 
Summary: ON THE RICCI TENSOR IN THE COMMON SECTOR OF TYPE II
STRING THEORY
I. AGRICOLA, T. FRIEDRICH, P.A. NAGY, AND C. PUHLE
Abstract. Let # be a metric connection with totally skewsymmetric torsion T on
a Riemannian manifold. Given a spinor field # and a dilaton function #, the basic
equations in the common sector of type II string theory are
## = 0 , #(T) = a · d# T , T · # = b · d# · # + µ · #
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 # 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 # is nontrivial
and if the conditions
(d# T) T = 0 , # # (dT) · # = 0
hold, then the energymomentum tensor is divergencefree. 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 energymomentum tensor vanishes
if and only if one condition # # (dT) · # = 0 holds. Strong models (dT = 0) have this
property, but there are examples with # # (dT) #= 0 and # # (dT) · # = 0.
