Categorified Symplectic Geometry and the String Lie 2-Algebra

Details Categorified Symplectic Geometry and the String Lie 2-Algebra Categorified Symplectic Geometry and the String Lie 2-Algebra

2009-01-29

arxiv.org/abs/0901.4721v1

Homology, Homotopy and Applications 12 (2010), 221-236

Physics

Mathematical Physics

16 pages

Scientific paper

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.

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