Kavli Affiliate: J. S. Villasenor

| First 5 Authors: Juan Margalef-Bentabol, Eduardo J. S. VillaseĆ±or, , ,

| Summary:

We analyze in full-detail the geometric structure of the covariant phase

space (CPS) of any local field theory defined over a space-time with boundary.

To this end, we introduce a new frame: the "relative bicomplex framework". It

is the result of merging an extended version of the "relative framework"

(initially developed in the context of algebraic topology by R.~Bott and

L.W.~Tu in the 1980s to deal with boundaries) and the variational bicomplex

framework (the differential geometric arena for the variational calculus). The

relative bicomplex framework is the natural one to deal with field theories

with boundary contributions, including corner contributions. In fact, we prove

a formal equivalence between the relative version of a theory with boundary and

the non-relative version of the same theory with no boundary. With these tools

at hand, we endow the space of solutions of the theory with a (pre)symplectic

structure canonically associated with the action and which, in general, has

boundary contributions. We also study the symmetries of the theory and

construct, for a large group of them, their Noether currents, and charges.

Moreover, we completely characterize the arbitrariness (or lack thereof for

fiber bundles with contractible fibers) of these constructions. This clarifies

many misconceptions about the role of the boundary terms in the CPS description

of a field theory. Finally, we provide what we call the CPS-algorithm to

construct the aforementioned (pre)symplectic structure and apply it to some

relevant examples.

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