Skip to main content
SpringerLink
Log in

Two-Dimensional State Sum Models and Spin Structures

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

The state sum models in two dimensions introduced by Fukuma, Hosono and Kawai are generalised by allowing algebraic data from a non-symmetric Frobenius algebra. Without any further data, this leads to a state sum model on the sphere. When the data is augmented with a crossing map, the partition function is defined for any oriented surface with a spin structure. An algebraic condition that is necessary for the state sum model to be sensitive to spin structure is determined. Some examples of state sum models that distinguish topologically-inequivalent spin structures are calculated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Fukuma M., Hosono S., Kawai H.: Lattice topological field theory in two dimensions. Commun. Math. Phys. 161, 157–175 (1994). doi:10.1007/BF02099416

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Lauda A.D., Pfeiffer H.: State sum construction of two-dimensional open-closed topological quantum field theories. J. Knot Theory Ram. 16(09), 1121–1163 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baxter R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, New York (1982)

    MATH  Google Scholar 

  4. Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2-D gravity and random matrices. Phys. Rep. 254, 1–133 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  5. Kauffman Louis H.: State models and the Jones polynomial. Topology 26(3), 395–407 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. Oeckl R., Pfeiffer H.: The dual of pure non-Abelian lattice gauge theory as a spin foam model. Nucl. Phys. B 598, 400–426 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  7. Barrett, J.W.: State sum models for quantum gravity. In: Fokas, E. (ed.) XIIIth International Congress of Mathematical Physics. International Press, Boston (2001). arXiv:gr-qc/0010050

  8. Levin M.A., Wen X.-G.: String net condensation: a physical mechanism for topological phases. Phys. Rev. B 71, 045110 (2005)

    Article  ADS  Google Scholar 

  9. Witten E.: On quantum gauge theories in two-dimensions. Commun. Math. Phys. 141, 153–209 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Fuchs J., Runkel I., Schweigert C.: Conformal correlation functions, Frobenius algebras and triangulations. Nucl. Phys. B 624, 452–468 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Major S.A.: A spin network primer. Am. J. Phys. 67, 972–980 (1999)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Kauffman, L.H.: Knot diagrammatics. In: Thistlethwaite, M., Menasco, W. (eds.) Handbook of Knot Theory, pp. 233–318. Elsevier, USA (2005)

  13. Barrett J.W., Kerr S., Louko J.: A topological state sum model for fermions on the circle. J. Phys. A46, 185201 (2013)

    ADS  MathSciNet  Google Scholar 

  14. Cimasoni D., Reshetikhin N.: Dimers on surface graphs and spin structures. I. Commun. Math. Phys. 275, 187–208 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Novak, S.: Topological field theory on spin surfaces. In: The 11th Annual Simons Summer Workshop in Mathematics and Physics—Defects (2013)

  16. Novak, S., Runkel, I.: State sum construction of two-dimensional topological quantum field theories on spin surfaces (2014). arXiv:1402.2839

  17. Douglas, C.L., Schommer-Pries, C., Snyder, N.: Dualizable tensor categories (2013). arXiv:1312.7188

  18. Pachner U.: P.l. homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12(2), 129–145 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Lickorish, W.B.R.: Simplicial moves on complexes and manifolds. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998), vol. 2 of Geom. Topol. Monogr., pp. 299–320. Geom. Topol. Publ., Coventry (1999)

  20. Coward A., Lackenby M.: An upper bound on Reidemeister moves. Am. J. Math. 136, 1023–1066 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Bröcker, T., tom Dieck, T.: Representations of compact Lie groups. In: Graduate Texts in Mathematics. Springer, New York (1985)

  22. Witten E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141, 153–209 (1991). doi:10.1007/BF02100009

    Article  ADS  MATH  MathSciNet  Google Scholar 

  23. Davydov, A., Kong, L., Runkel, I.: Field theories with defects and the centre functor. In: Math. Found. of Quant. Field and Perturbative String Theory—Proceedings of Symposia in Pure Mathematics, vol. 83 (2011)

  24. Kadison, L., Stolin, A.A.: Separability and Hopf algebras. In: Algebra and its Applications (Athens, OH, 1999), vol. 259 of Contemp. Math., pp. 279–298. Am. Math. Soc., Providence (2000)

  25. Aguiar, M.: A note on strongly separable algebras. Bol. Acad. Nac. Cienc. (Córdoba) 65, 51–60 (2000). [Colloquium on Homology and Representation Theory (Spanish) (Vaquerías, 1998)]

  26. Kock, J.: Frobenius algebras and 2-d topological quantum field theories. In: London Mathematical Society Student Texts. Cambridge University Press, Cambridge(2003)

  27. Barrett J.W., Westbury B.W.: Spherical categories. Adv. Math. 143(2), 357–375 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Hirsch M.W.: Smooth regular neighborhoods. Ann. Math. 76(2), 524–530 (1962)

    Article  MATH  Google Scholar 

  29. Haefliger A., Poenaru V.: La classification des immersions combinatoires. Inst. Hautes Études Sci. Publ. Math. 23, 75–91 (1964)

    Article  MathSciNet  Google Scholar 

  30. Pinkall U.: Regular homotopy classes of immersed surfaces. Topology 24(4), 421–434 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  31. Kauffman L.H., Manturov V.O.: Virtual knots and links. Proc. Steklov Inst. Math. 252(1), 104–121 (2006)

    Article  MathSciNet  Google Scholar 

  32. Kauffman L.H.: An invariant of regular isotopy. Trans. Am. Math. Soc. 318, 417–471 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  33. Freyd P.J., Yetter D.N.: Braided compact closed categories with applications to low-dimensional topology. Adv. Math. 77(2), 156–182 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  34. Yetter D.N.: Category theoretic representations of knotted graphs in s3. Adv. Math. 77(2), 137–155 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  35. Reshetikhin N.Yu., Turaev V.G.: Ribbon graphs and their invariants derived from quantum groups. Commun. Math. Phys. 127(1), 1–26 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Bahturin Yu.A., Parmenter M.M.: Generalized commutativity in group algebras. Can. Math. Bull. 46(1), 14–25 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Barrett J.W.: Skein spaces and spin structures. Math. Proc. Camb. Philos. Soc. 126, 267–275 (1999)

    Article  MATH  Google Scholar 

  38. Kirby, R.C.: The Topology of 4-manifolds. In: Lecture Notes In Mathematics: Subseries. Springer, New York (1989)

  39. Lickorish W.B.R.: A finite set of generators for the homeotopy group of a 2-manifold. Proc. Camb. Philos. Soc. 60, 769–778 (1964)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  40. Lickorish W.B.R.: Corrigendum: on the homeotopy group of a 2-manifold. Proc. Camb. Philos. Soc. 62, 679–681 (1966)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Street, R.: Frobenius algebras and monoidal categories. In: Lecture at the Annual Meeting of the Australian Math. Soc. (2004). http://maths.mq.edu.au/~street/Publications.htm

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, University of Nottingham, Nottingham, UK

    John W. Barrett & Sara O. G. Tavares

Authors
  1. John W. Barrett
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sara O. G. Tavares
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John W. Barrett.

Additional information

Communicated by N. Reshetikhin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barrett, J.W., Tavares, S.O.G. Two-Dimensional State Sum Models and Spin Structures. Commun. Math. Phys. 336, 63–100 (2015). https://doi.org/10.1007/s00220-014-2246-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2246-z

Keywords

Search

Navigation