An Index for Non-relativistic Superconformal Field Theories

TL;DR

This work develops a highest-weight framework for the non-relativistic ${ m SSCH}_2$ superconformal algebra in $(1+2)$ dimensions and defines a robust superconformal index invariant under exact marginal deformations. It analyzes both bosonic and ${ m N}=2$ supersymmetric representations, classifies unitary multiplets (vacuum, chiral, anti-chiral, long), and derives shortening conditions that constrain the spectrum. The index is constructed via two conjugations, with the BPZ-type conjugation yielding $I(x)=\mathrm{Tr}(-1)^F e^{-eta\Delta} x^{R-2J}$ and $\Delta=-\tfrac12(iD-J+\tfrac32 R)$, so only short multiplets contribute, ensuring deformation invariance. The paper then computes explicit NR indices for the non-relativistic limit of Abelian CS-matter theory and ABJM theory in the large-$N$ limit, illustrating the method with exact product representations and matrix-integral expressions, and discusses future connections to relativistic theories and potential gravity duals in NRAdS/CFT.

Abstract

We study the highest-weight representation of N=2 supersymmetric Schrodinger algebra which appears in non-relativistic superconformal field theories in (1+2) dimension. We define the index for the non-relativistic superconformal field theories and study its properties. As a concrete example, we compute the index for the non-relativistic limit of N=6 superconformal Chern-Simons-matter theory recently proposed by Aharony et al.