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@book{60889, author = {Khavkine, Igor}, keywords = {local observables; differential invariants; QFT in curved spacetime; Pseudo-Riemannian geometry; Constant curvature; Differential complex; Sheaf cohomology; Linearized gravity; cosmology; equivalence up to isometry; IDEAL characterization; Schwarzschild; Tangherlini; warped product; Birkhoff's theorem; linearized gravity; gauge-invariant observables; FLRW}, language = {eng}, title = {Invariants in Relativity and Gauge Theory}, year = {2019} }
TY - ID - 60889 AU - Khavkine, Igor PY - 2019 TI - Invariants in Relativity and Gauge Theory KW - local observables KW - differential invariants KW - QFT in curved spacetime KW - Pseudo-Riemannian geometry KW - Constant curvature KW - Differential complex KW - Sheaf cohomology KW - Linearized gravity KW - cosmology KW - equivalence up to isometry KW - IDEAL characterization KW - Schwarzschild KW - Tangherlini KW - warped product KW - Birkhoff's theorem KW - linearized gravity KW - gauge-invariant observables KW - FLRW N2 - In mathematical physics, one of the goals in the study of classical eld theory (essentially, a variational partial di erential equation (PDE)) with gauge symmetries (a.k.a gauge theories) is a precise and rigorous construction of the corresponding reduced phase space [6, 12, 22, 23]: the space of solutions, endowed with the canonical Poisson structure, and quotiented by the gauge transformations. A large number of technical problems stands in the way, including but not limited to describing the solutions of a non-linear PDE as an in nite dimensional space with some kind of smooth structure, specifying a su ciently regular class of functions on this space on which the Poisson structure is well-de ned, explicitly describing the structure of the quotient, or alternatively the structure of the functions invariant under gauge symmetries. The functions in the latter class are referred to as gauge-invariant observables (or invariants). The scope of this Habilitation Thesis is to address some purely geometric problems that arise in the study of gauge-invariant observables and can be attacked using tools from the theory of di erential invariants [30] and the theory of formal integrability of overdetermined PDEs [31, 35]. The main focus is on General Relativity (GR) as a non-trivial representative example, but the perspective is such that the tools used would also apply to other gauge theories, of which Electrodynamics, Yang-Mills, Chern-Simons, Supergravity and many other models used in fundamental theoretical physics and geometry [12], are prominent examples. The papers collected in this Thesis consist of [25], [27], [26], [10], [28], [29], all of which have been published with the exception of [29], which has been submitted for publication to Communications in Mathematical Physics. In the remainder of the Introduction, we give a brief summary of relevant geometric notions(Section 0.1) and summarize the main problems addressed and results obtained in the above papers, grouped by theme in Sections 0.2, 0.3 and 0.4. ER -
KHAVKINE, Igor. \textit{Invariants in Relativity and Gauge Theory}. 2019.
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