Periodic Reporting for period 3 - NP-QFT (Non-perturbative dynamics of quantum fields: from new deconfined phases of matter to quantum black holes)
Berichtszeitraum: 2023-10-01 bis 2025-03-31
When the degrees of freedom that constitute a quantum physical system are strongly coupled among each other, their collective behaviour are low energies or large distances can exhibit a plethora of exotic, surprising and unconventional phenomena. At the same time, however, our most sophisticated tool to describe the quantum world — namely, quantum field theory — becomes extremely difficult to use. This problem appears across the board in many areas of physics, from particle physics, to condensed matter physics, to astrophysics: strong coupling is an intrinsic complexity of quantum systems, whose solution can benefit disparate fields. A large variety of examples is provided by the so-called deconfined quantum states of matter, in which the collective behaviour of atoms, electrons and photons gives rise to emergent low-energy degrees of freedom, often strongly coupled, that do not exist as fundamental particles but “appear” at lower energies. Another context in which decrypting strong coupling can be the key to a breakthrough is quantum gravity: by the celebrated AdS/CFT correspondence, we can describe gravity in Anti-de-Sitter space in a fully-consistent quantum fashion, in terms of an ordinary — but strongly coupled — quantum field theory in one dimension less. It follows that our ability to compute at strong coupling can then grant us access to the mysteries of quantum gravity.
The ambitious goal of this project is twofold. First, I plan to develop innovative techniques to tame strong coupling. Second, I aim to exploit those techniques in order to discover new deconfined phases of matter on one side, and to unravel mysteries of quantum gravity and the quantum physics of black holes on the other side.
Within this project I will follow several avenues in the quest for new computational tools at strong coupling, such as refining the concept of symmetry, developing supersymmetric localisation, and probing Borel summability of certain gauge theories. Applying these and other methods, I plan to systematically explore three-dimensional gauge theories with bosons and fermions, landscaping their phase diagrams and identifying new interesting deconfined critical points. Meanwhile, I will study how to extract the quantum entropy and other quantum thermodynamic properties of black holes, exploring signatures of quantum effects to be compared with future experiments.
The ambitious goal of this project is twofold. First, I plan to develop innovative techniques to tame strong coupling. Second, I aim to exploit those techniques in order to discover new deconfined phases of matter on one side, and to unravel mysteries of quantum gravity and the quantum physics of black holes on the other side.
Within this project I will follow several avenues in the quest for new computational tools at strong coupling, such as refining the concept of symmetry, developing supersymmetric localisation, and probing Borel summability of certain gauge theories. Applying these and other methods, I plan to systematically explore three-dimensional gauge theories with bosons and fermions, landscaping their phase diagrams and identifying new interesting deconfined critical points. Meanwhile, I will study how to extract the quantum entropy and other quantum thermodynamic properties of black holes, exploring signatures of quantum effects to be compared with future experiments.
Symmetry is one of the most powerful concepts in physics, and in particular in quantum field theory (QFT). Traditionally, symmetry is the property of a system to remain unchanged under a set of transformations, that form the mathematical structure of a symmetry group. In the last decade, the concept of symmetry in QFT has been revolutionised by the observation that symmetry transformations are implemented by a specific type of topological extended operators, whose topological nature reflects the fact that charges are conserved. The new idea is to regard any extended topological operator as a “symmetry,” even those that do not implement any transformation under which the system is invariant. The topological character guarantees that those operators behave as a sort of symmetry: they organise the spectrum of the QFT into representations, they can play the role of order parameters for phases of matter in which spontaneous breaking takes place, they can be gauged to give rise to new theories, they have associated anomalies that are exact observables of the theory.
In this ERC project we have been studying explicit examples of these generalised symmetries, called non-invertible symmetries, in gauge theories in four spacetime dimensions. We have in particular analysed some of their properties, how they appear in the context of holography (described below), and what mathematical structures other than a group they form. A particularly interesting result was the identification of renormalisation group (RG) flows that preserve the non-invertible symmetries: along such flows the gauge coupling is under control, even at strong coupling, and one can infer novel phases of matter at low energies, as described at the end.
