---
_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: '8325'
abstract:
- lang: eng
  text: "Let \U0001D439:ℤ2→ℤ be the pointwise minimum of several linear functions.
    The theory of smoothing allows us to prove that under certain conditions there
    exists the pointwise minimal function among all integer-valued superharmonic functions
    coinciding with F “at infinity”. We develop such a theory to prove existence of
    so-called solitons (or strings) in a sandpile model, studied by S. Caracciolo,
    G. Paoletti, and A. Sportiello. Thus we made a step towards understanding the
    phenomena of the identity in the sandpile group for planar domains where solitons
    appear according to experiments. We prove that sandpile states, defined using
    our smoothing procedure, move changeless when we apply the wave operator (that
    is why we call them solitons), and can interact, forming triads and nodes. "
acknowledgement: We thank Andrea Sportiello for sharing his insights on perturbative
  regimes of the Abelian sandpile model which was the starting point of our work.
  We also thank Grigory Mikhalkin, who encouraged us to approach this problem. We
  thank an anonymous referee. Also we thank Misha Khristoforov and Sergey Lanzat who
  participated on the initial state of this project, when we had nothing except the
  computer simulation and pictures. We thank Mikhail Raskin for providing us the code
  on Golly for faster simulations. Ilia Zharkov, Ilia Itenberg, Kristin Shaw, Max
  Karev, Lionel Levine, Ernesto Lupercio, Pavol Ševera, Yulieth Prieto, Michael Polyak,
  Danila Cherkashin asked us a lot of questions and listened to us; not all of their
  questions found answers here, but we are going to treat them in subsequent papers.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Nikita
  full_name: Kalinin, Nikita
  last_name: Kalinin
- first_name: Mikhail
  full_name: Shkolnikov, Mikhail
  id: 35084A62-F248-11E8-B48F-1D18A9856A87
  last_name: Shkolnikov
  orcid: 0000-0002-4310-178X
citation:
  ama: Kalinin N, Shkolnikov M. Sandpile solitons via smoothing of superharmonic functions.
    <i>Communications in Mathematical Physics</i>. 2020;378(9):1649-1675. doi:<a href="https://doi.org/10.1007/s00220-020-03828-8">10.1007/s00220-020-03828-8</a>
  apa: Kalinin, N., &#38; Shkolnikov, M. (2020). Sandpile solitons via smoothing of
    superharmonic functions. <i>Communications in Mathematical Physics</i>. Springer
    Nature. <a href="https://doi.org/10.1007/s00220-020-03828-8">https://doi.org/10.1007/s00220-020-03828-8</a>
  chicago: Kalinin, Nikita, and Mikhail Shkolnikov. “Sandpile Solitons via Smoothing
    of Superharmonic Functions.” <i>Communications in Mathematical Physics</i>. Springer
    Nature, 2020. <a href="https://doi.org/10.1007/s00220-020-03828-8">https://doi.org/10.1007/s00220-020-03828-8</a>.
  ieee: N. Kalinin and M. Shkolnikov, “Sandpile solitons via smoothing of superharmonic
    functions,” <i>Communications in Mathematical Physics</i>, vol. 378, no. 9. Springer
    Nature, pp. 1649–1675, 2020.
  ista: Kalinin N, Shkolnikov M. 2020. Sandpile solitons via smoothing of superharmonic
    functions. Communications in Mathematical Physics. 378(9), 1649–1675.
  mla: Kalinin, Nikita, and Mikhail Shkolnikov. “Sandpile Solitons via Smoothing of
    Superharmonic Functions.” <i>Communications in Mathematical Physics</i>, vol.
    378, no. 9, Springer Nature, 2020, pp. 1649–75, doi:<a href="https://doi.org/10.1007/s00220-020-03828-8">10.1007/s00220-020-03828-8</a>.
  short: N. Kalinin, M. Shkolnikov, Communications in Mathematical Physics 378 (2020)
    1649–1675.
date_created: 2020-08-30T22:01:13Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2023-08-22T09:00:03Z
day: '01'
department:
- _id: TaHa
doi: 10.1007/s00220-020-03828-8
ec_funded: 1
external_id:
  arxiv:
  - '1711.04285'
  isi:
  - '000560620600001'
intvolume: '       378'
isi: 1
issue: '9'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1711.04285
month: '09'
oa: 1
oa_version: Preprint
page: 1649-1675
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '291734'
  name: International IST Postdoc Fellowship Programme
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: Sandpile solitons via smoothing of superharmonic functions
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 378
year: '2020'
...