Implication of the Jacobian map for the structure of the Euclidean space-time

Question

I'm listening to Alain Connes "On the Fine-Structure of Space-Time" around minute 23 saying that it was disappoing that the solution $Y$ to the equation

$$ \langle Y[D,Y]^{2m} \rangle= \gamma $$

with $D$ a Dirac operator on a (spin Riemannian) manifold $\mathcal{M}$ and $\langle \rangle$ a trace, defines a map

$$\mathcal{M} \xrightarrow{\text{Y}} \mathcal{S}^{2m} $$

to the sphere and the Jacobian of this map $Y$ never vanishes. The explanation is that, when you pull-back a volume form on the unit sphere, you get the volume form of the Riemannian manifold. This is not clear to me and moreover he adds that it means that you have a covering (namely, the map $Y$ is a covering). Is it derived by the nLab definition of cover? Now the sphere is simply connected (which is discussed also here) but I don't understand the disappointment and the physical interpretation as of a sphere of Planck size, even if I see some clarifications in the paper "Geometry and the Quantum" of Chamseddine

Theorem 1 gives a concrete realization of this quantization of the volume by interpreting the integer $k$ as the number of geometric quantas forming the Riemannian geometry $M$. Each geometric quantum is a sphere of arbitrary shape and unit volume (in Planck units).

Can someone shed more light on on this passage ("at the moment we're only able to find Euclidean space-time which looks like big collections of bubbles")?

Answer 1

The Heisenberg-like commutation relation $$ \langle Y[D,Y]^{2m} \rangle= \gamma $$ (with $\gamma $ the chirality operator) appeared first in a preprint then a PRL article.

It was devised by A. Chamseddine, A. Connes and S. Mukhanov, aiming at using it as a tentative equation of motion of some field theory on a manifold in the spectral Non Commutative Geometric framework.

To have some context one needs to know NCG can be seen as a sound mathematical apparatus built to generalise Riemannian geometry and go beyond the classical notion of a manifold. Its operator theoretic language has been largely inspired by quantum mechanics (statistical and relativistic quantum gauge field theories). In general the data of a manifold, its spin and metric structures are encoded by spectral NCG in terms that avoid local charts, using instead relations between an algebra of "position" ($Y$) and Dirac ($D$) operators in a Hilbert space setting. In their original paper Chamseddine, Connes and Mukhanov explain how to use the Feynman slash of real scalar fields to built the proper $Y$ operators and show how their work is a starting point for a quantization of geometry with quanta corresponding to irreducible representations of their Heisenberg-like commutation relation.

Now to understand mathematically the connection between the Heisenberg-like equation and the quantization of the volume of the manifold "parametrized" by $Y$ and $D$ (roughly speaking), it's worth reading the paragraph XI "Spectral Manifold" in this short review article. To build some physical intuition one may also watch this informative video by Ali Chamseddine.

To answer the question about the "disappointment" Connes talks about in the video it suffices to know that it came from the fact that the Heisenberg-like commutation relation implies the only manifolds where the proper $Y$ and $D$ can "live" are necessary unions of disconnected spheres which is not a promising model for a Euclidean spacetime.

… Hopefully for them they explain in the same article how to refine their equation incorporating a refinement of NCG devised by Connes in this article named "the real structure". They built then a new commutation relation (refered as a"two-sided" equation) which makes it possible to get any 4D spin Riemannian manifold with quantized volume but connected: a more encouraging result for their quantum gravity research programme one can guess.

Remark: the "real structure" proposed by Alain Connes is inspired by the deep Tomita-Takesaki theory and plays a fundamental role in his vision of NCG. The Tomitat-Takesaki theory is presented by Edward Witten in his pedagogical Notes on Some Entanglement Properties of Quantum Field Theory