The Lie derivative as a canonical morphism

Abstract A classical Ed(d)-invariant Hamiltonian formulation of world-volume theories of half-BPS p-branes in type IIb and eleven-dimensional supergravity is proposed, extending known results to d ≤ 6. It consists of a Hamiltonian, characterised by a generalised metric, and a current algebra constructed s.t. it reproduces the Ed(d) generalised Lie derivative. Ed(d)-covariance necessitates the introduction of so-called charges, specifying the type of p-brane and the choice of section. For p > 2, currents of p-branes are generically non- geometric due to the imposition of U-duality, e.g. the M5-currents contain coordinates associated to the M2-momentum.A derivation of the Ed(d)-invariant current algebra from a canonical Poisson structure is in general not possible. At most, one can derive a current algebra associated to para-Hermitian exceptional geometry.The membrane in the SL(5)-theory is studied in detail. It is shown that in a generalised frame the current algebra is twisted by the generalised fluxes. As a consistency check, the double dimensional reduction from membranes in M-theory to strings in type IIa string theory is performed. Many features generalise to p-branes in SL(p + 3) generalised geometries that form building blocks for the Ed(d)-invariant currents.

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Gauged double field theory as an L∞ algebra

Journal of High Energy Physics ◽

10.1007/jhep06(2021)058 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Eric Lescano ◽

Martín Mayo

Keyword(s):

Field Theory ◽

Algebraic Structure ◽

Classical Field ◽

Field Theories ◽

Lie Derivative ◽

Linear Perturbation ◽

Double Field Theory ◽

Generalized Frame ◽

Symmetry Transformations ◽

Classical Field Theories

Abstract L∞ algebras describe the underlying algebraic structure of many consistent classical field theories. In this work we analyze the algebraic structure of Gauged Double Field Theory in the generalized flux formalism. The symmetry transformations consist of a generalized deformed Lie derivative and double Lorentz transformations. We obtain all the non-trivial products in a closed form considering a generalized Kerr-Schild ansatz for the generalized frame and we include a linear perturbation for the generalized dilaton. The off-shell structure can be cast in an L3 algebra and when one considers dynamics the former is exactly promoted to an L4 algebra. The present computations show the fully algebraic structure of the fundamental charged heterotic string and the $$ {L}_3^{\mathrm{gauge}} $$ L 3 gauge structure of (Bosonic) Enhanced Double Field Theory.

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Some Properties of Symplectic Manifolds S on the Projective Tangent Bundle PTM

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.187.483 ◽

2011 ◽

Vol 187 ◽

pp. 483-486

Author(s):

Yong He ◽

Xiao Ying Lu ◽

Wei Na Lu

Keyword(s):

Vector Field ◽

Symplectic Manifold ◽

Tangent Bundle ◽

Fiber Bundle ◽

Symplectic Manifolds ◽

Lie Derivative ◽

The Relationship

In this paper, we show the relationship between 2-form of the two projective tangent bundle and the relationship between 2-form on projective tangent bundle and 1-form on by using the theory of fiber bundle and the properties of symplectic manifold of the projective tangent bundle . Moreover, we derived a simpler formula of Lie derivative of a special vector field, which is on the projective tangent bundle.

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