Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 102, 51 pages      arXiv:2312.01946      https://doi.org/10.3842/SIGMA.2024.102

The Evaluation of Graphs on Surfaces for State-Sum Models with Defects

Julian Farnsteiner and Christoph Schweigert
Fachbereich Mathematik, Universität Hamburg, Bereich Algebra und Zahlentheorie, Bundesstraße 55, 20146 Hamburg, Germany

Received December 10, 2023, in final form November 06, 2024; Published online November 18, 2024

Abstract
The evaluation of graphs on 2-spheres is a central ingredient of the Turaev-Viro construction of three-dimensional topological field theories. In this article, we introduce a class of graphs, called extruded graphs, that is relevant for the Turaev-Viro construction with general defect configurations involving defects of various dimensions. We define the evaluation of extruded graphs and show that it is invariant under a set of moves. This ensures the computability and uniqueness of our evaluation.

Key words: state-sum; vertex evaluation; topological quantum field theory; $6j$ symbols; defects.

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References

  1. Balsam B., Turaev-Viro theory as an extended TQFT, Ph.D. Thesis, State University of New York at Stony Brook, 2010, arXiv:1004.1533.
  2. Barrett J.W., Meusburger C., Schaumann G., Gray categories with duals and their diagrams, Adv. Math. 450 (2024), 109740, 129 pages, arXiv:1211.0529.
  3. Barrett J.W., Westbury B.W., Spherical categories, Adv. Math. 143 (1999), 357-375, arXiv:hep-th/9310164.
  4. Bartlett B., Douglas C., Schommer-Pries C., Vicary J., Modular categories as representations of the 3-dimensional bordism 2-category, arXiv:1509.06811.
  5. Carqueville N., Orbifolds of topological quantum field theories, arXiv:2307.16674.
  6. Carqueville N., Meusburger C., Schaumann G., 3-dimensional defect TQFTs and their tricategories, Adv. Math. 364 (2020), 107024, 58 pages, arXiv:1603.01171.
  7. Carqueville N., Müller L., Orbifold completion of 3-categories, arXiv:2307.06485.
  8. Carqueville N., Runkel I., Schaumann G., Orbifolds of $n$-dimensional defect TQFTs, Geom. Topol. 23 (2019), 781-864, arXiv:1705.06085.
  9. Carqueville N., Zotto M.D., Runkel I., Topological defects, in Encyclopedia of Mathematical Physics, Vol. 5, Elsevier, 2025, 621-647, arXiv:2311.02449.
  10. Costello K., Topological conformal field theories and Calabi-Yau categories, Adv. Math. 210 (2007), 165-214, arXiv:math.QA/0412149.
  11. Davydov A., Kong L., Runkel I., Field theories with defects and the centre functor, in Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proc. Sympos. Pure Math., Vol. 83, American Mathematical Society, Providence, RI, 2011, 71-128, arXiv:1107.0495.
  12. Douglas C.L., Schommer-Pries C., Snyder N., The balanced tensor product of module categories, Kyoto J. Math. 59 (2019), 167-179, arXiv:1406.4204.
  13. Douglas C.L., Schommer-Pries C., Snyder N., Dualizable tensor categories, Mem. Amer. Math. Soc. 268 (2020), vii+88 pages, arXiv:1312.7188.
  14. Etingof P., Gelaki S., Nikshych D., Ostrik V., Tensor categories, Math. Surveys Monogr., Vol. 205, American Mathematical Society, Providence, RI, 2015.
  15. Etingof P., Nikshych D., Ostrik V., Fusion categories and homotopy theory, Quantum Topol. 1 (2010), 209-273, arXiv:0909.3140.
  16. Farnsteiner J., Towards Turaev-Viro topological field theories on stratified manifolds. Evaluation at vertices and local moves, Ph.D. Thesis, Universität Hamburg, 2023, https://ediss.sub.uni-hamburg.de/handle/ediss/10494.
  17. Fuchs J., Priel J., Schweigert C., Valentino A., On the Brauer groups of symmetries of abelian Dijkgraaf-Witten theories, Comm. Math. Phys. 339 (2015), 385-405, arXiv:1404.6646.
  18. Fuchs J., Schaumann G., Schweigert C., A trace for bimodule categories, Appl. Categ. Structures 25 (2017), 227-268, arXiv:1412.6968.
  19. Fuchs J., Schaumann G., Schweigert C., Eilenberg-Watts calculus for finite categories and a bimodule Radford $S^4$ theorem, Trans. Amer. Math. Soc. 373 (2020), 1-40, arXiv:1612.04561.
  20. Fuchs J., Schaumann G., Schweigert C., A modular functor from state sums for finite tensor categories and their bimodules, Theory Appl. Categ. 38 (2022), 436-594, arXiv:1911.06214.
  21. Fuchs J., Schweigert C., Yang Y., String-net models for pivotal bicategories, arXiv:2302.01468.
  22. Kapustin A., Saulina N., Surface operators in 3d topological field theory and 2d rational conformal field theory, in Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proc. Sympos. Pure Math., Vol. 83, American Mathematical Society, Providence, RI, 2011, 175-198, arXiv:1012.0911.
  23. Koppen V., Mulevičius V., Runkel I., Schweigert C., Domain walls between 3d phases of Reshetikhin-Turaev TQFTs, Comm. Math. Phys. 396 (2022), 1187-1220, arXiv:2105.04613.
  24. MacLane S., Categories for the working mathematician, Grad. Texts in Math., Vol. 5, Springer, New York, 1971.
  25. Meusburger C., State sum models with defects based on spherical fusion categories, Adv. Math. 429 (2023), 109177, 97 pages, arXiv:2205.06874.
  26. Moore G.W., Segal G., D-branes and K-theory in 2D topological field theory, arXiv:hep-th/0609042.
  27. Ng S.-H., Schauenburg P., Higher Frobenius-Schur indicators for pivotal categories, in Hopf Algebras and Generalizations, Contemp. Math., Vol. 441, American Mathematical Society, Providence, RI, 2007, 63-90, arXiv:math.QA/0503167.
  28. Schaumann G., Duals in tricategories and in the tricategory of bimodule categories, Ph.D. Thesis, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2013.
  29. Schaumann G., Traces on module categories over fusion categories, J. Algebra 379 (2013), 382-425, arXiv:1206.5716.
  30. Turaev V., Virelizier A., Monoidal categories and topological field theory, Progr. Math., Vol. 322, Birkhäuser, Cham, 2017.

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