But, how did I practically approach the problem? The BV construction is made of several steps, whose key point is the introduction of the so-called ghost fields. Indeed, the core idea of this construction is that our problem can be (surprisingly) simplified by the introduction of extra (non-physical) particles. Hence, given an initial gauge theory, one has to:
- determine how many and which kind of ghost fields to introduce;
- which extra auxiliary fields add;
- how to perform the so-called gauge-fixing procedure;
- what is the classical BRST cohomology induced by the extended theory;
- what is the gauge-fixed version of this BRST complex.
Indeed, what makes the BV construction so interesting is that it provides a method to determine the so-called BRST complex for a very large class of theories. This BRST complex is a completely mathematical object with encodes several physically relevant information such as the space of observables for the theory (that is, the quantities we can be measured via experiments).
So, I started considering an initial physical theory naturally induced by a spectral triple, that is a theory which comes from the mathematical context of noncommutative geometry. The question that I asked and hence answered was if it would be possible to encode all the steps of the BV construction in terms of noncommutative geometrical objects. The result is very positive: first of all, also the extended theory, obtained by the introduction of ghost fields, can be described using an appropriate extended spectral triple. Moreover, this provides the ghost fields with a geometrical interpretation. In addition, also the auxiliary field and the gauge-fixing procedure perfectly fit within this framework. Finally, I succeeded in relating both the classical as well as the gauge-fixed BRST complex to other structures already known in the mathematical context (namely the Hochschild complex of an algebra). This offers a new prospective to approach the challenging task of computing these complexes, by applying already known mathematical techniques.
These results have been achieved in collaboration with some of the worldwide leading figures in the field: in particular, Prof. Dr. G. Landi (Trieste University) and Prof. Dr. M. Marcolli (Caltech and University of Toronto).
Because any discovery becomes meaningful only when it contributed to the development of the whole field, I did my best to present my results to the scientific community through articles in peer-reviewed journals as well as international conferences. In particular, I have been asked to present my research in several thematic conferences, attended by all the major mathematicians working in this area. Moreover, I have also been an invited speaker to the XIX International Congress of Mathematical Physics: this congress is organized each three years and gathers together hundreds of scientists interested in the field of mathematical physics. During this congress, all groundbreaking results achieved in the field are presented and all future challenges are discussed, determining the main directions the whole research area will follow in the upcoming years.
In order to interest the new generation of PhD students to this promising area of research, I organized and delivered a Masterclass on “Noncommutative geometry: spaces, bundles and connections”. All the lessons are available on the website of my research group, so that not only the students attending the event and coming from several European as well as extra-European countries, but also all the persons who might get interest to the topic could beneficiate from that.
Finally, I strongly believe in the importance of trying to explain the relevance of fundamental research. Indeed, while more applied area of research can be easily motivated by presenting the immediate impact that their results can have on our daily life, for more fundamental research more time is needed to go from a theoretical understanding to a practical implementation. However, behind any major achievement, there is always a deep understanding of the fundamentals. Hence, I made a YouTube video, where I explain that, to fully understand the world we all life in, physics is crucial but not sufficient: it was only when mathematics was used to precisely formulate the physical concepts that the main results could be achieved and the scientific revolution started. But we surely need more advanced mathematics if we want to be able to conceive a more advance future.
