7-sphere x 4-sphere manifold and its physical significance [closed]

Question

I am looking for sources about this manifold 7-sphere*4-sphere and its relations to theoretical physics.

Where to go to read about 7-sphereX4-sphere manifold and its physical significance?

Answer 1

I suppose that you don't really mean for both spaces to be spheres, so I will interpret your question about the (4,7) split. The context here is eleven-dimensional supergravity and the studies of such geometries date from the time (predating M-theory) of "Kaluza-Klein supergravity", when eleven-dimensional supergravity was of phenomenological interest. A good place to read this, albeit written for physicists, is the Physics Report "Kaluza-Klein supergravity" by Duff, Nilsson and Pope.

The name of the game back then was the construction of solutions of the (bosonic) field equations of eleven-dimensional supergravity. The dynamical fields are an eleven-dimensional lorentzian metric and a closed 4-form, satisfying equations reminiscent of Einstein-Maxwell theory, except that the Maxwell-like equation is nonlinear, reflecting a Chern-Simons term in the action.

Solutions were sought where the metric is (at least locally) a product of a four-dimensional lorentzian metric and a seven-dimensional strictly riemannian metric. The idea being that the four-dimensional lorentzian metric is to be interpreted as the metric in our physical spacetime and the seven-dimensional riemannian metric is that of an internal space, assumed compact and small, since it is not observed. With such a split there is a natural (geometric) choice for the 4-form: namely the volume form of the four-dimensional lorentzian metric. The first such solution was found by Freund and Rubin in Dynamics of dimensional reduction and are now known as Freund-Rubin compactifications.

One may wonder why the initial interest in Kaluza-Klein supergravity. In trying to geometrise the standard model of particle physics and unify it with gravity, it is natural to derive inspiration from the Kaluza-Klein idea of viewing pure gravity in five dimensions as unifying four-dimensional gravity and Maxwell theories. In this context, the gauge group of the standard model ought to be realised as isometries of the higher-dimensional geometry. The smallest dimension of a riemannian manifold in which the standard model gauge group, which is isomorphic to $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$, can act effectively and isometrically is $7$. Since we live in a four-dimensional spacetime, this suggests that the minimum dimension we need is $4+7=11$, which coincidentally is the highest dimension in which we can construct a (Poincaré) supergravity theory. This was the lucky coincidence which launched a thousand ships, so to speak.