Abstract
Using generalized bosons, we construct the fuzzy sphere S 2 F and monopoles on S 2 F in a reducible representation of SU(2). The corresponding quantum states are naturally obtained using the GNS-construction. We show that there is an emergent nonabelian unitary gauge symmetry which is in the commutant of the algebra of observables. The quantum states are necessarily mixed and have non-vanishing von Neumann entropy, which increases monotonically under a bistochastic Markov map. The maximum value of the entropy has a simple relation to the degeneracy of the irreps that constitute the reducible representation that underlies the fuzzy sphere.
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References
J. Hoppe, Quantum theory of a massless relativistic surface and a two-dimensional bound state problem, Ph.D. thesis, M.I.T., Cambridge U.S.A. (1982).
J. Madore, The fuzzy sphere, Class. Quant. Grav. 9 (1992) 69 [INSPIRE].
H. Grosse, C. Klimčík and P. Prešnajder, Topologically nontrivial field configurations in noncommutative geometry, Commun. Math. Phys. 178 (1996) 507 [hep-th/9510083] [INSPIRE].
H. Grosse, C. Klimčík and P. Prešnajder, Field theory on a supersymmetric lattice, Commun. Math. Phys. 185 (1997) 155 [hep-th/9507074] [INSPIRE].
H. Grosse, C. Klimčík and P. Prešnajder, On finite 4D quantum field theory in noncommutative geometry, Commun. Math. Phys. 180 (1996) 429 [hep-th/9602115] [INSPIRE].
M.R. Douglas and C.M. Hull, D-branes and the noncommutative torus, JHEP 02 (1998) 008 [hep-th/9711165] [INSPIRE].
S. Baez, A.P. Balachandran, B. Ydri and S. Vaidya, Monopoles and solitons in fuzzy physics, Commun. Math. Phys. 208 (2000) 787 [hep-th/9811169] [INSPIRE].
A.P. Balachandran and S. Vaidya, Instantons and chiral anomaly in fuzzy physics, Int. J. Mod. Phys. A 16 (2001) 17 [hep-th/9910129] [INSPIRE].
S. Vaidya, Scalar multi solitons on the fuzzy sphere, JHEP 01 (2002) 011 [hep-th/0109102] [INSPIRE].
N. Acharyya and S. Vaidya, Monopoles on S 2 F from the fuzzy conifold, JHEP 06 (2013) 034 [arXiv:1302.2754] [INSPIRE].
A. Basu and J.A. Harvey, The M 2-M 5 brane system and a generalized Nahm’s equation, Nucl. Phys. B 713 (2005) 136 [hep-th/0412310] [INSPIRE].
H. Nastase, C. Papageorgakis and S. Ramgoolam, The fuzzy S 2 structure of M 2-M 5 systems in ABJM membrane theories, JHEP 05 (2009) 123 [arXiv:0903.3966] [INSPIRE].
A.Y. Alekseev, A. Recknagel and V. Schomerus, Brane dynamics in background fluxes and noncommutative geometry, JHEP 05 (2000) 010 [hep-th/0003187] [INSPIRE].
K. Hashimoto and K. Krasnov, D-brane solutions in noncommutative gauge theory on fuzzy sphere, Phys. Rev. D 64 (2001) 046007 [hep-th/0101145] [INSPIRE].
D. Tomino, N =2 3D matrix integral with Myers term, JHEP 01 (2004) 062 [hep-th/0309264] [INSPIRE].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
J. Polchinski and M.J. Strassler, The string dual of a confining four-dimensional gauge theory, hep-th/0003136 [INSPIRE].
D.P. Jatkar, G. Mandal, S.R. Wadia and K.P. Yogendran, Matrix dynamics of fuzzy spheres, JHEP 01 (2002) 039 [hep-th/0110172] [INSPIRE].
Y. Kimura, Myers effect and tachyon condensation, Nucl. Phys. B 692 (2004) 394 [hep-th/0309082] [INSPIRE].
R.A. Brandt and O.W. Greenberg, Generalized Bose operators in the Fock space of a single Bose operator, J. Math. Phys. 10 (1969) 1168.
N. Acharyya, N. Chandra and S. Vaidya, Noncommutative vortices and instantons from generalized Bose operators, JHEP 12 (2011) 110 [arXiv:1109.3703] [INSPIRE].
A.P. Balachandran, A.R. de Queiroz and S. Vaidya, Entropy of quantum states: ambiguities, Eur. Phys. J. Plus 128 (2013) 112 [arXiv:1212.1239] [INSPIRE].
A.P. Balachandran, A.R. de Queiroz and S. Vaidya, Quantum entropic ambiguities: ethylene, Phys. Rev. D 88 (2013) 025001 [arXiv:1302.4924] [INSPIRE].
A.P. Balachandran, S. Kurkcuoglu and S. Vaidya, Lectures on fuzzy and fuzzy SUSY physics, World Scientific, Singapore (2007) [hep-th/0511114] [INSPIRE].
H. Grosse and J. Madore, A noncommutative version of the Schwinger model, Phys. Lett. B 283 (1992) 218 [INSPIRE].
J. Katriel and G. Adam, Equivalence of different representations of the generalized Bose operator, J. Math. Phys. 12 (1971) 1352.
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Acharyya, N., Chandra, N. & Vaidya, S. Quantum entropy for the fuzzy sphere and its monopoles. J. High Energ. Phys. 2014, 78 (2014). https://doi.org/10.1007/JHEP11(2014)078
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DOI: https://doi.org/10.1007/JHEP11(2014)078