[Paper Review] Regularization of Automorphic Functions of Manifolds with Special Kähler Geometry

N. Vanegas|ArXiv.org|Jun 3, 1999

Advanced Algebra and Geometry15 references3 citations

TL;DR

This paper introduces a regularization method for automorphic functions on manifolds with special Kähler geometry using zeta functions to tame divergent infinite products over duality-invariant lattices. The approach yields duality-invariant functions linked to Eisenstein series, which model BPS state contributions in string amplitudes, with a concrete realization on the coset space SU(1,n)/(SU(n)×U(1)) and connections to Calabi-Yau moduli spaces and Abelian varieties.

ABSTRACT

In this paper we find automorphic functions of coset manifolds with special Kähler geometry. We use ζ-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) imes 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

Motivation & Objective

• To develop a regularization procedure for infinite products over duality-invariant lattices in manifolds with special Kähler geometry.
• To construct automorphic functions that are invariant under duality symmetries, relevant to string theory amplitudes.
• To establish connections between automorphic forms on SU(1,n)/(SU(n)×U(1)) and physical quantities such as threshold corrections in gauge couplings.
• To explore links between the theory of Abelian varieties and moduli spaces of Calabi-Yau manifolds via automorphic functions.

Proposed method

• Utilizes zeta-function regularization to control divergent infinite products over integers in a duality-invariant lattice.
• Constructs automorphic functions via Eisenstein series associated with the coset manifold SU(1,n)/(SU(n)×U(1)) as a representative example.
• Applies techniques from automorphic forms and special Kähler geometry to ensure duality invariance of the resulting functions.
• Employs the structure of symmetric spaces and Heisenberg-type groups to define the automorphic data on the coset manifold.
• Relies on the correspondence between automorphic forms and string theory amplitudes that receive contributions from BPS states.
• Generalizes the construction beyond the specific example to other manifolds with special Kähler geometry.

Experimental results

Research questions

• RQ1 How can divergent infinite products over duality-invariant lattices be regularized to yield well-defined automorphic functions on special Kähler manifolds?
• RQ2 What is the role of zeta-function regularization in constructing duality-invariant automorphic forms from lattice products?
• RQ3 How do the resulting automorphic functions relate to Eisenstein series and physical string amplitudes involving BPS states?
• RQ4 What connections exist between automorphic forms on SU(1,n)/(SU(n)×U(1)) and moduli spaces of Calabi-Yau manifolds or Abelian varieties?
• RQ5 Can the construction be generalized beyond the SU(1,n)/(SU(n)×U(1)) example to other special Kähler manifolds?

Key findings

• The zeta-function regularization successfully tames the infinite product over the duality-invariant lattice, yielding a convergent automorphic function.
• The resulting function is duality-invariant and corresponds to an Eisenstein series, which models string amplitude contributions from BPS states.
• The construction is explicitly realized on the coset manifold SU(1,n)/(SU(n)×U(1)), providing a concrete example of the regularization method.
• The automorphic functions derived are shown to be relevant for computing threshold corrections in gauge coupling constants in string theory.
• The paper establishes non-trivial connections between the theory of Abelian varieties and moduli spaces of Calabi-Yau manifolds through the lens of automorphic forms.
• The method provides a systematic framework for constructing duality-invariant functions on special Kähler manifolds, extendable to broader classes of symmetric spaces.