Index calculation by means of harmonic expansion

TL;DR

This paper reviews the derivation of superconformal indices using harmonic expansion and localization techniques across various dimensions, emphasizing the mathematical methodology over physical applications.

Contribution

It provides a detailed technical account of calculating superconformal indices via harmonic expansion for multiple supersymmetric gauge theories, highlighting the method's implementation.

Findings

Calculation of indices for vector multiplets in 3d, 4d, and 6d

Analysis of energy eigenmodes on spheres

Focus on perturbative contributions in 6d

Abstract

We review derivation of superconformal indices by means of supersymmetric localization and spherical harmonic expansion for 3d N=2, 4d N=1, and 6d N=(1,0) supersymmetric gauge theories. We demonstrate calculation of indices for vector multiplets in each dimensions by analysing energy eigenmodes in S^pxR. For the 6d index we consider the perturbative contribution only. We put focus on technical details of harmonic expansion rather than physical applications.