SL(2,R) covariant conditions for N=1 flux vacua

TL;DR

This paper reformulates the supersymmetry conditions for N=1 flux vacua in type IIB supergravity to explicitly display SL(2,R) symmetry, simplifying the analysis of F-theory compactifications and classifying solutions into charged and chargeless types.

Contribution

It introduces an SL(2,R) covariant formulation of pure spinor equations, enabling a unified treatment of various supersymmetric flux vacua in F-theory.

Findings

Derived covariant supersymmetry conditions for chargeless solutions.

Classified solutions into charged and chargeless categories.

Simplified the geometric analysis of F-theory compactifications.

Abstract

Four-dimensional supersymmetric N = 1 vacua of type IIB supergravity are elegantly described by generalized complex geometry. However, this approach typically obscures the SL(2, R) covariance of the underlying theory. We show how to rewrite the pure spinor equations of Grana, Minasian, Petrini and Tomasiello (hep-th/0505212) in a manifestly SL(2,R) covariant fashion. Solutions to these equations fall into two classes: "charged" solutions, such as those containing D5-branes, and "chargeless" solutions, such as F-theory solutions in the Sen limit and AdS4 solutions. We derive covariant supersymmetry conditions for the chargeless case, allowing general SU(3)xSU(3) structure. The formalism presented here greatly simplifies the study of the ten-dimensional geometry of general supersymmetric compactifications of F-theory.