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

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.