Arrangements of hyperplanes I: Rational functions and Jeffrey-Kirwan residue

Abstract

Consider the space $R_{\Delta }$ of rational functions of several variables with poles on a fixed arrangement $\Delta$ of hyperplanes. We obtain a decomposition of $R_{\Delta }$ as a module over the ring of differential operators with constant coefficients. We generalize to the space $R_{\Delta }$ the notions of principal part and of residue, and we describe its relations to Laplace transforms of locally polynomial functions. This explains algebraic aspects of work by L. Jeffreys and F. Kirwan about integrals of equivariant cohomology classes on Hamiltonian manifolds. As another application, we will construct multidimensional versions of Eisenstein series in a subsequent article, and we will obtain another proof of a residue formula of A. Szenes for Witten zeta functions.

arXiv comment33 pages, LaTEX2e, to appear in the Annales Scientifiques de l'Ecole Normale Superieure

Cited by in corpus (10)

• Elliptic genera of 2d N=2 gauge theories
• Witten Index and Wall Crossing
• Comments on twisted indices in 3d supersymmetric gauge theories
• Seifert fibering operators in 3d $\mathcal{N}=2$ theories
• Fundamental Vortices, Wall-Crossing, and Particle-Vortex Duality
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• A-twisted correlators and Hori dualities
• Local B-model Yukawa couplings from A-twisted correlators
• The colored Jones polynomials as vortex partition functions
• Mutation, Witten Index, and Quiver Invariant