A free field perspective of λ-deformed coset CFT’s

Abstract

We continue our study of λ-deformed σ-models by setting up a 1 k perturbative expansion around the free field point for cosets, in particular for the λ-deformed SU(2)/U(1) coset CFT. We construct an interacting field theory in which all deformation effects are manifestly encoded in the interaction vertices. Using this we reproduce the known βfunction and the anomalous dimension of the composite operator perturbing away from the conformal point. We introduce the λ-dressed parafermions which have an essential Wilsonlike phase in their expressions. Subsequently, we compute their anomalous dimension, as well as their four-point functions, as exact functions of the deformation and to leading order in the k expansion. Correlation functions with an odd number of these parafermions vanish as in the conformal case.

$λ$-deformations in the upper-half plane

“…The reason is that in such CFTs, the building blocks are parafermions which have more complicated operator product expansions than currents and as a result they contain Wilson-like phases in their expressions in terms of target space fields. However, we can still use the free field expansion as it was done in [40] for the full plane. In these case one might expect the anomalous dimension of the single parafermion may still stay intact as it is governed by the short-distance behavior.…”

“…This approach utilizes the expansion of (1.2) around the free field point, thus manifestly taking into account the exact dependence of the interaction vertices on the deformation parameter λ. The above approach was further extended for λ-deformations based on coset CFTs in [40].…”

A generalized method for all-loop results in λ-models

“…Then, the β-functions (3.13), (3.14) up to O(λ) and O(1/k) read 19) where the components of g andg are the ones of (3.18). As expected, β ab has the same form as (2.11) of [10], corresponding to the β-function of the doubly deformed asymmetric model of (2.4).…”

“…components of (4.6) with c G = 4, c H = 0 and c G/H = 0, corresponding to the SU(2)/U(1) symmetric coset. However, the SU(2)/U(1) coset limit defined by taking equal levels and λ 3 → 1, λ 1 = λ 2 = λ (where now λ 3 = λ H corresponds to the subgroup deformation) is not well defined here for the reasons explained in the previous section, and thus, the corresponding result of [19] cannot be recovered via a limiting procedure. Moreover, in the IR fixed point (λ 1 = λ 2 = λ 3 = λ 0 ), (5.3) equals with 4 k 2 −k 1 , while the anomalous dimension of the anti-chiral current is zero, recovering the results of [6].…”

“…Then, as expected, one of the well-defined eigenvalues of the above 9 × 9 matrix is identical with eq. (4.34) of [19], yielding…”