Annales Henri Poincaré

, Volume 15, Issue 9, pp 1801–1865 | Cite as

Itsy Bitsy Topological Field Theory

Article

Abstract

We construct an elementary, combinatorial kind of topological quantum field theory (TQFT), based on curves, surfaces, and orientations. The construction derives from contact invariants in sutured Floer homology and is essentially an elaboration of a TQFT defined by Honda–Kazez–Matić. This topological field theory stores information in binary format on a surface and has “digital” creation and annihilation operators, giving a toy-model embodiment of “it from bit”.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baez, J., Stay, M.: Physics topology, logic and computation: a Rosetta Stone. In: New Structures for Physics. Lecture Notes in Physics, vol. 813, pp. 95–172. Springer, Berlin (2011)Google Scholar
  2. 2.
    Bennequin, D.: Entrelacements et équations de Pfaff. Third Schnepfenried geometry conference, vol. 1 (Schnepfenried, 1982). Astérisque, vol. 107, pp. 87–161. Soc. Math. France, Paris (1983)Google Scholar
  3. 3.
    Eliashberg Y.: Contact 3-manifolds twenty years since J. Martinet’s Work. Ann. Inst. Fourier (Grenoble) 42(1–2), 165–192 (1992)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Finkelstein D.: Space-time code. Phys. Rev. (2) 184, 1261–1271 (1969)CrossRefMATHMathSciNetADSGoogle Scholar
  5. 5.
    Freedman, M.H., Kitaev, A., Larsen, M.J., Wang, Z.: Topological quantum computation. Bull. Am. Math. Soc. (N.S.) 40(1), 31–38 (2003) (electronic). Mathematical challenges of the 21st century (Los Angeles, CA, 2000)Google Scholar
  6. 6.
    Frenkel I.B., Khovanov Mikhail G.: Canonical bases in tensor products and graphical calculus for \({U_q(\mathfrak{sl}_2)}\). Duke Math. J. 87(3), 409–480 (1997)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Giroux E.: Convexité en topologie de contact. Comment. Math. Helv. 66(4), 637–677 (1991)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Golovko, R.: The Embedded Contact Homology of Sutured Solid Tori I. (2009). http://arxiv.org/abs/0911.0055
  9. 9.
    Golovko, R.: The Cylindrical Contact Homology of Universally Tight Sutured Contact Solid Tori. (2010). http://arxiv.org/abs/1006.4073
  10. 10.
    Honda, K.: On the classification of tight contact structures. I. Geom. Topol. 4, 309–368 (2000) (electronic)Google Scholar
  11. 11.
    Honda, K., Kazez, W.H., Matić, G.: Contact Structures, Sutured Floer Homology and TQFT. (2008). http://arxiv.org/abs/0807.2431
  12. 12.
    Honda K., Kazez W.H., Matić G.: On the contact class in Heegaard Floer homology. J. Differ. Geom. 83(2), 289–311 (2009)MATHGoogle Scholar
  13. 13.
    Juhász, A.: Holomorphic discs and sutured manifolds. Algebr. Geom. Topol. 6, 1429–1457 (2006) (electronic)Google Scholar
  14. 14.
    Juhász András: Floer homology and surface decompositions. Geom. Topol. 12(1), 299–350 (2008)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kauffman, L.H., Lins, S.L.: Temperley–Lieb recoupling theory and invariants of 3-manifolds. In: Annals of Mathematics Studies, vol. 134. Princeton University Press, Princeton (1994)Google Scholar
  16. 16.
    Kock, J.: Frobenius algebras and 2D topological quantum field theories. In: London Mathematical Society Student Texts, vol. 59. Cambridge University Press, Cambridge (2004)Google Scholar
  17. 17.
    Major Seth A.: A spin network primer. Am. J. Phys. 67(11), 972–980 (1999)CrossRefMATHADSGoogle Scholar
  18. 18.
    Massot, P.: Infinitely Many Universally Tight Torsion Free Contact Structures with Vanishing OzsvÁTh–SZabÓ Contact Invariants. (2009). http://arxiv.org/abs/0912.5107
  19. 19.
    Mathews, D.: Chord Diagrams, Contact-Topological Quantum Field Theory, and Contact Categories. Ph.D. thesis, Stanford University (2009). http://www.danielmathews.info/research
  20. 20.
    Mathews D.: Chord diagrams, contact-topological quantum field theory, and contact categories. Algebraic Geom. Topol. 10(4), 2091–2189 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Mathews, D.: Sutured Floer Homology, Sutured TQFT and Non-Commutative QFT. (2010). http://arxiv.org/abs/1006.5433
  22. 22.
    Nakamoto A.: Diagonal transformations in quadrangulations of surfaces. J. Graph Theory 21(3), 289–299 (1996)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Nakamoto A., Suzuki Y.: Diagonal slides and rotations in quadrangulations on the sphere. Yokohama Math. J. 55(2), 105–112 (2010)MATHMathSciNetGoogle Scholar
  24. 24.
    Negami S., Nakamoto A.: Diagonal transformations of graphs on closed surfaces. Sci. Rep. Yokohama Nat. Univ. Sect. I Math. Phys. Chem. 40, 71–97 (1993)MathSciNetGoogle Scholar
  25. 25.
    Ozsváth, P., Szabó, Z.: Heegaard Floer homology and contact structures. Duke Math. J. 129(1), 39–61 (2005)Google Scholar
  26. 26.
    Penner R.C.: The decorated Teichmüller space of punctured surfaces. Comm. Math. Phys. 113(2), 299–339 (1987)CrossRefMATHMathSciNetADSGoogle Scholar
  27. 27.
    Penrose, R.: Angular momenum: an approach to combinatorial space-time. In: Quantum Theory and Beyond. Cambridge University Press, Cambridge (1971)Google Scholar
  28. 28.
    Wendl, C.: A hierarchy of local symplectic filling obstructions for contact 3-manifolds. (2010). http://arxiv.org/abs/1009.2746
  29. 29.
    Wheeler, J.A.: Information, physics, quantum: the search for links. In: Foundations of Quantum Mechanics in the Light of New Technology (Tokyo, 1989), pp. 354–368. Phys. Soc. Japan, Tokyo (1990)Google Scholar
  30. 30.
    Witten E.: Topological quantum field theory. Comm. Math. Phys. 117(3), 353–386 (1988)CrossRefMATHMathSciNetADSGoogle Scholar
  31. 31.
    Zarev, R.: Bordered Floer Homology for Sutured Manifolds. (2009). http://arxiv.org/abs/0908.1106
  32. 32.
    Zarev, R.: Joining and Gluing Sutured Floer Homology. (2010). http://arxiv.org/abs/1010.3496

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesMonash UniversityVictoriaAustralia

Personalised recommendations