Communications in Mathematical Physics

, Volume 304, Issue 1, pp 69–93 | Cite as

Colored Group Field Theory

  • Razvan Gurau


Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to higher dimensional topological spaces. The perturbative development of the usual GFT’s is rather involved combinatorially and plagued by topological singularities (which we discuss in great detail in this paper), thus very difficult to control and unsatisfactory.

Both these problems simplify greatly for the “colored” GFT (CGFT) model we introduce in this paper. Not only this model is combinatorially simpler but also it is free from the worst topological singularities. We establish that the Feynman graphs of our model are combinatorial cellular complexes dual to manifolds or pseudomanifolds, and study their cellular homology. We also relate the amplitude of CGFT graphs to their fundamental group.


Quantum Gravity Homology Group Euler Characteristic Colored Graph Feynman Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Boulatov D.V.: A Model of three-dimensional lattice gravity. Mod. Phys. Lett. A 7, 1629 (1992) [arXiv:hep-th/9202074]CrossRefMATHADSMathSciNetGoogle Scholar
  2. 2.
    Freidel L.: Group field theory: An overview. Int. J. Theor. Phys. 44, 1769 (2005) [arXiv:hep-th/0505016]CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Oriti, D.: The group field theory approach to quantum gravity, in [15]. [arXiv:gr-qc/0607032]Google Scholar
  4. 4.
    Oriti, D.: Quantum gravity as a quantum field theory of simplicial geometry. In: Fauser, B., Tolksdorf, J., Zeidler, E. (eds.) Quantum Gravity. Birkhaeuser, Basel (2007). [arXiv:gr-qc/0512103]Google Scholar
  5. 5.
    Magnen, J., Noui, K., Rivasseau, V., Smerlak, M.: Scaling behaviour of three-dimensional group field theory. Class. Quant. Grav. 26, 185012 (2009) [arXiv:0906.5477[hep-thGoogle Scholar
  6. 6.
    David F.: A model of random surfaces with nontrivial critical behavior. Nucl. Phys. B 257, 543 (1985)CrossRefADSGoogle Scholar
  7. 7.
    Ginsparg, P.H.: Matrix models of 2-d gravity. [arXiv:hep-th/9112013]Google Scholar
  8. 8.
    Gross M.: Tensor models and simplicial quantum gravity in >  2-D. Nucl. Phys. Proc. Suppl. 25A, 144 (1992)CrossRefMATHADSGoogle Scholar
  9. 9.
    Ambjorn J., Durhuus B., Jonsson T.: Three-dimensional simplicial quantum gravity and generalized matrix models. Mod. Phys. Lett. A 6, 1133 (1991)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Sasakura N.: Tensor model for gravity and orientability of manifold. Mod. Phys. Lett. A 6, 2613 (1991)CrossRefMATHADSMathSciNetGoogle Scholar
  11. 11.
    Williams R.: Quantum Regge calculus, in Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge University Press, Cambridge (2009)Google Scholar
  12. 12.
    Ambjorn J., Jurkiewicz J., Loll R.: Reconstructing the universe. Phys. Rev. D 72, 064014 (2005) [arXiv:hep-th/0505154]CrossRefADSGoogle Scholar
  13. 13.
    Ambjorn J., Jurkiewicz J., Loll R.: The universe from scratch. Contemp. Phys. 47, 103–117 (2006) [arXiv:hep-th/0509010]CrossRefADSGoogle Scholar
  14. 14.
    Oriti D.: Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity. Rept. Prog. Phys. 64, 1703 (2001) [arXiv:gr-qc/0106091]CrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Perez A.: Spin foam models for quantum gravity. Class. Quant. Grav. 20, R43 (2003) [arXiv:gr-qc/0301113]CrossRefMATHADSGoogle Scholar
  16. 16.
    Oriti, D. (ed.): Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter. Cambridge University Press, Cambridge (2009)MATHGoogle Scholar
  17. 17.
    De Pietri R., Petronio C.: Feynman diagrams of generalized matrix models and the associated manifolds in dimension 4. J. Math. Phys. 41, 6671 (2000) [arXiv:gr-qc/0004045]CrossRefMATHADSMathSciNetGoogle Scholar
  18. 18.
