Boundary Structure Constants Research Articles

Boundary Liouville theory and 2D quantum gravity

We study the boundary correlation functions in Liouville theory and in solvable statistical models of 2D quantum gravity. In Liouville theory we derive functional identities for all fundamental boundary structure constants, similar to the one obtained for the boundary two-point function by Fateev, Zamolodchikov and Zamolodchikov. All these functional identities can be written as difference equations with respect to one of the boundary parameters. Then we switch to the microscopic realization of 2D quantum gravity as a height model on a dynamically triangulated disc and consider the boundary correlation functions of electric, magnetic and twist operators. By cutting open the sum over surfaces along a domain wall, we derive difference equations identical to those obtained in Liouville theory. We conclude that there is a complete agreement between the predictions of Liouville theory and the discrete approach.

Structure constants for the D-series Virasoro minimal models

In this paper expressions are given for the bulk and boundary structure constants of D-series Virasoro minimal models on the upper half plane. It is the continuation of an earlier work on the A-series. The solution for the boundary theory is found first and then extended to the bulk. The modular invariant bulk field content is recovered as the maximal set of bulk fields consistent with the boundary theory. It is found that the structure constants are unique up to redefinition of the fields and in the chosen normalisation exhibit a manifest Z_2-symmetry associated to the D-diagram. The solution has been subjected to random numerical tests against the constraints it has to fulfill.

Boundary structure constants for the A-series Virasoro minimal models

We consider A-series modular invariant Virasoro minimal models on the upper half plane. From Lewellen's sewing constraints a necessary form of the bulk and boundary structure constants is derived. Necessary means that any solution can be brought to the given form by a rescaling of the fields. All constants are expressed essentially in terms of fusing (F-)matrix elements and the normalisations are chosen such that they are real and no square roots appear. It is not shown in this paper that the given structure constants solve the sewing constraints, however random numerical tests show no contradiction and agreement of the bulk structure constants with Dotsenko and Fateev. In order to facilitate numerical calculations a recursion relation for the F-matrices is given.

LOCAL WZW THEORY ON A MANIFOLD WITH BOUNDARY

This letter studies a boundary variation of the well-known WZW theory. The corresponding "extended" boundary Ward identity, the analog of the Knizhnik–Zamolodchikov equation, and the boundary structure constants for the group SU(2) are derived.

QUANTUM LIOUVILLE FIELD THEORY ON A MANIFOLD WITH BOUNDARY

This letter studies the quantum Liouville field theory on a manifold with boundary. The boundary conformal Ward identity (CWI) is written and its semiclassical approximation is analyzed. This establishes a method of finding the accessory parameters of the theory with boundary. The boundary structure constants of the theory are defined and the functional equations which determine them are derived.

SUPERCONFORMAL FIELD THEORY WITH BOUNDARY: FERMIONIC MODEL

Fermionic model of superconformal field theory with boundary is considered. The "boundary" Ward identity for this theory is written and the boundary states for fermionic and spin models are constructed. For this model, the "bootstrap" equations for boundary structure constants are derived.