Some remarks on strong ${\mathrm{G}}_2$-structures with torsion

Abstract

A ${\mathrm{G}}_2$-structure on a $7$-manifold $M$ is called a ${\mathrm{G}}_{2}T$-structure if $M$ admits a ${\mathrm{G}}_2$-connection ${\nabla }^T$ with totally skew-symmetric torsion $T_{\phi }$. If furthermore, $T_{\phi }$ is closed then it is called a strong ${\mathrm{G}}_{2}T$-structure. In this paper we investigate the geometry of (strong) ${\mathrm{G}}_{2}T$-manifolds in relation to its curvature, $S^1$ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of ${\nabla }^T$ and give an equivalent characterisation in terms of geometric properties of the ${\mathrm{G}}_2$ Lee form. Analogous results are also obtained for almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the $S^1$ reduction by the dual of the ${\mathrm{G}}_2$ Lee form, we show that Ricci-flat strong ${\mathrm{G}}_{2}T$-structures correspond to solutions of the $\mathrm{SU}(3)$ heterotic system on certain almost Hermitian half-flat $6$-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong ${\mathrm{G}}_{2}T$-structures with ${\nabla }^T$ not Ricci flat. Lastly, we classify ${\mathrm{G}}_2$-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.

39 pages

References in corpus (20)

• Superstrings with Intrinsic Torsion
• Vanishing Theorems and String Backgrounds
• Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds
• The Ricci tensor of SU(3)-manifolds
• Kaehler reduction of metrics with holonomy G_2
• Supersymmetric Domain Walls from Metrics of Special Holonomy
• Soliton solutions for the Laplacian coflow of some $G_2$-structures with symmetry
• SU(3)-instantons and $G_2,Spin(7)$-heterotic string solitons
• Short-time behaviour of a modified Laplacian coflow of G2-structures
• The infinitesimal moduli space of heterotic $G_2$ systems
• Half-flat structures on S^3xS^3
• Harmonic flow of geometric structures
• $SU(2{)}^2$-invariant $G_2$-instantons
• Estimates and monotonicity for a heat flow of isometric G2-structures
• Heterotic G_2-manifold compactifications with fluxes and fermionic condensates
• The heterotic ${\mathrm{G}}_2$ system on contact Calabi--Yau $7$-manifolds
• The twisted G${}_2$ equation for strong G${}_2$-structures with torsion
• CYT and SKT manifolds with parallel Bismut torsion
• Flows of $G_2$-structures, II: Curvature, torsion, symbols, and functionals
• Flows of SU(2)-structures

Cited by in corpus (1)

• ${\mathrm{G}}_2$-structures with parallel skew-symmetric torsion