Superpotentials for M-theory on a G_2 holonomy manifold and Triality symmetry

TL;DR

This paper investigates the superpotential in M-theory on a G_2 manifold, revealing its invariance under triality symmetry and uncovering connections to hyperbolic geometry and flat bundles similar to Seiberg/Witten theory.

Contribution

It demonstrates that the superpotential respects triality symmetry and introduces a hyperbolic geometric framework involving flat bundles related to the Heisenberg group.

Findings

Superpotential is invariant under triality symmetry.

A flat bundle related to the Heisenberg group appears in the moduli space.

Connections to hyperbolic geometry and Seiberg/Witten theory are established.

Abstract

For $M$-theory on the $G_2$ holonomy manifold given by the cone on ${\bf S^3}\x {\bf S^3}$ we consider the superpotential generated by membrane instantons and study its transformations properties, especially under monodromy transformations and triality symmetry. We find that the latter symmetry is, essentially, even a symmetry of the superpotential. As in Seiberg/Witten theory, where a flat bundle given by the periods of an universal elliptic curve over the $u$-plane occurs, here a flat bundle related to the Heisenberg group appears and the relevant universal object over the moduli space is related to hyperbolic geometry.