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

TL;DR

This paper explores the geometric structures of moduli spaces of Hermitian-Einstein and instanton connections on manifolds with torsion, revealing symmetries, models, and applications to theoretical physics.

Contribution

It introduces new geometric models for moduli spaces with torsion and analyzes their symmetries, extending understanding of instanton and Hermitian-Einstein moduli spaces.

Findings

Vector fields induce invariant actions on moduli spaces.

Moduli spaces can be modeled on holomorphic toric principal bundles.

Geometry of specific instanton moduli spaces relates to principal bundles over QKT manifolds.

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 K\"ahler 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 K\"ahler) and hyper-K\"ahler 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 K\"ahler 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.