3d superconformal indices and isomorphisms of M2-brane theories

TL;DR

The paper investigates quantum isomorphisms among M2-brane theories by computing and comparing 3d superconformal indices for $U(N)_k\times U(N)_{-k}$ ABJM, $(SU(N)_k\times SU(N)_{-k})/{\mathbb Z}_N$ BLG, and ABJ variants, across several $N$ and $k$. Using localization, it derives explicit index expressions with fixed topological charge $T$ for ABJ(M) and flavor fugacity $z$ for BLG, including the effects of monopole charges and the ${\mathbb Z}_k$ identifications. The results confirm the isomorphism $U(2)_k\times U(2)_{-k}$ ABJM with the ${\mathbb Z}_k$ quotient of $(SU(2)_k\times SU(2)_{-k})/{\mathbb Z}_2$ BLG for odd $k$, and support the conjecture that the $N=3$ case holds when gcd$(N,k)=1$, up to the computed order in $x$; they also find no evidence for extending these isomorphisms to higher $k$ in the tested pairs. Overall, the index matches provide nontrivial quantum checks of these isomorphisms beyond classical moduli-space analyses, sharpening the map between ABJM/ABJ(M) and BLG descriptions of multiple M2-branes.

Abstract

We test several expected isomorphisms between the U(N)xU(N) ABJM theory and (SU(N)xSU(N))/Z_N theory including the BLG theory by comparing their superconformal indices. From moduli space analysis, it is expected that this equivalence can hold if and only if the rank N and Chern-Simons level k are coprime. We also calculate the index of the ABJ theory and investigate whether some theories with identical moduli spaces are isomorphic or not.