On The Consistent Supersymmetric Reduction Of Heterotic Supergravity With Geometrically Arising Yang-Mills Symmetries

In the paper arXiv:2501.04771, a novel compactification of heterotic supergravity on a warped product of $\mathbb{R}\times T^{1,1}$ was constructed, where $T^{1,1}$ is a five-dimensional coset space $(SU(2)\times SU(2))\mathrm{/}U(1)$. It was shown that this admits a four dimensional Minkowski vacuum solution with $\mathcal{N}=1$ supersymmetry, and furthermore that in the bosonic sector there exists a remarkable fully consistent truncation in which the gauge bosons of the $SU(2)\times SU(2)$ isometries of the $T^{1,1}$ are retained. In this paper, we examine this reduction further, and show that the consistency can be extended to include the fermionic sector also. Thus the heterotic theory admits a consistent reduction to give an ungauged $\mathcal{N}=1$ supergravity coupled to $SU(2)\times SU(2)$ Yang-Mills multiplets and a scalar multiplet.