In compactifications over smooth geometrical spaces, closed differential forms can take on a prominent role. For instance, closed forms can represent the geometrical structure of special holonomy manifolds and also fluxes that are present in the compactifications. In this talk, we will describe novel geometrical invariants that arise on manifolds with a distinguished closed form. In particular, we will show that there are natural cohomologies of mapping cone type that in general are dependent on the distinguished closed form. These cohomologies provide another tool to help count the massless scalars that arise in compactifications.

Single field models of inflation capable to produce primordial black holes usually require a significant departure from the standard, perturbative slow-roll regime. In fact, in many of these scenarios, the size of the slow-roll parameter eta becomes larger than one during a short phase of inflationary evolution. In order to develop an analytical control on these systems, I explore the limit of eta large, and promote 1/eta to a small quantity to be used for perturbative expansions. Formulas simplify, and analytic expressions for the two and three point functions of curvature fluctuations are obtained. I will then discuss the behaviour of loop corrections to inflationary observables in this framework

General relativity makes spectacular predictions about our world, predictions which have captured the popular imagination more than any other part of physics: gravitational waves, black holes, spacetime singularities. For the mathematician, however, perhaps the most spectacular thing about these predictions is not their exoticness, but, on the contrary, the fact that they all correspond to well-defined mathematical concepts: Indeed, it was precisely through mathematics that these predictions of general relativity were first discovered—originally to much controversy and objection!—and the qualitative mathematical analysis of the Einstein equations remains one of the most powerful ways to understand the great conceptual questions of the theory. This talk will describe some past contributions of mathematics to general relativity and some of the big open conjectures which mathematics hopes to answer in the future.

We study Chern-Simons theories at large N with either bosonic or fermionic matter in the fundamental representation. The most fundamental operators in these theories are mesonic line operators, the simplest example being Wilson lines ending on fundamentals. We classify the conformal line operators along an arbitrary smooth path as well as the spectrum of conformal dimensions and transverse spins of their boundary operators at finite 't Hooft coupling. These line operators are shown to satisfy first-order chiral evolution equations, in which a smooth variation of the path is given by a factorized product of two line operators. We argue that this equation, together with the spectrum of boundary operators, are sufficient to determine these operators' expectation values uniquely. We demonstrate this by bootstrapping the two-point function of the displacement operator on a straight line. We show that the line operators in the theory of bosons and the theory of fermions satisfy the same evolution equation and have the same spectrum of boundary operators.

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I will describe how non-invertible global symmetries act on operators in a quantum field theory. The various possible actions are called generalized charges. This provides a stepping stone for understanding physical applications of non-invertible symmetries, as will be exemplified in the case of Ising symmetry. One of the surprising findings of this endeavor is that there exist new and unexplored generalized charges already for ordinary invertible global symmetries! These generalized charges are described by higher-representations of the symmetry group, generalizing the ordinary charges described by ordinary representations of the symmetry group.

The locality in space of interactions between elementary particles is a key property of our universe. This locality is hardwired into quantum field theoretic descriptions of nature. However, locality and indeed space itself are likely not fundamental concepts. In holographic duality, local interactions on a dynamical spacetime emerge from "large N" matrices where no locality need be manifest in the microscopic Hamiltonian. The emergence of locality from matrix theories is well-established but not well-understood. In recent years it has been appreciated that locally is closely tied up with so-called "area law" entanglement of the microscopic degrees of freedom. I will discuss a particularly robust notion of entanglement in matrix theories that is rooted in an underlying Gauss law constraint and show how simple models of matrix, or 'fuzzy' geometry contain area law entanglement.

