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

SIGMA 7 (2011), 094, 22 pages      arXiv:0911.1700

Four-Dimensional Spin Foam Perturbation Theory

João Faria Martins a and Aleksandar Mikovic b, c
a) Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
b) Departamento de Matemática, Universidade Lusófona de Humanidades e Tecnologia, Av do Campo Grande, 376, 1749-024 Lisboa, Portugal
c) Grupo de Física Matemática da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, 1649-003 Lisboa, Portugal

Received June 03, 2011, in final form September 23, 2011; Published online October 11, 2011

We define a four-dimensional spin-foam perturbation theory for the BF-theory with a BB potential term defined for a compact semi-simple Lie group G on a compact orientable 4-manifold M. This is done by using the formal spin foam perturbative series coming from the spin-foam generating functional. We then regularize the terms in the perturbative series by passing to the category of representations of the quantum group Uq(g) where g is the Lie algebra of G and q is a root of unity. The Chain-Mail formalism can be used to calculate the perturbative terms when the vector space of intertwiners Λ⊗Λ→A, where A is the adjoint representation of g, is 1-dimensional for each irrep Λ. We calculate the partition function Z in the dilute-gas limit for a special class of triangulations of restricted local complexity, which we conjecture to exist on any 4-manifold M. We prove that the first-order perturbative contribution vanishes for finite triangulations, so that we define a dilute-gas limit by using the second-order contribution. We show that Z is an analytic continuation of the Crane-Yetter partition function. Furthermore, we relate Z to the partition function for the FF theory.

Key words: spin foam models; BF-theory; spin networks; dilute-gas limit; Crane-Yetter invariant; spin-foam perturbation theory.

pdf (558 Kb)   tex (222 Kb)


  1. Baez J., An introduction to spin foam models of quantum gravity and BF theory, in Geometry and Quantum Physics (Schladming, 1999), Lecture Notes in Phys., Vol. 543, Springer, Berlin, 2000, 25-93, gr-qc/9905087.
  2. Baez J., Spin foam perturbation theory, in Diagrammatic Morphisms and Applications (San Francisco, CA, 2000), Contemp. Math., Vol. 318, Amer. Math. Soc., Providence, RI, 2003, 9-21, gr-qc/9910050.
  3. Barrett J.W., Faria Martins J., García-Islas J.M., Observables in the Turaev-Viro and Crane-Yetter models, J. Math. Phys. 48 (2007), 093508, 18 pages, math.QA/0411281.
  4. Broda B., Surgical invariants of four-manifolds, hep-th/9302092.
  5. Cooper D., Thurston W., Triangulating 3-manifolds using 5 vertex link types, Topology 27 (1988), 23-25.
  6. Crane L., Yetter D.A., A categorical construction of 4D topological quantum field theories, in Quantum Topology, Ser. Knots Everything, Vol. 3, World Sci. Publ., River Edge, NJ, 1993, 120-130, hep-th/9301062.
  7. Eguchi T., Gilkey P.B., Hanson A.J., Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980), 213-393.
  8. Faria Martins J., Mikovic A., Invariants of spin networks embedded in three-manifolds, Comm. Math. Phys. 279 (2008), 381-399, gr-qc/0612137.
  9. Faria Martins J., Mikovic A., Spin foam perturbation theory for three-dimensional quantum gravity, Comm. Math. Phys. 288 (2009), 745-772, arXiv:0804.2811.
  10. Freidel L., Krasnov K., Spin foam models and the classical action principle, Adv. Theor. Math. Phys. 2 (1999), 1183-1247, hep-th/9807092.
  11. Freidel L., Starodubtsev A., Quantum gravity in terms of topological observables, hep-th/0501191.
  12. Gompf R.E., Stipsicz A.I., 4-manifolds and Kirby calculus, Graduate Studies in Mathematics, Vol. 20, American Mathematical Society, Providence, RI, 1999.
  13. Kauffman L.H., Lins S.L., Temperley-Lieb recoupling theory and invariants of 3-manifolds, Annals of Mathematics Studies, Vol. 134, Princeton University Press, Princeton, NJ, 1994.
  14. Kirby R.C., The topology of 4-manifolds, Lecture Notes in Mathematics, Vol. 1374, Springer-Verlag, Berlin, 1989.
  15. Lickorish W.B.R., The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), 171-194.
  16. Mackaay M., Spherical 2-categories and 4-manifold invariants, Adv. Math. 143 (1999), 288-348, math.QA/9805030.
  17. Mackaay M., Finite groups, spherical 2-categories, and 4-manifold invariants, Adv. Math. 153 (2000), 353-390, math.QA/9903003.
  18. Mikovic A., Spin foam models of Yang-Mills theory coupled to gravity, Classical Quantum Gravity 20 (2003), 239-246, gr-qc/0210051.
  19. Mikovic A., Quantum gravity as a deformed topological quantum field theory, J. Phys. Conf. Ser. 33 (2006), 266-270, gr-qc/0511077.
  20. Mikovic A., Quantum gravity as a broken symmetry phase of a BF theory, SIGMA 2 (2006), 086, 5 pages, hep-th/0610194.
  21. Mizoguchi S., Tada T., Three-dimensional gravity from the Turaev-Viro invariant, Phys. Rev. Lett. 68 (1992), 1795-1798, hep-th/9110057.
  22. Reshetikhin N., Turaev V.G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597.
  23. Roberts J., Skein theory and Turaev-Viro invariants, Topology 34 (1995), 771-787.
  24. Rourke C.P., Sanderson B.J., Introduction to piecewise-linear topology, Reprint, Springer Study Edition, Springer-Verlag, Berlin - New York, 1982.
  25. Smolin L., Linking topological quantum field theory and nonperturbative quantum gravity, J. Math. Phys. 36 (1995), 6417-6455, gr-qc/9505028.
  26. Turaev V.G., Quantum invariants of knots and 3-manifolds, de Gruyter Studies in Mathematics, Vol. 18, Walter de Gruyter & Co., Berlin, 1994.
  27. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.

Previous article   Next article   Contents of Volume 7 (2011)