A gravitational action with stringy $Q$ and $R$ fluxes via deformed differential graded Poisson algebras

TL;DR

This paper develops a novel deformation of a graded Poisson algebra incorporating stringy fluxes, linking it to generalized geometry and constructing an action functional for a gravity theory with fluxes.

Contribution

It introduces a new deformation framework for graded Poisson algebras using string theory fluxes, connecting it to generalized geometry and formulating a related gravitational action.

Findings

Deformation of Poisson algebra using B-field and bivector f4 fields.

Connection to generalized geometry via Courant algebroid and higher gauge theory.

Construction of an action functional encoding flux dynamics in gravity.

Abstract

We study a deformation of a $2$-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a $2$-form $B$-field and a bivector $\Pi$, that we consider as gauge fields of the geometric and non-geometric fluxes $H$, $f$, $Q$ and $R$ arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a $U(1)$ gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: With an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle $E \cong TM \oplus T^{*}M$, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and $Q$ and $R$ fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and $H$-flux.