Categorified Symplectic Geometry and the Classical String

• Published in: Communications in mathematical physics 2010-02, Vol.293 (3), p.701-725
• Main Authors: Baez, John C., Hoffnung, Alexander E., Rogers, Christopher L.
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-009-0951-9

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.