Cyclicity of non-associative products on D-branes

TL;DR

This paper explores the cyclicity of non-associative star products in string theory's non-commutative geometry, extending previous work to second order and revealing modifications to the Kontsevich formula in curved backgrounds.

Contribution

It extends the analysis of non-associative products on D-branes to second order, demonstrating cyclicity with respect to the Born--Infeld measure and identifying a logarithmic correction to the Kontsevich formula.

Findings

Cyclicity is established at second order in derivative expansion.

A logarithmic correction modifies the Kontsevich formula in curved backgrounds.

The generalized Maxwell equation ensures divergence-free non-commutative parameters.

Abstract

The non-commutative geometry of deformation quantization appears in string theory through the effect of a B-field background on the dynamics of D-branes in the topological limit. For arbitrary backgrounds, associativity of the star product is lost, but only cyclicity is necessary for a description of the effective action in terms of a generalized product. In previous work we showed that this property indeed emerges for a non-associative product that we extracted from open string amplitudes in curved background fields. In the present note we extend our investigation through second order in a complete derivative expansion. We establish cyclicity with respect to the Born--Infeld measure and find a logarithmic correction that modifies the Kontsevich formula in an arbitrary background satisfying the generalized Maxwell equation. This equation is the physical equivalent of a divergence-free non-commutative parameter, which is required for cyclicity already in the associative case.