Regularization of Automorphic Functions of Manifolds with Special K\"ahler Geometry

TL;DR

This paper develops a method to regularize automorphic functions on special K"ahler manifolds using zeta-functions, linking them to Eisenstein series and string theory amplitudes involving BPS states.

Contribution

It introduces a novel regularization approach for automorphic functions on special K"ahler manifolds using zeta-functions, applicable to string theory and moduli space analysis.

Findings

Regularized automorphic functions correspond to Eisenstein series.

The approach applies to coset manifolds like SU(1,n)/SU(n)×U(1).

Connections established between Abelian varieties and Calabi-Yau moduli spaces.

Abstract

In this paper we find automorphic functions of coset manifolds with special K\"ahler geometry. We use \zeta-functions to regularize an infinite product over integers which belong to a duality-invariant lattice, this product is known to produce duality-invariant functions. In turn these functions correspond to Eisenstein series which can be understood as string theory amplitudes that receive contributions from BPS states. The Ansatz is constructed using the coset manifold SU(1,n)\over SU(n) \times U(1) as an example but it can be generalized. Automorphic functions play an important role in the calculation of threshold corrections to gauge coupling and other stringy phenomena. We also find some connections between the theory of Abelian varieties and moduli spaces of Calabi-Yau manifolds