Open strings in Lie groups and associative products

TL;DR

This paper explores the semi-classical limit of open strings on D-branes in Lie groups, leading to associative products and matrix algebra descriptions, with extensions to quantum groups and connections to boundary conformal field theories.

Contribution

It introduces a new formalism for the semi-classical limit of open strings on Lie groups, resulting in associative products and matrix algebra frameworks, including extensions to quantum groups.

Findings

Construction of associative products for open strings in Lie groups.

Explicit analysis of SU(2) and SL(2,R) cases.

Connection to boundary vertex operator algebras and quantization methods.

Abstract

Firstly, we generalize a semi-classical limit of open strings on D-branes in group manifolds. The limit gives rise to rigid open strings, whose dynamics can efficiently be described in terms of a matrix algebra. Alternatively, the dynamics is coded in group theory coefficients whose properties are translated in a diagrammatical language. In the case of compact groups, it is a simplified version of rational boundary conformal field theories, while for non-compact groups, the construction gives rise to new associative products. Secondly, we argue that the intuitive formalism that we provide for the semi-classical limit, extends to the case of quantum groups. The associative product we construct in this way is directly related to the boundary vertex operator algebra of open strings on symmetry preserving branes in WZW models, and generalizations thereof, e.g. to non-compact groups. We treat the groups SU(2) and SL(2,R) explicitly. We also discuss the precise relation of the semi-classical open string dynamics to Berezin quantization and to star product theory.