Abstract
We rewrite the equations for ten-dimensional supersymmetry in a way formally identical to a necessary and sufficient G-structure system in N=2 gauged supergravity, where all four-dimensional quantities are replaced by combinations of pure spinors and fluxes in the internal space. This provides a way to look for lifts of BPS solutions without having to reduce or even rewrite the ten-dimensional action. In particular this avoids the problem of consistent truncation, and the introduction of unphysical gravitino multiplets.
Highlights
In the context of string theory compactifications, four-dimensional effective actions are often a useful way to capture and organize the relevant physics
We rewrite the equations for ten-dimensional supersymmetry in a way formally identical to a necessary and sufficient G-structure system in N = 2 gauged supergravity, where all four-dimensional quantities are replaced by combinations of pure spinors and fluxes in the internal space
The pure spinor equations for vacua are promoted to first-order equations that dictate the evolution of the internal geometry with the point in spacetime, and we show how they correspond to the equations in four dimensions dictating the evolution of the scalars of the theory. (In black hole applications, these equations would become the attractor equations mentioned earlier.) Most of our equations can be matched with four-dimensional ones in a way consistent with [5, 22]
Summary
In the context of string theory compactifications, four-dimensional effective actions are often a useful way to capture and organize the relevant physics. Sometimes the reduced action misses important subtleties: for example, the truncation of modes that defines it is often “nonconsistent”, in that it misses some equations of motion of the ten-dimensional action This means that lifting a solution to ten dimensions is not guaranteed to work. The pure spinors are differential forms φ± on M6 that define an SU(3) × SU(3) structure on the “generalized tangent bundle” T ⊕ T ∗; the supersymmetry equations constrain them so that, for the Minkowski case, M6 needs to be a generalized complex manifold [17]. Each of these sections has a concluding subsection summarizing its main results.
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