Periodic Reporting for period 1 - EXTENDED (Extended degrees of freedom in QFT)
Okres sprawozdawczy: 2023-09-01 do 2026-02-28
Quantum field theory (QFT) is the modern theoretical framework that underlies particle physics and results from combining the laws of quantum mechanics with Einstein’s theory of special relativity. Its predictions help us understand nature from the smallest to the largest of scales, ranging from the interactions observed at colliders, to the properties of materials, to the origins of matter in the early universe. In situations where fundamental particles interact weakly with one another (for example, in the interactions of light with charged particles) there exists a set of techniques in QFT that rely on perturbation theory and have been extraordinarily successful at explaining observed physical phenomena. On the other hand, many fundamental phenomena, including phase transitions and nuclear interactions, are described by strongly interacting systems for which perturbative techniques are insufficient and a rigorous, predictive theoretical formulation is lacking.
There are reasons to expect that a satisfactory reformulation of QFT which is valid beyond weak coupling requires developing a novel understanding of its fundamental degrees of freedom. This is the overarching goal of my project. In particular, there is evidence that the dynamics of strongly interacting systems cannot be understood solely in terms of the interactions between fundamental particles. Rather, extended degrees of freedom like strings or membranes also play an essential role. A prototypical example of this are flux tubes which stretch between quarks and bind them within the nucleus.
I aim to obtain a precise description of the extended degrees of freedom in a variety of QFTs and use this knowledge to improve our understanding of strongly coupled QFTs beyond what can be achieved by more conventional means. A major component of this is to identify the algebraic and geometric structures which encode the physics of extended degrees of freedom. Achieving this enables us to tackle a number of fundamental questions, including understanding how these mathematical structures manifest themselves in the physics of strongly coupled QFTs, how they can be used to perform exact computations beyond perturbation theory, and how they constrain the space of possible QFTs. In the course of the project we have found applications beyond what was expected at the outset: in particular, we have discovered a novel way to employ extended degrees of freedom to study QFTs that interact with gravity.
There are reasons to expect that a satisfactory reformulation of QFT which is valid beyond weak coupling requires developing a novel understanding of its fundamental degrees of freedom. This is the overarching goal of my project. In particular, there is evidence that the dynamics of strongly interacting systems cannot be understood solely in terms of the interactions between fundamental particles. Rather, extended degrees of freedom like strings or membranes also play an essential role. A prototypical example of this are flux tubes which stretch between quarks and bind them within the nucleus.
I aim to obtain a precise description of the extended degrees of freedom in a variety of QFTs and use this knowledge to improve our understanding of strongly coupled QFTs beyond what can be achieved by more conventional means. A major component of this is to identify the algebraic and geometric structures which encode the physics of extended degrees of freedom. Achieving this enables us to tackle a number of fundamental questions, including understanding how these mathematical structures manifest themselves in the physics of strongly coupled QFTs, how they can be used to perform exact computations beyond perturbation theory, and how they constrain the space of possible QFTs. In the course of the project we have found applications beyond what was expected at the outset: in particular, we have discovered a novel way to employ extended degrees of freedom to study QFTs that interact with gravity.
Work performed to this point has focused mainly on two lines of research.
The first line of research concerns six-dimensional superconformal field theories (SCFTs) on singular spacetimes and their two-dimensional extended objects, the BPS strings. Published results include: a thorough understanding of the properties of the BPS strings, and of the role they play in the 6d SCFTs; for the first time, the computation of important physical quantities, the partition functions, which depend on the topology of the spacetime; the discovery of novel connections with mathematical structures that play important roles in the context of differential geometry; the explanation of a relation between two a priori distinct types of mathematical objects associated to four-dimensional QFTs: Donaldson-Witten and Vafa-Witten invariants. The latter result is an example of how progress in higher-dimensional QFTs can lead to a better understanding of theories in 4d, which is the number of dimensions relevant for describing our universe. Additional ongoing work includes a detailed study of the behavior of BPS strings at low energies, which has further ramifications in the study of QFTs in 4 and 5 dimensions; as well as generalizations of the work discussed above to other classes of QFTs and more general choices of spacetime geometries.
The second line of research was not anticipated at the outset of the project. We discovered an unexpected and far-reaching application of extended degrees of freedom in the context of QFTs that interact with gravity (QG theories), a topic at the forefront of research in theoretical high-energy physics. Our achievements include the discovery that the BPS strings of certain six-dimensional QG theories are described by a very special type of algebraic structure, which allowed us to obtain a very precise understanding of their properties. We also discovered that different configurations of the BPS string coincide with different possible QG theories, which gives us an unprecedented handle to study the landscape of consistent 6d QG theories and the connections between them. Finally, we have found evidence for the existence of 6d QG theories that admit no other known realization but can be studied in detail by our techniques.
