Week 19.03.2023 – 25.03.2023

“Special” geometries, such as Calabi-Yau manifolds, play a central role in multiple areas of string theory, as well as gravitational theories more generally. The goal of these lectures is to introduce some of the formalism and tools useful for characterising such geometries, pitched at the level of a starting PhD student. We will start with purely geometrical backgrounds using the general notions of a G-structure and special holonomy and then will go on to describe backgrounds that also have non-trivial fluxes. We will be guided by applications to string phenomenology and the AdS/cft correspondence.

I will present a powerful new method that for the first time allows us to compute the Kaluza-Klein spectrum of a large class of string theory compactifications, including those arising in maximal gauged supergravities and beyond. This includes geometries with little to no remaining (super-)symmetries, completely inaccessible by previous methods. I will show how these insights can be used to holographically compute the anomalous dimensions of protected and unprotected operators in strongly-coupled CFTs, as well as to study global properties of their conformal manifolds. I will also show how the method can be used to determine the perturbative stability of non-supersymmetric AdS vacua. We will see the importance of higher Kaluza-Klein modes to the physics of string compactifications, e.g. in realising the compactness of moduli spaces, restoring supersymmetry that is lost in a consistent truncation, and in destabilising non-supersymmetric vacua that appear to stable in lower-dimensional supergravities.

Abstract: I will present a way to promote the anomalous axial U(1) in 4d QED to an exact symmetry, with the price of losing its invertibility. I will then discuss some applications of this non-invertible U(1) symmetry. In particular, I will show how to couple this non-invertible symmetry to a gauge field. By taking this gauge field to be dynamical, we get a new type of gauge theory with unconventional interactions and constraints. By taking this gauge field to be background, we can study 't-Hooft anomalies of the non-invertible symmetry.

I will present a powerful new method that for the first time allows us to compute the Kaluza-Klein spectrum of a large class of string theory compactifications, including those arising in maximal gauged supergravities and beyond. This includes geometries with little to no remaining (super-)symmetries, completely inaccessible by previous methods. I will show how these insights can be used to holographically compute the anomalous dimensions of protected and unprotected operators in strongly-coupled CFTs, as well as to study global properties of their conformal manifolds. I will also show how the method can be used to determine the perturbative stability of non-supersymmetric AdS vacua. We will see the importance of higher Kaluza-Klein modes to the physics of string compactifications, e.g. in realising the compactness of moduli spaces, restoring supersymmetry that is lost in a consistent truncation, and in destabilising vacua that appear to stable in lower-dimensional supergravities.

In this talk we will explore a class of 3d N=2 theories labeled by graphs and related to quivers in an unusual way. Unlike quiver gauge theories --- a class of Lagrangian field theories widely used in modern QFT --- theories that we consider are non-Lagrangian, in a sense that they can be defined as IR fixed points of gauge fields coupled to non-linear matter as in the Skyrme models of nuclei. Just like the physics of the Skyrme model is intimately tied to symmetries of QCD, generalized symmetries play an important role in these 3d N=2 theories. The connection to quivers, on the other hand, arises in a way that is not standard in modern particle physics, but is standard in the study of logarithmic CFTs and motivic DT invariants. Download Mathematica demonstration here: theory.caltech.edu/~gukov/Plumbed.nb 3d Modularity software: https://github.com/d-passaro/pySeifert