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 $δ^{\nabla}(d \T) \cdot Ψ= 0$.