Localizations on Moduli Spaces and Free Field Realizations of Feynman Rules

TL;DR

This paper proves a conjecture linking string theory free energy in toric geometries with Wess-Zumino-Witten models, using localization, free boson realizations, and Hodge integral formulas.

Contribution

It introduces a novel approach connecting localization techniques, free field realizations, and Hodge integrals to analyze string theory free energy.

Findings

Validated Iqbal's conjecture.

Established a new Feynman rule realization via free bosons.

Derived explicit formulas for Hodge integrals.

Abstract

We prove Iqbal's conjecture on the relationship between the free energy of closed string theory in local toric geometry and the Wess-Zumino-Witten model. This is achieved by first reformulating the calculations of the free energy by localization techniques in terms of suitable Feynman rule, then exploiting a realization of the Feynman rule by free bosons. We also use a formula of Hodge integrals conjectured by the author and proved jointly with Chiu-Chu Melissa Liu and Kefeng Liu.