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2010 Categorified symplectic geometry and the string Lie 2-algebra
John C. Baez, Christopher L. Rogers
Homology Homotopy Appl. 12(1): 221-236 (2010).

Abstract

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.

Citation

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John C. Baez. Christopher L. Rogers. "Categorified symplectic geometry and the string Lie 2-algebra." Homology Homotopy Appl. 12 (1) 221 - 236, 2010.

Information

Published: 2010
First available in Project Euclid: 28 January 2011

zbMATH: 1277.70031
MathSciNet: MR2638872

Subjects:
Primary: 53D05, 53Z05, 70S05, 81T30

Rights: Copyright © 2010 International Press of Boston

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