On the Ricci tensor in the common sector of type II string theory
Abstract
Let ∇ 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 the common sector of type II string theory are \fl
\nabla \Psi = 0 , \qquad \delta({\ensuremath{T}}) = a \cdot (d \Phi {\mathbin{\hbox to 6pt {\vrule height0.4pt width5pt depth0pt \kern-.4pt \vrule height6pt width0.4pt depth0pt\hss}}} {T}), \qquad {T} \cdot \Psi = b \cdot d \Phi \cdot \Psi + \mu \cdot \Psi for some auxiliary parameters a, b, μ. We derive some relations between the length ||T||2 of the torsion form, the scalar curvature of ∇, the dilaton function Φ and the parameters a, b, μ. We show that for constant dilaton and μ = 0 (the physically relevant case), there cannot be even local solutions to this system of equations with vanishing scalar curvature. The main results deal with the divergence of the Ricci tensor Ric∇ of the connection. In particular, if the supersymmetry Ψ is non-trivial and if the conditions (d \Phi {\mathbin{\hbox to 6pt {\vrule height0.4pt width5pt depth0pt \kern-.4pt \vrule height6pt width0.4pt depth0pt\hss}}} {\ensuremath{T}}) {\mathbin{\hbox to 6pt {\vrule height0.4pt width5pt depth0pt \kern-.4pt \vrule height6pt width0.4pt depth0pt\hss}}} {T} = 0, \qquad \delta^{\nabla}(d {\ensuremath{T}}) \cdot \Psi = 0 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∇(dT) sdot Ψ = 0 holds. Strong models (dT = 0) have this property, but there are examples with δ∇(dT) ≠ 0 and δ∇(dT) sdot Ψ = 0. Supported by the SFB 647 'Raum, Zeit, Materie' and the SPP 1154 'Globale Differentialgeometrie' of the DFG as well as the Volkswagen Foundation.- Publication:
-
Classical and Quantum Gravity
- Pub Date:
- July 2005
- DOI:
- 10.1088/0264-9381/22/13/003
- arXiv:
- arXiv:hep-th/0412127
- Bibcode:
- 2005CQGra..22.2569A
- Keywords:
-
- High Energy Physics - Theory;
- Differential Geometry
- E-Print:
- 9 pages, Latex2e