S1 reduction of 4D $$ \mathcal{N} $$ = 3 SCFTs and squashing independence of ABJM theories

Abstract

Abstract We study the compactification of 4D $$ \mathcal{N} $$ N = 3 superconformal field theories (SCFTs) on S1, focusing on the relation between the 4D superconformal index and 3D partition function on the squashed sphere $$ {S}_b^3 $$ S b 3 . Since the center $$ \mathfrak{u} $$ u (1) of the $$ \mathfrak{u} $$ u (3) R-symmetry of the 4D theory can mix with an $$ \mathcal{N} $$ N = 6 abelian flavor symmetry in three dimensions, the precise 4D/3D relation for the global symmetry is not obvious. Focusing on the case in which the 3D theory is the ABJM theory, we demonstrate that the above R-symmetry mixing can be precisely identified by considering the Schur limit (and/or its $$ \mathcal{N} $$ N = 3 cousin) of the 4D index. As a result, we generalize to the ABJM theories recent discussions on the connection between supersymmetry enhancement of the 4D index and squashing independence of the $$ {S}_b^3 $$ S b 3 partition function.