A note on singular D-branes in group manifolds

TL;DR

This paper classifies and constructs all untwisted D-branes in Lie groups of A-D-E series within WZW models, revealing their positions, moduli space, and stability conditions.

Contribution

It provides a systematic classification and construction method for untwisted D-branes in A-D-E Lie groups using Dynkin diagram techniques.

Findings

D-branes are classified by their positions in the maximal torus.

The moduli space corresponds to a unit cell in the weight space.

Quantum stability restricts D-branes to discrete positions.

Abstract

After reviewing D-branes as conjugacy classes and various charge quantizations (modulo $k$) in WZW model, we develop the classification and systematic construction of all possible untwisted D-branes in Lie groups of A-D-E series. D-branes are classified according to their positions in the maximal torus. The moduli space of D-branes is naturally identified with a unit cell in the weight space which is exponentiated to be the maximal torus. However, for the D-brane classification, one may consider only the fundamental Weyl domain that is surrounded by the hyperplanes defined by Weyl reflections. We construct all the D-branes by the method of iterative deletion in the Dynkin diagram. The dimension of a D-brane always becomes an even number and it reduces as we go from a generic point of the fundamental domain to its higher co-dimensional boundaries. Quantum mechanical stability requires that only D-branes at discrete positions are allowed.