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 type II B string theory areWe 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. 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 (dΦ T) T = 0 , δ ∇ (dT) • Ψ = 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) • Ψ = 0 holds. Strong models (dT = 0) have this property, but there are examples with δ ∇ (dT) = 0 and δ ∇ (dT) • Ψ = 0.

Killing and twistor spinors with torsion

“…10.1]. In particular, in dimension 6 there exists a positive constant τ 0 = 0 such that (see [31,7,11])…”

“…Notice that the existence of such spinors on nearly parallel G 2 -manifolds has been conjectured in [3], and this was part of our motivation. We finally remark that these weak holonomy structures are both solutions of the equations for the common sector of type II superstring theory: ∇ c ϕ = 0, T · ϕ = γ · ϕ, δ(T ) = 0 and ∇ c Ric c = 0 (see [31,1,11,2]). The supersymmetries of the model are interpreted by the ∇ c -parallel spinors.…”

Spin (7)-Manifolds with Parallel Torsion Form

“…The case iso (T c ) = R ⊕ su (2). Here the characteristic torsion form T c is an element of the 1-parameter family…”

“…Furthermore this connection, denoted by ∇ c , is unique, preserves the underlying Spin (7)-structure and makes a nontrivial spinor field parallel. The more general point of view of [2,12,18] indicates that structures with parallel torsion form T c ,…”

The Ricci tensor of SU(3)-manifolds

“…[2], [3], [4], [5], [10], [11], [12] , [13], [14], [20] and the references therein). New directions in this field were suggested by the work of Hitchin [22].…”