The universe has turned out to be simpler than expected on small and large scales. This encourages us to build unified theories connecting particle physics to the LCDM model. Instead of postulating an ``attractor” phase such as inflation, prior to the hot big bang, we extrapolate the observed universe all the way back to the initial singularity. If the hot plasma in the early universe is perfectly conformal radiation, the singularity is only conformal and one can analytically extend cosmic spacetime and matter through it into a ``mirror” universe on the other side. The universe is then CPT symmetric. We calculate the gravitational entropy for cosmologies with radiation, matter, Lambda and space curvature, finding that thermodynamics favours flat, homogeneous and isotropic universes like ours. To maintain conformal symmetry we include unusual Dim-0 (dimension zero) fields, whose unique physical state is the vacuum. They improve the Standard Model’s (SM’s) coupling to gravity, by cancelling the SM’s vacuum energy and two local “Weyl” anomalies due to gauge fields and fermions. They also cancel the acausal, nonanalytic behaviour introduced into the graviton propagator by loops of SM particles. Cancellation requires (and predicts) precisely 3 generations of SM fermions, each with a RH neutrino, and that the Higgs is composite. One of the RH neutrinos, if stable, is then the simplest-yet proposed viable candidate for the dark matter. Galaxy surveys including EUCLID and LSST will allow precise tests soon. Finally, and most exciting, Dim-0 fields have scale-invariant fluctuations in the vacuum. These source curvature perturbations in the early universe. We recently calculated their power spectrum, ab initio, in terms of Standard Model couplings at the Planck scale. Subject to some theoretical assumptions, the amplitude and spectral tilt closely match the observations, with no free parameters. (See arXiv:2302.00344and references therein).

Over the past few years, it has been shown that, when integrating out the spacetime dependence with a certain integration measure, some four-point correlation functions in N=4 super Yang-Mills (SYM) can be computed exactly. These physical observables are often called integrated correlators, which are functions of Yang-Mills coupling \tau, and transform under S-duality of N=4 SYM. In this talk, I will review some of the recent developments regarding these integrated correlators. In particular, I will discuss the so-called Laplace-difference equations that determine the integrated correlators recursively. I will also present the generating functions of the integrated correlators that resum the ranks of the gauge group and the charges of the operators, from which we will further determine the large-N and large-charge properties of the integrated correctors.

Free or interacting UV fixed points play a key role in the fundamental definition of QFT. In this talk, I give a broad overview of weakly and strongly interacting fixed points in 3d and 4d QFTs including models of particle physics with or without supersymmetry, and fermionic theories. Further, I explain methods and ideas to search for fixed points in 4d quantum gravity. Implications from the viewpoint of CFTs and higher-spin gauge theories through the AdS/CFT conjecture are also discussed.

We consider vacuum transitions by bubble nucleation among vacua with different values and signs of the cosmological constant Lambda, including both up and down tunnelings. Following the Hamiltonian formalism in four dimensions, we explicitly compute the decay rates for all possible combinations of initial and final values of Lambda and find that up-tunneling may be allowed starting not only from pure dS spacetime but also from pure AdS and Minkowski spacetimes. We trace the difference with the Euclidean approach, for which these transitions are found not to be allowed, to the difference of assigning either vanishing or infinite entropy to both pure AdS and Minkowski. We find that, in all allowed cases, detailed balance is satisfied. Also in the formal limit Lambda -> -infinity, the transition rates for AdS to dS agree with the Hartle-Hawking and Vilenkin amplitudes for the creation of dS from nothing. This is consistent with a proposal of Brown and Dahlen to define 'nothing' as AdS in this limit. We generalise our results to include black hole backgrounds for which transitions are allowed only in certain regimes of the black hole mass M but detailed balance is not satisfied, except for Schwarzschild de Sitter (SdS) to another SdS for which the transition is allowed and detailed balance satisfied. We compute the bubble trajectory after nucleation and find that, contrary to the M = 0 case, the trajectory is not a geodesic for the open universe slicing of dS. We briefly discuss the relevance of our results to the string landscape.

The Quantum Spectral Curve (QSC) is a powerful integrability-based formalism capable of computing the non-perturbative spectrum of planar N=4 SYM. The success and utility of QSC motivate trying to extend it beyond N=4 to other instances of the AdS/CFT correspondence where integrability is expected to be present. This has been successfully accomplished for AdS4/CFT3 and a curve has been conjectured for AdS3/CFT2. I will review the basics of the QSC framework in the well-understood AdS5 case and then turn to low-dimensional versions of the QSC. I will discuss the conjectured curve for AdS3 and how it differs from previous iterations of the QSC. Furthermore, I will discuss recent perturbative results with a peculiar structure.

