TY - JOUR A1 - Benini, Marco T1 - Optimal space of linear classical observables for Maxwell k-forms via spacelike and timelike compact de Rham cohomologies JF - Journal of mathematical physics N2 - Being motivated by open questions in gauge field theories, we consider non-standard de Rham cohomology groups for timelike compact and spacelike compact support systems. These cohomology groups are shown to be isomorphic respectively to the usual de Rham cohomology of a spacelike Cauchy surface and its counterpart with compact support. Furthermore, an analog of the usual Poincare duality for de Rham cohomology is shown to hold for the case with non-standard supports as well. We apply these results to find optimal spaces of linear observables for analogs of arbitrary degree k of both the vector potential and the Faraday tensor. The term optimal has to be intended in the following sense: The spaces of linear observables we consider distinguish between different configurations; in addition to that, there are no redundant observables. This last point in particular heavily relies on the analog of Poincare duality for the new cohomology groups. Published by AIP Publishing. Y1 - 2016 U6 - https://doi.org/10.1063/1.4947563 SN - 0022-2488 SN - 1089-7658 VL - 57 SP - 1249 EP - 1279 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Benini, Marco A1 - Capoferri, Matteo A1 - Dappiaggi, Claudio T1 - Hadamard States for Quantum Abelian Duality JF - Annales de l'Institut Henri Poincaré N2 - Abelian duality is realized naturally by combining differential cohomology and locally covariant quantum field theory. This leads to a -algebra of observables, which encompasses the simultaneous discretization of both magnetic and electric fluxes. We discuss the assignment of physically well-behaved states on this algebra and the properties of the associated GNS triple. We show that the algebra of observables factorizes as a suitable tensor product of three -algebras: the first factor encodes dynamical information, while the other two capture topological data corresponding to electric and magnetic fluxes. On the former factor and in the case of ultra-static globally hyperbolic spacetimes with compact Cauchy surfaces, we exhibit a state whose two-point correlation function has the same singular structure of a Hadamard state. Specifying suitable counterparts also on the topological factors, we obtain a state for the full theory, ultimately implementing Abelian duality transformations as Hilbert space isomorphisms. Y1 - 2017 U6 - https://doi.org/10.1007/s00023-017-0593-y SN - 1424-0637 SN - 1424-0661 VL - 18 SP - 3325 EP - 3370 PB - Springer CY - Basel ER - TY - GEN A1 - Benini, Marco A1 - Schenkel, Alexander T1 - Quantum field theories on categories fibered in groupoids T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 895 KW - C-asterisk-algebra KW - observables KW - covariance KW - locality Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-431541 SN - 1866-8372 IS - 895 ER - TY - JOUR A1 - Benini, Marco A1 - Schenkel, Alexander T1 - Quantum Field Theories on Categories Fibered in Groupoids JF - Communications in mathematical physics N2 - We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories. Y1 - 2017 U6 - https://doi.org/10.1007/s00220-017-2986-7 SN - 0010-3616 SN - 1432-0916 VL - 356 SP - 19 EP - 64 PB - Springer CY - New York ER - TY - JOUR A1 - Benini, Marco A1 - Schenkel, Alexander T1 - Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos JF - Annales de l'Institut Henri Poincaré Y1 - 2017 U6 - https://doi.org/10.1007/s00023-016-0533-2 SN - 1424-0637 SN - 1424-0661 VL - 18 SP - 1435 EP - 1464 PB - Springer CY - Basel ER - TY - JOUR A1 - Benini, Marco A1 - Schenkel, Alexander A1 - Schreiber, Urs T1 - The Stack of Yang-Mills Fields on Lorentzian Manifolds JF - Communications in mathematical physics N2 - We provide an abstract definition and an explicit construction of the stack of non-Abelian Yang-Mills fields on globally hyperbolic Lorentzian manifolds. We also formulate a stacky version of the Yang-Mills Cauchy problem and show that its well-posedness is equivalent to a whole family of parametrized PDE problems. Our work is based on the homotopy theoretical approach to stacks proposed in Hollander (Isr. J. Math. 163:93-124, 2008), which we shall extend by further constructions that are relevant for our purposes. In particular, we will clarify the concretification of mapping stacks to classifying stacks such as BG (con). Y1 - 2018 U6 - https://doi.org/10.1007/s00220-018-3120-1 SN - 0010-3616 SN - 1432-0916 VL - 359 IS - 2 SP - 765 EP - 820 PB - Springer CY - New York ER - TY - JOUR A1 - Becker, Christian A1 - Benini, Marco A1 - Schenkel, Alexander A1 - Szabo, Richard J. T1 - Cheeger-Simons differential characters with compact support and Pontryagin duality JF - Communications in analysis and geometry N2 - By adapting the Cheeger-Simons approach to differential cohomology, we establish a notion of differential cohomology with compact support. We show that it is functorial with respect to open embeddings and that it fits into a natural diagram of exact sequences which compare it to compactly supported singular cohomology and differential forms with compact support, in full analogy to ordinary differential cohomology. We prove an excision theorem for differential cohomology using a suitable relative version. Furthermore, we use our model to give an independent proof of Pontryagin duality for differential cohomology recovering a result of [Harvey, Lawson, Zweck - Amer. J. Math. 125 (2003), 791]: On any oriented manifold, ordinary differential cohomology is isomorphic to the smooth Pontryagin dual of compactly supported differential cohomology. For manifolds of finite-type, a similar result is obtained interchanging ordinary with compactly supported differential cohomology. Y1 - 2019 U6 - https://doi.org/10.4310/CAG.2019.v27.n7.a2 SN - 1019-8385 SN - 1944-9992 VL - 27 IS - 7 SP - 1473 EP - 1522 PB - International Press of Boston CY - Somerville ER -