D-branes and Deformation Quantization

TL;DR

This paper demonstrates how D-brane world-volume geometries can be reconstructed from boundary conformal field theory, revealing their non-commutative algebraic structures and connecting to deformation quantization via the Moyal product.

Contribution

It provides a microscopic derivation of non-commutative D-brane algebras from boundary CFT, including perturbative calculations matching Kontsevich's deformation quantization.

Findings

Derived non-commutative algebras from open string operator products.

Connected the operator product expansion to the Moyal product.

Extended analysis to fermionic fields and discussed curved backgrounds.

Abstract

In this note we explain how world-volume geometries of D-branes can be reconstructed within the microscopic framework where D-branes are described through boundary conformal field theory. We extract the (non-commutative) world-volume algebras from the operator product expansions of open string vertex operators. For branes in a flat background with constant non-vanishing B-field, the operator products are computed perturbatively to all orders in the field strength. The resulting series coincides with Kontsevich's presentation of the Moyal product. After extending these considerations to fermionic fields we conclude with some remarks on the generalization of our approach to curved backgrounds.