Abstract
The generalized Fradkin-Tseytlin counterterm for the (type I) Green-Schwarz superstring is determined for background fields satisfying the generalized supergravity equations (GSE). For this purpose, we revisit the derivation of the GSE based upon the requirement of kappa-symmetry of the superstring action. Lifting the constraint of vanishing bosonic torsion components, we are able to make contact to several different torsion constraints used in the literature. It is argued that a natural geometric interpretation of the GSE vector field that generalizes the dilaton is as the torsion vector, which can combine with the dilatino spinor into the torsion supervector. To find the counterterm, we use old results for the one-loop effective action of the heterotic sigma model. The counterterm is covariant and involves the worldsheet torsion for vanishing curvature, but cannot be constructed as a local functional in terms of the worldsheet metric. It is shown that the Weyl anomaly cancels without imposing any further constraints on the background fields. In the case of ordinary supergravity, it reduces to the Fradkin-Tseytlin counterterm modulo an additional constraint.
Highlights
Xa which, in the special case of supergravity, are given by χα = ∇αΦ and Xa = ∇aΦ, respectively
For supergravity backgrounds with non-trivial fermionic components, the situation is a bit more subtle, because of issues connected to the quantization of the GS superstring [30,31,32,33,34,35,36]. For example, it has been shown [37] that the Fradkin-Tseytlin term cancels the Weyl anomaly under the assumption of a constraint on the fermionic fields, which was argued to be necessary, because an analogous constraint was used to gauge-fix the fermions in the semi-light-cone gauge calculation of the one-loop effective action
We have revisited the recent derivation of the generalized supergravity equations (GSE) based upon the requirement of invariance of the GS sigma model under kappa-symmetry transformations
Summary
We shall obtain the generalized supergravity equations. We closely follow the calculation by Tseytlin and Wulff [12] and adopt their notation. We shall refer to (2.3) and (2.4) as the torsion Bianchi identity (TBI) and curvature Bianchi identity (RBI), respectively. It is a classical result [46] that all curvature components are determined by the TBI in terms of the torsion and its covariant derivatives, because the curvature is a structure-group valued two-form, Rαa = 0 = Raα , Rαβ. To remove spurious fermionic degrees of freedom, the superstring action must be invariant under kappa-symmetry transformations [48]. This constrains the background fields of dimension. In order to obtain the remaining components, one must solve the superspace Bianchi identities, which we will do
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