Walter H. Baron

We study modular properties of the ${\mathrm{AdS}}_3$ Wess-Zumino-Novikov-Witten model. Although the Euclidean partition function is modular invariant, the characters on the Euclidean torus diverge and the regularization proposed in the literature removes information on the spectrum and the usual one to one map between characters and representations of rational models is lost. Reconsidering the characters defined on the Lorentzian torus and focusing on their structure as distributions, we obtain expressions that recover those properties. We study their modular transformations and find a generalized $S$ matrix, depending on the sign of the real modular parameters, which has two diagonal blocks and one off-diagonal block, mixing discrete and continuous representations, that we fully determine. We then explore the relations among the modular transformations, the fusion algebra and the boundary states. We explicitly construct Ishibashi states for the maximally symmetric $D$-branes and show that the generalized $S$ matrix defines the one-point functions associated to pointlike and ${\mathrm{H}}_2$-branes as well as the fusion rules of the degenerate representations of $sl(2,\mathbb{R}$) appearing in the open string spectrum of the pointlike $D$-branes, through a generalized Verlinde theorem. DOI: http://dx.doi.org/10.1103/PhysRevD.83.106010 © 2011 American Physical Society

We study the operator product expansion in the ${\mathrm{AdS}}_3$ Wess-Zumino-Novikov-Witten (WZNW) model. The operator-product expansion of primary fields and their spectral flow images is computed from the analytic continuation of the expressions in the ${\mathrm{H}}_3^{+}$ WZNW model, adding spectral flow. We argue that the symmetries of the affine algebra require a truncation which establishes the closure of the fusion rules on the Hilbert space of the theory. Although the physical mechanism determining the decoupling is not completely understood, we present several consistency checks on the results. A preliminary analysis of factorization allows to obtain some properties of four-point functions involving fields in generic sectors of the theory, to verify that they agree with the spectral flow selection rules and to show that the truncation must be realized in physical amplitudes for consistency.