Geometric construction of D-branes in 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\, $F=-R$.Comment: 23 pages, discussion of limitations of the gluing condition approach adde

D-branes with Lorentzian signature in the Nappi-Witten model

“…Points in N can be parameterized by the string endpoints coordinates x µ (τ ) = X µ (τ, σ) ∂Σ , so we will write g(x). The D-brane can be specified [9,12,13,18] by (i) An isometry F of Ω, that in general may depend on g, and a condition…”

“…In Section 2, we review the semiclassical characterization of D-branes in a WZW model. The material presented there can be found elsewhere [11][12][13], though it emphasizes some points [18] concerning the rôle of Frobenius theorem and involutivity that have gone somewhat unnoticed in the literature. Section 3 contains a brief account of the Nappi-Witten model, including a complete characterization of its Lie algebra isometries.…”

“…More generally, since WZW models are the building blocks of many string backgrounds, one expects to learn about D-branes and their noncommutative field theories by looking at open strings on group manifolds. This entails as a first step the characterization of D-branes in WZW models [9][10][11][12][13][14][15][16][17][18]. Such characterization is well understood in some cases.…”

“…In particular, it is known that the metrically nondegenerate R-twined conjugacy classes of a WZW group manifold are D-branes for all Lie algebra metric-preserving automorphisms R. These twined conjugacy classes are obtained as the solutions to a gluing condition J + = RJ − that matches the chiral currents J + and J − of the model at the D-brane. Very little is known, however, if in the gluing condition, instead of a Lie algebra automorphism, an arbitrary isometry F of the Lie algebra metric is considered [18]. This is due to the fact that involutivity (required for the solution to the gluing condition to define a submanifold) holds trivially for Lie algebra automorphisms, whereas for general isometries it usually does not.…”

“…In this paper we apply geometric characterization of D-branes in nonsemisimple Lie groups along the lines of ref. [18] to the Nappi-Witten model. There are other ways to approach the study of D-branes in WZW models.…”