Doubled Aspects of Vaisman Algebroid and Gauge Symmetry in Double Field Theory

Abstract

The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'd double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a para-Hermitian manifold play Dirac-like structures in the Vaisman algebroid. We find that an algebraic origin of the strong constraint in DFT is traced back to the compatibility condition needed for $(L,\tilde{L})$ be a Lie bialgebroid. The analysis provides a foundation toward the "coquecigrue problem" for the gauge symmetry in DFT.

arXiv comment49 pages, version published in JMP

References in corpus (37)

• Double Field Theory
• Generalized metric formulation of double field theory
• Superspace Duality in Low-Energy Superstrings
• Two-Vierbein Formalism for String-Inspired Axionic Gravity
• The gauge algebra of double field theory and Courant brackets
• Courant algebroids, derived brackets and even symplectic supermanifolds
• Derived brackets
• Gauged Double Field Theory
• Large Gauge Transformations in Double Field Theory
• Comments on double field theory and diffeomorphisms
• Doubled aspects of generalised dualities and integrable deformations
• Branes are Waves and Monopoles
• Global aspects of double geometry
• On the geometry of double field theory
• Finite Gauge Transformations and Geometry in Double Field Theory
• Towards a double field theory on para-Hermitian manifolds
• Worldsheet Instanton Corrections to 522-brane Geometry
• Strings and Branes are Waves
• Gauged Linear Sigma Model for Exotic Five-brane
• Double Field Theory and Membrane Sigma-Models
• Generalised fluxes, Yang-Baxter deformations and the O(d,d) structure of non-abelian T-duality
• Non-geometric branes are DFT monopoles
• Towards an invariant geometry of double field theory
• A Unique Connection for Born Geometry
• The geometry behind double geometry
• Even symplectic supermanifolds and double field theory
• Generalised Kinematics for Double Field Theory
• Algebroid Structures on Para-Hermitian Manifolds
• Para-Hermitian Geometry, Dualities and Generalized Flux Backgrounds
• Extended Riemannian Geometry I: Local Double Field Theory
• Integration of Exact Courant Algebroids
• Worldsheet Instanton Corrections to Five-branes and Waves in Double Field Theory
• A note on large gauge transformations in double field theory
• Doubling, T-Duality and Generalized Geometry: a Simple Model
• Finite Transformations in Doubled and Exceptional Space
• DFT in supermanifold formulation and group manifold as background geometry
• Semi-doubled Gauged Linear Sigma Model for Five-branes of Codimension Two

Cited by in corpus (10)

• The Geometry, Branes and Applications of Exceptional Field Theory
• U-duality extension of Drinfel'd double
• Born sigma model for branes in exceptional geometry
• Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality
• Global Aspects of Doubled Geometry and Pre-rackoid
• More on Doubled Aspects of Algebroids in Double Field Theory
• Complex Structures, T-duality and Worldsheet Instantons in Born Sigma Models
• A note on Faddeev--Popov action for doubled-yet-gauged particle and graded Poisson geometry
• World-volume Effective Theories of Locally Non-geometric Branes
• Extended Doubled Structures of Algebroids for Gauged Double Field Theory