Abstract
Using generalized bosons, we construct the fuzzy sphere $S_F^2$ and monopoles on $S_F^2$ 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 non-abelian 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.
Highlights
JHEP11(2014)078 κ is a (Chern-Simons) coupling constant and φi’s (i = 1, 2, 3) are N × N matrices
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
This is a puzzling circumstance as the fuzzy spheres in the reducible representation have higher energy
Summary
There is a natural chain of descent C2F → SF3 → SF2. The algebra of C2F is described by the algebra of a pair of independent harmonic oscillators (see [24]). These oscillators acts on the Fock space F = span{|n1, n2 }. Is the noncommutative version of Hopf map [25]. This subspace is the carrier space of (n + 1) dimensional UIR of SU(2). Φ’s are (l + 1) × (n + 1) rectangular matrices and are element of a noncommutative bi-module Hnl — it is a left Al-module and a right An-module. Φ is the noncommutative analogue of a section of the complex line bundle with topological charge κ
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