## Periodic Reporting for period 1 - BVCGA (The BV Construction: a Geometric Approach)

Reporting period: 2017-05-01 to 2019-04-30

Nowadays, physics is taking gigantic steps in decoding the deep structure of the world: particle accelerators and advanced detectors have finally verified theoretical predictions, of which the Higgs boson and gravitational waves are monumental. Nevertheless, many aspects of nature are still mysterious, such as the quantization of gauge theories.

Historically, the concept of quantization was introduced at the beginning of the 20th century, when physicists discovered that the behavior of particles is ruled by probabilistic laws. However, even if more than a century has passed, we are still searching for a (mathematically rigorous) method to quantize a crucial class of theories, called gauge theories. What makes solving this problem so important is that, mathematically, all fundamental interactions, such as the gravitational or the electromagnetic ones, are governed by this type of theories.

Over the years, several attempts have been made to solve this problem. Lately, the BV (Batalin-Vilkovisky) quantization procedure proved to be a good candidate. Within this scenario, my research aims to find a new geometric formulation of the BV construction, within the framework of a recently developed mathematical theory called noncommutative geometry. Even though the deep relation existing between gauge theories and noncommutative geometry, by the means of its key notion of spectral triple, has always been evident, the possibility of using this mathematical language to investigate the geometric meaning of the BV construction was still highly unexplored when I started this project. But now, I proved that by approaching the BV quantization procedure from the prospective of noncommutative geometry, one opens completely new scenarios.

Historically, the concept of quantization was introduced at the beginning of the 20th century, when physicists discovered that the behavior of particles is ruled by probabilistic laws. However, even if more than a century has passed, we are still searching for a (mathematically rigorous) method to quantize a crucial class of theories, called gauge theories. What makes solving this problem so important is that, mathematically, all fundamental interactions, such as the gravitational or the electromagnetic ones, are governed by this type of theories.

Over the years, several attempts have been made to solve this problem. Lately, the BV (Batalin-Vilkovisky) quantization procedure proved to be a good candidate. Within this scenario, my research aims to find a new geometric formulation of the BV construction, within the framework of a recently developed mathematical theory called noncommutative geometry. Even though the deep relation existing between gauge theories and noncommutative geometry, by the means of its key notion of spectral triple, has always been evident, the possibility of using this mathematical language to investigate the geometric meaning of the BV construction was still highly unexplored when I started this project. But now, I proved that by approaching the BV quantization procedure from the prospective of noncommutative geometry, one opens completely new scenarios.

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.

- 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.

Of course, at it holds for all interesting and challenging topics, also the results I reached should be seen as the first promising steps in a long project. But how the progress I made allow us to go beyond the state-of-arts? And what would be the potential impact of that? While answering the second question would be more complicated, due to the difficulty of foreseeing the effect that a very theoretical research could have, over few decades or more, in our daily life, I can surely answer the first question. Indeed, this project proved how rich and fruitful can be the synergy of the mathematical framework provided by noncommutative geometry with the physical content of the BV construction. Moreover, now that we reached this goal, we can use all the richness of this mathematical framework to go even further: indeed, when we speak about quantization, we are actually considering phenomena that happen at terribly small scales (i.e. the Planck scale: 1.6 × 10−35 m). At these scales, we surely cannot rely on empirical guidance, but we must look to mathematical formulation of the physical concept of quantization to suggest the way forward.