Gaugino mass term for D-branes and Generalized Complex Geometry

Mariana Graña,Ander Retolaza,Nicolás Kovensky

doi:10.1007/jhep06(2020)047

Abstract

We compute the four-dimensional gaugino mass for a Dp-brane extended in spacetime and wrapping a cycle on the internal geometry in a warped compactification with fluxes. Motivated by the backreaction of gaugino bilinear VEVs, we use Generalized Complex Geometry to characterize the internal geometry as well as the cycle wrapped by the brane. We find that the RR fluxes and the non-closure of the generalized complex structures combine in the gaugino mass terms in the same form as they do in the bulk superpotential, while for the NSNS fluxes there is a crucial minus sign in the component normal to the brane. Our expression extends the known result for D3 and D7-branes in Calabi-Yau manifolds, where the gaugino masses are induced respectively by the imaginary anti-self dual and imaginary self-dual components of the complex 3-form flux G3.

Highlights

Motivated by the backreaction of gaugino bilinear VEVs, we use Generalized Complex Geometry to characterize the internal geometry as well as the cycle wrapped by the brane
We find that the RR fluxes and the non-closure of the generalized complex structures combine in the gaugino mass terms in the same form as they do in the bulk superpotential, while for the NSNS fluxes there is a crucial minus sign in the component normal to the brane
Same as fluxes, gaugino bilinear VEVs back-react on the geometry: the internal manifold cannot be Calabi-Yau, since it does not support AdS4 vacua

Summary

Compactifications and GCG

We start with type II superstring theory on a warped product of an extended and maximally symmetric four-dimensional manifold (Mink, AdS4 or dS4) and a compact internal six-. In type II string theory, the most generic situation is to consider two globally defined spinors η1 and η2, which can become parallel at certain loci of the manifold. By using these spinors, one can build two polyforms or pure spinors which characterise the background geometry, and are defined as i Ψ+ = − ||η||2. Since the internal manifold has two globally defined spinors, it is natural to use them in the decomposition of the supersymmetry parameter into an external and an internal spinor, namely.

D-branes in GCG

Fermionic D-brane action

Dimensional reduction: general idea

Dimensional reduction: details of calculations

Dimensional reduction: summary of results and analysis

Spinor conventions

Other conventions and definitions

Useful identities