Additional research that has not yet reached publication stage focuses on other classes of QFTs and their extended degrees of freedom including four-dimensional membranes in 6d SCFTs, and magnetically charged strings of 5d QFTs.
The first line of research concerns six-dimensional superconformal field theories (SCFTs) on singular spacetimes and their two-dimensional extended objects, the BPS strings. Published results include: a thorough understanding of the properties of the BPS strings, and of the role they play in the 6d SCFTs; for the first time, the computation of important physical quantities, the partition functions, which depend on the topology of the spacetime; the discovery of novel connections with mathematical structures that play important roles in the context of differential geometry; the explanation of a relation between two a priori distinct types of mathematical objects associated to four-dimensional QFTs: Donaldson-Witten and Vafa-Witten invariants. The latter result is an example of how progress in higher-dimensional QFTs can lead to a better understanding of theories in 4d, which is the number of dimensions relevant for describing our universe. Additional ongoing work includes a detailed study of the behavior of BPS strings at low energies, which has further ramifications in the study of QFTs in 4 and 5 dimensions; as well as generalizations of the work discussed above to other classes of QFTs and more general choices of spacetime geometries.
The second line of research was not anticipated at the outset of the project. We discovered an unexpected and far-reaching application of extended degrees of freedom in the context of QFTs that interact with gravity (QG theories), a topic at the forefront of research in theoretical high-energy physics. Our achievements include the discovery that the BPS strings of certain six-dimensional QG theories are described by a very special type of algebraic structure, which allowed us to obtain a very precise understanding of their properties. We also discovered that different configurations of the BPS string coincide with different possible QG theories, which gives us an unprecedented handle to study the landscape of consistent 6d QG theories and the connections between them. Finally, we have found evidence for the existence of 6d QG theories that admit no other known realization but can be studied in detail by our techniques.
Additional research that has not yet reached publication stage focuses on other classes of QFTs and their extended degrees of freedom including four-dimensional membranes in 6d SCFTs, and magnetically charged strings of 5d QFTs.
Our results on 6d SCFTs are a substantial advance beyond the state of the art in understanding non-perturbative degrees of freedom of supersymmetric quantum field theories in higher dimensions. We have been able to compute for the first time the partition functions of 6d on geometries of nontrivial topology, generalizing and helping explain analogous computations that have been performed in four dimensional QFTs. In particular in our most recent publication we have developed a systematic approach to computing partition functions of 6d theories on singular geometries. Prior to this work, almost all computations had been restricted to the simplest class of singularities. Our work brings to light general features of the non-perturbative degrees of freedom of 6d SCFTs which had not previously been appreciated, most notably the existence of interactions between BPS strings and other degrees of freedom which are important for understanding QFTs on nontrivial spacetimes in dimensions 4, 5 and 6.
Our work on QG theories is a significant advance in the context of understanding theories of quantum gravity in higher dimensions, a topic which has received much attention in recent years. We shed new light on a class of 6d theories which has been difficult to analyze by more conventional means. In particular, we found strong evidence that there exist theories that cannot be realized by any other known method. Extended degrees of freedom turned out to be essential to understand these theories. In particular we found that this class of theories possesses among its spectrum a specific BPS string, the hyperplane string, whose possible distinct configurations correspond to the possible inequivalent QG theories. This provides a very powerful tool to determine which 6d QG theories within this class are consistent, regardless of whether they have a known string theory origin, and to study transitions between them. Moreover, we found that the theories that are the farthest from admitting a conventional string theory realization are also the ones whose hyperplane string is the easiest to study. A very exciting possibility we are exploring is that the algebraic structure underlying the hyperplane string may be constraining enough that we may be able to determine the entire spectrum of the string, giving us access to information about non-supersymmetric sectors in 6d QG theories at strong coupling, which would be in itself a groundbreaking result.
Our work on QG theories is a significant advance in the context of understanding theories of quantum gravity in higher dimensions, a topic which has received much attention in recent years. We shed new light on a class of 6d theories which has been difficult to analyze by more conventional means. In particular, we found strong evidence that there exist theories that cannot be realized by any other known method. Extended degrees of freedom turned out to be essential to understand these theories. In particular we found that this class of theories possesses among its spectrum a specific BPS string, the hyperplane string, whose possible distinct configurations correspond to the possible inequivalent QG theories. This provides a very powerful tool to determine which 6d QG theories within this class are consistent, regardless of whether they have a known string theory origin, and to study transitions between them. Moreover, we found that the theories that are the farthest from admitting a conventional string theory realization are also the ones whose hyperplane string is the easiest to study. A very exciting possibility we are exploring is that the algebraic structure underlying the hyperplane string may be constraining enough that we may be able to determine the entire spectrum of the string, giving us access to information about non-supersymmetric sectors in 6d QG theories at strong coupling, which would be in itself a groundbreaking result.