$n$-plectic Maxwell Theory

L-infinity algebras from multisymplectic geometry

A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, we showed that just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L-infinity algebras: graded vector spaces which are equipp…

Automorphisms of Strong Homotopy Lie Algebras of Local Observables

There is a well-established procedure of assigning a strong homotopy Lie algebra of local observables to a multisymplectic manifold which can be regarded as part of a categorified Poisson structure. For a 2-plectic manifold, the resulting Lie 2-algebra is isomorphic to a sub Lie 2-algebra of a natural Lie 2-algebra structure on an exact Courant algebroid. We generalize this statement to arbitrary n-plectic manifolds and study automorphisms on the arising Lie n-algebras. Our observations may be useful in studying the quantization problem on multisymplectic manifolds.

Multisymplectic Geometry Method for Maxwell's Equations and Multisymplectic Scheme

In this paper we discussed the self-adjointness of the Maxwell's equations with variable coefficients $ε$ and $μ$. Three different Lagrangian are attained. By the Legendre transformation, a multisymplectic Bridge's (Hamilton) form is obtained. Based on the multisymplectic structure, the multisymplectic conservation law of the system is derived and a nine-point Preissman multisymplectic scheme which preserve the multisymplectic conservation law is given for the Maxwell's equations in an inhomogeneous, isotropic and lossless medium. At last a numerical example is illustrated.

（预）多重辛流形上李代数胚的相容 E-微分形式

Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

本文在（预）多重辛流形上的李代数胚中引入相容 E-n-形式，统一刻画扭曲泊松结构、动量映射及其同伦推广等结构。

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. 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 t…