Skip to main content

Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

Abstract

A family of invariants of smooth, oriented four-dimensional manifolds is defined via handle decompositions and the Kirby calculus of framed link diagrams. The invariants are parametrised by a pivotal functor from a spherical fusion category into a ribbon fusion category. A state sum formula for the invariant is constructed via the chain-mail procedure, so a large class of topological state sum models can be expressed as link invariants. Most prominently, the Crane-Yetter state sum over an arbitrary ribbon fusion category is recovered, including the nonmodular case. It is shown that the Crane-Yetter invariant for nonmodular categories is stronger than signature and Euler invariant. A special case is the four-dimensional untwisted Dijkgraaf–Witten model. Derivations of state space dimensions of TQFTs arising from the state sum model agree with recent calculations of ground state degeneracies in Walker-Wang models. Relations to different approaches to quantum gravity such as Cartan geometry and teleparallel gravity are also discussed.

References

  1. Akbulut, S.: 4-Manifolds. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2016). ISBN: 9780191827136

  2. Baez J.C.: An introduction to spin foam models of quantum gravity and BF theory. Lect. Notes Phys. 543, 25–94 (2000). https://doi.org/10.1007/3-540-46552-9_2. arXiv:gr-qc/9905087 [gr-qc]

    ADS  Article  MATH  Google Scholar 

  3. Baez J.C., Wise D.K.: Teleparallel gravity as a higher Gauge theory. Commun. Math. Phys. 333(1), 153–186 (2015) arXiv:1204.4339 [gr-qc]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  4. Barrett J.W., Faria Martins J., García-Islas J.M.: Observables in the Turaev-Viro and Crane-Yetter models. J. Math. Phys. 48(9), 093508 (2007). https://doi.org/10.1063/1.2759440. arXiv:math/0411281

    ADS  MathSciNet  Article  MATH  Google Scholar 

  5. Barrett, J.W., Meusburger, C., Schaumann, G.: Gray categories with duals and their diagrams. In: ArXiv e-prints (2012). arXiv:1211.0529 [math.QA]

  6. Barrett J.W.: Quantum gravity as topological quantum field theory. J. Math. Phys. 36, 6161–6179 (1995). https://doi.org/10.1063/1.531239. arXiv:gr-qc/9506070 [gr-qc]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. Barrett J.W., Crane L.: Relativistic spin networks and quantum gravity. J. Math. Phys. 39(6), 3296–3302 (1998)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. Barrett J.: Geometrical measurements in three-dimensional quantum gravity. Int. J. Mod. Phys. A 18, 97–113 (2003) arXiv:gr-qc/0203018 [gr-qc]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  9. Broda, B.: Surgical invariants of four manifolds. In: Quantum Topology: Proceedings, pp. 45–50 (1993). arXiv:hep-th/9302092 [hep-th]

  10. Bruguiéres, A.: Catégories prémodulaires, modularisations et invariants des variétés de dimension 3 (French). In: Mathematische Annalen 316.2, pp. 215-236 (2000). https://doi.org/10.1007/s002080050011 . ISSN: 0025-5831

  11. Cheng E., Gurski N.: The periodic table of n-categories for low dimensions II: degenerate tricategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques 52, 82–125 (2011) arXiv:0706.2307 [math.CT]

    MathSciNet  MATH  Google Scholar 

  12. Crane, L., Kauffman, L.H., Yetter, D.N.: On the classicality of Broda’s SU(2) invariants of four manifolds. In: ArXiv e-prints (1993).

  13. Crane L., Yetter D.N., Kauffman L.: State-sum invariants of 4-manifolds. J. Knot Theory Ramif. 6(2), 177–234 (1997) arXiv:hep-th/9409167 [hep-th]

    MathSciNet  Article  MATH  Google Scholar 

  14. von Keyserlingk C.W., Burnell F.J., Simon S.H.: Three-dimensional topological lattice models with surface anyons. Phys. Rev. B 87(4), 045107 (2013). https://doi.org/10.1103/PhysRevB.87.045107. arXiv:1208.5128 [cond-mat. str-el]

    ADS  Article  Google Scholar 

  15. Davydov, A.A.: Quasitriangular structures on cocommutative Hopf algebras. In: ArXiv e-prints (1997). arXiv:q-alg/9706007 [q-alg]

  16. Deligne P.: Catégories tensorielles. Mosc. Math. J. 2, 227–248 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Drinfeld, V. et al.: (2010) On braided fusion categories I (English). Sel. Math. 16.1:1-119. https://doi.org/10.1007/s00029-010-0017-z. ISSN: 1022-1824, arXiv:0906.0620 [math.QA]

  18. Engle J. et al.: LQG vertex with finite Immirzi parameter. Nucl. Phys. B 799, 136–149 (2008). https://doi.org/10.1016/j.nuclphysb.2008.02.018. arXiv:0711.0146 [gr-qc]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  19. Etingof, P., Nikshych, D., Ostrik, V.: On fusion categories (English). Ann. Math. Second Ser. 162.2:581−642 (2005). https://doi.org/10.4007/annals.2005.162.581. ISSN: 0003-486X, 1939-8980/e.

