I. Agricola | Searcharxiv

arXiv2005

Solvmanifolds with integrable and non-integrable G_2 structures

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G_2 can admit non-integrable G_2 structures of type R + S^2_0(R^7) + R^7 in the sense of Fernández and Gray. This relies on the construction of some G_2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family of metric connections with non-zero skew-symmetric torsion that has parallel spinors as well. The family turns out to be a deformation of the Levi-Civita connection. This is in contrast with the case of compact scalar-flat Riemannian spin manifolds, where any metric connection with closed torsion admitting parallel spinors has to be torsion-free.

math.DG↗

arXiv2004

On the Ricci tensor in type II B string theory

Let $\nabla$ be a metric connection with totally skew-symmetric torsion $\T$ on a Riemannian manifold. Given a spinor field $Ψ$ and a dilaton function $Φ$, the basic equations in type II B string theory are \bdm \nabla Ψ= 0, \quad δ(\T) = a \cdot \big(d Φ\haken \T \big), \quad \T \cdot Ψ= b \cdot d Φ\cdot Ψ+ μ\cdot Ψ. \edm We derive some relations between the length $||\T||^2$ of the torsion form, the scalar curvature of $\nabla$, the dilaton function $Φ$ and the parameters $a,b,μ$. The main results deal with the divergence of the Ricci tensor $\Ric^{\nabla}$ of the connection. In particular, if the supersymmetry $Ψ$ is non-trivial and if the conditions \bdm (d Φ\haken \T) \haken \T = 0, \quad δ^{\nabla}(d \T) \cdot Ψ= 0 \edm hold, then the energy-momentum tensor is divergence-free. We show that the latter condition is satisfied in many examples constructed out of special geometries. A special case is $a = b$. Then the divergence of the energy-momentum tensor vanishes if and only if one condition $δ^{\nabla}(d \T) \cdot Ψ= 0$ holds. Strong models ($d \T = 0$) have this property, but there are examples with $δ^{\nabla}(d \T) \neq 0$ and $δ^{\nabla}(d \T) \cdot Ψ= 0$.

hep-th↗