D-instanton, threshold corrections, and topological string

TL;DR

This paper establishes a connection between the one-loop pfaffian of non-perturbative superpotentials in type II string compactifications and topological string theory, enabling direct computation of these quantum corrections.

Contribution

It proves that the one-loop pfaffian is determined by a moduli integral of a supersymmetric index, linking it to topological string theory and related computational methods.

Findings

The pfaffian can be computed via topological string theory, Chern-Simons theory, matrix models, or holomorphic anomaly equations.

The relation between the pfaffian, threshold corrections, and open topological string partition functions is clarified.

Abstract

In this note, we prove that the one-loop pfaffian of the non-perturbative superpotential generated by Euclidean D-branes in type II compactifications on orientifolds of Calabi-Yau threefolds is determined by the moduli integral of the new supersymmetric index defined by Cecotti, Fendley, Intriligator, and Vafa. As this quantity can be computed via topological string theory, Chern-Simons theory, matrix models, or by solving the holomorphic anomaly equation, this result provides a method to directly compute the one-loop pfaffian of the non-perturbative superpotential. The relation between the one-loop pfaffian, threshold corrections to the gauge coupling, and the one-loop partition function of open topological string theory is discussed.