A peculiar type of symmetry is supersymmetry, that relates bosons and fermions. What is special about this symmetry is that it leads to cancelation among quantum effects, and it allows us to perform exact nonperturbative computations of certain observables at strong coupling -- which is in general impossible without supersymmetry. In this project we have analysed a variety of such supersymmetric observables, with a focus on grand canonical partition functions (which enumerate quantum states of the system according to their total charge), in the limit in which the number of degrees of freedom is large -- known as the large N limit. The reason to study these observables is that from them we can learn about the quantum properties of black holes.
Indeed, one of the most interesting applications we have found of the aforementioned results is in the context of gravity. The so-called AdS/CFT correspondence provides us with a nonperturbative definition of quantum gravity, in terms of an ordinary QFT in one dimension less, but at strong coupling and at large N. We have thus exploited our results on the partition functions of supersymmetric theories to answer the question: what is a black hole made of? A black hole has a humongous entropy, meaning that it should be made of a huge number of quantum states. From the large N limit of the partition function of the boundary QFT have been able to derived the Bekenstein-Hawking entropy of supersymmetric (zero temperature) black holes, as well as to compute a variety of different quantum corrections: higher derivative, quantum loops, nonperturbative coming from complex saddles of the gravitational path integral, and nonperturbative coming from stringy instantons.
A particularly difficult long-standing problem is how to go beyond supersymmetry and study black holes with non-vanishing temperature, since supersymmetric localisation methods cannot be applied. We have devised a new methodology consisting in a reduction (or compactification) of the boundary theory on the spatial manifold of the event horizon. This reduction produces a quantum mechanical system with a large number of couplings, whose values appear as quasi-random numbers. This resembles quantum mechanical models with random disorder known as Sachdev-Ye-Kitaev (SYK), which are amenable to both analytical and numerical computations at non-vanishing temperature, and have been noticed to display very similar physics to black holes. Our methodology promises to provide a direct and derivable link between the physics of black holes and of SYK-like models.
In the same context of holography, we have investigated the relations between gravity and the emergence of chaos and averaging in QFT. Indeed, the holographic computation of partition functions in terms of semiclassical gravity produces answers that are typical of averaged systems. An open question in holography had been whether low-dimensional examples of AdS/CFT correspond to averages over ensembles of theories. Working in a toy, but solvable, model of 3d gravity based on Chern-Simons theory, we showed that when gravity is UV complete it is dual to a well-defined, specific and unitary theory, while it is when using a low-energy effective theory that the system behaves as an average over unitary quantum systems. As a byproduct, we constructed a simple but UV complete, unitary, and solvable model of 3d quantum gravity.
In the context of QFT, one of the major goals of this ERC project is to enlarge our understanding and classification of quantum states of matter, in particular identifying new ones. The non-invertible generalised symmetries are a prime tool to do that. For instance, we identified gapped phases of 4d gauge theories characterised by the spontaneous breaking of non-invertible symmetries. Other interesting phases of matter arise from QFTs at finite density. At least for 3d supersymmetric models, our newly developed technique of reduction to disordered quantum mechanics promises to deliver new computable signatures in those systems (as confirmed by preliminary computations).
In this ERC project we have been studying explicit examples of these generalised symmetries, called non-invertible symmetries, in gauge theories in four spacetime dimensions. We have in particular analysed some of their properties, how they appear in the context of holography (described below), and what mathematical structures other than a group they form. A particularly interesting result was the identification of renormalisation group (RG) flows that preserve the non-invertible symmetries: along such flows the gauge coupling is under control, even at strong coupling, and one can infer novel phases of matter at low energies, as described at the end.
A peculiar type of symmetry is supersymmetry, that relates bosons and fermions. What is special about this symmetry is that it leads to cancelation among quantum effects, and it allows us to perform exact nonperturbative computations of certain observables at strong coupling -- which is in general impossible without supersymmetry. In this project we have analysed a variety of such supersymmetric observables, with a focus on grand canonical partition functions (which enumerate quantum states of the system according to their total charge), in the limit in which the number of degrees of freedom is large -- known as the large N limit. The reason to study these observables is that from them we can learn about the quantum properties of black holes.
