# Symplectic Three-Algebra Unifying N=5,6 Superconformal Chern-Simons-Matter Theories

## TL;DR

This paper introduces a symplectic 3-algebra framework that unifies various N=5 and N=6 superconformal Chern-Simons-matter theories, including ABJM, by defining specific algebraic structures and demonstrating supersymmetry enhancement.

## Contribution

The paper develops a symplectic 3-algebra that unifies multiple superconformal theories and shows how supersymmetry can be enhanced within this algebraic framework.

## Key findings

- Unified description of N=5 and N=6 theories using symplectic 3-algebra
- Demonstration of supersymmetry enhancement from N=5 to N=6
- Description of known theories like ABJM within the new algebraic framework

## Abstract

We define a 3-algebra with structure constants being symmetric in the first two indices. We also introduce an invariant anti-symmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The general N=5 superconformal Chern-Simons-matter (CSM) theory with SO(5) R-symmetry in three dimensions is constructed by using this algebraic structure. We demonstrate that the supersymmetry can be enhanced to N=6 if the sympelctic 3-algebra and the fields are decomposed in a proper fashion. By specifying the 3-brackets, some presently known N=5, 6 superconformal theories are described in terms of this unified 3-algebraic framework. These include the N=5, Sp(2N) X O(M) CSM theory with SO(5) R-symmetry , the N=6, Sp(2N) X U(1) CSM theory with SU(4) R-symmetry, as well as the ABJM theory as a special case of U(M) X U(N) theory with SU(4) R-symmetry.