On the Ricci tensor in type II B string theory

TL;DR

This paper investigates the Ricci tensor in type II B string theory with skew-symmetric torsion, deriving conditions under which the energy-momentum tensor is divergence-free, especially in special geometric configurations and for particular torsion properties.

Contribution

It establishes relations between torsion, scalar curvature, and dilaton in type II B string theory, and identifies conditions ensuring divergence-free energy-momentum tensor in various geometric settings.

Findings

Divergence of Ricci tensor related to torsion and dilaton.

Energy-momentum tensor divergence-free under specific geometric conditions.

Examples of geometries satisfying the divergence-free conditions.

Abstract

Let $\nabla$ be a metric connection with totally skew-symmetric torsion $\T$ on a Riemannian manifold. Given a spinor field $\Psi$ and a dilaton function $\Phi$, the basic equations in type II B string theory are \bdm \nabla \Psi = 0, \quad \delta(\T) = a \cdot \big(d \Phi \haken \T \big), \quad \T \cdot \Psi = b \cdot d \Phi \cdot \Psi + \mu \cdot \Psi . \edm We derive some relations between the length $||\T||^2$ of the torsion form, the scalar curvature of $\nabla$, the dilaton function $\Phi$ and the parameters $a,b,\mu$. The main results deal with the divergence of the Ricci tensor $\Ric^{\nabla}$ of the connection. In particular, if the supersymmetry $\Psi$ is non-trivial and if the conditions \bdm (d \Phi \haken \T) \haken \T = 0, \quad \delta^{\nabla}(d \T) \cdot \Psi = 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 $\delta^{\nabla}(d \T) \cdot \Psi = 0$ holds. Strong models ($d \T = 0$) have this property, but there are examples with $\delta^{\nabla}(d \T) \neq 0$ and $\delta^{\nabla}(d \T) \cdot \Psi = 0$.