Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds

TL;DR

This paper formulates a class of four-dimensional gravity theories with scalar and gauge fields using advanced geometric structures, extending supergravity models to include globally non-trivial and potentially non-geometric configurations.

Contribution

It introduces a global geometric framework for gravity coupled to scalars and gauge fields, incorporating symplectic duality bundles and twisted self-duality, extending supergravity theories.

Findings

Global formulation of gravity with scalar and gauge fields using fiber bundles.

Identification of conditions under which theories reduce to Einstein-Scalar-Maxwell models.

Description of solutions as classical U-folds, including non-geometric cases.

Abstract

We give the global mathematical formulation of a class of generalized four-dimensional theories of gravity coupled to scalar matter and to Abelian gauge fields. In such theories, the scalar fields are described by a section of a surjective pseudo-Riemannian submersion $\pi$ over space-time, whose total space carries a Lorentzian metric making the fibers into totally-geodesic connected Riemannian submanifolds. In particular, $\pi$ is a fiber bundle endowed with a complete Ehresmann connection whose transport acts through isometries between the fibers. In turn, the Abelian gauge fields are "twisted" by a flat symplectic vector bundle defined over the total space of $\pi$. This vector bundle is endowed with a vertical taming which locally encodes the gauge couplings and theta angles of the theory and gives rise to the notion of twisted self-duality, of crucial importance to construct the theory. When the Ehresmann connection of $\pi$ is integrable, we show that our theories are locally equivalent to ordinary Einstein-Scalar-Maxwell theories and hence provide a global non-trivial extension of the universal bosonic sector of four-dimensional supergravity. In this case, we show using a special trivializing atlas of $\pi$ that global solutions of such models can be interpreted as classical "locally-geometric" U-folds. In the non-integrable case, our theories differ locally from ordinary Einstein-Scalar-Maxwell theories and may provide a geometric description of classical U-folds which are "locally non-geometric".