Research Interests

• Quantum hamiltonian reduction and W-algebras

Quantum hamiltonian reduction is a process which takes certain vertex algebras and produces new vertex algebras, called W-algebras.

It also defines a 'reduction functor' mapping (complicated) representations of one vertex algebra to (simpler) representations of the W-algebra. This process can be partially reversed using so called 'inverse quantum hamiltonian reduction' functors.

I am interested in how we can using these functors to study vertex algebras and their representation theories. This work has included determining the reduction of various classes of modules (relaxed modules, logarithmic modules, etc). I also study how reduction functors behave when twisted by automorphisms (spectral flow and conjugation) and compatibility conditions between reduction and inverse reduction.

• Logarithmic conformal field theory and monoidal categories

Logarithmic CFT involves studying reducible but indecomposable representations of vertex operator algebras.

Mathematically, they give examples of non-semisimple categories with a monoidal structure (fusion).

Physically they are used to model a vast array of things. This includes modelling particles with substructure, non-semisimple TQFTs, and models with irreversible phenomena like particle decay or percolation.

I work on developing methods for constructing and classifying such logarithmic modules, and for studying the monoidal categories they form.

• Deformed W-algebras, cluster algebras and quiver realisations

Deformed W-algebras are notoriously difficult to construct, and very few examples are known at the level of generators and relations. Often the generators and relations are described using 'free field realisations', although these are messy to work with. 

Recently, techniques have been developed to encode these realisations into elegant quivers with an underlying cluster algebra structure. By taking an appropriate limit, one also obtains quivers and cluster algebras encoding realisations of quantum groups and deformed finite W-algebras.

Currently I am studying how these tools can be used to generalise the coset construction, quantum hamiltonian reduction and inverse reduction to the q-deformed setting. This would provide a new way to construct deformed W-algebras and their representations.

• Non-semisimple TQFTs

I am interested in the study of TQFTs built from non-semisimple monoidal categories. In my undergraduate honours thesis, I studied Chern-Simons theory, its relation to WZW models (affine vertex algebras at positive integer level) and how they lead to topological invariants like the Jones polynomial. During my PhD, I have extensively studied affine vertex algebras at non-positive integer level. In this case, there are categories of modules (believed to correspond to CFTs) which are non-semisimple. However (as far as I know) it is still an open problem to explicitly construct and describe the corresponding TQFTs.

While I am not actively working on this problem right now, I would like to investigate this in the future. It is also one reason I am interested in constructing non-semisimple monoidal categories of VOA-modules. There are also hints that there are topological invariants described by such TQFTs with surface defects implementing the reduction functor.

• Geometric representation theory for logarithmic modules

Vertex algebras and their representations have many different ties to algebraic geometry and geometric representation theory.

While not currently an active focus of mine, in the future I would like to better understand the link between these areas and my work. Topics I would like to learn more about include:

• Conformal blocks and moduli space in the non-rational setting,

• Quantum Langlands in the non-rational setting,

• Multiplicative Poisson geometry foundations for q-deformed vertex algebras.

I am also interested in better understanding how algebraic tools (screening operators, spectral flow, localisation, reduction, inverse reduction, Zhu's algebra, etc) relate to the geometry in these settings, and how these extend to the non-rational setting.