Quantum fields in a stable universe

In this project, I propose a mathematical language to study geometric moduli of quantum field theories, crucially, in a setting that encodes renormalization group flows.

This will enable the identification of new computational tools by applying state-of-the-art algebro-geometric and -topological tools to highly structured geometric objects that encode the moduli of QFTs.

More generally, these geometries will give us a broader understanding of QFT, which, in the long term, will be crucial in unifying quantum theory with gravity. IWe will begin by analysing so-called invertible theories.

Preliminary work shows that there is an affine spectral scheme associated to a so-called Thom spectrum which classifies the theory in the stable homotopy category. This construction extends to the important case of anomalous theories, realised as a class of relative theories.

We will then move to theories without the invertibility assumption, which will more properly exhibit the geometric structures associated with renormalization group flow.

We will study the general features of such models, as well as extracting RG flow invariants through the study of the K-theory of such models.

An important class of examples, using all the machinery that we develop, will be two dimensional supersymmetric theories.

We can compare these with extant results and can hope to make contact with the influential conjecture of Stolz-Teichner regarding the classification of such theories.