The geometry of WZW branes
G. Felder, J. Fröhlich, J. Fuchs, C. Schweigert
TL;DR
The paper addresses how conformally invariant boundary conditions in WZW models correspond to brane geometry on a group manifold and how the bulk-boundary correspondence encodes the embedding of branes in target space. It introduces a computable observable from bulk-boundary couplings, analyzes it with a non-abelian Peter–Weyl Fourier transform on the level-$k$ function space ${\cal F}_k(G)$, and shows that in the large level limit the brane world volume supports a commutative algebra of functions with open strings between different conjugacy classes decoupled; at finite level the branes are smeared into a fuzzy neighborhood of regular conjugacy classes. The paper further shows that boundary structure constants in any rational CFT coincide with entries of the fusing matrices ${\sf F}$, linking RCFT boundary data to bulk fusion data. For symmetry-breaking boundaries, endpoints lie on twined conjugacy classes ${\cal C}_G^\omega(h)$, and the analysis extends to non-simply connected groups, revealing fractional branes and related subtleties. Overall, the work provides a geometric and algebraic framework for D-branes in curved backgrounds, clarifying how non-commutative geometry emerges in WZW target spaces.
Abstract
The structures in target space geometry that correspond to conformally invariant boundary conditions in WZW theories are determined both by studying the scattering of closed string states and by investigating the algebra of open string vertex operators. In the limit of large level, we find branes whose world volume is a regular conjugacy class or, in the case of symmetry breaking boundary conditions, a `twined' version thereof. In particular, in this limit one recovers the commutative algebra of functions over the brane world volume, and open strings connecting different branes disappear. At finite level, the branes get smeared out, yet their approximate localization at (twined) conjugacy classes can be detected unambiguously. As a by-product, it is demonstrated how the pentagon identity and tetrahedral symmetry imply that in any rational conformal field theory the structure constants of the algebra of boundary operators coincide with specific entries of fusing matrices.