Non-Perturbative Superpotentials In String Theory

TL;DR

This paper discusses how non-perturbative superpotentials are calculated in string theory compactifications, highlighting conditions under which they vanish or are generated, with implications for various string theories and instanton contributions.

Contribution

It provides a unified framework for calculating non-perturbative superpotentials across different string theories using M-theory and F-theory insights.

Findings

Superpotentials can be zero or non-zero depending on the Calabi-Yau four-fold.

Non-perturbative effects include space-time and world-sheet instantons.

In simple cases, contributions come from finitely many instanton numbers.

Abstract

The non-perturbative superpotential can be effectively calculated in $M$-theory compactification to three dimensions on a Calabi-Yau four-fold $X$. For certain $X$, the superpotential is identically zero, while for other $X$, a non-perturbative superpotential is generated. Using $F$-theory, these results carry over to certain Type IIB and heterotic string compactifications to four dimensions with $N=1$ supersymmetry. In the heterotic string case, the non-perturbative superpotential can be interpreted as coming from space-time and world-sheet instantons; in many simple cases contributions come only from finitely many values of the instanton numbers.