Relating U(N)xU(N) to SU(N)xSU(N) Chern-Simons Membrane theories

TL;DR

This paper demonstrates the equivalence between U(N)xU(N) and SU(N)xSU(N) Chern-Simons membrane theories under certain conditions, clarifying their relationship in M2-brane models and identifying specific cases of equivalence.

Contribution

It shows that U(n)xU(n) ABJM theory reduces to a Z_k quotient of (SU(n)xSU(n))/Z_n theory when n and k are coprime, and proposes specific equivalences for low-rank cases.

Findings

U(n)xU(n) ABJM at level k is equivalent to a Z_k quotient of (SU(n)xSU(n))/Z_n for coprime n and k.

The k=1 ABJM model for two M2-branes is equivalent to the N=8 (SU(2)xSU(2))/Z_2 theory.

Conjecture that the U(2)xU(2) ABJM model at k=2 is equivalent to the N=8 SU(2)xSU(2) theory.

Abstract

By integrating out the U(1)_B gauge field, we show that the U(n)xU(n) ABJM theory at level k is equivalent to a Z_k identification of the (SU(n)xSU(n))/Z_n Chern-Simons theory, but only when n and k are coprime. As a consequence, the k=1 ABJM model for two M2-branes in R^8 can be identified with the N=8 (SU(2)xSU(2))/Z_2 theory. We also conjecture that the U(2)xU(2) ABJM model at k=2 is equivalent to the N=8 SU(2)xSU(2)-theory.