Gauge-Dressed Complex Geometry and T-duality in Heterotic String Theories

We study T-duality of $(p,q)$-hermitian geometries in backgrounds with non-Abelian gauge fields $A$ in heterotic string theories. We introduce a gauge-dressed complex geometry characterized by a shifted metric $\bar{g}=g+\frac{1}{2}\mathrm{T}\mathrm{r}(A^2)$, the closed 2-form $\omega$ and a quasi complex structure satisfying ${\bar{J}}^2<0$, but not necessarily ${\bar{J}}^2=-1$. Utilizing the positive and negative chirality half generalized complex-like structures constructed by $(\bar{g},\bar{J})$, we derive a heterotic Buscher-like rule for geometric quantities. We also demonstrate that the gauge-dressed structures can be used to construct an extended Born geometry that satisfies algebras of hypercomplex numbers.