng
KW - Multiple Bernoulli series; wall crossing formulae; moduli spaces of flat connections; multiple zeta series; splines; multiple Bernoulli series
UR - http://eudml.org/doc/251122
ER -

## References

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9. P-E. Paradan, Wall-crossing formulas in Hamiltonian geometry Zbl1253.53083
10. P-E. Paradan, Localization of the Riemann-Roch character, J. Funct. Anal. 187 (2001), 442-509 Zbl1001.53062MR1875155
11. P-E. Paradan, The moment map and equivariant cohomology with generalized coefficients, Topology 39 (2001), 401-444 Zbl0941.37050MR1722000
12. A. Szenes, Iterated Residues and Multiple Bernoulli Polynomials, International Mathematics Research Notices 18 (1998), 937-956 Zbl0968.11015MR1653791
13. A. Szenes, Residue theorem for rational trigonometric sums and Verlinde’s formula, Duke Math. J. 118 (2003), 189-227 Zbl1042.14030MR1980993
14. A. Szenes, M. Vergne, $\left[Q,R\right]=0$ and Kostant partition functions
15. A. Szenes, M. Vergne, Residue formulae for vector partitions and Euler-MacLaurin sums, Advances in Applied Mathematics 30 (2003), 295-342 Zbl1067.52014MR1979797
16. M. Vergne, A Remark on the Convolution with Box Splines Zbl1248.41024
17. E. Witten, On quantum gauge theories in two dimensions, Commun. Math. Phys. 141 (1991), 153-209 Zbl0762.53063MR1133264
18. E. Witten, Two dimensional gauge theories revisited, J. Geom. Phys. 9 (1992), 303-368 Zbl0768.53042MR1185834

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