On the Ricci tensor in type II B string theory

Let $\mathrm{\nabla }$ be a metric connection with totally skew-symmetric torsion $\text{T}$ on a Riemannian manifold. Given a spinor field $\mathrm{\Psi }$ and a dilaton function $\mathrm{\Phi }$, 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 $\mathrm{\mid }\mathrm{\mid }\text{T}\mathrm{\mid }{\mathrm{\mid }}^2$ of the torsion form, the scalar curvature of $\mathrm{\nabla }$, the dilaton function $\mathrm{\Phi }$ and the parameters $a,b,\mu$. The main results deal with the divergence of the Ricci tensor ${\text{Ric}}^{\mathrm{\nabla }}$ of the connection. In particular, if the supersymmetry $\mathrm{\Psi }$ 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 ${\delta }^{\mathrm{\nabla }}(d\text{T})\cdot \mathrm{\Psi }=0$ holds. Strong models ($d \T = 0$)havethisproperty,butthereareexampleswith$\delta^{\nabla}(d \T) \neq 0$and$\delta^{\nabla}(d \T) \cdot \Psi = 0$.