@article{BeckerBeniniSchenkeletal.2019, author = {Becker, Christian and Benini, Marco and Schenkel, Alexander and Szabo, Richard J.}, title = {Cheeger-Simons differential characters with compact support and Pontryagin duality}, series = {Communications in analysis and geometry}, volume = {27}, journal = {Communications in analysis and geometry}, number = {7}, publisher = {International Press of Boston}, address = {Somerville}, issn = {1019-8385}, doi = {10.4310/CAG.2019.v27.n7.a2}, pages = {1473 -- 1522}, year = {2019}, abstract = {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.}, language = {en} } @article{BeniniSchenkel2017, author = {Benini, Marco and Schenkel, Alexander}, title = {Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos}, series = {Annales de l'Institut Henri Poincar{\´e}}, volume = {18}, journal = {Annales de l'Institut Henri Poincar{\´e}}, publisher = {Springer}, address = {Basel}, issn = {1424-0637}, doi = {10.1007/s00023-016-0533-2}, pages = {1435 -- 1464}, year = {2017}, language = {en} } @article{BeniniCapoferriDappiaggi2017, author = {Benini, Marco and Capoferri, Matteo and Dappiaggi, Claudio}, title = {Hadamard States for Quantum Abelian Duality}, series = {Annales de l'Institut Henri Poincar{\´e}}, volume = {18}, journal = {Annales de l'Institut Henri Poincar{\´e}}, publisher = {Springer}, address = {Basel}, issn = {1424-0637}, doi = {10.1007/s00023-017-0593-y}, pages = {3325 -- 3370}, year = {2017}, abstract = {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.}, language = {en} } @article{BeniniSchenkel2017, author = {Benini, Marco and Schenkel, Alexander}, title = {Quantum Field Theories on Categories Fibered in Groupoids}, series = {Communications in mathematical physics}, volume = {356}, journal = {Communications in mathematical physics}, publisher = {Springer}, address = {New York}, issn = {0010-3616}, doi = {10.1007/s00220-017-2986-7}, pages = {19 -- 64}, year = {2017}, abstract = {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.}, language = {en} } @article{Benini2016, author = {Benini, Marco}, title = {Optimal space of linear classical observables for Maxwell k-forms via spacelike and timelike compact de Rham cohomologies}, series = {Journal of mathematical physics}, volume = {57}, journal = {Journal of mathematical physics}, publisher = {American Institute of Physics}, address = {Melville}, issn = {0022-2488}, doi = {10.1063/1.4947563}, pages = {1249 -- 1279}, year = {2016}, abstract = {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.}, language = {en} }