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Communications in Mathematical Physics

, Volume 360, Issue 2, pp 663–714 | Cite as

Dichromatic State Sum Models for Four-Manifolds from Pivotal Functors

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Article

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. Akb16.
    Akbulut, S.: 4-Manifolds. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford (2016). ISBN: 9780191827136Google Scholar
  2. Bae00.
    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]ADSCrossRefMATHGoogle Scholar
  3. BW15.
    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]ADSMathSciNetCrossRefMATHGoogle Scholar
  4. BFG07.
    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 ADSMathSciNetCrossRefMATHGoogle Scholar
  5. BMS12.
    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. Bar95.
    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]ADSMathSciNetCrossRefMATHGoogle Scholar
  7. BC98.
    Barrett J.W., Crane L.: Relativistic spin networks and quantum gravity. J. Math. Phys. 39(6), 3296–3302 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. Bar03.
    Barrett J.: Geometrical measurements in three-dimensional quantum gravity. Int. J. Mod. Phys. A 18, 97–113 (2003) arXiv:gr-qc/0203018 [gr-qc]ADSMathSciNetCrossRefMATHGoogle Scholar
  9. Bro93.
    Broda, B.: Surgical invariants of four manifolds. In: Quantum Topology: Proceedings, pp. 45–50 (1993). arXiv:hep-th/9302092 [hep-th]
  10. Bru00.
    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. CG11.
    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]MathSciNetMATHGoogle Scholar
  12. CKY93.
    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).Google Scholar
  13. CYK97.
    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]MathSciNetCrossRefMATHGoogle Scholar
  14. CBS13.
    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]ADSCrossRefGoogle Scholar
  15. Dav97.
    Davydov, A.A.: Quasitriangular structures on cocommutative Hopf algebras. In: ArXiv e-prints (1997). arXiv:q-alg/9706007 [q-alg]
  16. Del02.
    Deligne P.: Catégories tensorielles. Mosc. Math. J. 2, 227–248 (2002)MathSciNetMATHGoogle Scholar
  17. Dri+10.
    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. Eng+08.
    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]ADSMathSciNetCrossRefMATHGoogle Scholar
  19. ENO05.
    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. Fre+05.
    Freedman M.H. et al.: Universal manifold pairings and positivity. Geom. Topol. 9(4), 2303–2317 (2005) arXiv:math/0503054 MathSciNetCrossRefMATHGoogle Scholar
  21. GS99.
    Gompf, R.E., Stipsicz, A.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics. American Mathematical Society (1999). ISBN: 9780821809945Google Scholar
  22. HPT16.
    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]MathSciNetMATHGoogle Scholar
  23. Kir89.
    Kirby, R.C.: The Topology of 4-Manifolds. Lecture Notes in Mathematics, vol. 1374, pp. vi+108. Springer, Berlin (1989). ISBN: 3-540-51148-2Google Scholar
  24. Kir11.
    Kirillov Jr., A.: String-net model of Turaev-Viro invariants. In: ArXiv e-prints (2011). arXiv:1106.6033 [math.AT]
  25. Lic93.
    Lickorish W.: The skein method for three-manifold invariants. J. Knot Theory Ramif. 2(2), 171–194 (1993).  https://doi.org/10.1142/S0218216593000118 MathSciNetCrossRefMATHGoogle Scholar
  26. ML63.
    Mac Lane, S.: Natural associativity and commutativity. Rice Univ. Stud. 49.4:28-46 (1963). ISSN: 0035-4996Google Scholar
  27. Mac00.
    Mackaay M.: Finite groups, spherical 2-categories, and 4-manifold invariants. Adv. Math. 153(2), 353–390 (2000) arXiv:math/9903003 MathSciNetCrossRefMATHGoogle Scholar
  28. Maj00.
    Majid S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  29. Mu00.
    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. Mu03a.
    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. Mu03b.
    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. Pet08.
    Petit J.: The dichromatic invariants of smooth 4-manifolds. Glob. J. Pure Appl. Math. 4(3), 1–16 (2008)ADSGoogle Scholar
  33. Pfe09.
    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. Rob95.
    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. Rob97.
    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)Google Scholar
  36. Sa79.
    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)Google Scholar
  37. SP11.
    Schommer-Pries, C.: The classification of two-dimensional extended topological field theories. In: ArXiv e-prints (2011). arXiv:1112.1000 [math.AT]
  38. Sel10.
    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. Shu94.
    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. Sok97.
    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. TV92.
    Turaev V.G., Viro O.Y.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992)MathSciNetCrossRefMATHGoogle Scholar
  42. WW12.
    Walker K., Wang Z.: (3+1)-TQFTs and topological insulators. Front. Phys. 7, 150–159 (2012) arXiv:1104.2632 [cond-mat.str-el]CrossRefGoogle Scholar
  43. Wis10.
    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]ADSMathSciNetCrossRefMATHGoogle Scholar
  44. Wit89.
    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. Yet92.
    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 MathSciNetCrossRefMATHGoogle Scholar

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© The Author(s) 2017

Open AccessThis 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.

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.School of Mathematical SciencesUniversity of NottinghamUniversity Park, NottinghamUK

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