Abstract
We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as -equivariant projections of quantized coadjoint orbits of . We show that they arise as solutions in Yang–Mills matrix models, which naturally leads to higher-spin gauge theories on S4. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions , which is expected to entail 2 + 1 generations.
Highlights
We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions
The mathematical structure of the novel class of spaces is as follows: Fuzzy SΛ4 is a quantization of a certain coadjoint orbit OΛ of SO(6), defined in terms of irreducible representations with highest weight Λ
It is very encouraging to observe that the contribution from GμTν gives a large contribution O(N 4) to the t space (6.8b), in contrast to the O(N 2) contribution from both embeddings to the x space. This suggests that a large separation of scales seems to arise naturally between the base S4 and the fuzzy extra dimensions in the (Y, T ) embedding, leading to a suppression of the corresponding harmonics in t space
Summary
We are interested in fuzzy 4-spheres which are covariant under SO(5). They will be defined in terms of five hermitian matrices Xa, a = 1, . . . , 5 acting on some finite-dimensional. Such relations constitute a covariant quantum 4-sphere Particular realizations of such fuzzy 4-spheres are obtained from generators Mab, a, b = 1, . This class of quantum spheres was considered in [1] as a promising basis for a higher-spin theory including gravity We study their fuzzy geometry in more detail, and provide new embeddings in matrix models, which resolve some of the internal structure. The price to pay is that the algebra of “coordinates” Xa does not close, but involves extra generators Θab Their proper geometric interpretation allows to proceed with the construction of physical theories on such spaces via matrix models, leading to fully covariant higher-spin theories with large gauge symmetry, including a gauged version of SO(5)
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