Higher dimensional geometries related to fuzzy odd-dimensional spheres

TL;DR

This paper explores higher-dimensional geometries of fuzzy odd-dimensional spheres using $SO(m)$ covariant matrix realizations, revealing their structure as cosets and implications for matrix models and brane physics.

Contribution

It introduces new matrix algebra realizations of fuzzy odd spheres as cosets, connecting them to higher-dimensional geometries and matrix models.

Findings

Fuzzy odd spheres relate to higher-dimensional cosets with hermitian symmetric spaces.

Matrix actions admitting fuzzy spheres as solutions are proposed.

Insights into five-brane entropy scaling in matrix theory contexts.

Abstract

We study $SO(m)$ covariant Matrix realizations of $ \sum_{i=1}^{m} X_i^2 = 1 $ for even $m$ as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy $S^{2k-1} $ is related to a higher dimensional coset $ {SO(2k) \over U(1) \times U(k-1)}$. These cosets are bundles where base and fibre are hermitian symmetric spaces. The detailed form of the generators and relations for the Matrix algebras related to the fuzzy three-spheres suggests Matrix actions which admit the fuzzy spheres as solutions. These Matrix actions are compared with the BFSS, IKKT and BMN Matrix models as well as some others. The geometry and combinatorics of fuzzy odd spheres lead to some remarks on the transverse five-brane problem of Matrix theories and the exotic scaling of the entropy of 5-branes with the brane number.