Publications, Grants and Awards
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I study the mathematical structures that underlie my favourite physical theories. These include vertex operator algebras / conformal field theories and their relations with string theory, integrable models, representation theory and number theory (and anything else that I can think of). At the moment, this means logarithmic conformal field theory, Kac-Moody superalgebras, non-semisimple tensor categories and mock modular forms. I'm especially interested in the appearance of indecomposable (but reducible) representations in physics, but pretty much anything that involves cool math is fine with me. Currently, I'm studying examples of conformal field theories in which the correlation functions exhibit logarithmic singularities. In representation-theoretic terms, these logarithmic theories are built from representations which are indecomposable but not irreducible. Such theories arise naturally when considering so-called non-local observables (crossing probabilities, fractal dimensions) in the conformal limit of many exactly solvable lattice models (percolation, Ising). They are also relevant to string-theoretic considerations, especially when the target space admits fermionic directions, AdS/CFT, and perhaps even to black hole holography. In mathematics, there is a tantalising suggestion that logarithmic conformal field theory and Schramm-Loewner Evolution may be equivalent in some sense. The corresponding logarithmic vertex operator algebras also suggest natural generalisations of the notion of a modular tensor category. Finally, the characters of the modules of a vertex operator algebra tend to have nice modular properties. Some of the examples that I study are related to false / partial theta functions and mock / quantum modular forms.
Past Postgraduate Supervision
||"Coset construction for the N=2 and osp(1/2) minimal models"
||"Singular vectors for the WN algebras and the BRST cohomology for relaxed highest-weight Lk(sl(2)) modules"
Past Honours & MSc Students
||"Wess-Zumino-Witten models and the Knizhnik-Zamolodchikov equations"
||"The Structure Constants of the Minimal Models"
||"On the twisted sector of Wess-Zumino-Witten models"
||"Vertex operator algebras, modular tensor categories and a Kazhdan-Lusztig correspondence at a non-negative integral level"
Recent Grant History
|2020 - 2023
||Logarithmic conformal field theory and the 4D/2D correspondence
|2016 - 2018
||Towards higher rank logarithmic conformal field theories
|2010 - 2014
||Indecomposable structure in representation theory and logarithmic conformal field theory
- Course advisors - Upper Undergraduate
- Postgraduate Programs Committee