Abstract
We derive the conditions for unbroken supersymmetry for a Mink2, (2, 0) vacuum, arising from Type II supergravity on a compact eight-dimensional manifold ℳ8. When specialized to internal manifolds enjoying SU(4) × SU(4) structure the resulting system is elegantly rewritten in terms of generalized complex geometry. This particular class of vacua violates the correspondence between supersymmetry conditions and calibrations conditions of D branes (supersymmetry-calibrations correspondence). Our analysis includes and extends previous results about the failure of the supersymmetry-calibrations correspondence, and confirms the existence of a precise relation between such a failure and a subset of the supersymmetry conditions.
Highlights
Generalized complex geometry [1, 2] proved to be a powerful tool in order to study the more complicated story of vacuum solutions in presence of fluxes
In that paper the authors found that the conditions for unbroken supersymmetry for four-dimensional N = 1 vacua are elegantly rewritten in terms of firstorder differential equations involving a pair of pure spinors on the generalized internal tangent bundle T6 ⊕ T6∗; this fancy formulation allowed to find a large number of explicit vacuum solutions
In this paper we have obtained the conditions for unbroken supersymmetry for a Mink2, N = (2, 0) vacuum in terms of generalized complex geometry
Summary
We will discuss how the ten-dimensional SUSY parameters 1 and 2 decompose in order to have an N = (2, 0), Mink vacuum, namely a configuration of the form Mink2 × M8 (with M8 compact) enjoying the maximal symmetry of Mink and where two real supercharges are preserved. As usual for vacuum solutions the manifold is given by a simple product M10 = Mink2 × M8, and the external part of the metric depends on the internal coordinates via the so-called warping factor A(y) only. We are interested in N = (2, 0) configurations, i.e. configurations like (2.1) preserving supersymmetry for any given two-dimensional, complex, Weyl spinor ζ. It is worth noting that a spinorial decomposition like (2.2) is not compatible with an AdS2 vacuum: in this case the Killing spinor equation (2.4) becomes. Where ζ+ (ζ−) is a spinor of positive (negative) chirality, and μ is a constant proportional to the cosmological constant; it can be shown that ζ and ζc have the same chiralities and so we conclude that the spinorial Ansatz (2.2) is not compatible with (2.5)
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