Koba-Nielsen local zeta functions, convex subsets, and generalized Selberg-Mehta-Macdonald and Dotsenko-Fateev-like integrals

TL;DR

This paper extends the theory of Koba-Nielsen local zeta functions to integrals over convex subsets, providing explicit meromorphic continuations and polar loci, with applications to generalized Selberg-Mehta-Macdonald and Dotsenko-Fateev integrals.

Contribution

It introduces a new class of Koba-Nielsen local zeta functions over convex sets and describes their meromorphic continuations explicitly.

Findings

Meromorphic continuation of integrals over convex subsets is achieved.

Polar loci are explicitly described using embedded resolution techniques.

Integrals are expressed as weighted sums of Gamma functions evaluated at linear combinations of parameters.

Abstract

The Koba-Nielsen local zeta functions are integrals depending on several complex parameters, used to regularize the Koba-Nielsen string amplitudes. These integrals are convergent and admit meromorphic continuations in the complex parameters. In the original case, the integration is carried out on the n-dimensional Euclidean space. In this work, the integration is over a variety of (bounded or unbounded) convex subsets; the resulting integrals also admit meromorphic continuations in the complex parameters. We describe the meromorphic continuation's polar locus explicitly, using the technique of embedded resolution. This result can be reinterpreted as saying that the meromorphic continuations are weighted sums of Gamma functions, evaluated at linear combinations of the complex parameters, where the weights are holomorphic functions. The integrals announced in the title of this paper occur as a particular case of these new Koba-Nielsen local zeta functions, or of a further generalization to arbitrary hyperplane arrangements.