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Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations. (English) Zbl 1390.83426
Summary: 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 \(\phi\) (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 [G. Arutyunov et al., Nucl. Phys., B 903, 262–303 (2016; Zbl 1332.81167)]. These equations depend on two vectors \(\mathrm{X}_a\) and \(K_a\) subject to 1st order differential relations (with the equations in the NS-NS sector depending only on the combination \(X_a=\mathrm{X}_a+K_a\)). In the special case of \(K_a=0\) one finds that \(\mathrm{X}_a=\partial_a\phi\) 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 \(\eta\)-deformations of \(\mathrm{AdS}_n\times S^n\) 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.

83E50 Supergravity
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