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

François, Jordan; Parrini, Noémie; Boulanger, Nicolas

doi:10.1007/jhep12(2021)186

Cited by 8 publications

(10 citation statements)

References 87 publications

(187 reference statements)

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“…This can be achieved introducing a particular set of extra dynamical fields [327][328][329][330], and thus working on a specific extended phase space. These extra phase-space variables come in gravitational theories from the embedding of corners, and are known as edge modes [327,[331][332][333][334][335][336], which are the core of a novel, thriving, area of investigation [320][321][322][337][338][339][340][341][342][343][344][345][346][347][348][349]. One of the merits of this method to treat integrability is that the charge algebra is then realized by the standard Poisson bracket.…”

Section: Pos(modave2022)002mentioning

confidence: 99%

From Asymptotic Symmetries to the Corner Proposal

Ciambelli¹

2023

Proceedings of Modave Summer School in Mathematical Physics — PoS(Modave2022)

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These notes are a transcript of lectures given by the author in the XVIII Modave summer school in mathematical physics. The introduction is devoted to a detailed review of the literature on asymptotic symmetries, flat holography, and the corner proposal. It covers much more material than needed, for it is meant as a lamppost to help the reader in navigating the vast existing literature. The notes then consist of three main parts. The first is devoted to Noether's theorems and their underlying framework, the covariant phase space formalism, with special focus on gauge theories. The surface-charges algebra is shown to projectively represent the asymptotic symmetry algebra. Issues arising in the gravitational case, such as conservation, finiteness, and integrability, are addressed. In the second part, we introduce the geometric concept of corners, and show the existence of a universal asymptotic symmetry group at corners. A careful treatment of corner embeddings provides a resolution to the issue of integrability, by extending the phase space. In the last part we bridge asymptotic symmetries and corners by formulating the corner proposal. In essence, the latter focuses on the central question of extracting from classical gravity universal results that are expected to hold in the quantum realm. After reviewing the coadjoint orbit method and Atiyah Lie algebroids, we apply these concepts to the corner proposal. Exercises are solved in the notes, to elucidate the arguments exposed.

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Section: Pos(modave2022)002mentioning

confidence: 99%

From Asymptotic Symmetries to the Corner Proposal

Ciambelli¹

2023

Proceedings of Modave Summer School in Mathematical Physics — PoS(Modave2022)

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“…For a series of diverse approaches that involve actual extensions of the phase space of a gauge theory, either in the bulk or at the boundary, we refer the reader to e.g. [FM72; RT74; GZ80; IK85; BCM96; DF16; DR19], and also [FPB21]. ♦ 3.1.…”

Section: Geometric Setup For Gauge Theories With Cornersmentioning

confidence: 99%

Hamiltonian gauge theory with corners: constraint reduction and flux superselection

