Area-Dependent Quantum Field Theory
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Communications in
Mathematical Physics
Area-Dependent Quantum Field Theory Ingo Runkel1 , Lóránt Szegedy2 1 Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany.
E-mail: [email protected]
2 Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria.
E-mail: [email protected] Received: 23 November 2018 / Accepted: 24 August 2020 © The Author(s) 2020
Abstract: Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number—interpreted as area—which behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We also provide a state sum construction for area-dependent theories. Our main example is two-dimensional Yang–Mills theory with compact gauge group, which we treat in detail.
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Introduction and Summary . . . . . . . . . . . . . . . . . . . Regularised Frobenius Algebras . . . . . . . . . . . . . . . . 2.1 Definition of regularised algebras and Frobenius algebras 2.2 Properties of RFAs . . . . . . . . . . . . . . . . . . . . 2.3 Classification of Hermitian RFAs . . . . . . . . . . . . . 2.4 Examples of RFAs . . . . . . . . . . . . . . . . . . . . Area-Dependent QFTs as Functors . . . . . . . . . . . . . . 3.1 Bordisms with area and aQFTs . . . . . . . . . . . . . . 3.2 Equivalence of aQFTs and commutative RFAs . . . . . . State Sum Construction of aQFTs . . . . . . . . . . . . . . . 4.1 PLCW decompositions with area . . . . . . . . . . . . . 4.2 State sum construction . . . . . . . . . . . . . . . . . . Example: 2d Yang–Mills Theory . . . . . . . . . . . . . . . 5.1 Two RFAs from a compact group G . . . . . . . . . . . 5.2 State sum construction of 2d Yang–Mills theory . . . . .
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I. Runkel, L. Szegedy
A Proof of Joint Continuity References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction and Summary Area-dependent quantum field theory (aQFT1 ) is a modification of 2-dimensional topological quantum field theory (TQFT): we consider the category of bordisms with area Bor d2ar ea and symmetric monoidal functors from it to the category of Hilbert spaces Hilb which depend continuously on the area. We can think of the area as a positive number attached to each connected component of a surface, additive under composition.2 In order to have identities in Bor d2ar ea , we allow zero area on cylinders. The main change when passing from TQFTs to aQFTs, and indeed the m
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