Harmonic Analysis of the Instanton Prepotential

TL;DR

This paper explores how symmetries in Calabi-Yau moduli spaces constrain the instanton expansion of the 4D $ abla=2$ Type IIA prepotential, revealing a spectral decomposition linked to eigenfunctions of a Laplace-Beltrami operator.

Contribution

It demonstrates that the prepotential's organization into Coxeter-invariant functions corresponds to eigenfunctions of a Laplace-Beltrami operator, providing a spectral perspective on the Gromov-Witten expansion.

Findings

Prepotential functions are eigenfunctions of a Laplace-Beltrami operator.

Spectral decomposition explains the appearance of special functions like Bessel and theta functions.

Spectral representations converge efficiently within the moduli space.

Abstract

Discrete symmetries of Calabi-Yau moduli spaces, generated by isomorphic flops, constrain the instanton expansion of the 4D $\mathcal{N}=2$ Type~IIA prepotential. We show that the Coxeter-invariant functions into which the prepotential organizes are eigenfunctions of a Laplace-Beltrami operator built from the Coxeter-invariant symmetric bilinear form on the moduli space. This means that the Gromov-Witten expansion can be interpreted as a superposition of waves propagating on the Coxeter quotient of the moduli space, and its resummation is the corresponding spectral decomposition. For the dihedral Coxeter groups, separation of variables in the eigenvalue equation explains from first principles why special modified Bessel functions, ordinary Bessel functions and Jacobi theta functions appear as the natural building blocks of the prepotential, depending on whether the Coxeter rotation acts hyperbolically, elliptically, or parabolically. The resulting spectral representations converge efficiently in the interior of the moduli space, complementing the standard large-volume instanton expansion.