The interpretation of LHC data, and the assessment of possible hints for new physics in the experimental signals, require the precise knowledge of the proton subnuclear structure in terms of its elementary constituents, quarks and gluons. In this talk I will present the fascinating precision frontier that phenomenologists are facing an will describe the challenges behind the determination of proton structure, involving precise perturbative QCD calculations and machine learning techniques. I will show how global fits of the proton might inadvertently ‘fit away’ signals of new physics in the high-energy tails of the distributions that are experimentally measured. A new physics scenario in which the fit of the proton’s structure may completely absorb such signs of new physics is showcased. Strategies to single out such effects and disentangle the inconsistencies that stem from new physics signals are discussed.

In this talk I will discuss N=(2,2) susy generalisation of Hull's doubled sigma model. The doubled formulation of the worldsheet provides a description of string theory in which T-duality is promoted to a manifest symmetry. Formulation via N=(2,2) superspace provides a doubled formulation for bi-Hermitian/generalised Kahler target spaces. The theory is described by a single function, a doubled-generalised Kahler potential, supplemented with a manifestly N=(2,2) constraint. If time permits I will ilustrate some of the concepts developed on examples.

In this talk, we start by introducing massive spin-2 theories and reviewing some of their key features. Then, using massive spinor helicity variables, we review an on-shell superspace formalism for massive particles in four dimensions. Finally, we apply this formalism to massive spin-2 amplitudes, deriving all spin-2 cubic vertices that are compatible with supersymmetry and exploring the constraints that adding more supersymmetry has on these vertices. Additionally, we discuss how the massive graviton supermultiplets and cubic superamplitudes can be constructed via a double copy of massive Yang—Mills supermultiplets and cubic superamplitudes. We conclude by commenting on possible future directions such as computing the 4-point massive supersymmetric spin-2 amplitudes and the issues that can arise in a massive double copy for higher point amplitudes.

In this talk, I explore the set of line defects supported by 3D N=4 theories, and their significance in the context of generalised symmetries. I begin by discussing mirror symmetry of line defects using the example of Sp(k) SQCD and its two mirror theories. I then introduce rational Q-systems, a powerful technique borrowed from spin-chains/integrability, for evaluating twisted indices and studying line operator correlation functions. Finally, I highlight the role of line defects in realising mirror symmetry in the presence of non-trivial higher symmetries

There has been some recent progress in understanding the microscopic derivation of the entropy of supersymmetric AdS black holes using holography and a localisation computation in the dual quantum field theory. In this talk, after a general discussion of the overall picture, I discuss the large N factorization properties of supersymmetric partition functions for CFT with a holographic dual in various dimensions. I show how this factorization is related to a universal gluing formula for the entropy functionals of known AdS black holes in terms of elementary objects called gravitational blocks.

A natural way to extend Einstein's General Relativity in the high-energy regime is to introduce higher-order curvature terms in the gravitational Lagrangian. Indeed, by working in the framework of perturbative QFT one can show that quadratic-curvature gravity in four dimensions is strictly renormalizable. The quadratic-curvature terms are multiplied by dimensionless parameters that are related to the masses of the additional gravitational degrees of freedom and to the interaction couplings. In this talk, after having motivated Renormalizable Quantum Gravity, we will study the limits in which those dimensionless parameters tend to zero or to infinity, and show that different types of decoupling can occur. In particular, it will be shown that the presence of a non-zero cosmological constant affects the decoupling in a non-trivial way in the limit where the coefficient of the Weyl-squared term tends to infinity. We will discuss possible physical implications of this mathematical analysis for the high-energy behavior of the spin-2 massive ghost and for the classical limit of the theory. Several concepts that have been developed in the context of massive gravity will naturally emerge in this talk, sometimes with different relevance.