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

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.

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