Perturbative Corrections to Kahler Moduli Spaces

TL;DR

This work identifies a universal, perturbative $\alpha'$-correction structure for the Kähler potential on the quantum Kähler moduli space of Calabi–Yau $n$-folds in the large-volume limit. The corrections are encoded by the log Gamma class $\Lambda_X$, implemented via a generalized Mukai map and its complex Gamma class $\widehat{\Gamma}^{\mathbb{C}}_X$, and are corroborated by a 2d $\mathcal{N}=(2,2)$ sphere partition function that computes the perturbative piece of the Kähler potential. The authors derive explicit expressions for these corrections using GLSMs for toric CY hypersurfaces, matching known four-loop results and predicting higher-loop terms; mirror symmetry arguments further support the proposed form by relating to the complex structure moduli of the mirror. The framework provides a principled path to extract Gromov–Witten invariants from sphere partition data and extends to Calabi–Yau $n$-folds beyond threefolds, facilitating a broader understanding of quantum corrections in Kähler moduli spaces.

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.