Geometry of the moduli space of Hermitian-Einstein connections on manifolds with a dilaton

TL;DR

This work analyzes the moduli space $\mathscr{M}^*_{HE}(M^{2n})$ of Hermitian-Einstein connections on Hermitian manifolds endowed with a dilaton $\Phi$, showing that a Gauduchon-like condition $D^i(e^{-2\Phi}K^\flat_i)=0$ with $K^\flat=\theta+2d\Phi$ yields a strong KT structure on the moduli space. It develops a dilaton-adjusted inner product and Hermitian form, proves the horizontality operator is invertible under the stated condition, and demonstrates that holomorphic and Killing vector fields on the base manifold lift to holomorphic and Killing symmetries on $\mathscr{M}^*_{HE}$. The paper further extends these symmetry results to geometries with dilaton and applies them to KT, CYT, HKT, and heterotic backgrounds, uncovering rich holomorphic/Killing symmetry algebras on the moduli spaces with potential implications for string compactifications and gauge theory moduli. Overall, it provides a unified framework linking dilaton geometry, KT-type structures, and moduli-space symmetries, offering tools for exploring gauge theories in non-Gauduchon settings and their string-theoretic applications.

Abstract

We demonstrate that the moduli space of Hermitian-Einstein connections $\text{M}^*_{HE}(M^{2n})$ of vector bundles over compact non-Gauduchon Hermitian manifolds $(M^{2n}, g, ω)$ that exhibit a dilaton field $Φ$ admit a strong Kähler with torsion structure provided a certain condition is imposed on their Lee form $θ$ and the dilaton. We find that the geometries that satisfy this condition include those that solve the string field equations or equivalently the gradient flow soliton type of equations. In addition, we demonstrate that if the underlying manifold $(M^{2n}, g, ω)$ admits a holomorphic and Killing vector field $X$ that leaves $Φ$ also invariant, then the moduli spaces $\text{M}^*_{HE}(M^{2n})$ admits an induced holomorphic and Killing vector field $α_X$. Furthermore, if $X$ is covariantly constant with respect to the compatible connection $\hat\nabla$ with torsion a 3-form on $(M^{2n}, g, ω)$, then $α_X$ is also covariantly constant with respect to the compatible connection $\hat D$ with torsion a 3-form on $\text{M}^*_{HE}(M^{2n})$ provided that $K^\flat\wedge X^\flat$ is a $(1,1)$-form with $K^\flat=θ+2dΦ$ and $Φ$ is invariant under $X$ and $IX$, where $I$ is the complex structure of $M^{2n}$.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Remark
• Theorem 2.2
• proof
• Remark
• proof
• Remark