[{"page":"467-532","ddc":["530"],"quality_controlled":"1","doi":"10.4171/qt/193","article_processing_charge":"Yes","publisher":"European Mathematical Society","date_updated":"2024-01-09T09:27:46Z","_id":"14756","type":"journal_article","publication":"Quantum Topology","status":"public","date_published":"2023-10-16T00:00:00Z","acknowledgement":"N.C. is supported by the DFG Heisenberg Programme.\r\nWe are grateful to Tobias Dyckerhoff, Lukas Müller, Ingo Runkel, and Christopher Schommer-Pries for helpful discussions.","year":"2023","keyword":["Geometry and Topology","Mathematical Physics"],"has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"text":"We prove the r-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer r: the 2-groupoid of 2-dimensional fully extended r-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced Spin 2r -action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the rth power of their Serre automorphisms. For r=1, we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to r=2.\r\nTo construct examples, we explicitly describe Spin 2r​-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau–Ginzburg models gives rise to fully extended spin TQFTs and that half of these do not factor through the oriented bordism 2-category.","lang":"eng"}],"intvolume":"        14","publication_identifier":{"issn":["1663-487X"]},"publication_status":"published","file_date_updated":"2024-01-09T09:25:34Z","author":[{"full_name":"Carqueville, Nils","last_name":"Carqueville","first_name":"Nils"},{"last_name":"Szegedy","full_name":"Szegedy, Lorant","id":"7943226E-220E-11EA-94C7-D59F3DDC885E","orcid":"0000-0003-2834-5054","first_name":"Lorant"}],"day":"16","scopus_import":"1","title":"Fully extended r-spin TQFTs","oa_version":"Published Version","volume":14,"article_type":"original","date_created":"2024-01-08T13:14:48Z","oa":1,"language":[{"iso":"eng"}],"issue":"3","citation":{"ieee":"N. Carqueville and L. Szegedy, “Fully extended r-spin TQFTs,” <i>Quantum Topology</i>, vol. 14, no. 3. European Mathematical Society, pp. 467–532, 2023.","short":"N. Carqueville, L. Szegedy, Quantum Topology 14 (2023) 467–532.","ama":"Carqueville N, Szegedy L. Fully extended r-spin TQFTs. <i>Quantum Topology</i>. 2023;14(3):467-532. doi:<a href=\"https://doi.org/10.4171/qt/193\">10.4171/qt/193</a>","apa":"Carqueville, N., &#38; Szegedy, L. (2023). Fully extended r-spin TQFTs. <i>Quantum Topology</i>. European Mathematical Society. <a href=\"https://doi.org/10.4171/qt/193\">https://doi.org/10.4171/qt/193</a>","mla":"Carqueville, Nils, and Lorant Szegedy. “Fully Extended R-Spin TQFTs.” <i>Quantum Topology</i>, vol. 14, no. 3, European Mathematical Society, 2023, pp. 467–532, doi:<a href=\"https://doi.org/10.4171/qt/193\">10.4171/qt/193</a>.","chicago":"Carqueville, Nils, and Lorant Szegedy. “Fully Extended R-Spin TQFTs.” <i>Quantum Topology</i>. European Mathematical Society, 2023. <a href=\"https://doi.org/10.4171/qt/193\">https://doi.org/10.4171/qt/193</a>.","ista":"Carqueville N, Szegedy L. 2023. Fully extended r-spin TQFTs. Quantum Topology. 14(3), 467–532."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"10","department":[{"_id":"MiLe"}],"file":[{"access_level":"open_access","content_type":"application/pdf","success":1,"file_name":"2023_QuantumTopol_Carqueville.pdf","checksum":"b0590aff6e7ec89cc149ba94d459d3a3","relation":"main_file","creator":"dernst","date_updated":"2024-01-09T09:25:34Z","file_size":707344,"date_created":"2024-01-09T09:25:34Z","file_id":"14764"}]},{"volume":381,"date_created":"2020-11-29T23:01:17Z","article_type":"original","scopus_import":"1","day":"01","author":[{"first_name":"Ingo","full_name":"Runkel, Ingo","last_name":"Runkel"},{"full_name":"Szegedy, Lorant","id":"7943226E-220E-11EA-94C7-D59F3DDC885E","last_name":"Szegedy","orcid":"0000-0003-2834-5054","first_name":"Lorant"}],"oa_version":"Published Version","title":"Area-dependent quantum field theory","file_date_updated":"2021-02-03T15:00:30Z","publication_identifier":{"eissn":["14320916"],"issn":["00103616"]},"publication_status":"published","has_accepted_license":"1","intvolume":"       381","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"abstract":[{"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.","lang":"eng"}],"department":[{"_id":"MiLe"}],"file":[{"relation":"main_file","checksum":"6f451f9c2b74bedbc30cf884a3e02670","success":1,"file_name":"2021_CommMathPhys_Runkel.pdf","access_level":"open_access","content_type":"application/pdf","file_id":"9081","date_created":"2021-02-03T15:00:30Z","file_size":790526,"creator":"dernst","date_updated":"2021-02-03T15:00:30Z"}],"month":"01","citation":{"mla":"Runkel, Ingo, and Lorant Szegedy. “Area-Dependent Quantum Field Theory.” <i>Communications in Mathematical Physics</i>, vol. 381, no. 1, Springer Nature, 2021, pp. 83–117, doi:<a href=\"https://doi.org/10.1007/s00220-020-03902-1\">10.1007/s00220-020-03902-1</a>.","apa":"Runkel, I., &#38; Szegedy, L. (2021). Area-dependent quantum field theory. <i>Communications in Mathematical Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00220-020-03902-1\">https://doi.org/10.1007/s00220-020-03902-1</a>","chicago":"Runkel, Ingo, and Lorant Szegedy. “Area-Dependent Quantum Field Theory.” <i>Communications in Mathematical Physics</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s00220-020-03902-1\">https://doi.org/10.1007/s00220-020-03902-1</a>.","ista":"Runkel I, Szegedy L. 2021. Area-dependent quantum field theory. Communications in Mathematical Physics. 