Categorified Symplectic Geometry and the Classical String

A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and t...

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Bibliographic Details
Published in:arXiv.org 2008-08
Main Authors: Baez, John C, Hoffnung, Alexander E, Rogers, Christopher L
Format: Article
Language:English
Subjects:
Algebra
Charged particles
Classical mechanics
Electromagnetic fields
Field theory
Isomorphism
Lie groups
Manifolds (mathematics)
Quantum theory
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Summary:A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the classical mechanics of point particles, the phase space is often a symplectic manifold, and the Poisson bracket of functions on this space gives a Lie algebra of observables. Multisymplectic geometry describes an n-dimensional field theory using a phase space that is an "n-plectic manifold": a finite-dimensional manifold equipped with a closed nondegenerate (n+1)-form. Here we consider the case n = 2. For any 2-plectic manifold, we construct a Lie 2-algebra of observables. We then explain how this Lie 2-algebra can be used to describe the dynamics of a classical bosonic string. Just as the presence of an electromagnetic field affects the symplectic structure for a charged point particle, the presence of a B field affects the 2-plectic structure for the string.
ISSN:2331-8422
DOI:10.48550/arxiv.0808.0246