Abstract
All non-trivial two dimensional conformal field theories (2d CFT) have an infinite set of mutually commuting conserved charges related to the KdV integrable hierarchy. These conserved charges can be inserted with a corresponding chemical potential into the characters of the 2d CFT to obtain generalised Gibbs ensembles (GGEs). These generalised Gibbs ensembles have important applications in condensed matter systems as well as interesting connections to black holes in three dimensional Anti de Sitter gravity. Mathematically it is known that the characters of a large class of 2d CFTs without the additional conserved charges are vector valued, weight 0 modular forms and the thermal correlation functions of the charges are quasi modular forms. A natural question is: what are the modular properties of the full GGE?The majority of this thesis is focused on answering this question. We study the problem in the c = 1/2 minimal model which has a realisation as free fermions. In this model the KdV charges can all be written in terms of bilinears in the fermion modes. This allows us to explicitly diagonalise the charges and write the GGE in a closed form. One can then use the thermodynamic Bethe ansatz to derive a conjecture for the correct transformed expression for the GGE. This conjecture can be proved using recent results in the mathematics literature on power partitions. Finally we provide a physical interpretation for this transformed GGE as a system with a line defect.
In this thesis we also study the modular properties of massive two dimensional quantum field theories. The presence of a mass breaks conformal invariance and so it is not expected that the partition functions and correlators will be modular forms. However if the mass is appropriately transformed under a modular transformation the partition functions and correlators do have nice modular properties. We study the correlators for an infinite set of conserved currents in the massive free fermion model. We find that if the mass transforms as a weight (1, 1) modular form then these correlators are non-holomorphic modular forms.
Furthermore they fit into the framework of massive Maass and Jacobi forms introduced by Berg, Bringmann and Gannon.
| Date of Award | 1 Nov 2024 |
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| Original language | English |
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| Supervisor | Gerard Watts (Supervisor) |