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Categorified symplectic geometry and the classical string. (English) Zbl 1192.81208
Summary: 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.

81S10 Geometry and quantization, symplectic methods
53D05 Symplectic manifolds (general theory)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
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