Renormalisation interfaces in two-dimensional quantum field theory

Supervisor: A. Konechny

Description: Renormalisation group (RG) is as fundamental concept in Quantum Field Theory (QFT) that describes how the physics changes under the change of energy scale. A typical renormalisation group trajectory starts from one fixed point and drives the theory to a different fixed point. In two dimensions the fixed points are described by conformal field theories that possess an infinite-dimensional symmetry algebra. While a lot is known about the end points of renormalisation group flows very little is known about the global structure of the space of flows linking the end points. The project is centred around the use of renormalisation domain walls or renormalisation interfaces. In two dimensions this object is a line of contact between two different conformal field theories on each side. The project involves studying such objects for concrete  RG flows both analytically and numerically. The aims are to learn how to construct such objects, what information they encode about the RG flows and how could they be used in gaining control over the space of flows.

Topological interfaces and renormalisation group flows

Supervisor: A. Konechny

Description: Topological interfaces can be thought  of as generalisations of conserved charges  in quantum field theories. In the context of two-dimensional conformal field theories (CFTs) the set of topological interfaces forms an interesting algebraic structure called fusion  category. This structure proved to be a useful tool in analysing various aspects of 2D CFTs.  Moreover in certain situations they can be used to constrain RG flows triggered by perturbing a 2D CFT by relevant operators. The project aims at studying this type of  constraints for particular RG flows and developing new methods of deriving such constraints. 

Twistors, quantum Donaldson-Thomas invariants and dispersionful integrable systems

Supervisor: R. Szabo

Description: This project develops a novel geometric framework for understanding the twistor geometry underlying quantum Donaldson-Thomas invariants and dispersionful integrable systems, based on the Moyal-deformed version of Plebanski's second heavenly equation for self-dual gravity.

Instantons and Donaldson-Thomas theory on Calabi-Yau 4-folds

Supervisor: R. Szabo

Description: This project explores the computation of Donaldson-Thomas invariants of toric Calabi-Yau 4-folds by enumerating BPS states in an 8-dimensional cohomological gauge theory. Specific goals are to extend the known calculations beyond flat space and local orbifolds to more general curved backgrounds, and to study the moduli space structures of these theories including their wall-crossing behaviour.

Homotopical descriptions of higher-form symmetries

Supervisor: R. Szabo

Description: This project explores various mathematical structures which underpin higher-form symmetries and their symmetry topological field theories (SymTFTs) using techniques based on groupoids of field configurations, the modern homotopical incarnation of the Batalin-Vilkovisky formalism based on factorisation algebras, and differential cohomology.

Braided homotopy algebras and noncommutative field theories

Supervisor: R. Szabo

Description: Several projects are available under this general theme, which formulates noncommutative field theories with braided symmetries in terms of a braided version of the Batalin-Vilkovisky formalism. Among the goals is to reach a novel homotopy double copy realisation of twisted noncommutative gravity in terms of noncommutative Yang-Mills theory.