Bootstrapping fuzzy scalar field theory

  • Christian Sämann
Open Access
Regular Article - Theoretical Physics


We describe a new way of rewriting the partition function of scalar field theory on fuzzy complex projective spaces as a solvable multitrace matrix model. This model is given as a perturbative high-temperature expansion. At each order, we present an explicit analytic expression for most of the arising terms; the remaining terms are computed explicitly up to fourth order. The method presented here can be applied to any model of hermitian matrices. Our results confirm constraints previously derived for the multitrace matrix model by Polychronakos. A further implicit expectation about the shape of the multitrace terms is however shown not to be true.


Matrix Models Non-Commutative Geometry Field Theories in Lower Dimensions 


Open Access

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  1. [1]
    J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    H. Grosse, C. Klimčík and P. Prešnajder, Towards finite quantum field theory in noncommutative geometry, Int. J. Theor. Phys. 35 (1996) 231 [hep-th/9505175] [INSPIRE].CrossRefzbMATHGoogle Scholar
  3. [3]
    X. Martin, A matrix phase for the ϕ 4 scalar field on the fuzzy sphere, JHEP 04 (2004) 077 [hep-th/0402230] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    F. Garcia Flores, D. O’Connor and X. Martin, Simulating the scalar field on the fuzzy sphere, PoS(LAT2005)262 [hep-lat/0601012] [INSPIRE].
  5. [5]
    M. Panero, Numerical simulations of a non-commutative theory: the scalar model on the fuzzy sphere, JHEP 05 (2007) 082 [hep-th/0608202] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    F. Garcia Flores, X. Martin and D. O’Connor, Simulation of a scalar field on a fuzzy sphere, Int. J. Mod. Phys. A 24 (2009) 3917 [arXiv:0903.1986] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    B. Ydri, New algorithm and phase diagram of noncommutative ϕ 4 on the fuzzy sphere, JHEP 03 (2014) 065 [arXiv:1401.1529] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    H. Steinacker, A non-perturbative approach to non-commutative scalar field theory, JHEP 03 (2005) 075 [hep-th/0501174] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  9. [9]
    D. O’Connor and C. Sämann, Fuzzy scalar field theory as a multitrace matrix model, JHEP 08 (2007) 066 [arXiv:0706.2493] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C. Sämann, The multitrace matrix model of scalar field theory on fuzzy CP n, SIGMA 6 (2010) 050 [arXiv:1003.4683] [INSPIRE].zbMATHGoogle Scholar
  11. [11]
    M. Ihl, C. Sachse and C. Sämann, Fuzzy scalar field theory as matrix quantum mechanics, JHEP 03 (2011) 091 [arXiv:1012.3568] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A.P. Polychronakos, Effective action and phase transitions of scalar field on the fuzzy sphere, Phys. Rev. D 88 (2013) 065010 [arXiv:1306.6645] [INSPIRE].ADSGoogle Scholar
  13. [13]
    J. Tekel, Uniform order phase and phase diagram of scalar field theory on fuzzy CP n, JHEP 10 (2014) 144 [arXiv:1407.4061] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    B. Ydri, A multitrace approach to noncommutative Φ 24, arXiv:1410.4881 [INSPIRE].
  15. [15]
    V.P. Nair, A.P. Polychronakos and J. Tekel, Fuzzy spaces and new random matrix ensembles, Phys. Rev. D 85 (2012) 045021 [arXiv:1109.3349] [INSPIRE].ADSGoogle Scholar
  16. [16]
    F.A. Berezin, General concept of quantization, Commun. Math. Phys. 40 (1975) 153 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. thesis,, MIT, U.S.A. (1982).

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Maxwell Institute for Mathematical Sciences, Department of MathematicsHeriot-Watt UniversityEdinburghUnited Kingdom

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