Abstract
We provide an elegant homological construction of the extended phase space for linear Yang–Mills theory on an oriented and time-oriented Lorentzian manifold M with a time-like boundary partial M that was proposed by Donnelly and Freidel (JHEP 1609:102, 2016). This explains and formalizes many of the rather ad hoc constructions for edge modes appearing in the theoretical physics literature. Our construction also applies to linear Chern–Simons theory, in which case we obtain the extended phase space introduced by Geiller (Nucl Phys B 924:312, 2017).
Highlights
The topic of edge modes is a time-honored one in the study of gauge theories on manifolds with boundary
A noteworthy reaction to their work is [24,25,26,27], which observes that the notion of boundary in [20] is ambiguous between a ‘fiducial’ boundary, meaning a non-physical boundary that does not in any way influence the field content and which disappears upon gluing along the boundary, and a ‘physical’ boundary, meaning a boundary that influences the field content in some way, e.g., by carrying a defect theory or a Higgs field. (This ambiguity is heightened by the fact that [20] do not associate any action to the edge modes.) For pure gauge fields, the study in [25,27] uses a certain Singer–De Witt connection form on field space, which they interpret as a geometric generalization of ghost fields, to show that fiducial boundaries cannot carry charged edge modes
The goal of this paper is to provide an elegant and rigorous construction of extended phase spaces as in [20] for two simple cases: linear Yang–Mills theory on a globally hyperbolic Lorentzian manifold M with a time-like boundary ∂ M and linear Chern– Simons theory on a three-dimensional product manifold M = R × with boundary ∂ M = R × ∂
Summary
Let M be an oriented and time-oriented Lorentzian manifold with a smooth boundary ∂ M. The groupoid of fields F(M) is obtained by identifying the restriction of the bulk principal R-bundle with the fixed principal R-bundle on ∂ M, i.e., we implement a boundary condition via the homotopy pullback diagram (2.5). Manm−1 which implements the metric-independent boundary condition that the boundary ∂ M of the bulk manifold M is ‘the same as’ the fixed m − 1-dimensional manifold B Computing this homotopy pullback via Proposition A.1, we obtain that (i) objects in Fgravity(M) are pairs (g, X ), where g is a metric on M and X : B → ∂ M is a diffeomorphism between B and the boundary ∂ M, and (ii) morphisms f : (g, X ) → (g , X ) in Fgravity(M) are all diffeomorphisms f : M → M preserving the metrics, i.e., f ∗(g ) = g, and satisfying f∂ ◦ X = X. The signs in (2.18) are a consequence of graded commutativity of the ∧-product
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