Geometric construction of D-branes in WZW models

TL;DR

This paper advances the geometric understanding of D-branes in WZW models by analyzing boundary conditions via linear maps, establishing conditions for their existence, and validating certain classes of D-branes through isometries.

Contribution

It provides a detailed geometric framework for D-branes in WZW models, including conditions on the gluing map and validation of R-twined conjugacy classes as D-branes.

Findings

Frobenius integrability constrains the gluing map F to be an isometry.

Metrically degenerate R-twined conjugacy classes are confirmed as D-branes.

No D-branes exist for constant F=-R in semisimple WZW models.

Abstract

The geometric description of D-branes in WZW models is pushed forward. Our starting point is a gluing condition\, $J_{+}=FJ_-$ that matches the model's chiral currents at the worldsheet boundary through a linear map $F$ acting on the WZW Lie algebra. The equivalence of boundary and gluing conditions of this type is studied in detail. The analysis involves a thorough discussion of Frobenius integrability, shows that $F$ must be an isometry, and applies to both metrically degenerate and nondegenerate D-branes. The isometry $F$ need not be a Lie algebra automorphism nor constantly defined over the brane. This approach, when applied to isometries of the form $F=R$ with $R$ a constant Lie algebra automorphism, validates metrically degenerate $R$-twined conjugacy classes as D-branes. It also shows that no D-branes exist in semisimple WZW models for constant\, $F=-R$.