Generalized geometry and non-symmetric gravity

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the 2-graded symplectic manifold T * [2]T [1]M that are based on a metric g, a closed Neveu-Schwarz 3-form H (locally expressed in terms of a Kalb-Ramond 2-form B) and a scalar dilaton φ. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symmetries, and yields a connection with non-trivial curvature and torsion on the generalized "doubled" tangent bundle E ∼ = T M ⊕ T * M . Projecting onto T M with the help of a natural non-isotropic splitting of E, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10 dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles. A byproduct is a unique Koszul-type formula for the torsionful connection naturally associated to a non-symmetric metric, which resolves ambiguity problems and inconsistencies of traditional approaches to non-symmetric gravity theories.

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“…The case of T M , analyzed extensively in many other works e.g. [33], is presented just for reference and is regarded as dual to the case of T * M , which is the relevant one for the description of a gravitational field paired to the local expressions of the stringy non-geometric fluxes. As subspaces of E, T M and T * M are isotropic with respect to the natural pairing η of vector fields with forms but not with respect to the metric G ≡ •, • ′ .…”

Section: Connections On Tangent Space and On Cotangent Space For The ...mentioning

confidence: 99%

“…The Ricci tensor of that connection, integrated against g −1 • (g − B) • g −1 and by means of Stokes' theorem in the integration, is known to reproduce the effective action functional of the low-energy closed strings, i.e. the Ricci scalar and the square of the field strength H [33], [31],…”

Section: Gravitational Actions With Fluxesmentioning

confidence: 99%

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

Boffo,

Schupp

2021

Preprint

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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 Π, 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 [1], 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 ∼ = T M ⊕ 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.deformation 6 2.3 Gauge symmetries 9 2.4 Deformation and open-closed string relation 10 3 Courant algebroids and graded Poisson structures 11 3.1 Correspondence between 2-graded QP-manifolds and exact Courant algebroids 12 3.2 Courant algebroid for a non-canonical QP-manifold 13 4 Differential geometry of T M ⊕ T * M 14 4.1 Generalized Lie bracket, connection and torsion tensor 14 4.2 Connections on tangent space and on cotangent space for the given deformation 17 4.3 Curvature tensor 19 5 Gravitational actions with fluxes 20 5.1 Invariance under gauge symmetries of Π 22

show abstract

“…The case of T M , analyzed extensively in many other works e.g. [39], is presented just for reference and is regarded as dual to the case of T * M , which is the relevant one for the description of a gravitational field paired to the local expressions of the stringy non-geometric fluxes. As subbundles of E, T M and T * M are isotropic with respect to the natural pairing η of vector fields with forms but not with respect to the metric G ≡ •, • .…”

Section: Connections On Tangent Space and On Cotangent Space For The Given Deformationmentioning

confidence: 99%

“…The Ricci tensor of that connection, integrated against g −1 • (g − B) • g −1 , and by means of Stokes' theorem in the integration, is known to reproduce the effective action functional of the low-energy closed strings, i.e. the Ricci scalar and the square of the field strength H [2,39],…”

Section: Gravitational Actions With Fluxesmentioning

confidence: 99%

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

Boffo

¹

,

Schupp

²

2021

J. High Energ. Phys.

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1

0

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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 Π, 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 ≅ TM ⊕ 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.

show abstract

Generalized geometry and non-symmetric gravity

Cited by 3 publications

References 12 publications

Deformed graded Poisson structures, generalized geometry and supergravity

Deformed graded Poisson structures, generalized geometry and supergravity

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

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

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