Indeed, one of the most interesting applications we have found of the aforementioned results is in the context of gravity. The so-called AdS/CFT correspondence provides us with a nonperturbative definition of quantum gravity, in terms of an ordinary QFT in one dimension less, but at strong coupling and at large N. We have thus exploited our results on the partition functions of supersymmetric theories to answer the question: what is a black hole made of? A black hole has a humongous entropy, meaning that it should be made of a huge number of quantum states. From the large N limit of the partition function of the boundary QFT have been able to derived the Bekenstein-Hawking entropy of supersymmetric (zero temperature) black holes, as well as to compute a variety of different quantum corrections: higher derivative, quantum loops, nonperturbative coming from complex saddles of the gravitational path integral, and nonperturbative coming from stringy instantons.
A particularly difficult long-standing problem is how to go beyond supersymmetry and study black holes with non-vanishing temperature, since supersymmetric localisation methods cannot be applied. We have devised a new methodology consisting in a reduction (or compactification) of the boundary theory on the spatial manifold of the event horizon. This reduction produces a quantum mechanical system with a large number of couplings, whose values appear as quasi-random numbers. This resembles quantum mechanical models with random disorder known as Sachdev-Ye-Kitaev (SYK), which are amenable to both analytical and numerical computations at non-vanishing temperature, and have been noticed to display very similar physics to black holes. Our methodology promises to provide a direct and derivable link between the physics of black holes and of SYK-like models.
In the same context of holography, we have investigated the relations between gravity and the emergence of chaos and averaging in QFT. Indeed, the holographic computation of partition functions in terms of semiclassical gravity produces answers that are typical of averaged systems. An open question in holography had been whether low-dimensional examples of AdS/CFT correspond to averages over ensembles of theories. Working in a toy, but solvable, model of 3d gravity based on Chern-Simons theory, we showed that when gravity is UV complete it is dual to a well-defined, specific and unitary theory, while it is when using a low-energy effective theory that the system behaves as an average over unitary quantum systems. As a byproduct, we constructed a simple but UV complete, unitary, and solvable model of 3d quantum gravity.
In the context of QFT, one of the major goals of this ERC project is to enlarge our understanding and classification of quantum states of matter, in particular identifying new ones. The non-invertible generalised symmetries are a prime tool to do that. For instance, we identified gapped phases of 4d gauge theories characterised by the spontaneous breaking of non-invertible symmetries. Other interesting phases of matter arise from QFTs at finite density. At least for 3d supersymmetric models, our newly developed technique of reduction to disordered quantum mechanics promises to deliver new computable signatures in those systems (as confirmed by preliminary computations).
There are three major achievements that we expect to accomplish by the end of the project.
First, to progress towards a mathematically compelling formulation of non-invertible symmetries in three and four spacetime dimensions, in particular outlining the structure that controls the intersection and fusion of extended topological operators and that substitutes the concept of group. Besides, we expect to identify topical examples of physical phenomena controlled and explained by non-invertible symmetries, and to develop a unified description of the observables known as anomalies.
Second, to derive the spectrum of quantum states of black holes at small but non-vanishing temperature, so as to push our understanding beyond supersymmetry. The new methodology of reduction promises to deliver this result for magnetically charged black holes in four-dimensional AdS space. In order to understand other setups, we aim at a more universal description of the density of states in conformal field theories at finite density and small but non-vanishing temperature.
Third, to advance our understanding of the structure of perturbative loop corrections in gauge theories with fermions in three spacetime dimensions, which are argued to be Borel resummable.
First, to progress towards a mathematically compelling formulation of non-invertible symmetries in three and four spacetime dimensions, in particular outlining the structure that controls the intersection and fusion of extended topological operators and that substitutes the concept of group. Besides, we expect to identify topical examples of physical phenomena controlled and explained by non-invertible symmetries, and to develop a unified description of the observables known as anomalies.
Second, to derive the spectrum of quantum states of black holes at small but non-vanishing temperature, so as to push our understanding beyond supersymmetry. The new methodology of reduction promises to deliver this result for magnetically charged black holes in four-dimensional AdS space. In order to understand other setups, we aim at a more universal description of the density of states in conformal field theories at finite density and small but non-vanishing temperature.
Third, to advance our understanding of the structure of perturbative loop corrections in gauge theories with fermions in three spacetime dimensions, which are argued to be Borel resummable.