  20. Freedman M.H. et al.: Universal manifold pairings and positivity. Geom. Topol. 9(4), 2303–2317 (2005) arXiv:math/0503054

    MathSciNet  Article  MATH  Google Scholar 

  21. Gompf, R.E., Stipsicz, A.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics. American Mathematical Society (1999). ISBN: 9780821809945

  22. Henriques A., Penneys D., Tener J.: Categorified trace for module tensor categories over braided tensor categories. Doc. Math. 21, 1089–1149 (2016) arXiv:1509.02937 [math.QA]

    MathSciNet  MATH  Google Scholar 

  23. Kirby, R.C.: The Topology of 4-Manifolds. Lecture Notes in Mathematics, vol. 1374, pp. vi+108. Springer, Berlin (1989). ISBN: 3-540-51148-2

  24. Kirillov Jr., A.: String-net model of Turaev-Viro invariants. In: ArXiv e-prints (2011). arXiv:1106.6033 [math.AT]

  25. Lickorish W.: The skein method for three-manifold invariants. J. Knot Theory Ramif. 2(2), 171–194 (1993). https://doi.org/10.1142/S0218216593000118

    MathSciNet  Article  MATH  Google Scholar 

  26. Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49.4:28-46 (1963). ISSN: 0035-4996

  27. Mackaay M.: Finite groups, spherical 2-categories, and 4-manifold invariants. Adv. Math. 153(2), 353–390 (2000) arXiv:math/9903003

    MathSciNet  Article  MATH  Google Scholar 

  28. Majid S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  29. Müger, M.: Galois theory for braided tensor categories and the modular closure. Adv. Math. 150.2:151−201 (2000). http://dx.doi.org/10.1006/aima.1999.1860. ISSN: 0001-8708, arXiv:math/9812040, http://www.sciencedirect.com/science/article/pii/S0001870899918601

  30. Müger, M.: From subfactors to categories and topology. I. Frobenius algebras in and Morita equivalence of tensor categories. J. Pure Appl. Algebra 180.1-2:81−157 (2003). https://doi.org/10.1016/S0022-4049(02)00247-5. ISSN: 0022-4049

  31. Müger, M.: On the structure of modular categories. Proc. Lond. Math. Soc. 87.2:291-308 (2003). https://doi.org/10.1112/S0024611503014187. http://plms.oxfordjournals.org/content/87/2/291.full.pdf+html

  32. Petit J.: The dichromatic invariants of smooth 4-manifolds. Glob. J. Pure Appl. Math. 4(3), 1–16 (2008)

    ADS  Google Scholar 

  33. Pfeiffer, H.: Finitely semisimple spherical categories and modular categories are selfdual. Adv. Math. 221.5:1608-1652 (2009). https://doi.org/10.1016/j.aim.2009.03.002. ISSN: 0001-8708

  34. Roberts, J.: Skein theory and Turaev-Viro invariants. Topology 34.4:771-787 (1995). https://doi.org/10.1016/0040-9383(94)00053-0. ISSN: 0040-9383

  35. Roberts, J.: Refined state-sum invariants of 3- and 4-manifolds. In: Geometric Topology (Athens, GA, 1993), Vol. 2. AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI, pp. 217-234 (1997)

  36. de Sá, E.C.: A link calculus for 4-manifolds. In: Topology of Low-Dimensional Manifolds, Proceedings of the Second Sussex Conference, Lecture Notes in Mathematics, vol. 722, pp. 16–30 (1979)

  37. Schommer-Pries, C.: The classification of two-dimensional extended topological field theories. In: ArXiv e-prints (2011). arXiv:1112.1000 [math.AT]

  38. Selinger, P.: A survey of graphical languages for monoidal categories. In: New Structures for Physics, pp. 289-355. Springer (2010). arXiv:0908.3347 [math.CT]

  39. Shum, M.C.: Tortile tensor categories. J. Pure Appl. Algebra 93.1:57-110 (1994). https://doi.org/10.1016/0022-4049(92)00039-T. ISSN: 0022-4049, http://www.sciencedirect.com/science/article/pii/002240499200039T

  40. Sokolov, M.V.: Which lens spaces are distinguished by Turaev-Viro invariants. Math. Notes 61.3:384−387 (1997). https://doi.org/10.1007/BF02355426.. ISSN: 1573–8876.

  41. Turaev V.G., Viro O.Y.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  42. Walker K., Wang Z.: (3+1)-TQFTs and topological insulators. Front. Phys. 7, 150–159 (2012) arXiv:1104.2632 [cond-mat.str-el]

    Article  Google Scholar 

  43. Wise D.: MacDowell–Mansouri gravity and cartan geometry. Class. Quantum Gravity 27, 155010 (2010). https://doi.org/10.1088/0264-9381/27/15/155010. arXiv:gr-qc/0611154 [gr-qc]

    ADS  MathSciNet  Article  MATH  Google Scholar 

  44. Witten, E.: Topology-changing amplitudes in 2 + 1 dimensional gravity. Nucl. Phys. B 323.1:113-140 (1989). https://doi.org/10.1016/0550-3213(89)90591-9. ISSN: 0550-3213

  45. Yetter D.N.: Topological quantum field theories associated to finite groups and crossed G-sets. J. Knot Theory Ramif. 1, 1–20 (1992). https://doi.org/10.1142/S0218216592000021

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Manuel Bärenz.

Additional information

Communicated by C. Schweigert

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bärenz, M., Barrett, J. Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors. Commun. Math. Phys. 360, 663–714 (2018). https://doi.org/10.1007/s00220-017-3012-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-017-3012-9