Week 11.11.2024 – 17.11.2024

Scattering amplitudes provide crucial theoretical input in collider and gravitational wave physics, and at the same time exhibit a remarkable mathematical structure. These lectures will introduce essential concepts and modern techniques exploiting this structure so as to efficiently compute amplitudes and their building blocks, Feynman integrals, in perturbation theory. We will start by decomposing gauge theory amplitudes into simpler pieces based on colour and helicity information. Focusing on tree level, we will then show how these may be determined from their analytic properties with the help of Britto-Cachazo-Feng-Witten recursion. Moving on to loop level, we will define the the class of polylogarithmic functions amplitudes and integrals often evaluate to, and explain their properties as well as relate them to the universal framework for predicting their singularities, known as the Landau equations. Time permitting, we will also summarise the state of the art in the calculation of the aforementioned singularities, and their intriguing relation to mathematical objects known as cluster algebras.

Tidal Love numbers quantify the deformability and dissipative properties of compact gravitating objects. However, even in classical GR, they undergo renormalization group running due to the nonlinearity of gravity. In this talk I will explain some exact results about their running, which can be extracted by matching calculations of scattering amplitudes in black hole perturbation theory and point-particle effective theories. Due to the universality of EFT, the results have applications to the physics of black holes, neutron stars, and even binary systems. For the specific case of black holes, our matching calculation also provides the precise values of both static and dynamical Love numbers in various dimensions.

I will present arXiv: 2203.17157 with N. Drukker and G. Sakkas and the paper to appear with N. Drukker and P. Kravchuk. Symmetry-breaking is innate to defects. There is a distinguished set of defect operators that keeps track of the symmetries in the parent conformal field theory broken by the defect insertion, such as the tilt operators and displacement operators. We find identities of such defect operators between their 2-pt functions and integrated 4-pt functions. These identities are derived either from the geometric properties of the defect conformal manifold which is the symmetry-breaking coset, or from the Lie algebra of the corresponding broken symmetry generators. I will demonstrate these integral identities in the case of the 1/2 BPS Maldacena-Wilson loop in N = 4 SYM as an example.

For small values of k and N, this theory describes various experimentally relevant systems in condensed matter, and is also conjectured to be part of a web of non-supersymmetric dualities. We compute the scaling dimensions of monopole operators in a large N and k expansion, which appears to be extremely accurate even down to the smallest values of N and k, and allows us to find dynamical evidence for these dualities and make predictions about the phase transitions. For instance, we combine these estimates with the conformal bootstrap to predict that the notorious Neel-VBS transition (QED3 with 2 scalars) is tricritical, which was recently confirmed by independent lattice simulations. Lastly, we propose a novel phase diagram for QED3 with 2 fermions, including duality with the O(4) Wilson-Fisher fixed point.

Four-point correlation functions are observables of significant interest in holographic quantum field theories. In this talk I will describe the computation of a family of four-point correlation functions of operators in short multiplets of 4D N=4 super Yang-Mills theory, by studying the quadratic fluctuations around non-trivial supergravity backgrounds. The supergravity backgrounds are supersymmetric smooth geometries in the family derived by Lin, Lunin and Maldacena. For generic parameters, the supergravity backgrounds are dual to heavy CFT states. However I will also discuss the limit in which the dual CFT states become light single-particle states. The resulting all-light four-point correlators are related by superconformal Ward identities to previously known four-point correlators of half-BPS chiral primary operators. By verifying that the Ward identities are satisfied, we confirm the validity of the supergravity method.