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

SIGMA 17 (2021), 040, 22 pages      arXiv:1911.10147

CFT Correlators for Cardy Bulk Fields via String-Net Models

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

Received November 01, 2020, in final form April 12, 2021; Published online April 21, 2021

We show that string-net models provide a novel geometric method to construct invariants of mapping class group actions. Concretely, we consider string-net models for a modular tensor category ${\mathcal C}$. We show that the datum of a specific commutative symmetric Frobenius algebra in the Drinfeld center $Z(\mathcal{C})$ gives rise to invariant string-nets. The Frobenius algebra has the interpretation of the algebra of bulk fields of the conformal field theory in the Cardy case.

Key words: two-dimensional conformal field theory; string-net models; correlators; Cardy case.

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  1. Bakalov B., Kirillov Jr. A., On the Lego-Teichmüller game, Transform. Groups 5 (2000), 207-244, arXiv:math.GT/9809057.
  2. Bakalov B., Kirillov Jr. A., Lectures on tensor categories and modular functors, University Lecture Series, Vol. 21, Amer. Math. Soc., Providence, RI, 2001.
  3. Balsam B., Turaev-Viro theory as an extended TQFT II, arXiv:1010.1222.
  4. Balsam B., Turaev-Viro theory as an extended TQFT III, arXiv:1012.0560.
  5. Etingof P., Nikshych D., Ostrik V., On fusion categories, Ann. of Math. 162 (2005), 581-642, arXiv:math.QA/0203060.
  6. Felder G., Fröhlich J., Fuchs J., Schweigert C., Correlation functions and boundary conditions in rational conformal field theory and three-dimensional topology, Compositio Math. 131 (2002), 189-237, arXiv:hep-th/9912239.
  7. Fjelstad J., Fuchs J., Runkel I., Schweigert C., TFT construction of RCFT correlators. V. Proof of modular invariance and factorisation, Theory Appl. Categ. 16 (2006), 16, 342-433, arXiv:hep-th/0503194.
  8. Fröhlich J., Fuchs J., Runkel I., Schweigert C., Correspondences of ribbon categories, Adv. Math. 199 (2006), 192-329, arXiv:math.CT/0309465.
  9. Fuchs J., Gannon T., Schaumann G., Schweigert C., The logarithmic Cardy case: boundary states and annuli, Nuclear Phys. B 930 (2018), 287-327, arXiv:1712.01922.
  10. Fuchs J., Runkel I., Schweigert C., TFT construction of RCFT correlators. I. Partition functions, Nuclear Phys. B 646 (2002), 353-497, arXiv:hep-th/0204148.
  11. Fuchs J., Runkel I., Schweigert C., TFT construction of RCFT correlators. II. Unoriented world sheets, Nuclear Phys. B 678 (2004), 511-637, arXiv:hep-th/0306164.
  12. Fuchs J., Runkel I., Schweigert C., TFT construction of RCFT correlators. III. Simple currents, Nuclear Phys. B 694 (2004), 277-353, arXiv:hep-th/0403157.
  13. Fuchs J., Runkel I., Schweigert C., TFT construction of RCFT correlators. IV. Structure constants and correlation functions, Nuclear Phys. B 715 (2005), 539-638, arXiv:hep-th/0412290.
  14. Fuchs J., Schweigert C., Consistent systems of correlators in non-semisimple conformal field theory, Adv. Math. 307 (2017), 598-639, arXiv:1604.01143.
  15. Goosen G., Oriented 123-TQFTs via string-nets and state-sums, Ph.D. Thesis, Stellenbosch University, 2018.
  16. Hatcher A., Lochak P., Schneps L., On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521 (2000), 1-24.
  17. Kirillov Jr. A., String-net model of Turaev-Viro invariants, arXiv:1106.6033.
  18. Kirillov Jr. A., Balsam B., Turaev-Viro invariants as an extended TQFT, arXiv:1004.1533.
  19. Koenig R., Kuperberg G., Reichardt B.W., Quantum computation with Turaev-Viro codes, Ann. Physics 325 (2010), 2707-2749, arXiv:1002.2816.
  20. Kong L., Runkel I., Morita classes of algebras in modular tensor categories, Adv. Math. 219 (2008), 1548-1576, arXiv:0708.1897.
  21. Kong L., Runkel I., Cardy algebras and sewing constraints. I, Comm. Math. Phys. 292 (2009), 871-912, arXiv:0807.3356.
  22. Levin M.A., Wen X.-G., String-net condensation: a physical mechanism for topological phases, Phys. Rev. B 71 (2005), 045110, 21 pages, arXiv:cond-mat/0404617.
  23. Ng S.-H., Schauenburg P., Higher Frobenius-Schur indicators for pivotal categories, in Hopf algebras and generalizations, Contemp. Math., Vol. 441, Amer. Math. Soc., Providence, RI, 2007, 63-90, arXiv:math.QA/0503167.
  24. Shimizu K., Non-degeneracy conditions for braided finite tensor categories, Adv. Math. 355 (2019), 106778, 36 pages, arXiv:1602.06534.
  25. Traube M., Cardy algebras, sewing constraints and string-nets, arXiv:2009.11895.
  26. Turaev V., Virelizier A., On two approaches to 3-dimensional TQFTs, arXiv:1006.3501.

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