Riello¹,

Schiavina²

2022

Preprint

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We study the Hamiltonian formulation of gauge field theories on manifolds with corners, and develop a method to characterize their symplectic reduction whenever the theory admits a local momentum map for the gauge group action. This is achieved by adapting reduction by stages to the case of Hamiltonian actions of gauge groups associated to gauge subgroups that emerge from the presence of corners.We start from a decomposition of the local momentum map into a bulk term called constraint map, and a boundary term called flux map. The first stage, or constraint reduction, views the constraint as the zero locus of a momentum map for a normal subgroup G• ⊂ G (called constraint gauge group). The second stage, or flux superselection, interprets the flux map as the momentum map for the residual action of the flux gauge group G/G•. Equivariance is controlled by cocycles of the flux gauge group.Whereas the only physically admissible value of the constraint map is zero, the flux map is (largely) unconstrained. Consequently, the reduction by stages generally outputs a Poisson space with symplectic leaves corresponding to level sets of the flux map, and thus to the coadjoint (or affine) orbits of the flux gauge group. We call such symplectic leaves flux superselection sectors, for they provide a classical analogue of (and a road map to) the phenomenon of quantum superselection. Finally, we construct a symplectic (action) Lie algebroid over a Poisson manifold of corner fields, A ∂ → P ∂ . This encodes the corner/flux gauge symmetries and provides a description of flux superselection intrinsic to the corner submanifold which in many cases of interest is local.We work out in detail the example of Yang-Mills theory and briefly discuss other applications to Chern-Simons theory and BF theory. ContentsA. RIELLO AND M. SCHIAVINA 3.4. Local momentum forms 19 3.5. Equivariance and corner cocycles 21 3.6. Central extension by a nontrivial cocycle 24 3.7. Yang-Mills theory: general setup 25 4. Constraint and Flux momentum maps 28 4.1. Constraint/flux decomposition: Main assumptions 28 4.2. Yang-Mills theory: constraint/flux decomposition 32 4.3. The flux map 33 4.4. Flux annihilators 34 4.5. Yang-Mills theory: flux annihilators 37 4.6. The constraint and flux gauge groups 40 4.7. Yang-Mills theory: flux gauge groups 45 5. Locally Hamiltonian symplectic reduction with corners 46 5.1. First stage symplectic reduction: the constraint 46 5.2. Yang-Mills theory: constraint-reduced phase space 50 5.3. Second stage symplectic reduction: flux superselections 53 5.4. Yang-Mills theory: flux analysis 58 6. Corner data 61 6.1. Off-shell corner data 62 6.2. The Poisson manifold of off-shell corner data and superselections 70 6.3. Equivalence with on-shell superselection sectors 76 6.4. An alternative set of pre-corner data 78 6.5. Yang-Mills theory: corner Poisson structure and superselections 79 7. Applications 84 7.1. Chern-Simons theory 84 7.2. Corner ambiguity and θ-Yang-Mills theory 87 7.3. BF theory 89 Appendix A. Remarks on infinite dimensional...

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“…Its applications range from the construction of twistors and tractors in conformal Cartan geometry [5,6], to reformulation of electroweak physics dispensing with the notion of spontaneous symmetry breaking (SSB) [7]. More recently, it was shown in [8,9] that it is the geometric underpinning of the notion of edge modes introduced in recent years in the study of the presymplectic structure of gauge theories over bounded regions [10][11][12][13][14][15][16][17][18]-which was first hinted at in [19].…”

Section: Introductionmentioning

confidence: 99%

“…Yang-Mills theories and gauge theories of gravity formulated via Cartan geometry. The interested reader can find self-contained expositions in sections 3 and 4.3.1 of [8], section 2.3 of [9], or chapter 5 of [7]. See also [20][21][22] for other mathematical developments and applications.…”

Section: Introductionmentioning

confidence: 99%

The dressing field method for diffeomorphisms: a relational framework

André

2024

J. Phys. A: Math. Theor.

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The dressing field method (DFM) is a tool to reduce gauge symmetries. Here we extend it to cover the case of diffeomorphisms. The resulting framework is a systematic scheme to produce Diff ( M ) -invariant objects, which has a natural relational interpretation. Its precise formulation relies on a clear understanding of the bundle geometry of field space. By detailing it, among other things we stress the geometric nature of field-independent and field-dependent diffeomorphisms, and highlight that the heuristic ‘extended bracket’ for field-dependent vector fields often featuring in the covariant phase space literature can be understood as arising from the Frölicher-Nijenhuis bracket. Furthermore, by articulating this bundle geometry with the covariant phase space approach, we give a streamlined account of the elementary objects of the (pre)symplectic structure of a Diff ( M ) -theory: Noether charges and their bracket, as induced by the standard prescription for the presymplectic potential and 2-form. We give conceptually transparent expressions allowing to read the integrability conditions and the circumstances under which the bracket of charge is Lie, and the resulting Poisson algebras of charges are central extensions of the Lie algebras of field-independent ( diff ( M ) ) and field-dependent vector fields. We show that, applying the DFM, one obtains a Diff ( M ) -invariant and manifestly relational formulation of a general relativistic field theory. Relying on results just mentioned, we easily derive the ‘dressed’ (relational) presymplectic structure of the theory. This reproduces or extends results from the gravitational edge modes and gravitational dressings literature. In addition to simplified technical derivations, the conceptual clarity of the framework supplies several insights and allows us to dispel misconceptions.

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

Cited by 8 publications

References 87 publications

From Asymptotic Symmetries to the Corner Proposal

From Asymptotic Symmetries to the Corner Proposal

Hamiltonian gauge theory with corners: constraint reduction and flux superselection

The dressing field method for diffeomorphisms: a relational framework

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