Week 02.03.2025 – 08.03.2025

Quantum Chromodynamics (QCD) has been a profound source of inspiration for theoretical physics, driving the development of key concepts such as string theory, effective field theories, instantons, anomalies, and lattice gauge theories. In these lectures, I will explore two distinct regimes of QCD - its infrared (IR) and ultraviolet (UV) limits - and the theoretical tools used to study them. In the IR regime, where perturbative techniques break down, Effective Field Theories (EFTs) provide a powerful framework. I will introduce the pion EFT as a tool to study non-linearly realized symmetries and soft theorems. In the UV regime, where QCD becomes amenable to perturbative analysis, I will discuss the Operator Product Expansion and renormalization group equations, focusing on their application to deep inelastic scattering, a cornerstone in the discovery of quarks and gluons. These two regimes illustrate the richness of QCD and its pivotal role in shaping our understanding of fundamental physics.

About fifty years ago, Gibbons and Hawking argued that the Euclidean gravitational path integral with suitable boundary conditions can be interpreted as a grand-canonical partition function. Classical gravitational solutions, including black holes, arise as saddles of this path integral, and from the saddle-point action one can extract the black hole entropy. In the talk, I will discuss some recent developments of these ideas. Working in five dimensions, we will see how imposing supersymmetric boundary conditions converts the partition function into an index. Then we will construct a class of saddles of this index which interpolates between supersymmetric black holes and horizonless microstate geometries. I will discuss how the saddle point action can be computed via equivariant localization. Finally, I will comment on the relevance of these findings for black hole microstate counting and holography.

We propose a novel analytical framework for incompressible Navier-Stokes (NS) turbulence, revealing a duality between classical fluid dynamics and one-dimensional nonlinear dynamics in loop space. This reformulation leads to a universal momentum loop equation, which excludes finite-time blow-ups, establishing a \textit{No Explosion Theorem} for turbulent flows with finite initial noise. Decaying turbulence emerges as a solution to this equation and is interpreted as a solvable string theory with a discrete target space composed of regular star polygons. The derived decay spectrum exhibits excellent agreement with experimental data and direct numerical simulations (DNS), replacing classical Kolmogorov scaling laws with universal functions derived from number theory. These results suggest a deeper mathematical structure underlying turbulence, uniting fluid dynamics, quantum mechanics, and number theory.

Topological field theories, or what is otherwise commonly known as topological orders, can be given a rigorous axiomization in terms of higher fusion categories with some extra structure. The extra structure is necessary to incorporate the relevant physical properties of topological orders. With this categorical framework we can give classifications of topological orders in low dimensions. I will explain how this is done for (3+1)d topological orders that are fermionic in nature. with Pedagogical Intro by Juven Wang

Since its discovery by Berkovits 25 years ago, the pure-spinor formulation of the superstring has proven to be a very useful tool in the calculation of scattering amplitudes, both at tree- and loop-level. However, almost all of its applications are confined to the scattering of massless states. Computation of massive string amplitudes is possible in principle, but difficult to perform within the usual pure-spinor prescription. In this talk, I will report progress on computing manifestly super-Poincare invariant amplitudes through an alternative procedure, which does not explicitly involve the pure-spinor variable and should apply equally well to massless and massive external states.