Categorified Symplectic Geometry and the Classical String
Abstract
A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures analogous to 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.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- February 2010
- DOI:
- 10.1007/s00220-009-0951-9
- arXiv:
- arXiv:0808.0246
- Bibcode:
- 2010CMaPh.293..701B
- Keywords:
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- Mathematical Physics;
- 70S05;
- 81T30;
- 53Z05 (Primary) 53D05 (Secondary)
- E-Print:
- 28 pages