Indices for Supersymmetric Quantum Field Theories in Four Dimensions
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Authors
Ehrhardt, Mathieu
Advisors
Osborn, Hugh
Date
2012-07-21Awarding Institution
University of Cambridge
Author Affiliation
Department of Applied Mathematics and Theoretical Physics
Qualification
Doctor of Philosophy (PhD)
Language
English
Type
Thesis
Metadata
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Ehrhardt, M. (2012). Indices for Supersymmetric Quantum Field Theories in Four Dimensions (Doctoral thesis). https://doi.org/10.17863/CAM.21444
Abstract
In this thesis, we investigate four dimensional supersymmetric indices. The motivation for studying such objects lies in the physics of Seiberg's electric-magnetic duality in supersymmetric field theories. In the first chapter, we first define the index and underline its cohomological nature, before giving a first computation based on representation theory of free superconformal field theories. After listing all representations of the superconformal algebra based on shortening conditions, we compute the associated Verma module characters, from which we can extract the index in the appropriate limit. This approach only provides us with the free field theory limit for the index and does not account for the values of the $R$-charges away from free field theories. To circumvent this limitation, we then study a theory on $\mathbb{R}\times S^3$ which allows for a computation of the superconformal index for multiplets with non-canonical $R$-charges. We expand the fields in harmonics and canonically quantise the theory to analyse the set of quantum states, identifying the ones that contribute to the index. To go beyond free field theory on $\mathbb{R}\times S^3$, we then use the localisation principle to compute the index exactly in an interacting theory, regardless of the value of the coupling constant. We then show that the index is independent of a particular geometric deformation of the underlying manifold, by squashing the sphere. In the final chapter, we show how the matching of the index can be used in the large $N$ limit to identify the $R$-charges for all fields of the electric-magnetic theories of the canonical Seiberg duality. We then conclude by outlining potential further work.
Keywords
supersymmetric index, Seiberg duality, superconformal field theories, harmonics, electric-magnetic duality, indices
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