Emergence of longitudinal 7-branes and fuzzy $S^4$ in compactification scenarios of M(atrix) theory

In M(atrix) theory, there exist membranes and longitudinal 5-branes (L5-branes) as extended objects. Transverse components of these brane solutions are known to be described by fuzzy ${\mathbf{C}\mathbf{P}}^k$ ($k=1$, 2), where $k=1$ and $k=2$ correspond to spherical membranes and L5-branes of ${\mathbf{C}\mathbf{P}}^2\times S^1$ world-volume geometry, respectively. In addition to these solutions, we here show the existence of L7-branes of ${\mathbf{C}\mathbf{P}}^3\times S^1$ geometry, introducing extra potentials to the M(atrix) theory Lagrangian. As in the cases of $k=1$, 2, the L7-branes (corresponding to $k=3$) also break the supersymmetries of M(atrix) theory. The extra potentials are introduced such that the energy of a static L7-brane solution becomes finite in the large $N$ limit where $N$ represents the matrix dimension of fuzzy ${\mathbf{C}\mathbf{P}}^3$. As a consequence, fluctuations from the L7-branes are suppressed, which effectively describes compactification of M(atrix) theory down to 7 dimensions. We show that one of the extra potentials can be considered as a matrix-valued 7-form. The presence of the 7-form in turn supports a possibility of Freund-Rubin type compactification. This suggests that our modification of M(atrix) theory can also lead to a physically interesting matrix model in four dimensions. In hope of such a possibility, we further consider compactification of M(atrix) theory down to fuzzy $S^4$ which can be defined in terms of fuzzy ${\mathbf{C}\mathbf{P}}^3$. Along the way, we also find a new L5-brane solution to M(atrix) theory which has purely spherical geometry in the transverse directions.