Fields are physical entities that extend throughout spacetime. Our best theories of physics so far describe the elementary constituents of the universe in terms of elementary excitations of said fields. They also play a fundamental role in the effective description of systems with a large number of constituents, such as those found in condensed matter physics.

1. I am interested in understanding in a mathematically rigorous fashion the quantum (QFT) and statistical behavior of fields. I believe that, much like it has happened with past theories, understanding the rigorous mathematics of field theory will force us to develop deeper physical insights on what fields are.

   • One of the ways in which I am approaching this objective is by trying to understand the relationship between two complementary approaches to the perturbative aspects of quantum field theory. One of them is the approach extending algebraic quantum field theory (see for example the book by Kasia Rejzner or the one by Michael Dütsch). The other is the approach based on factorization algebras and effective field theory (see for example the book by Kevin Costello or the books by Costello and Owen Gwilliam ). One of the points in common of both approaches is the BV formalism, which is used to treat in a cohomological way the presence of gauge redundancies. During my master's thesis I did a review of this formalism.

   • I have also examined the non-perturbative aspects of QFT. For example, with my collaborator Francisco Calderón we have examined the nature of the overall normalization of path integrals in this paper. We have in particular proposed a method of normalizing path integrals. It is based on an extension of an observation due to Stephen Hawking where certain path integrals self-normalize. We can then cut and paste these integrals to find the normalization of others that do not.

   • I am interested in the more classic approaches to axiomatic QFT as well. In particular, I am interested in the use of the methods of operator algebras to understand the locality in physics that fields implement (see for example the book by Rudolf Haag for the quantum aspects and the book by Ola Bratteli and Derek Robinson for the statistical, or the reviews written by Edward Witten or by Cristopher Fewster and Kasia Rejzner). During my bachelor's thesis I reviewed the aspect of this formalism which connects the thermal equilibrium and the dynamical evolution. With Aiyalam Balachandran, Andrés F. Reyes Lega and Souad Maria Tabban Sabbagh we explored the space of extensions of a state in the Hilbert space of its canonical (GNS) purification in a finite dimensional toy model to QFT. We interpreted this space by identifying an emergent gauge symmetry in the description which could be traversed using entropy increasing quantum operations. The results can be found in this paper (see for example this paper by Xi Dong, Daniel Harlow and Donald Marolf, for physical implementations of this gauge symmetry in the exploration of fixed-area states in quantum gravity).

2. I want to understand the relationship between the equilibrium statistics and the quantum dynamics of fields. The main method of study that I use for this problem is the Schwinger-Keldysh formalism. I am particularly interested in obtaining from this a deeper understanding of the underpinnings of quantum mechanics. For example, one might expect that a better understanding of the Schwinger-Keldysh formalism would reveal connections between quantum measure theory (see for example the paper by Fay Dowker, Steven Johnston and Rafael Sorkin) and usual measure theory.

3. I am also interested in the use of extended operators, also known as defects, to probe QFTs.

   • The chiral Wess-Zumino-Witten (WZW) model $(E_8)_1$ is an important two-dimensional QFT with conformal symmetry. It has discrete symmetries which can be gauged, producing related theories known as orbifolds. Duality defects are defects supported on lines which separate the original theory on one side to its orbifold on the other. With Justin Kulp and Jonas Neuser we classified the duality defects of this theory in this paper .

   • An important technique used to study the flow of information in relativistic theories, has been to couple QFTs to simple quantum mechanical probes, particle detectors. With Tales Rick Perche and Bruno de Souza Leão Torres we reinterpreted these detectors as line defects by integrating them out withing the path integral formalism. In particular, we found a simple generalization of the Unruh-DeWitt model to gauge theories where the detector couples via the derivative of an associated Wilson line. The paper can be found here.

4. A round sphere and a potato are, in a sense, the same (see this lecture by Frederic Schuller). They can be obtained from one another by continuous deformations. Topological field theories (TFT) are particular examples of QFTs which are invariant under such deformations of the spacetime in which they are defined (see for example the lectures by Nathan Seiberg). What truly differentiates the round sphere from a potato, is the choice of a metric. As we learned from General Relativity, this metric contains the gravitational information of a system. Accordingly, all theories of dynamical gravity, and in particular, of quantum gravity, must be, in a certain sense, TFTs. Indeed, they are defined on spacetimes that do not carry any metric information since, at the end of the day, the metric is precisely the dynamical variable of the theory! This is made precise by studying gravity in low dimensions, where it is related to other well-known TFTs. These, in turn, are examples of holographic theories. The version of 2D gravity known as JT gravity is dual to BF theory, which in turn is holographically related to the SYK model (for a thorough study of this, see the thesis of Andreas Blommaert). In 3D, gravity is dual to a close cousin, Chern-Simons theory, which is in turn related to WZW models (see for example the article of Edward Witten). I am interested in understanding the way that these TFTs should be understood in the gravitational setting and how to recover gravitational observables from the usual observables of these theories.

Throughout history, the necessity of describing increasingly abstract and complex phenomena has prompted the development of new mathematics. However, field theory has also shown to be a new way of doing new mathematics (see for example the article by Mina Aganagic). By this I mean that the techniques developed to explore the physics of fields have provided deep insights into mathematical phenomena. These insights are at times so deep that they provide complete proofs of evasive theorems. The fundamental reason for the power of physics in mathematics remains mysterious. However the lectures of David Tong do point out some aspects of it:

This seems unfair, like physicists have some kind of secret weapon that mathematicians are unable to wield. And we do. In fact, we have two. The first is the path integral. The second, a wilful disregard for rigour.

As I already described above, I believe the lack of rigor can be mended. However, I think that the path integral is absolutely fundamental to the development of new mathematics.

1. We can use TFTs to probe the topology of the spacetimes in which they are embedded. A deep problem in topology is that of differentiating knots. Given two complicated knots it is a difficult task to know whether one can deform one of the knots into the other continuously. A powerful technique to solve these classification problems is the use of knot invariants. These are mathematical objects assigned to each knot that are preserved under continuous deformations. In particular, if the knot invariant assigns two different objects to two different knots, we can be certain that the two knots cannot be deformed into one another. The task is then to produce better and better invariants that allow us to distinguish more and more knots. Edward Witten provided a way of doing this using a simple though experiment. One places a knot in a spacetime filled with a quantum field and allows for a particle detector to move along the knot. The response of the detector can be used as a mathematical object that describes the knot. In particular, if the field theory is topological, the resulting object will be a knot invariant. In the work mentioned above with Tales Rick Perche and Bruno de Souza Leão Torres, we showed that the obvious coupling of Unruh-DeWitt detectors to Chern-Simons theory, which is a particular example of TFT, cannot provide a physical realization of these knot invariants.

2. Another way to study topology is to allow supersymmetric fields to permeate the geometry we are interested in studying. Supersymmetry is a very constraining symmetry. On the one hand, we can exploit it to be able to complete explicit computations of the path integral. On the other, it allows us to control the way in which the behavior of our theories changes under small deformations of the data used to define it. I recommend taking a look at the lectures of David Tong cited above to get a flavor for the way that we can use supersymmetry to rediscover several important results in geometry.