Starting with a Dirac operator on a configuration space of SU(2) gauge connections we consider its fluctuations with inner automorphisms. We show that a certain type of twisted inner fluctuations leads to a Dirac operator whose square gives the Hamiltonian of Yang-Mills quantum field theory coupled to a fermionic sector that consist of one-form fermions. We then show that if a metric exists on the underlying three-dimensional manifold then there exists a change of basis of the configuration space for which the transformed fermionic sector consists of fermions that are no-longer one-forms.
From the author's newsletter:
A fluctuated Dirac operator on a configuration space
In our new paper we present an alternative — and possibly better — solution. What we have found is that if we apply the mechanism of unification, which Chamseddine and Connes found, to our setting with a configuration space, then we obtain a new and powerful tool of unification that appears to tie together (at least in part) the three fundamental pillars of contemporary high-energy physics that I mentioned above. In particular, it involves quantum theory and quantised fields at a fundamental level⁶.
So what we do is to rotate our Dirac operator (the original Dirac operator on the configuration space, not the Bott-Dirac operator) and thereby obtain a new so-called fluctuated Dirac operator. And what we have found is that this operator gives us the building blocks of bosonic and fermionic quantum field theory when we take its square without us having to add any additional terms.
Now, there are some technical details that I need to mention here (and if anyone already feels exhausted from too much math-lingo then I think it’s best to skip this part). First of all, it turns out that the rotation of the Dirac operator, that we need, is not exactly the same as what Chamseddine and Connes did. In order to get the right result we need to add a certain ‘twist’ to the rotation⁷. This twist is a mathematically very subtle thing that we do not yet fully understand the significance of. I will not attempt to explain this here but simply refer the interested reader to our paper. Secondly, it has to be emphasised that the entire geometrical construction on the configuration space is a highly delicate matter that still remains to be fully explored. We have outlined how such a construction can be accomplished and proven that it exists rigorously in a few special cases, but much work remains to be done.