Categorified Symplectic Geometry and the Classical String

TL;DR

This paper develops a Lie 2-algebra framework for 2-plectic manifolds to model classical string dynamics, extending symplectic geometry concepts from point particles to strings in field theory.

Contribution

It constructs a Lie 2-algebra of observables for 2-plectic manifolds and applies it to describe classical bosonic string dynamics.

Findings

Constructed a Lie 2-algebra for 2-plectic manifolds

Connected 2-plectic geometry to classical string dynamics

Extended symplectic structures to higher-dimensional field theories

Abstract

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.