Abstract
We conclude our analysis of bubble divergences in the flat spinfoam model. In Bonzom and Smerlak, Comm. Math. Phys., (submitted), we showed that the divergence degree of an arbitrary 2-complex Γ can be evaluated exactly by means of twisted cohomology. Here, we specialize this result to the case where Γ is the 2-skeleton of the cell decomposition of a pseudomanifold, and sharpen it with a careful analysis of the cellular and topological structures involved. Moreover, we explain in detail how this approach reproduces all the previous powercounting results for the Boulatov–Ooguri (colored) tensor models, and sheds light on algebraic-topological aspects of Gurau’s 1/N expansion.
Article PDF
Similar content being viewed by others
References
Bonzom, V., Smerlak, M.: Bubble divergences from cellular cohomology. Lett. Math. Phys. series 93(3), 295–305. arXiv:1004.5196
Bonzom, V., Smerlak, M.: Bubble divergences from twisted cohomology. Comm. Math. Phys. (submitted)
Rovelli C.: Quantum Gravity. Cambridge University Press, Cambridge (2004)
Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Spectroscopic and group theoretical methods in physics. North-Holland, New York (1968)
Ooguri H.: Topological lattice models in four dimensions. Mod. Phys. Lett. series A 7, 2799–2810 (1992) arXiv:hep-th/9205090
Boulatov D.V.: A Model of three-dimensional lattice gravity. Mod. Phys. Lett. series A 7, 1629–1646 (1992) arXiv:hep-th/9202074
Freidel L.: Group field theory: An overview. Int. J. Theor. Phys. series 44, 1769–1783 (2005) arXiv:hep-th/0505016
Oriti, D.: The group field theory approach to quantum gravity. arXiv:gr-qc/0607032
Rivasseau, V.: Non-commutative renormalization. In: Duplantier, B., et al. (eds.) Poincare Seminar (2007). arXiv:0705.0705 [hep-th]
Ben Geloun, J., Bonzom, V.: Radiative corrections in the Boulatov–Ooguri tensor model: The 2-point function. arXiv:1101.4294
Barrett J.W., Naish-Guzman I.: The Ponzano–Regge model. Class. Quant. Grav. series 26, 155014 (2009) arXiv:0803.3319
Frohman, C., Kania-Bartoszynska, J.: Dubois’ Torsion, A-polynomial and Quantum Invariants. arxiv:1101.2695
Dubois, J.: Torsion de Reidemeister non abélienne et forme volume sur l’espace des représentations du groupe d’un noeud. Ph.D. thesis, Université Blaise Pascal, Clermont-Ferrand (2003). arXiv:quant-phys/9606031
Perez A., Rovelli C.: A spin foam model without bubble divergences. Nucl. Phys. series B 599, 255–282 (2001) arXiv:gr-c/0006107
De Pietri R., Petronio C.: Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. series 41, 6671–6688 (2000) arXiv:gr-qc/0004045
Smerlak, M.: Comment on ‘Lost in Translation: Topological Singularities in Group Field Theory’. arXiv:1102.1844 [hep-th]
Gurau, R.: Colored Group Field Theory. arXiv:0907.2582 [hep-th]
Matveev S.: Algorithmic Topology and Classification of 3-Manifolds. Springer, Berlin (2007)
Freidel L., Louapre D.: Ponzano–Regge model revisited I: Gauge fixing, observables and interacting spinning particles. Class. Quant. Grav. series 21, 5685–5726 (2004) arXiv:hep-th/0401076 [hep-th]
Freidel, L., Louapre, D.: Ponzano–Regge model revisited II: Equivalence with Chern-Simons. arXiv:gr-qc/0410141 [gr-qc]
Freidel, L., Gurau, R., Oriti, D.: Group field theory renormalization—the 3d case: power counting of divergences. arXiv:0905.3772 [hep-th]
Magnen J., Noui K., Rivasseau V., Smerlak M.: Scaling behaviour of three-dimensional group field theory. Class. Quant. Grav. series 26, 185012 (2009) arXiv: 0906.5477 [hep-th]
Ben Geloun J., Krajewski T., Magnen J., Rivasseau V.: Linearized Group Field Theory and Power Counting Theorems. Class. Quant. Grav. series 27, 155012 (2010) arXiv:1002.3592 [hep-th]
Gurau, R.: Lost in Translation: Topological Singularities in Group Field Theory. Class. Quant. Grav. 27, 235023 (2010). arXiv:1006.0714 [hep-th]
Gurau R.: The 1/N expansion of colored tensor models. arXiv:1011.2726 [gr-qc]
Gurau, R., Rivasseau, V.: The 1/N expansion of colored tensor models in arbitrary dimension. arXiv:1101.4182 [gr-qc]
Baratin, A., Girelli, F., Oriti, D.: Diffeomorphisms in group field theories. arXiv:1101.0590 [hep-th]
Bonzom, V., Smerlak, M.: Discrete BF theory (in preparation)
Ben Geloun J., Magnen J., Rivasseau V.: Bosonic Colored Group Field Theory. Eur. Phys. J. series C 70, 1119–1130 (2010) arXiv:0911.1719 [hep-th]
Gurau, R.: The complete 1/N expansion of colored tensor models in arbitrary dimension. arXiv:1102.5759 [gr-qc]
Cavicchioli A., Meschiari M.: A homology theory for colored graphs. Discr. Math 137(1–3), 99–136 (1995)
Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Carlo Rovelli.
Rights and permissions
About this article
Cite this article
Bonzom, V., Smerlak, M. Bubble Divergences: Sorting out Topology from Cell Structure. Ann. Henri Poincaré 13, 185–208 (2012). https://doi.org/10.1007/s00023-011-0127-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-011-0127-y