Integrable Boundaries and Universal TBA Functional Equations

Nuclear Physics B, 2000

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalization of the formalism of Cardy-Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk-boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints. 2

Journal of Physics A: Mathematical and Theoretical, 2018

We explain how to incorporate the action of local integrals of motion into the fermionic basis for the sine-Gordon model and its UV CFT. The examples up to the level 4 are presented. Numerical computation support the results. Possible applications are discussed.

Nuclear Physics B, 2002

We construct integrable boundary conditions for sℓ(2) coset models with central charges c = 3 2 -12 m(m+2) and m = 3, 4, . . . The associated cylinder partition functions are generating functions for the branching functions but these boundary conditions manifestly break the superconformal symmetry. We show that there are additional integrable boundary conditions, satisfying the boundary Yang-Baxter equation, which respect the superconformal symmetry and lead to generating functions for the superconformal characters in both Ramond and Neveu-Schwarz sectors. We also present general formulas for the cylinder partition functions. This involves an alternative derivation of the superconformal Verlinde formula recently proposed by Nepomechie.

Journal of Physics A: Mathematical and General, 2004

2010

We consider an infinite family of non-rational conformal field theories in the presence of a conformal boundary. These theories, which have been recently proposed in , are parameterized by two real numbers (b, m) in such a way that the corresponding central charges c (b,m) are given by c (b,m) = 3 + 6(b + b −1 (1 − m)) 2 . For the disk geometry, we explicitly compute the expectation value of a bulk vertex operator in the case m ∈ Z, such that the result reduces to the Liouville one-point function when m = 0. We perform the calculation of the disk one-point function in two different ways, obtaining results in perfect agreement, and giving the details of both the path integral and the free field derivations.

Physics Letters B, 1998

The generalization to N=1 superconformal minimal models of the relation between the modular transformation matrix and the fusion rules in rational conformal field theories, the Verlinde theorem, is shown to provide complete information about the fusion rules, including their fermionic parity. The results for the superconformal Tricritical Ising and Ashkin-Teller models agree with the known rational conformal formulation. The Coulomb gas description of correlation functions in the Ramond sector of N=1 minimal models is also discussed and a previous formulation is completed.

1995

Functional equations, in the form of fusion hierarchies, are studied for the transfer matrices of the fused restricted A (1) n−1 lattice models of Jimbo, Miwa and Okado. Specifically, these equations are solved analytically for the finite-size scaling spectra, central charges and some conformal weights. The results are obtained in terms of Rogers dilogarithm and correspond to coset conformal field theories based on the affine Lie algebra A (1) n−1 hep-th/9502067 1

International Journal of Modern Physics B, 2006

We review the recent progress on the construction of the determinant representations of the correlation functions for the integrable supersymmetric fermion models. The factorizing F-matrices (or the so-called F-basis) play an important role in the construction. In the F-basis, the creation (and the annihilation) operators and the Bethe states of the integrable models are given in completely symmetric forms. This leads to the determinant representations of the scalar products of the Bethe states for the models. Based on the scalar products, the determinant representations of the correlation functions may be obtained. As an example, in this review, we give the determinant representations of the two-point correlation function for the Uq(gl(2|1)) (i.e. q-deformed) supersymmetric t-J model. The determinant representations are useful for analyzing physical properties of the integrable models in the thermodynamical limit.

NATO ASI Series, 1993

We find an infinite set of new noncommuting conserved charges in a specific class of perturbed CFT's and present a criterion for their existence. They appear to be higher momenta of the already known commuting conserved currents. The algebra they close consists of two noncommuting W ∞ algebras. We find various Virasoro subalgebras of the full symmetry algebra. It is shown on the examples of the perturbed Ising and Potts models that one of them plays an essencial role in the computation of the correlation functions of the fields of the theory.

Physical Review D

Exact conformal field theories (CFTs) are obtained by using the approach of Poisson-Lie (PL) T-duality in the presence of spectators. We explicitly construct some non-Abelian T-dual σ-models (here as the PL T-duality on a semi-Abelian double) on 2 + 2-dimensional target manifolds M ≈ O×G andM ≈ O×G, where G andG as two-dimensional real non-Abelian and Abelian Lie groups act freely on M andM , respectively, while O is the orbit of G in M. The findings of our study show that the original models are equivalent to Wess-Zumino-Witten (WZW) models based on the Heisenberg (H4) and GL(2, R) Lie groups. In this way, some new T-dual backgrounds for these WZW models are obtained. For one of the duals of the H4 WZW model, we show that the model is self-dual. In the case of the GL(2, R) WZW model it is observed that the duality transformation changes the asymptotic behavior of solutions from AdS3 × R to flat space. Then, the structure and asymptotic nature of the dual spacetime of this model including the horizon and singularity are determined. We furthermore get the non-critical Bianchi type III string cosmological model with a non-vanishing field strength from T-dualizable σ-models and show that this model describes an exact CFT (equivalent to the GL(2, R) WZW model). After that, the conformal invariance of T-dual models up to two-loop order (first order in α ′) is discussed.