Scale and Conformal Invariance in Heterotic $\sigma$-Models

TL;DR

This paper proves that all perturbative scale invariant heterotic sigma models with compact target spaces are conformally invariant, using geometric flow techniques, and explores the geometric conditions for conformal invariance in special holonomy target spaces.

Contribution

It establishes a proof of conformal invariance for a class of heterotic sigma models and analyzes the geometric restrictions needed for conformal invariance in special holonomy target spaces.

Findings

All perturbative scale invariant heterotic sigma models are conformally invariant.

Target space geometry must be either conformally balanced or have reduced holonomy for conformal invariance.

Examples of special holonomy geometries satisfying these conditions are provided.

Abstract

We demonstrate that all perturbative scale invariant heterotic sigma models with a compact target space $M^D$ are conformally invariant. The proof, presented in detail for up to and including two loops, utilises a geometric analogue of the $c$-theorem based on a generalisation of the Perelman's results on geometric flows. Then, we present examples of scale invariant heterotic sigma models with target spaces that exhibit special geometry, which is characterised by the holonomy of the connection with torsion a 3-form, and explore the additional conditions that are necessary for such sigma models to be conformally invariant. For this, we find that the geometry of the target spaces is further restricted to be either conformally balanced or the a priori holonomy of the connection with torsion reduces. We identify the pattern of holonomy reduction in the cases that the holonomy is $SU(n)$ $(D=2n)$, $Sp(k)$ $D=4k)$, $G_2$ $(D=7)$ and $\mathrm{Spin}(7)$ $(D=8)$. We also investigate the properties of these geometries and present some examples.