381(1), 83–117.","short":"I. Runkel, L. Szegedy, Communications in Mathematical Physics 381 (2021) 83–117.","ieee":"I. Runkel and L. Szegedy, “Area-dependent quantum field theory,” <i>Communications in Mathematical Physics</i>, vol. 381, no. 1. Springer Nature, pp. 83–117, 2021.","ama":"Runkel I, Szegedy L. Area-dependent quantum field theory. <i>Communications in Mathematical Physics</i>. 2021;381(1):83–117. doi:<a href=\"https://doi.org/10.1007/s00220-020-03902-1\">10.1007/s00220-020-03902-1</a>"},"issue":"1","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"language":[{"iso":"eng"}],"_id":"8816","date_updated":"2023-08-04T11:13:35Z","type":"journal_article","article_processing_charge":"Yes (via OA deal)","doi":"10.1007/s00220-020-03902-1","publisher":"Springer Nature","quality_controlled":"1","page":"83–117","ddc":["510"],"year":"2021","isi":1,"external_id":{"isi":["000591139000001"]},"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).","date_published":"2021-01-01T00:00:00Z","status":"public","publication":"Communications in Mathematical Physics","project":[{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}]},{"arxiv":1,"month":"10","department":[{"_id":"MiLe"}],"article_number":"102302","oa":1,"language":[{"iso":"eng"}],"issue":"10","citation":{"ieee":"I. Runkel and L. Szegedy, “Topological field theory on r-spin surfaces and the Arf-invariant,” <i>Journal of Mathematical Physics</i>, vol. 62, no. 10. AIP Publishing, 2021.","short":"I. Runkel, L. Szegedy, Journal of Mathematical Physics 62 (2021).","ama":"Runkel I, Szegedy L. Topological field theory on r-spin surfaces and the Arf-invariant. <i>Journal of Mathematical Physics</i>. 2021;62(10). doi:<a href=\"https://doi.org/10.1063/5.0037826\">10.1063/5.0037826</a>","apa":"Runkel, I., &#38; Szegedy, L. (2021). Topological field theory on r-spin surfaces and the Arf-invariant. <i>Journal of Mathematical Physics</i>. AIP Publishing. <a href=\"https://doi.org/10.1063/5.0037826\">https://doi.org/10.1063/5.0037826</a>","mla":"Runkel, Ingo, and Lorant Szegedy. “Topological Field Theory on R-Spin Surfaces and the Arf-Invariant.” <i>Journal of Mathematical Physics</i>, vol. 62, no. 10, 102302, AIP Publishing, 2021, doi:<a href=\"https://doi.org/10.1063/5.0037826\">10.1063/5.0037826</a>.","ista":"Runkel I, Szegedy L. 2021. Topological field theory on r-spin surfaces and the Arf-invariant. Journal of Mathematical Physics. 62(10), 102302.","chicago":"Runkel, Ingo, and Lorant Szegedy. “Topological Field Theory on R-Spin Surfaces and the Arf-Invariant.” <i>Journal of Mathematical Physics</i>. AIP Publishing, 2021. <a href=\"https://doi.org/10.1063/5.0037826\">https://doi.org/10.1063/5.0037826</a>."},"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","author":[{"last_name":"Runkel","full_name":"Runkel, Ingo","first_name":"Ingo"},{"full_name":"Szegedy, Lorant","id":"7943226E-220E-11EA-94C7-D59F3DDC885E","last_name":"Szegedy","orcid":"0000-0003-2834-5054","first_name":"Lorant"}],"scopus_import":"1","day":"01","title":"Topological field theory on r-spin surfaces and the Arf-invariant","oa_version":"Preprint","volume":62,"article_type":"original","date_created":"2021-10-24T22:01:32Z","intvolume":"        62","abstract":[{"text":"We give a combinatorial model for r-spin surfaces with parameterized boundary based on Novak (“Lattice topological field theories in two dimensions,” Ph.D. thesis, Universität Hamburg, 2015). The r-spin structure is encoded in terms of ℤ𝑟-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state-sum construction of two-dimensional topological field theories on r-spin surfaces. We show that an example of such a topological field theory computes the Arf-invariant of an r-spin surface as introduced by Randal-Williams [J. Topol. 7, 155 (2014)] and Geiges et al. [Osaka J. Math. 49, 449 (2012)]. This implies, in particular, that the r-spin Arf-invariant is constant on orbits of the mapping class group, providing an alternative proof of that fact.","lang":"eng"}],"publication_status":"published","publication_identifier":{"issn":["00222488"]},"isi":1,"year":"2021","external_id":{"arxiv":["1802.09978"],"isi":["000755638500010"]},"status":"public","publication":"Journal of Mathematical Physics","date_published":"2021-10-01T00:00:00Z","acknowledgement":"We would like to thank Nils Carqueville, Tobias Dyckerhoff, Jan Hesse, Ehud Meir, Sebastian Novak, Louis-Hadrien Robert, Nick Salter, Walker Stern, and Lukas Woike for helpful discussions and comments. L.S. was supported by the DFG Research Training Group 1670 “Mathematics Inspired by String Theory and Quantum Field Theory.”","doi":"10.1063/5.0037826","article_processing_charge":"No","publisher":"AIP Publishing","date_updated":"2023-08-14T08:04:12Z","_id":"10176","type":"journal_article","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1802.09978"}],"quality_controlled":"1"}]