# zbMATH — the first resource for mathematics

Scalar and spinor field actions on fuzzy $$S^4$$: fuzzy $$\mathbb{C} P^3$$ as a $$S^2_F$$ bundle over $$S^4_F$$. (English) Zbl 1397.81417
Summary: We present a manifestly Spin(5) invariant construction of squashed fuzzy $$\mathbb{C} P^3$$ 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 $$S^4$$.

##### MSC:
 81T75 Noncommutative geometry methods in quantum field theory
Full Text:
##### References:
 [1] Grosse, H.; Klimčík, C.; Prešnajder, P., On finite 4 − $$D$$ quantum field theory in noncommutative geometry, Commun. Math. Phys., 180, 429, (1996) · Zbl 0872.58008 [2] Castelino, J.; Lee, S-M; Taylor, IW; Longitudinal five-branes as four spheres in matrix theory, No article title, Nucl. Phys., B 526, 334, (1998) · Zbl 1031.81594 [3] Abe, Y., Construction of fuzzy $$S$$\^{4}, Phys. Rev., D 70, 126004, (2004) [4] Sheikh-Jabbari, M.; Torabian, M., Classification of all 1/2 BPS solutions of the tiny graviton matrix theory, JHEP, 04, 001, (2005) [5] Kimura, Y., Noncommutative gauge theory on fuzzy four sphere and matrix model, Nucl. Phys., B 637, 177, (2002) · Zbl 0996.81101 [6] Behr, W.; Meyer, F.; Steinacker, H., Gauge theory on fuzzy $$S$$\^{2} × $$S$$\^{2} and regularization on noncommutative $$R$$\^{4}, JHEP, 07, 040, (2005) [7] P. Castro-Villarreal, R. Delgadillo-Blando and B. Ydri, Quantum effective potential for U(1) fields on $$S_L^2 × S_L^2$$, JHEP, 09, 066, (2005) [8] Ramgoolam, S., On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys., B 610, 461, (2001) · Zbl 0971.81039 [9] Medina, J.; O’Connor, D., Scalar field theory on fuzzy $$S$$\^{4}, JHEP, 11, 051, (2003) [10] Balachandran, A.; Dolan, BP; Lee, J-H; Martin, X.; O’Connor, D., Fuzzy complex projective spaces and their star products, J. Geom. Phys., 43, 184, (2002) · Zbl 1007.51007 [11] Dolan, BP; Huet, I.; Murray, S.; O’Connor, D., Noncommutative vector bundles over fuzzy CP\^{$$N$$} and their covariant derivatives, JHEP, 07, 007, (2007) [12] Dolan, BP; Huet, I.; Murray, S.; O’Connor, D., A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces, JHEP, 03, 029, (2008) [13] Huet, I., A projective Dirac operator on CP\^{2} within fuzzy geometry, JHEP, 02, 106, (2011) · Zbl 1294.81065 [14] Dolan, BP; O’Connor, D., A fuzzy three sphere and fuzzy tori, JHEP, 10, 060, (2003) [15] Salam, A.; Strathdee, J., On Kaluza-Klein theory, Annals Phys., 141, 316, (1982) [16] M. Hamermesh, Group theory and its application to physical problems, Dover Publications Inc., New York, U.S.A. (1962). · Zbl 0100.36704 [17] Balachandran, A.; Immirzi, G.; Lee, J.; Prešnajder, P., Dirac operators on coset spaces, J. Math. Phys., 44, 4713, (2003) · Zbl 1062.58032 [18] Balachandran, A.; Padmanabhan, P., Spin j Dirac operators on the fuzzy 2-sphere, JHEP, 09, 120, (2009) [19] Perelomov, A.; Popov, V., Eigenvalues of Casimir operators, Sov. J. Nucl. Phys., 7, 290, (1968) [20] Perelemov, AM; Popov, VS, Casimir operators for the orthogonal and symplectic groups, Sov. J. Nucl. Phys., 3, 819, (1968) [21] W. Fulton, J. Harris, Representation Theory. A First course, Springer Verlag, New York, U.S.A. (1991). · Zbl 0744.22001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.