---
_id: '14756'
abstract:
- lang: eng
  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."
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."
article_processing_charge: Yes
article_type: original
author:
- first_name: Nils
  full_name: Carqueville, Nils
  last_name: Carqueville
- first_name: Lorant
  full_name: Szegedy, Lorant
  id: 7943226E-220E-11EA-94C7-D59F3DDC885E
  last_name: Szegedy
  orcid: 0000-0003-2834-5054
citation:
  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>
  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>.
  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.
  ista: Carqueville N, Szegedy L. 2023. Fully extended r-spin TQFTs. Quantum Topology.
    14(3), 467–532.
  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>.
  short: N. Carqueville, L. Szegedy, Quantum Topology 14 (2023) 467–532.
date_created: 2024-01-08T13:14:48Z
date_published: 2023-10-16T00:00:00Z
date_updated: 2024-01-09T09:27:46Z
day: '16'
ddc:
- '530'
department:
- _id: MiLe
doi: 10.4171/qt/193
file:
- access_level: open_access
  checksum: b0590aff6e7ec89cc149ba94d459d3a3
  content_type: application/pdf
  creator: dernst
  date_created: 2024-01-09T09:25:34Z
  date_updated: 2024-01-09T09:25:34Z
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  file_name: 2023_QuantumTopol_Carqueville.pdf
  file_size: 707344
  relation: main_file
  success: 1
file_date_updated: 2024-01-09T09:25:34Z
has_accepted_license: '1'
intvolume: '        14'
issue: '3'
keyword:
- Geometry and Topology
- Mathematical Physics
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
page: 467-532
publication: Quantum Topology
publication_identifier:
  issn:
  - 1663-487X
publication_status: published
publisher: European Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Fully extended r-spin TQFTs
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 14
year: '2023'
...
---
_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. <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>
  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>.
  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.
  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.” <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>.
  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'
...
---
_id: '10176'
abstract:
- lang: eng
  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
    ℤ\U0001D45F-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."
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.”
article_number: '102302'
article_processing_charge: No
article_type: original
arxiv: 1
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. 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>
  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>.
  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.
  ista: Runkel I, Szegedy L. 2021. Topological field theory on r-spin surfaces and
    the Arf-invariant. Journal of Mathematical Physics. 62(10), 102302.
  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>.
  short: I. Runkel, L. Szegedy, Journal of Mathematical Physics 62 (2021).
date_created: 2021-10-24T22:01:32Z
date_published: 2021-10-01T00:00:00Z
date_updated: 2023-08-14T08:04:12Z
day: '01'
department:
- _id: MiLe
doi: 10.1063/5.0037826
external_id:
  arxiv:
  - '1802.09978'
  isi:
  - '000755638500010'
intvolume: '        62'
isi: 1
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1802.09978
month: '10'
oa: 1
oa_version: Preprint
publication: Journal of Mathematical Physics
publication_identifier:
  issn:
  - '00222488'
publication_status: published
publisher: AIP Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Topological field theory on r-spin surfaces and the Arf-invariant
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 62
year: '2021'
...