Moduli Spaces of Connections and B-fields from T-duality with H-flux

We study the geometry of mixed fields, consisting of a connection and a $B$-field on a principal circle bundle over a Riemann surface, from the perspective of gauge theory and T-duality. Motivated by the foundational work of Atiyah--Bott and Segal, we introduce a twisted Yang--Mills functional whose critical locus, in the flat case, is governed by the simultaneous vanishing of the curvature and the $H$-flux. We show that the gauge group is a semi-direct product of abelian groups parametrised by an integer $\lambda$. The moduli spaces are constructed by presymplectic reduction and shown to be Heisenberg contact manifolds for $\lambda \neq 0$, whose topology we characterise completely. We show that T-duality preserves the twisted Yang--Mills functional and acts on the configuration space of flat mixed fields. We identify the subgroups of gauge transformations that are compatible with the T-duality map and describe the induced action on the singular quotient of T-dualizable flat mixed fields. Precisely at $\lambda =1$ does T-duality descend to an involutive contactomorphism of the moduli space.