Symmetries and Covariant Poisson Brackets on Presymplectic Manifolds

As the space of solutions of the first-order Hamiltonian field theory has a presymplectic structure, we describe a class of conserved charges associated with the momentum map, determined by a symmetry group of transformations. A gauge theory is dealt with by using a symplectic regularization based on an application of Gotay’s coisotropic embedding theorem. An analysis of electrodynamics and of the Klein–Gordon theory illustrate the main results of the theory as well as the emergence of the energy–momentum tensor algebra of conserved currents.

Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions

In this note, we consider how the bundle geometry of field space interplays with the covariant phase space methods so as to allow to write results of some generality on the presymplectic structure of invariant gauge theories coupled to matter. We obtain in particular the generic form of Noether charges associated with field-independent and field-dependent gauge parameters, as well as their Poisson bracket. We also provide the general field-dependent gauge transformations of the presymplectic potential and 2-form, which clearly highlights the problem posed by boundaries in generic situations. We then conduct a comparative analysis of two strategies recently considered to evade the boundary problem and associate a modified symplectic structure to a gauge theory over a bounded region: namely the use of edge modes on the one hand, and of variational connections on the other. To do so, we first try to give the clearest geometric account of both, showing in particular that edge modes are a special case of a differential geometric tool of gauge symmetry reduction known as the “dressing field method”. Applications to Yang-Mills theory and General Relativity reproduce or generalise several results of the recent literature.

Presymplectic AKSZ formulation of Einstein gravity

Any local gauge theory can be represented as an AKSZ sigma model (upon parameterization if necessary). However, for non-topological models in dimension higher than 1 the target space is necessarily infinite-dimensional. The interesting alternative known for some time is to allow for degenerate presymplectic structure in the target space. This leads to a very concise AKSZ-like representation for frame-like Lagrangians of gauge systems. In this work we concentrate on Einstein gravity and show that not only the Lagrangian but also the full-scale Batalin-Vilkovisky (BV) formulation is naturally encoded in the presymplectic AKSZ formulation, giving an elegant supergeometrical construction of BV for Cartan-Weyl action. The same applies to the main structures of the respective Hamiltonian BFV formulation.

Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method

We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two restricting hypothesis. In particular, we derive the general field-dependent gauge transformations of the presymplectic potential and presymplectic 2-form in both cases. We point-out that a generalisation of the standard bundle geometry, called twisted geometry, arises naturally in the study of non-invariant gauge theories (e.g. non-Abelian Chern-Simons theory). These results prove that the well-known problem of associating a symplectic structure to a gauge theory over bounded regions is a generic feature of both classes. The edge modes strategy, recently introduced to address this issue, has been actively developed in various contexts by several authors. We draw attention to the dressing field method as the geometric framework underpinning, or rather encompassing, this strategy. The geometric insight afforded by the method both clarifies it and clearly delineates its potential shortcomings as well as its conditions of success. Applying our general framework to various examples allows to straightforwardly recover several results of the recent literature on edge modes and on the presymplectic structure of general relativity.

HAMILTONIAN FORMALISM FOR THE HIGHER KdV FLOWS ON THE SPACE OF CLOSED COMPLEX EQUICENTROAFFINE CURVES

We study the higher KdV flows on the space of closed complex equicentroaffine curves as Hamiltonian systems. Using a suitable presymplectic structure of the space, we give the Hamiltonian flows associated with the higher KdV equations and a map between the space and the space of closed curves in the complex plane, which induces the Miura transformation between the higher KdV equations and the higher mKdV equations.

GEOMETRY OF THE MODULI SPACES OF HARMONIC MAPS INTO LIE GROUPS VIA GAUGE THEORY OVER RIEMANN SURFACES

This paper aims to investigate the geometry of the moduli spaces of harmonic maps of compact Riemann surfaces into compact Lie groups or compact symmetric spaces. The approach here is to study the gauge theoretic equations for such harmonic maps and the moduli space of their solutions. We discuss the S1-action, the hyper-presymplectic structure, the energy function, the Hitchin map, the flag transforms on the moduli space, several kinds of subspaces in the moduli space, and the relationship among them, especially the structure of the critical point subset for the energy function on the moduli space. As results, we show that every uniton solution is a critical point of the energy function on the moduli space, and moreover we give a characterization of the fixed point subset fixed by S1-action in terms of a flag transform.