This week

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.

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Physics beyond relativistic invariance and without Lorentz (or Poincare) symmetry and the geometry underlying these non-Lorentzian structures have become very fashionable of late. This is primarily due to the discovery of uses of non-Lorentzian structures in various branches of physics, including condensed matter physics, classical and quantum gravity, fluid dynamics, cosmology, etc. In this talk, I will be talking about one such theory - Carrollian theory, where the Carroll group replaces the Poincare group as the symmetry group of interest. Interestingly, any null hypersurface is a Carroll manifold and the Killing vectors on the null manifold generate Carroll algebra. Historically, Carroll group was first obtained from the Poincare group via a contraction by taking the speed of light going to zero limit as a "degenerate cousin of the Poincare group". I will shed some light on Carrollian fermions, i.e. fermions defined on generic null surfaces. Due to the degenerate nature of the Carroll manifold, there exist two distinct Carroll Clifford algebras and, correspondingly, two different Carroll fermionic theories. I will discuss them in detail. Then, I will show some examples; when the dispersion relation becomes trivial, i.e. energy bands flatten out, there can be a possibility of the emergence of Carroll symmetry.

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.

While Feynman integrals and scattering amplitudes are often very difficult to compute directly, their underlying properties point to new approaches. Cuts of diagrams are a tool for their efficient computation, related to unitarity and singularities. I will present definitions and interpretations of cuts, and ways to embed them in new frameworks which can be used to construct the original integrals and amplitudes.

The systematic construction of string theory duals for (perturbative) field theories, such as Quantum Chromodynamics (QCD), would be a major advance in the quest to access non-perturbative physics. Symmetric orbifold theories in 2-dimensional conformal field theories (CFTs) provide a promising stage to explore new ideas for such a perturbative holography. While they can be controlled with the rich toolbox of 2D CFTs they do share some basic features with their higher dimensional gauge theory cousins. In my talk, I will show the engineering of dual string backgrounds at work in the context of some symmetric product orbifolds. The construction ensures that the amplitudes of the dual string theory inherently reproduce the correlation functions of the two-dimensional CFT, order-by-order in perturbation theory, without requiring explicit computation on either side of the correspondence.