Categorified symplectic geometry and the string Lie 2-algebra

• 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
• ISSN: 1532-0073; 1532-0081
• DOI: 10.4310/HHA.2010.v12.n1.a12

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.