  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 $H$.

High Energy Physics - Theory