D-branes and Deformation Quantization

TL;DR

The paper establishes that open-string boundary data in a flat background with a constant B-field generate a non-commutative, associative deformation of the D-brane world-volume algebra, encoded by a star product $f \star g$. Using conformal perturbation theory, the deformation is summed to all orders, yielding $f \star g = \sum_n (4\pi \alpha')^n w_n B_n(f,g)$ with $B_n(f,g)$ built from derivatives and the bi-vector ${\Theta}^{ij} = (B (1+B^2)^{-1})^{ij}$; the leading non-commutativity reproduces a Kontsevich/Moyal-type structure and, in the strong-field limit, aligns with a $B^{-1}$-directed deformation. The construction extends to the fermionic sector, deforming the Clifford algebra via $[\eta^i \stackrel{\star}{,} \eta^j]_+ = G^{ij} (1+B^2)^{-1}$, and suggests generalizations to curved backgrounds and coordinate-dependent B-fields. Overall, the work provides a concrete link between world-sheet open-string dynamics and deformation quantization on D-brane world-volumes, with implications for non-commutative effective theories and potential applications to WZW models and brane moduli spaces.

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.