D-Branes on Group Manifolds
Mitsuhiro Kato, Tomoharu Okada
TL;DR
This work classifies Dirichlet boundary states for WZW models with untwisted affine symmetry across all compact simple groups, showing that each consistent boundary condition arises from a ${f Z}_2$ automorphism of $G$ and yields a D-brane whose world-volume is the group manifold of a symmetric subgroup $H$. Consequently, the D-brane moduli space is the irreducible Riemannian symmetric space $G/H$, with abelian T-duality relating distinct boundary states within the same symmetry class. Notably, there is no D-particle on a compact simple group manifold, while if the world-volume contains an $S^1$ factor the moduli space becomes hermitian symmetric, allowing open-string world-sheet instantons and potential ${ m N}=2$ structure. The results connect D-brane geometry in curved backgrounds to symmetric-space structure and map the duality relations of boundary states in WZW theories.
Abstract
Possible Dirichlet boundary states for WZW models with untwisted affine super Kac-Moody symmetry are classified for all compact simple Lie groups. They are obtained by inner- and outer-automorphism of the group. D-brane world-volume turns out to be a group manifold of a symmetric subgroup, so that the moduli space of D-brane is a irreducible Riemannian symmetric space. It is also clarified how these D-branes are transformed to each other under abelian T-duality of WZW model. Our result implies, for example, there is no D-particle on the compact simple group manifold. When the D-brane world-volume contains $S^1$ factor, the D-brane moduli space becomes hermitian symmetric space and the open string world-sheet instantons are allowed.