Weyl quantization of degree 2 symplectic graded manifolds

Grützmann, Melchior; Michel, Jean‐Philippe; Xu, Ping

doi:10.48550/arxiv.1410.3346

Cited by 5 publications

(12 citation statements)

References 29 publications

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“…These operators, which are spin connections in the Spin bundle, indeed arise naturally from the derived bracket construction. The quantization issue of the underlying 2-graded symplectic manifolds was addressed in [26].…”

Section: Comparison With Other Definitions Of Torsionmentioning

confidence: 99%

Deformed graded Poisson structures, generalized geometry and supergravity

Boffo

Schupp

2020

J. High Energ. Phys.

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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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Section: Comparison With Other Definitions Of Torsionmentioning

confidence: 99%

Deformed graded Poisson structures, generalized geometry and supergravity

Boffo

Schupp

2020

J. High Energ. Phys.

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“…(2. 31) This means that the r.h.s. of (2.28) also vanishes and is another relation between the structure functions.…”

Section: Jacobi Identity On Tmmentioning

confidence: 99%

“…The Clifford algebra can be considered as a quantization of a graded symplectic manifold or, equivalently, a QP-manifold [31]. In [24], we have shown that certain algebraic relations in DFT can be formulated on a pre-QP-manifold.…”

Section: Derived Bracketmentioning

confidence: 99%

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Metric algebroid and Dirac generating operator in Double Field Theory

Carow-Watamura,

Miura,

Watamura

et al. 2020

Preprint

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We give a formulation of Double Field Theory (DFT) based on a metric algebroid. We derive a covariant completion of the Bianchi identities, i.e. the pre-Bianchi identity in torsion and an improved generalized curvature, and the pre-Bianchi identity including the dilaton contribution.The derived bracket formulation by the Dirac generating operator is applied to the metric algebroid. We propose a generalized Lichnerowicz formula and show that it is equivalent to the pre-Bianchi identities. The dilaton in this setting is included as an ambiguity in the divergence.The projected generalized Lichnerowicz formula gives a new formulation of the DFT action. The closure of the generalized Lie derivative on the spin bundle yields the Bianchi identities as a consistency condition. A relation to the generalized supergravity equations (GSE) is discussed.

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“…Our reference manifold is the (degree-)shifted cotangent space T * [2]T [1]M of the shifted tangent space to a manifold M . This is the minimal symplectic realization of a given graded manifold T [1]M ; for details on how to construct the bundle and relations to other graded manifolds we refer to [4]. We couple the particle to g − B, where g is a Riemannian metric on the base manifold and B a 2-form, and to (g + B)…”

Section: Introductionmentioning

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

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Weyl quantization of degree 2 symplectic graded manifolds

Cited by 5 publications

References 29 publications

Deformed graded Poisson structures, generalized geometry and supergravity

Deformed graded Poisson structures, generalized geometry and supergravity

Metric algebroid and Dirac generating operator in Double Field Theory

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

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