A Worldsheet Approach to $\mathbf{\mathcal{N}=1}$ Heterotic Flux Backgrounds

Heterotic backgrounds with torsion preserving minimal supersymmetry in four dimensions can be obtained as orbifolds of principal $T^{2}$ bundles over $K3$. We consider a worldsheet description of these backgrounds as gauged linear sigma-models (GLSMs) with $(0,2)$ supersymmetry. Such a formulation provides a useful framework in order to address the resolution of singularities of the orbifold geometries. We investigate the constraints imposed by discrete symmetries on the corresponding torsional GLSMs. In particular, the principal $T^{2}$ connection over $K3$ is inherited from $(0,2)$ vector multiplets. As these vectors gauge global scaling symmetries of products of projective spaces, the corresponding $K3$ geometry is naturally realized as an algebraic hypersurface in such a product (or as a branched cover of it). We outline the general construction for describing such orbifolds. We give explicit constructions for automorphisms of order two and three.