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

We demonstrate that the moduli space of Hermitian-Einstein connections ${\text{M}}_{HE}^{\ast }(M^{2n})$ of vector bundles over compact non-Gauduchon Hermitian manifolds $(M^{2n},g,\omega )$ that exhibit a dilaton field $\mathrm{\Phi }$ admit a strong Kähler with torsion structure provided a certain condition is imposed on their Lee form $\theta$ 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 $(M^{2n},g,\omega )$ admits a holomorphic and Killing vector field $X$ that leaves $\mathrm{\Phi }$ also invariant, then the moduli spaces ${\text{M}}_{HE}^{\ast }(M^{2n})$ admits an induced holomorphic and Killing vector field ${\alpha }_X$. Furthermore, if $X$ is covariantly constant with respect to the compatible connection $\widehat{\mathrm{\nabla }}$ with torsion a 3-form on $(M^{2n},g,\omega )$, then ${\alpha }_X$ is also covariantly constant with respect to the compatible connection $\widehat{D}$ with torsion a 3-form on ${\text{M}}_{HE}^{\ast }(M^{2n})$ provided that $K^{\mathrm{♭}}\wedge X^{\mathrm{♭}}$ is a $(1,1)$-form with $K^{\mathrm{♭}}=\theta +2d\mathrm{\Phi }$ and $\mathrm{\Phi }$ is invariant under both $X$ and $IX$, where $I$ is the complex structure of $M^{2n}$.