*-Trek II: *_n Operations, Open Wilson Lines and the Seiberg-Witten Map

TL;DR

This paper explores the mathematical structure of noncommutative gauge theories, linking n-ary star products to open Wilson lines, and applies this to derive gauge-invariant actions, couplings, and a Seiberg-Witten map.

Contribution

It introduces a new understanding of *_n operations via open Wilson lines and derives explicit gauge-invariant formulations in noncommutative gauge theories.

Findings

Gauge-invariant effective action for one-loop F^4 terms in noncommutative N=4 SYM.

Explicit gauge-invariant couplings between noncommutative modes and closed string modes.

A closed-form expression for the Seiberg-Witten map in the U(1) case.

Abstract

Generalizations of the *-product (e.g. n-ary *_n operations) appear in various places in the discussion of noncommutative gauge theories. These include the one-loop effective action of noncommutative gauge theories, the couplings between massless closed and open string modes, and the Seiberg-Witten map between the ordinary and noncommutative Yang-Mills fields. We propose that the natural way to understand the *_n operations is through the expansion of an open Wilson line. We establish the connection between an open Wilson line and the *_n operations and use it to: (I) write down a gauge invariant effective action for the one-loop F^4 terms in the noncommutative N=4 SYM theory; (II) find the gauge invariant couplings between the noncommutative SYM modes and the massless closed string modes in flat space; (III) propose a closed form for the Seiberg-Witten map in the U(1) case.