Quantum Boundary Currents for Nonsimply–laced Toda Theories

We study the quantum integrability of nonsimply–laced affine Toda theories defined on the half–plane and explicitly construct the first nontrivial higher–spin charges in specific examples. We find that, in contradistinction to the classical case, addition of total derivative terms to the ”bulk” current plays a relevant role for the quantum boundary conservation. IFUM–518–FT October 1995 Two–dimensional quantum field theories defined on a manifold with boundary are interesting for the description of various physical phenomena [1]. If the boundary system is quantum integrable, an exact scattering matrix can be constructed and the model is on–shell completely solvable [2]. The existence of an exact S matrix is guaranteed whenever the model possesses symmetries generated by high–spin conserved charges. Classical integrability has been studied for affine Toda theories based on simply–laced as well as nonsimply–laced Lie algebras [3, 4]. Recently we have addressed the issue of boundary conservation at the quantum level [5]. In particular we have considered the first relevant quantum currents for the sinh–Gordon model and the a n Toda systems defined on the half plane, perturbed by a boundary potential. Here we extend the analysis to the case of nonsimply–laced affine Toda theories. We have found that in order to ensure current conservation at the quantum level, total derivative terms need to be added to the currents. These terms, while irrelevant at the classical level, are crucial for the construction of exact quantum symmetries of the theory. We explicitly present the results for the spin–4 currents of the d (2) 3 and c (1) 2 theories. We work in euclidean space with the following notation for coordinates x = x0 + ix1 √ 2 x̄ = x0 − ix1 √ 2 (1) and derivatives ∂ ≡ ∂x = 1 √ 2 (∂0 − i∂1) ∂̄ ≡ ∂x̄ = 1 √ 2 (∂0 + i∂1) 2 = 2∂∂̄ (2) An affine Toda theory based on a Lie algebra G of rank N , has an exponential interaction of the form V = N