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).
