Star Products from Open Strings in Curved Backgrounds

TL;DR

This paper introduces a new non-commutative product for open strings in curved backgrounds, demonstrating its properties and conditions for associativity, with implications for string theory and non-commutative geometry.

Contribution

It defines a non-commutative product for open strings in arbitrary backgrounds, analyzing its properties and associativity without topological constraints, extending previous formulations.

Findings

The product has the trace property and is associative up to surface terms on-shell.

No on-shell conditions are needed for the functions inserted into the product.

Inclusion of the full Born-Infeld measure is essential for the formulation.

Abstract

We define a non-commutative product for arbitrary gauge and B-field backgrounds in terms of correlation functions of open strings. While off-shell correlations are, of course, not conformally invariant, it turns out that, at least to first derivative order, our product has the trace property and is associative up to surface terms if the background fields are put on-shell. No on-shell conditions for the inserted functions are needed, but it is essential to include the full contribution of the Born-Infeld measure. We work with a derivative expansion and avoid any topological limit, which would effectively constrain $H$.