In this video Alain Connes made a comment about the ,,quantum corrections'' of the geometry. I would like to understand this notion in some details since I haven't found anything about this in the literature. I'm guessing that this notion is different to the notion of so called ,,inner fluctuations'' of the metric (see for example this discussion) and unlike the inner fluctuations, quantum corrections is less understood. So making the long story short my question is

What are quantum corrections of a geometry?

EDIT: Let me be a little bit more precise. One of the highlights of noncommutative geometry is that it can reproduce the complicated Lagrangian of the Standard Model from the so called *spectral action* and can be seen as pure gravity on some slightly noncommutative space of the form $M \times F$. Here $F$ is not correctly defined as a set but manifests itself as a noncommutative algebra: the idea is that $F$ represents some noncommutative space. Since this noncommutativity is already present on the electroweak scale (i.e. distances approx. $10^{-16}\text{cm}$) I guess that the quantum corrections of geometry should be:

- already visible on this (much lower than Planck) energy scale
- understood within the context of noncommutative geometry i.e. Dirac operators and spectral triples (if we take the lesson from the Standard Model seriously).

So let me state the problem once again, this time more precisely:

How to understand quantum corrections of geometry within the context of noncommutative geometry?