Flux Vacua and Supermanifolds

TL;DR

This paper explores the construction of supermanifolds that incorporate RR fluxes in string theory, using supergeometry and super-Ricci flatness to extend bosonic manifolds with fermionic coordinates.

Contribution

It introduces a framework for constructing supermanifolds with RR fluxes, extending bosonic geometries with fermionic coordinates guided by super-Ricci flatness and superstring requirements.

Findings

Certain spaces admit multiple super-extensions.

Parameterization of supermanifolds with specific bosonic and fermionic coordinates.

Examples based on supergroup and coset supermanifolds.

Abstract

As been recently pointed out, physically relevant models derived from string theory require the presence of non-vanishing form fluxes besides the usual geometrical constraints. In the case of NS-NS fluxes, the Generalized Complex Geometry encodes these informations in a beautiful geometrical structure. On the other hand, the R-R fluxes call for supergeometry as the underlying mathematical framework. In this context, we analyze the possibility of constructing interesting supermanifolds recasting the geometrical data and RR fluxes. To characterize these supermanifolds we have been guided by the fact topological strings on supermanifolds require the super-Ricci flatness of the target space. This can be achieved by adding to a given bosonic manifold enough anticommuting coordinates and new constraints on the bosonic sub-manifold. We study these constraints at the linear and non-linear level for a pure geometrical setting and in the presence of p-form field strengths. We find that certain spaces admit several super-extensions and we give a parameterization in a simple case of d bosonic coordinates and two fermionic coordinates. In addition, we comment on the role of the RR field in the construction of the super-metric. We give several examples based on supergroup manifolds and coset supermanifolds.