Gerard Watts

A Study of W algebras using Jacobi identities

Abstract A theoretical framework in which to study the Jacobi identities on the commulators of W-algebra fields is established. W-algebras containing various specified combinations of fields of spin up to 8 are classified and constructed. In the cases studied, we show that there are closed W-algebraic structures, valid for all values of the central charge c , only for those field contents corresponding to the exponents of finite-dimensional semi-simple algebras.

A crossing probability for critical percolation in two dimensions

Langlands et al considered two crossing probabilities, and , in their extensive numerical investigations of critical percolation in two dimensions. Cardy was able to find the exact form of by treating it as a correlation function of boundary operators in the limit of the Q-state Potts model. We extend his results to find an analogous formula for which compares very well with the numerical results.

The conformal boundary states for SU(2) at level 1

Abstract For the case of the SU(2) WZW model at level one, the boundary states that only preserve the conformal symmetry are analysed. Under the assumption that marginal deformations of the usual Cardy boundary states are consistent, the most general conformal boundary states are determined. They are found to be parametrised by group elements in SL (2, C ) .

A non-rational CFT with c=1 as a limit of minimal models

We investigate the limit of minimal model conformal field theories where the central charge approaches one. We conjecture that this limit is described by a non-rational CFT of central charge one. The limiting theory is different from the free boson but bears some resemblance to Liouville theory. Explicit expressions for the three point functions of bulk fields are presented, as well as a set of conformal boundary states. We provide analytic and numerical arguments in support of the claim that this data forms a consistent CFT.

Reflection and transmission for conformal defects

We consider conformal defects joining two conformal field theories along a line. We define two new quantities associated to such defects in terms of expectation values of the stress tensors and we propose them as measures of the reflectivity and transmissivity of the defect. Their properties are investigated and they are computed in a number of examples. We obtain a complete answer for all defects in the Ising model and between certain pairs of minimal models. In the case of two conformal field theories with an enhanced symmetry we restrict ourselves to non-trivial defects that can be obtained by a coset construction.

One-point functions in perturbed boundary conformal field theories

Abstract We consider the one-point functions of bulk and boundary fields in the scaling Lee–Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip.

Generalisations of the Coleman-Thun mechanism and boundary reflection factors

Abstract We make a complete pole analysis of the reflection factors of the boundary scaling Lee-Yang model. In the process we uncover a number of previously unremarked mechanisms for the generation of simple poles in boundary reflection factors, which have implications for attempts to close the boundary bootstrap in more general models. We also explain how different boundary conditions can sometimes share the same fundamental reflection factor, by relating the phenomenon to potential ambiguities in the interpretation of certain poles. In the case discussed, this ambiguity can be lifted by specifying the sign of a bulk-boundary coupling. While the recipe we employ for the association of poles with general on-shell diagrams is empirically correct, we stress that a justification on the basis of more fundamental principles remains a challenge for future work.

Determinant formulae for extended algebras in two-dimensional conformal field theory

Abstract The determinant of the contravariant from on highest-weight Verma module representations of extended algebras related to su( n ) can be factorised. Some technical problems and results are presented which allow the representation theory of the su(3) algebra to be deduced in the non-degenerate case, 0 c

Minimal model boundary flows and c=1 CFT

Abstract We consider perturbations of unitary minimal models by boundary fields. Initially we consider the models in the limit as c→1 and find that the relevant boundary fields all have simple interpretations in this limit. This interpretation allows us to conjecture the IR limits of flows in the unitary minimal models generated by the fields φrr of ‘low’ weight. We check this conjecture using the truncated conformal space approach. In the process we find evidence for a new series of integrable boundary flows.

Null vectors of the W3 algebra

Abstract We construct W 3 null vectors of a restricted class explicitly in two different forms. The method we use is an extension of that of Bauer et al. in the Virasoro case. Our results are analogous to the formulae of Benoit and Saint-Aubin for the Virasoro null vectors. We derive in the Virasoro case some alternative formulae for the same null vectors involving only the L −1 and L −2 modes of the Virasoro algebra.