Non-geometric strings, symplectic gravity and differential geometry of Lie algebroids

TL;DR

This paper develops a differential geometry framework based on Lie algebroids for non-geometric fluxes in string theory, leading to a symplectic gravity action that parallels standard gravity and connects to string corrections.

Contribution

It introduces a novel differential geometry calculus combining diffeomorphisms and eta-diffeomorphisms, resulting in a bi-invariant Einstein-Hilbert type action for symplectic structures in string theory.

Findings

Constructed a bi-invariant symplectic gravity action similar to Einstein-Hilbert action.

Derived equations of motion that resemble standard gravity equations.

Discussed solutions including Calabi-Yau geometries.

Abstract

Based on the structure of a Lie algebroid for non-geometric fluxes in string theory, a differential-geometry calculus is developed which combines usual diffeomorphisms with so-called \beta-diffeomorphisms emanating from gauge symmetries of the Kalb-Ramond field. This allows to construct a bi-invariant action of Einstein-Hilbert type comprising a metric, a (quasi-)symplectic structure \beta and a dilaton. As a salient feature, this symplectic gravity action and the resulting equations of motion take a form which is similar to the standard action and field equations. Furthermore, the two actions turn out to be related via a field redefinition reminiscent of the Seiberg-Witten limit. Remarkably, this redefinition admits a direct generalization to higher-order \alpha'-corrections and to the additional fields and couplings appearing in the effective action of the superstring. Simple solutions to the equations of motion of the symplectic gravity action, including Calabi-Yau geometries, are discussed.