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

TL;DR

The paper develops a systematic framework connecting Hermitian-Einstein and instanton moduli spaces to KT/HKT/QKT geometries on KT manifolds. It shows that holomorphic, ∇-covariantly constant vector fields on the base KT manifolds induce Killing and holomorphic vector fields on HE moduli spaces, with extra conditions yielding covariant constancy with respect to the torsion, producing holomorphic torus fibrations and refined symmetry structures. The moduli spaces $ ext{M}^*_{ ext{HE}}(M^{2n})$ and $ ext{M}^*_{ ext{asd}}(M^4)$ are shown to carry strong KT, bi-KT, HKT, and bi-HKT structures in various settings, with explicit examples on $S^3 imes S^3$, $S^3 imes T^3$, and $S^3 imes S^1$ revealing torus- and quaternionic-Kähler-type base geometries (including QKT bases). The results furnish a toolkit to generate new geometric structures via moduli spaces and reveal rich symmetry algebras, including large $N=4$ superconformal algebras in sigma models with these targets, thereby linking differential-geometric moduli theory to AdS/CFT-inspired quantum field theory. Together, they illuminate how skew-symmetric torsion and generalized Kähler-type geometries arise naturally on physically relevant moduli spaces and provide concrete models for sigma-models with enhanced supersymmetry.

Abstract

We investigate the geometry of the moduli spaces $\mathscr{M}_{\HE}^*(M^{2n})$ of Hermitian-Einstein irreducible connections on a vector bundle $E$ over a Kähler with torsion (KT) manifold $M^{2n}$ that admits holomorphic and $\h\nabla$-covariantly constant vector fields, where $\h\nabla$ is the connection with skew-symmetric torsion $H$. We demonstrate that such vector fields induce an action on $\mathscr{M}_{\HE}^*(M^{2n})$ 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 $\h{\mathcal{ D}}$ on $\mathscr{M}_{\HE}^*(M^{2n})$. We demonstrate that in the presence of such vector fields, the geometry of $\mathscr{M}_{\HE}^*(M^{2n})$ 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 $\mathscr{M}_{\asd}^*(M^{4})$ 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 $\mathscr{M}_{\asd}^*(S^3\times S^1)$ 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.