Geometry and symmetries of Hermitian-Einstein and instanton connection moduli spaces

We investigate the geometry of the moduli spaces ${\mathcal{M}}_{\text{HE}}^{\ast }(M^{2n})$ of Hermitian-Einstein irreducible connections on a vector bundle $E$ over a Kähler with torsion (KT) manifold $M^{2n}$ that admits holomorphic and $\text{h}\mathrm{\nabla }$-covariantly constant vector fields, where $\text{h}\mathrm{\nabla }$ is the connection with skew-symmetric torsion $H$. We demonstrate that such vector fields induce an action on ${\mathcal{M}}_{\text{HE}}^{\ast }(M^{2n})$ that leaves both the metric and complex structure invariant. Moreover, if an additional condition is satisfied, the induced vector fields are covariantly constant with respect to the connection with skew-symmetric torsion $\text{h}\mathcal{D}$ on ${\mathcal{M}}_{\text{HE}}^{\ast }(M^{2n})$. We demonstrate that in the presence of such vector fields, the geometry of ${\mathcal{M}}_{\text{HE}}^{\ast }(M^{2n})$ can be modelled on that of holomorphic toric principal bundles with base space KT manifolds and give some examples. We also extend our analysis to the moduli spaces ${\mathcal{M}}_{\text{asd}}^{\ast }(M^4)$ of instanton connections on vector bundles over KT, bi-KT (generalised Kähler) and hyper-Kähler with torsion (HKT) manifolds $M^4$. We find that the geometry of ${\mathcal{M}}_{\text{asd}}^{\ast }(S^3\times S^1)$ can be modelled on that of principal bundles with fibre $S^3\times S^1$ over Quaternionic Kähler manifolds with torsion (QKT). In addition motivated by applications to AdS/CFT, we explore the (superconformal) symmetry algebras of two-dimensional sigma models with target spaces such moduli spaces.