D-branes on Group Manifolds and Deformation Quantization

TL;DR

This paper connects Kontsevich's deformation quantization formula to D-branes on group manifolds, deriving a specific star-product for the dual of a Lie algebra via correlation functions in Wess-Zumino-Witten theory.

Contribution

It provides a derivation of Kontsevich's star-product formula for the dual of a Lie algebra using D-branes on group manifolds and WZW correlation functions.

Findings

Derived the star-product formula for the dual of a Lie algebra.

Connected deformation quantization to D-branes and WZW theory.

Highlighted the role of the B-field in the derivation.

Abstract

Recently M. Kontsevich found a combinatorial formula defining a star-product of deformation quantization for any Poisson manifold. Kontsevich's formula has been reinterpreted physically as quantum correlation functions of a topological sigma model for open strings as well as in the context of D-branes in flat backgrounds with a Neveu-Schwarz B-field. Here the corresponding Kontsevich's formula for the dual of a Lie algebra is derived in terms of the formalism of D-branes on group manifolds. In particular we show that that formula is encoded at the two-point correlation functions of the Wess-Zumino-Witten effective theory with Dirichlet boundary conditions. The B-field entering in the formalism plays an important role in this derivation.