Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations

We determine the constraints imposed on the 10d target superspace geometry by the requirement of classical kappa-symmetry of the green-schwarz superstring. in the type i case we find that the background must satisfy a generalization of type i supergravity equations. these equations depend on an arbitrary vector x a and imply the one-loop scale invariance of the gs sigma model. in the special case when x a is the gradient of a scalar ϕ (dilaton) one recovers the standard type i equations equivalent to the 2d weyl invariance conditions of the superstring sigma model. in the type ii case we find a generalized version of the 10d supergravity equations the bosonic part of which was introduced in arxiv:1511.05795 . these equations depend on two vectors x a and k a subject to 1st order differential relations (with the equations in the ns-ns sector depending only on the combination x a = x a + k a ). in the special case of k a = 0 one finds that x a = ∂ a ϕ and thus obtains the standard type ii supergravity equations. new generalized solutions are found if k a is chosen to be a killing vector (and thus they exist only if the metric admits an isometry). non-trivial solutions of the generalized equations describe k-isometric backgrounds that can be mapped by t-duality to type ii supergravity solutions with dilaton containing a linear isometry-breaking term. examples of such backgrounds appeared recently in the context of integrable η-deformations of ads n × s n sigma models. the classical kappa-symmetry thus does not, in general, imply the 2d weyl invariance conditions for the gs sigma model (equivalent to type ii supergravity equations) but only weaker scale invariance type conditions.