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

SIGMA 6 (2010), 050, 23 pages      arXiv:1003.4683
Contribution to the Special Issue “Noncommutative Spaces and Fields”

The Multitrace Matrix Model of Scalar Field Theory on Fuzzy CPn

Christian Sämann a, b
a) Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK
b) Maxwell Institute for Mathematical Sciences, Edinburgh, UK

Received March 25, 2010, in final form June 03, 2010; Published online June 11, 2010

We perform a high-temperature expansion of scalar quantum field theory on fuzzy CPn to third order in the inverse temperature. Using group theoretical methods, we rewrite the result as a multitrace matrix model. The partition function of this matrix model is evaluated via the saddle point method and the phase diagram is analyzed for various n. Our results confirm the findings of a previous numerical study of this phase diagram for CP1.

Key words: matrix models; fuzzy geometry.

pdf (375 kb)   ps (280 kb)   tex (100 kb)


  1. Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174.
  2. Madore J., The fuzzy sphere, Classical Quantum Gravity 9 (1992), 69-87.
  3. Grosse H., Klimcík C., Presnajder P., Towards finite quantum field theory in noncommutative geometry, Internat. J. Theoret. Phys. 35 (1996), 231-244, hep-th/9505175.
  4. Steinacker H., A non-perturbative approach to non-commutative scalar field theory, J. High Energy Phys. 2005 (2005), no. 3, 075, 39 pages, hep-th/0501174.
    Steinacker H., Quantization and eigenvalue distribution of noncommutative scalar field theory, hep-th/0511076.
  5. O'Connor D., Sämann C., Fuzzy scalar field theory as a multitrace matrix model, J. High Energy Phys. 2007 (2007), no. 8, 066, 28 pages, arXiv:0706.2493.
    O'Connor D., Sämann C., A multitrace matrix model from fuzzy scalar field theory, arXiv:0709.0387.
  6. Garcia Flores F., O'Connor D., Martin X., Simulating the scalar field on the fuzzy sphere, PoS(LAT2005) (2005), 262, 6 pages, hep-lat/0601012.
    Garcia Flores F., Martin X., O'Connor D., Simulation of a scalar field on a fuzzy sphere, Internat. J. Modern Phys. A 24 (2009), 3917-3944, arXiv:0903.1986.
  7. Panero M., Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere, J. High Energy Phys. 2007 (2007), no. 5, 082, 20 pages, hep-th/0608202.
  8. Panero M., Quantum field theory in a non-commutative space: theoretical predictions and numerical results on the fuzzy sphere, SIGMA 2 (2006), 081, 14 pages, hep-th/0609205.
  9. Iuliu-Lazaroiu C., McNamee D., Sämann C., Generalized Berezin quantization, Bergman metrics and fuzzy Laplacians, J. High Energy Phys. 2008 (2008), no. 9, 059, 60 pages, arXiv:0804.4555.
  10. Creutz M., On invariant integration over SU(N), J. Math. Phys. 19 (1978), 2043-2046.
  11. Maekawa T., Formula for invariant integrations on SU(N), J. Math. Phys. 26 (1985), 1910-1913.
  12. Aubert S., Lam C.S., Invariant integration over the unitary group, J. Math. Phys. 44 (2003), 6112-6131, math-ph/0307012.
  13. Bars I., U(N) integral for the generating functional in lattice gauge theory, J. Math. Phys. 21 (1980), 2678-2681.
  14. Muskhelishvili N.I., Singular integral equations. Boundary problems of function theory and their application to mathematical physics, Noordhoff International Publishing, Leyden, 1977.
  15. Di Francesco P., Ginsparg P.H., Zinn-Justin J., 2D gravity and random matrices, Phys. Rep. 254 (1995), 1-133, hep-th/9306153.
  16. Brézin E., Itzykson C., Parisi G., Zuber J.B., Planar diagrams, Comm. Math. Phys. 59 (1978), 35-51.
  17. Das S.R., Dhar A., Sengupta A.M., Wadia S.R., New critical behavior in d=0 large N matrix models, Modern Phys. Lett. A 5 (1990), 1041-1056.
  18. Shishanin A.O., Phases of the Goldstone multitrace matrix model in the large-N limit, Theoret. and Math. Phys. 152 (2007), 1258-1265.
  19. Cicuta G.M., Molinari L., Montaldi E., Large-N spontaneous magnetization in zero dimensions, J. Phys. A: Math. Gen. 20 (1987), L67-L70.
  20. Gubser S.S., Sondhi S.L., Phase structure of non-commutative scalar field theories, Nuclear Phys. B 605 (2001), 395-424, hep-th/0006119.
  21. Das C.R., Digal S., Govindarajan T.R., Spontaneous symmetry breakdown in fuzzy spheres, Modern Phys. Lett. A 24 (2009), 2693-2701, arXiv:0801.4479.
  22. Fulton W., Harris J., Representation theory. A first course, Graduate Texts in Mathematics, Vol. 129, Springer-Verlag, New York, 1991.

Previous article   Next article   Contents of Volume 6 (2010)