    Turaev V.G., Viro O.Y.: State sum invariants of 3 manifolds and quantum 6j symbols. Topology 31, 865 (1992)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Barrett J.W., Naish-Guzman I.: The Ponzano-Regge model. Class. Quant. Grav. 26, 155014 (2009) [arXiv:0803.3319 [gr-qc]]CrossRefADSGoogle Scholar
  20. 20.
    Engle J., Pereira R., Rovelli C.: The loop-quantum-gravity vertex-amplitude. Phys. Rev. Lett. 99, 161301 (2007) [arXiv:0705.2388 [gr-qc]]CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Engle J., Pereira R., Rovelli C.: Flipped spinfoam vertex and loop gravity. Nucl. Phys. B 798, 251 (2008) [arXiv:0708.1236 [gr-qc]]CrossRefMATHADSMathSciNetGoogle Scholar
  22. 22.
    Livine E.R., Speziale S.: A new spinfoam vertex for quantum gravity. Phys. Rev. D 76, 084028 (2007) [arXiv:0705.0674 [gr-qc]]CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Freidel L., Krasnov K.: A new spin foam model for 4d gravity.. class. quant. grav. 25, 125018 (2008) [arXiv:0708.1595 [gr-qc]]CrossRefADSMathSciNetGoogle Scholar
  24. 24.
    Conrady F., Freidel L.: On the semiclassical limit of 4d spin foam models. Phys. Rev. D 78, 104023 (2008) [arXiv:0809.2280 [gr-qc]]CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    Bonzom V., Livine E.R., Smerlak M., Speziale S.: Towards the graviton from spinfoams: the complete perturbative expansion of the 3d toy model. Nucl. Phys. B 804, 507 (2008) [arXiv:0802.3983 [gr-qc]]CrossRefMATHADSMathSciNetGoogle Scholar
  26. 26.
    Grosse H., Wulkenhaar R.: Renormalisation of phi**4 theory on non-commutative R**4 in the matrix base. Commun. Math. Phys. 256, 305 (2005) [arXiv:hep-th/0401128]CrossRefMATHADSMathSciNetGoogle Scholar
  27. 27.
    Gurau R., Magnen J., Rivasseau V., Vignes-Tourneret F.: Renormalization of non-commutative phi**4(4) field theory in x space. Commun. Math. Phys. 267, 515 (2006) [arXiv:hep-th/0512271]CrossRefMATHADSMathSciNetGoogle Scholar
  28. 28.
    Disertori M., Gurau R., Magnen J., Rivasseau V.: Vanishing of beta function of non commutative phi(4)**4 theory to all orders. Phys. Lett. B 649, 95 (2007) [arXiv:hep-th/0612251]CrossRefADSMathSciNetGoogle Scholar
  29. 29.
    Geloun J.B., Gurau R., Rivasseau V.: Vanishing beta function for Grosse-Wulkenhaar model in a magnetic field. Phys. Lett. B 671, 284 (2009) [arXiv:0805.4362 [hep-th]]CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Thurston, W.P.: Three-Dimensional Geometry and Topology: Vol 1. University Presses of California, Columbia and Princeton, ISBN 13: 9780691083049, ISBN 10: 0691083045Google Scholar
  31. 31.
    Kozlov, D.: Combinatorial Algebraic Topology. Springer, Heidelberg, ISBN-10: 354071961X, ISBN-13: 978-3540719618Google Scholar
  32. 32.
    Freidel L., Gurau R., Oriti D.: Group field theory renormalization–the 3d case: power counting of divergences. Phys. Rev. D 80, 044007 (2009) [arXiv:0905.3772 [hep-th]]CrossRefADSGoogle Scholar
  33. 33.
    Mulazzani M.: Lins-Mandel graphs representing 3-manifolds. Discr. Math. 140, 107 (1995)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Alexandrov, P.S.: Combinatorial Topology. Dover Publications, ISBN-10: 0486401790, ISBN-13: 978-0486401799Google Scholar
  35. 35.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, ISBN: 0-521-79160-X, ISBN:0-521-79540-0Google Scholar
  36. 36.
    Mulase, M., Penkava, M.: Volume of representation varieties. [arXiv:math/0212012]Google Scholar
  37. 37.
    Fairbairn W.J., Livine E.R.: 3d spinfoam quantum gravity: matter as a phase of the group field theory. Class. Quant. Grav. 24, 5277 (2007) [arXiv:gr-qc/0702125]CrossRefMATHADSMathSciNetGoogle Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Perimeter Institute for Theoretical Physics WaterlooWaterlooCanada

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