Star products on graded manifolds and ${\alpha }^{\prime }$-corrections to Courant algebroids from string theory

Abstract

Deformation theory refers to an apparatus in many parts of math and physics for going from an infinitesimal (= first order) deformation to a full deformation, either formal or convergent appropriately. If the algebra being deformed is that of observables, the result is deformation quantization, independent of any realization in terms of Hilbert space operators. There are very important but rare cases in which a formula for a full deformation is known. For physics, the most important is the Moyal-Weyl star product formula. In this paper, we concentrate on deformations of Courant algebroid structures via star products on graded manifolds. In particular, we construct a graded version of the Moyal-Weyl star product. Recently, in Double Field Theory (DFT), deformations of the C-bracket and O(d,d)-invariant bilinear form to first order in the closed string sigma model coupling ${\alpha }^{\prime }$ were derived by analyzing the transformation properties of the Neveu-Schwarz B-field. By choosing a particular Poisson structure on the Drinfel'd double corresponding to the Courant algebroid structure of the generalized tangent bundle, we reproduce these deformations for a specific solution of the strong constraint of DFT as expansion of a graded version of the Moyal-Weyl star product.

28 pages, typos corrected, citations added

References in corpus (12)

• Double Field Theory
• Generalised Geometry for M-Theory
• The gauge algebra of double field theory and Courant brackets
• Generalized complex geometry
• T-duality and closed string non-commutative (doubled) geometry
• Double Field Theory at Order ${\alpha }^{\prime }$
• Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics
• Green-Schwarz mechanism and ${\alpha }^{\prime }$-deformed Courant brackets
• Non-associative Deformations of Geometry in Double Field Theory
• Even symplectic supermanifolds and double field theory
• A Note on the CFT Origin of the Strong Constraint of DFT
• Weyl quantization of degree 2 symplectic graded manifolds

Cited by in corpus (9)

• Doubled aspects of generalised dualities and integrable deformations
• Generalized Metric Formulation of Double Field Theory on Group Manifolds
• Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds
• Towards a Theory of Nonassociative Gravity
• Extended Riemannian Geometry II: Local Heterotic Double Field Theory
• The ${\alpha }^{\prime 2}$ correction from double field theory
• Courant-Dorfman algebras of differential operators and Dorfman connections of Courant algebroids
• Pre-NQ Manifolds and Correspondence Spaces: the Nilmanifold Example
• The standard cohomology of regular Courant algebroids