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Categorified symplectic geometry and the string Lie 2-algebra. (English) Zbl 1277.70031
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.

MSC:
70S05 Lagrangian formalism and Hamiltonian formalism in mechanics of particles and systems
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
53Z05 Applications of differential geometry to physics
53D05 Symplectic manifolds (general theory)
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