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Advanced studies in pure mathematics

Boundary Conditions in Conformal Field Theory

John Cardy

Published Jun 17, 2018• 17 citations
Publisher

Abstract

<!-- *** Custom HTML *** --> We consider conformal field theories on manifolds with a boundary, and the constraints placed by modular invariance on their partition functions. In particular, the partition functions on an annulus with particular boundary conditions are given by the fusion rules. This leads to a simple derivation of the Verlinde formula. We note the remarkable fact that, for some integrable models, these partition functions have the same form away from criticality, with the modular parameter $q$ of the annulus replaced by a temperature-like variable, and give a partial explanation of this in the case of the Ising model.

DOI
10.2969/aspm/01910127

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<!-- *** Custom HTML *** --> We consider conformal field theories on manifolds with a boundary, and the constraints placed by modular invariance on their partition functions. In particular, the partition functions on an annulus with particular boundary conditions are given by the fusion rules. This leads to a simple derivation of the Verlinde formula. We note the remarkable fact that, for some integrable models, these partition functions have the same form away from criticality, with the modular parameter $q$ of the annulus replaced by a temperature-like variable, and give a partial explanation of this in the case of the Ising model.
This lecture note covers topics on boundary conformal field theory, modular transformations and the Verlinde formula, and boundary logarithmic CFT. An introductory review on CFT with boundary and a discussion of its applications to logarithmic cases are given. LCFT at c=−2c=-2c=−2 is mainly discussed.
Abstract The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS3.
A bstract We use the Virasoro TQFT to derive an integral identity that we view as a non-rational generalization of the Verlinde formula for the Virasoro algebra with central charge c ≥ 25. The identity expresses the Virasoro fusion kernel as an integral over a ratio of modular S-kernels on the (punctured) torus. In particular, it shows that the one-point S-kernel diagonalizes the Virasoro 6 j symbol. After carefully studying the analytic properties of this ‘Virasoro-Verlinde formula’, we present three applications. In boundary Liouville CFT, the formula ensures the open-closed duality of the boundary one-point function on the annulus. In pure 3d gravity, it provides an essential step in computing the partition function on hyperbolic 3-manifolds that fiber over the circle. Lastly, in AdS 3 /CFT 2 , the formula computes a three-boundary torus wormhole, which leads to a prediction for the statistical correlation between the density of states and two OPE coefficients in the dual large- c CFT ensemble. We conclude by discussing the implications of our result for the fusion rules in generic non-rational 2d CFTs.
The author calculates the order parameters (local height probabilities) in the ordered phase of integrable SU(n) vertex models. The author shows that they have formally the same expression as the partition functions of the associated critical theory in a finite box with appropriate boundary conditions, once the distance to criticality in the former case is properly identified with the modular parameter in the latter. This points out a relation between off-critical integrable models and conformal theories in a finite geometry.