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.'...

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Bibliographic Details
Published in:Homology, homotopy, and applications homotopy, and applications, 2010, Vol.12 (1), p.221-236
Main Authors: Baez, John C., Rogers, Christopher L.
Format: Article
Language:English
Subjects:
53D05
53Z05
70S05
81T30
Categorification
multisymplectic geometry
string group
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Summary: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.
ISSN:1532-0073
1532-0081
DOI:10.4310/HHA.2010.v12.n1.a12