Abstract
We present a manifestly Spin(5) invariant construction of squashed fuzzy \(\mathbb{C}{{\text{P}}^3}\) as a fuzzy S 2 bundle over fuzzy S 4. We develop the necessary projectors and exhibit the squashing in terms of the radii of the S 2 and S 4. Our analysis allows us give both scalar and spinor fuzzy action functionals whose low lying modes are truncated versions of those of a commutative S 4.
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H. Grosse, C. Klimčík and P. Prešnajder, On finite 4 − D quantum field theory in noncommutative geometry, Commun. Math. Phys. 180 (1996) 429 [hep-th/9602115] [INSPIRE].
J. Castelino, S.-M Lee and I.W. Taylor, Longitudinal five-branes as four spheres in matrix theory, Nucl. Phys. B 526 (1998) 334 [hep-th/9712105] [INSPIRE].
Y. Abe, Construction of fuzzy S 4, Phys. Rev. D 70 (2004) 126004 [hep-th/0406135] [INSPIRE].
M. Sheikh-Jabbari and M. Torabian, Classification of all 1/2 BPS solutions of the tiny graviton matrix theory, JHEP 04 (2005) 001 [hep-th/0501001] [INSPIRE].
Y. Kimura, Noncommutative gauge theory on fuzzy four sphere and matrix model, Nucl. Phys. B 637 (2002) 177 [hep-th/0204256] [INSPIRE].
W. Behr, F. Meyer and H. Steinacker, Gauge theory on fuzzy S 2 × S 2 and regularization on noncommutative R 4, JHEP 07 (2005) 040 [hep-th/0503041] [INSPIRE].
P. Castro-Villarreal, R. Delgadillo-Blando and B. Ydri, Quantum effective potential for U(1) fields on \(S_L^2 \times S_L^2\), JHEP 09 (2005) 066 [hep-th/0506044] [INSPIRE].
S. Ramgoolam, On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B 610 (2001) 461 [hep-th/0105006] [INSPIRE].
J. Medina and D. O’Connor, Scalar field theory on fuzzy S 4, JHEP 11 (2003) 051 [hep-th/0212170] [INSPIRE].
A. Balachandran, B.P. Dolan, J.-H. Lee, X. Martin and D. O’Connor, Fuzzy complex projective spaces and their star products, J. Geom. Phys. 43 (2002) 184 [hep-th/0107099] [INSPIRE].
B.P. Dolan, I. Huet, S. Murray and D. O’Connor, Noncommutative vector bundles over fuzzy CP N and their covariant derivatives, JHEP 07 (2007) 007 [hep-th/0611209] [INSPIRE].
B.P. Dolan, I. Huet, S. Murray and D. O’Connor, A Universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces, JHEP 03 (2008) 029 [arXiv:0711.1347] [INSPIRE].
I. Huet, A projective Dirac operator on CP 2 within fuzzy geometry, JHEP 02 (2011) 106 [arXiv:1011.0647] [INSPIRE].
B.P. Dolan and D. O’Connor, A Fuzzy three sphere and fuzzy tori, JHEP 10 (2003) 060 [hep-th/0306231] [INSPIRE].
A. Salam and J. Strathdee, On Kaluza-Klein Theory, Annals Phys. 141 (1982) 316 [INSPIRE].
M. Hamermesh, Group theory and its application to physical problems, Dover Publications Inc., New York, U.S.A. (1962).
A. Balachandran, G. Immirzi, J. Lee and P. Prešnajder, Dirac operators on coset spaces, J. Math. Phys. 44 (2003) 4713 [hep-th/0210297] [INSPIRE].
A. Balachandran and P. Padmanabhan, Spin j Dirac Operators on the Fuzzy 2-Sphere, JHEP 09 (2009) 120 [arXiv:0907.2977] [INSPIRE].
A. Perelomov and V. Popov, Eigenvalues of Casimir operators, Sov. J. Nucl. Phys. 7 (1968) 290 [Yad. Fiz. 7 (1968) 460].
A.M. Perelemov and V.S. Popov, Casimir operators for the orthogonal and symplectic groups, Sov. J. Nucl. Phys. 3 (1968) 819.
W. Fulton, J. Harris, Representation Theory. A First course, Springer Verlag, New York, U.S.A. (1991).
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ArXiv ePrint: 1208.0348
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Medina, J., Huet, I., O’Connor, D. et al. Scalar and spinor field actions on fuzzy S 4: fuzzy \(\mathbb{C}{{\text{P}}^3}\) as a \(S_F^2\) bundle over \(S_F^4\) . J. High Energ. Phys. 2012, 70 (2012). https://doi.org/10.1007/JHEP08(2012)070
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DOI: https://doi.org/10.1007/JHEP08(2012)070