The Schur index of 4d N=2 superconformal field theories

Student thesis: Doctoral ThesisDoctor of Philosophy


We study in this thesis the superconformal index of 4-dimensional N 2 SUpNq
gauge eld theories on S1 S3. We focus in particular on a reduced version of
the index, known as the Schur index, with only one fugacity q for the space-time
symmetries. We start by studying circular quiver gauge theories which can be diagrammatically
represented in a way reminiscent of A^ Dynkin diagrams. The Schur
index of these theories is usually given in terms of a complex matrix model involving
various elliptic functions. We use an elliptic determinant identity to simplify the
integrand and express it in terms of determinants, allowing us to rewrite the whole
index as a weighted sum over partition functions of free Fermi gases living on a
circle. Each partition function is then studied in the grand canonical ensemble and
we nd exact compact expressions for the analogue of the grand partition function
which we dene as the grand index. For short quivers with only one or two nodes we
are able to analytically deduce the Schur index exaclty as a q-series, and for small
values of N we are able to express it in terms of the complete elliptic integrals of the
rst and second kind. For longer quivers, we are able to extract the Schur index in
the large N limit, up to non-perturbative corrections. We also investigate another
class of theories, namely ^D-type quivers, for which we are able to apply the same
techniques. We obtain the grand index as well as a simple and compact expression
for its leading term in the large chemical potential limit.

This thesis also contains a detailed review of the superconformal index, a comparison
of our results with some other previously known expressions for the Schur
index obtained through other formalisms, as well as various technical appendices.
Date of Award2017
Original languageEnglish
Awarding Institution
  • King's College London
SupervisorNadav Drukker (Supervisor) & Nikolay Gromov (Supervisor)

Cite this