Geometric formulation of the covariant phase space methods with boundaries

Abstract

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.

Conservation and integrability in lower-dimensional gravity

“…In the covariant phase space formalism [6][7][8][9][10][11][12]101], the infinitesimal charges associated with the asymptotic symmetry parameters ξ are computed by integrating some co-dimension 2…”

Chiral massive news: null boundary symmetries in topologically massive gravity

“…(5.41) 20 The warped solution is not "circular" in the sense that it does not have (t, θ) → (−t, −θ) symmetry. This is due to the presence of the CS term in the action.…”

“…This ambiguity may be (partially) fixed by different physical requirements. For example, this ambiguity was used in [11,12] to fix the central charge of the algebra of surface charges to a certain value, in [13][14][15][16][17] it was used to render surface charges computed at asymptotic boundaries finite and also to relate charges computed in different formulations of gravity [18,19]; see also [20,21]. In this work, as we will see the Y -ambiguity is crucially used in the charge integrability analysis.…”

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

“…A better geometrical approach would certainly have been to work on J 1 (A) or J r (A), the first and r th -jet spaces of A. In particular, it is worth comparing our results with the proposition of [120]. Taking for now this framework as a starting point, its domain of validity can be extended on several fronts, each will be the object of a forthcoming paper.…”

“…48 We point e.g. to [115] and [120] for mathematically more sophisticated approaches to the boundary problem. See also [69][70][71]116] for an approach using an Ehresmann connection on field space.…”