$\lambda$-deformations in the upper-half plane

TL;DR

This paper develops methods to compute correlation functions in $$-deformed $$-models with boundaries, revealing exact results in the deformation parameter and exploring integrability-preserving boundary conditions.

Contribution

It introduces two complementary approaches for exact correlation function calculations in boundary $$-deformed models, including boundary conditions that preserve integrability.

Findings

Exact correlation functions obtained in the deformation parameter $$ and large level $k$ limit.

Consistent results from conformal perturbation theory and free field expansion methods.

Insights into boundary conditions that preserve integrability.

Abstract

We formulate $\lambda$-deformed $\sigma$-models as QFTs in the upper-half plane. For different boundary conditions we compute correlation functions of currents and primary operators, exactly in the deformation parameter $\lambda$ and for large values of the level $k$ of the underlying WZW model. To perform our computations we use either conformal perturbation theory in association with Cardy's doubling trick, as well as meromorphicity arguments and a non-perturbative symmetry in the parameter space $(\lambda,k)$, or standard QFT techniques based on the free field expansion of the $\sigma$-model action, with the free fields obeying appropriate boundary conditions. Both methods have their own advantages yielding consistent and rich, compared to those in the absence of a boundary, complementary results. We pay particular attention, albeit not exclusively, to integrability preserving boundary conditions.