The geometry of N=4 twisted string

TL;DR

This paper compares N=2 and N=4 topological strings using sigma models, revealing that N=4 requires special geometric conditions like a covariantly constant holomorphic two-form, leading to Ricci-flat manifolds.

Contribution

It demonstrates that N=4 twisted string theories impose stricter geometric conditions than N=2, specifically requiring a covariantly constant holomorphic two-form and Ricci-flatness.

Findings

N=4 theories require a covariantly constant holomorphic two-form.

N=4 manifolds have holonomy subgroup of SU(1,1).

N=4 theories are potentially better at handling ultraviolet divergences.

Abstract

We compare N=2 string and N=4 topological string within the framework of the sigma model approach. Being classically equivalent on a flat background, the theories are shown to lead to different geometries when put in a curved space. In contrast to the well studied Kaehler geometry characterising the former case, in the latter case a manifold has to admit a covariantly constant holomorphic two-form in order to support an N=4 twisted supersymmetry. This restricts the holonomy group to be a subgroup of SU(1,1) and leads to a Ricci--flat manifold. We speculate that, the N=4 topological formalism is an appropriate framework to smooth down ultraviolet divergences intrinsic to the N=2 theory.