--- _id: '8816' abstract: - lang: eng text: 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. acknowledgement: The authors thank Yuki Arano, Nils Carqueville, Alexei Davydov, Reiner Lauterbach, Pau Enrique Moliner, Chris Heunen, André Henriques, Ehud Meir, Catherine Meusburger, Gregor Schaumann, Richard Szabo and Stefan Wagner for helpful discussions and comments. We also thank the referees for their detailed comments which significantly improved the exposition of this paper. LS is supported by the DFG Research Training Group 1670 “Mathematics Inspired by String Theory and Quantum Field Theory”. Open access funding provided by Institute of Science and Technology (IST Austria). article_processing_charge: Yes (via OA deal) article_type: original author: - first_name: Ingo full_name: Runkel, Ingo last_name: Runkel - first_name: Lorant full_name: Szegedy, Lorant id: 7943226E-220E-11EA-94C7-D59F3DDC885E last_name: Szegedy orcid: 0000-0003-2834-5054 citation: ama: Runkel I, Szegedy L. Area-dependent quantum field theory. Communications in Mathematical Physics. 2021;381(1):83–117. doi:10.1007/s00220-020-03902-1 apa: Runkel, I., & Szegedy, L. (2021). Area-dependent quantum field theory. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-020-03902-1 chicago: Runkel, Ingo, and Lorant Szegedy. “Area-Dependent Quantum Field Theory.” Communications in Mathematical Physics. Springer Nature, 2021. https://doi.org/10.1007/s00220-020-03902-1. ieee: I. Runkel and L. Szegedy, “Area-dependent quantum field theory,” Communications in Mathematical Physics, vol. 381, no. 1. Springer Nature, pp. 83–117, 2021. ista: Runkel I, Szegedy L. 2021. Area-dependent quantum field theory. Communications in Mathematical Physics. 381(1), 83–117. mla: Runkel, Ingo, and Lorant Szegedy. “Area-Dependent Quantum Field Theory.” Communications in Mathematical Physics, vol. 381, no. 1, Springer Nature, 2021, pp. 83–117, doi:10.1007/s00220-020-03902-1. short: I. Runkel, L. Szegedy, Communications in Mathematical Physics 381 (2021) 83–117. date_created: 2020-11-29T23:01:17Z date_published: 2021-01-01T00:00:00Z date_updated: 2023-08-04T11:13:35Z day: '01' ddc: - '510' department: - _id: MiLe doi: 10.1007/s00220-020-03902-1 external_id: isi: - '000591139000001' file: - access_level: open_access checksum: 6f451f9c2b74bedbc30cf884a3e02670 content_type: application/pdf creator: dernst date_created: 2021-02-03T15:00:30Z date_updated: 2021-02-03T15:00:30Z file_id: '9081' file_name: 2021_CommMathPhys_Runkel.pdf file_size: 790526 relation: main_file success: 1 file_date_updated: 2021-02-03T15:00:30Z has_accepted_license: '1' intvolume: ' 381' isi: 1 issue: '1' language: - iso: eng month: '01' oa: 1 oa_version: Published Version page: 83–117 project: - _id: B67AFEDC-15C9-11EA-A837-991A96BB2854 name: IST Austria Open Access Fund publication: Communications in Mathematical Physics publication_identifier: eissn: - '14320916' issn: - '00103616' publication_status: published publisher: Springer Nature quality_controlled: '1' scopus_import: '1' status: public title: Area-dependent quantum field theory tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 381 year: '2021' ...