3d partition functions & 4d superconformal indices from 1d free fermions

Student thesis: Doctoral ThesisDoctor of Philosophy


We study supersymmetric theories in three and four dimensions, focusing on two fundamental quantities - the partition function and the superconformal index. These observables give invaluable insight into the spectrum of supersymmetric theories and provide an important testing ground for dualities such as 3d mirror symmetry or AdS/CFT. As such, it is extremely desirable to evaluate them exactly. In recent years there has been immense progress in this direction: powerful exact techniques, such as supersymmetric localisation, have in many cases been able to reduce the calculation of these observables to the problem of evaluating finite dimensional matrix models. A great deal of effort has been applied to solving these matrix models, and already many results have been obtained. A particularly fruitful approach has been to manipulate these matrix models such that they resemble partition functions of free fermions in 1 dimension, so that one can apply powerful techniques from statistical mechanics.
We apply this `Fermi gas approach' to study the matrix models of 3d ^D quiver theories with UpNq gauge groups, as well as certain linear quiver theories, and we obtain large N evaluations of the partition function for a wide class of such theories. Along the way, we observe that 3d mirror symmetries - dualities that relate 3d supersymmetric theories that ow to the same infra-red fixed point - have a surprisingly elegant realisation in this setup, acting as linear canonical transformations on the Hamiltonians of the free fermions. This allows for extremely efficient derivations of the mirror maps - the relations between FI parameters and mass deformations on either side of the duality.
We also find a free fermion formulation for the matrix models corresponding to Schur indices of 4d A^ quiver theories with SUpNq gauge groups, which allows us to extract their leading order behaviour at large N. For some special examples, including N 4 SYM, we are able to go further and obtain exact, all order evaluations of the Schur index.
Date of Award2016
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorSakura Schafer-Nameki (Supervisor) & Nadav Drukker (Supervisor)

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