## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 12, Number 1 (2010), 221-236.

### Categorified symplectic geometry and the string Lie 2-algebra

John C. Baez and Christopher L. Rogers

#### 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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 12, Number 1 (2010), 221-236.

**Dates**

First available in Project Euclid: 28 January 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1296223828

**Mathematical Reviews number (MathSciNet)**

MR2638872

**Zentralblatt MATH identifier**

1277.70031

**Subjects**

Primary: 70S05: Lagrangian formalism and Hamiltonian formalism 81T30: String and superstring theories; other extended objects (e.g., branes) [See also 83E30] 53Z05: Applications to physics 53D05: Symplectic manifolds, general

**Keywords**

Categorification string group multisymplectic geometry

#### Citation

Baez, John C.; Rogers, Christopher L. Categorified symplectic geometry and the string Lie 2-algebra. Homology Homotopy Appl. 12 (2010), no. 1, 221--236. https://projecteuclid.org/euclid.hha/1296223828