Three-algebra for supermembrane and two-algebra for superstring

TL;DR

This paper develops a covariant supermembrane action in eleven dimensions using Nambu three-brackets, revealing a three-algebra structure, and demonstrates how it reduces to superstring actions and matrix models, advancing M-theory formulations.

Contribution

It introduces a supermembrane action explicitly formulated with Nambu three-brackets, connecting three-algebra structures to superstring and matrix models.

Findings

Supermembrane action expressed via Nambu three-brackets.

Double dimensional reduction yields type IIA string action.

Constructs covariant type IIB superstring action leading to IKKT matrix model.

Abstract

While string or Yang-Mills theories are based on Lie algebra or two-algebra structure, recent studies indicate that M-theory may require a one higher, three-algebra structure. Here we construct a covariant action for a supermembrane in eleven dimensions, which is invariant under global supersymmetry, local fermionic symmetry and worldvolume diffeomorphism. Our action is classically on-shell equivalent to the celebrated Bergshoeff-Sezgin-Townsend action. However, the novelty is that we spell the action genuinely in terms of Nambu three-brackets: All the derivatives appear through Nambu brackets and hence it manifests the three-algebra structure. Further the double dimensional reduction of our action gives straightforwardly to a type IIA string action featuring two-algebra. Applying the same method, we also construct a covariant action for type IIB superstring, leading directly to the IKKT matrix model.