The Symplectic Geometry of p-Form Gauge Fields

TL;DR

The paper develops a geometric, symplectic framework for interacting $q$-form gauge fields by using a democratic configuration space of dual field strengths $(F^a,G_a)$ and viewing the field equations and Bianchi identities as the intersection of a Lagrangian submanifold with a (co)isotropic counterpart in an infinite-dimensional setting, with the symplectic form $\omega=\int_{\mathcal{M}} \delta F^a \wedge \delta G_a$. It extends this structure to include local Gluing data and Chern-Simons interactions, introduces an extended configuration space with additional connections $(A^a, C_a)$, and demonstrates the construction in a D-dimensional Yang–Mills context and a detailed six-dimensional model with a closed 3-form coupled to Yang–Mills via CS$_3(a)$. Key contributions include the coisotropic reduction to cohomology, the emergence of a holomorphic symplectic structure when $D=2q-1$, and an explicit global treatment of CS-induced transition functions through locally defined $G_a$ and twisted forms; the framework is then instantiated in a 6D gauge system with a dual formulation and gauge-invariance-consistent transformations. Overall, this geometric, democratic approach provides a principled foundation for dualities in gauge theories and opens avenues toward quantization via coisotropic or Lagrangian submanifold quantization and potential boundary/topological constructions. The work broadens the toolkit for understanding gauge dynamics by recasting them as constrained intersections in a symplectic field space, potentially offering new insights into dualities and the role of global topological data.

Abstract

We formulate interacting antisymmetric tensor gauge theory in a configuration space consisting of a pair of dual field strengths which has a natural symplectic structure. The field equations are formulated as the intersection of a pair of submanifolds of this infinite-dimensional symplectic configuration space, one of which is a Lagrangian submanifold while the other is either a coisotropic or Lagrangian submanifold, depending on the topology. Chern-Simons interactions give the configuration space an interesting global structure. We consider in detail the example of a six-dimensional theory of a 3-form field strength coupled to Yang-Mills theory via a Chern-Simons interaction. Our approach applies to a broad class of gauge systems.