Categorified symplectic geometry and the string Lie 2-algebra

Multisymplectic geometry is a generalization of symplectic geometry suitable for $n$-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate $(n+1)$-form. The case $n=2$ is relevant to string theory: we call this “2-plectic geometry.” Just as the Poisson bracket makes the smooth functions on a symplectic manifold into a Lie algebra, the observables associated to a 2-plectic manifold form a “Lie 2-algebra,” which is a categorified version of a Lie algebra. Any compact simple Lie group $G$ has a canonical 2-plectic structure, so it is natural to wonder what Lie 2-algebra this example yields. This Lie 2-algebra is infinite-dimensional, but we show here that the sub-Lie-2-algebra of left-invariant observables is finite-dimensional, and isomorphic to the already known “string Lie 2-algebra” associated to $G$. So, categorified symplectic geometry gives a geometric construction of the string Lie 2-algebra.

Keywords

categorification, string group, multisymplectic geometry

2010 Mathematics Subject Classification

53D05, 53Z05, 70S05, 81T30

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Published 1 January 2010