Note on the bundle geometry of field space, variational connections, the dressing field method, & presymplectic structures of gauge theories over bounded regions. (English) Zbl 1521.81341
Summary: 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.
MSC:
| 81T45 | Topological field theories in quantum mechanics |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83D05 | Relativistic gravitational theories other than Einstein’s, including asymmetric field theories |
| 53C80 | Applications of global differential geometry to the sciences |
| 83C45 | Quantization of the gravitational field |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
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