Perturbative Corrections to Kahler Moduli Spaces

TL;DR

This paper introduces a universal formula for perturbative corrections to the Kahler potential on Calabi-Yau moduli spaces, linking geometric invariants with quantum field theory insights to enhance understanding of string compactifications.

Contribution

It proposes a universal form for perturbative alpha' corrections involving the log Gamma class, connecting mirror symmetry and quantum Kahler geometry.

Findings

Derived a general formula for perturbative corrections applicable to all Calabi-Yau n-folds.

Identified the log Gamma class as a key geometric invariant in these corrections.

Validated the formula by comparing with mirror symmetry predictions.

Abstract

We propose a general formula for perturbative-in-alpha' corrections to the Kahler potential on the quantum Kahler moduli space of Calabi-Yau n-folds, for any n, in their asymptotic large volume regime. The knowledge of such perturbative corrections provides an important ingredient needed to analyze the full structure of this Kahler potential, including nonperturbative corrections such as the Gromov-Witten invariants of the Calabi-Yau n-folds. We argue that the perturbative corrections take a universal form, and we find that this form is encapsulated in a specific additive characteristic class of the Calabi-Yau n-fold which we call the log Gamma class, and which arises naturally in a generalization of Mukai's modified Chern character map. Our proposal is inspired heavily by the recent observation of an equality between the partition function of certain supersymmetric, two-dimensional gauge theories on a two-sphere, and the aforementioned Kahler potential. We further strengthen our proposal by comparing our findings on the quantum Kahler moduli space to the complex structure moduli space of the corresponding mirror Calabi-Yau geometry.