An associative star-three-product and applications to M two/M five-brane theory

TL;DR

This paper introduces a star-three-product to extend the algebraic framework of star-products, enabling the realization of hermitian three-algebras and deconstruction of M-brane theories, including M2 and M5 branes, on fuzzy three-tori.

Contribution

It develops a novel star-three-product and applies it to construct hermitian three-algebras, facilitating the deconstruction of M-brane theories on fuzzy three-tori.

Findings

Defined a fuzzy three-torus and its algebraic structure.

Realized hermitian three-algebra of ABJM theory using star-three-product.

Deconstructed Abelian and non-Abelian M five-branes on fuzzy three-tori.

Abstract

The star-product between functions enable us to take the large $N$ limit in a controlled way. At finite $N$ it serves as a substitute for matrix multiplications. Non-abelian gauge theory can be deconstructed from lower dimensional gauge theories using star-products. In this paper we extend the star-product to a star-three-product. We then apply the star-three-product to realize hermitian three-algebra of ABJM theory. We define a fuzzy three-torus. We deconstruct Abelian M five-brane in a constant background three-form potential on a fuzzy three-torus. We deconstruct non-Abelian extensions which might be related with multiple M five branes. We also mention the fuzzy three-sphere case.