Factorization and generalized *-products

TL;DR

This paper explores the mathematical structure of generalized *-products in noncommutative gauge theories and string theory, providing a unified understanding and explicit formulas for these products, with potential extensions to more complex cases.

Contribution

It offers a uniform framework for understanding and computing $*_N$-products in noncommutative theories, including explicit formulas and insights into their factorization properties.

Findings

Unified understanding of $*_N$-products through string amplitude factorization

Explicit momentum space formulas for arbitrary N in non-Abelian cases

Discussion of possible extensions to ${}_M *_{N}$-products in multi-loop contexts

Abstract

The generalized *-products, or the $*_N$-products, appear both in the one-loop effective action of noncommutative Yang-Mills theories and in the coupling of a closed string to N open strings on a disk when the D-brane world-volume is noncommutative. Factorization of the string amplitudes provides a uniform understanding of the $*_N$-products and a hint to obtain a simple, explicit formula (in the momentum space) for arbitrary N in the non-Abelian case. Possible extension to a more general ${}_M *_{N}$-product in the M-loop context is discussed.