![](https://csdnimg.cn/release/download_crawler_static/12288240/bg6.jpg)
the relation via the Poisson-Lie duality was first understood in the bosonic case. It is thus
natural to first explore the question of quantum corrections by studying the bosonic models.
It is also worth emphasizing that the bosonic models are of interest in their own right
in the context of investigation of general integrable 2d theories. The η-model has played
an important role in generalizing the duality between the deformed O(3) and O(4) sigma
models and massive integrable QFTs [1–3] to higher-rank groups [4–6]. While the dual
theories are quantum-exact, the σ-model side of the duality is only understood so far
to leading order in the loop expansion. Therefore, after finding quantum corrections to
integrable σ-models consistent with renormalizability, studying their compatibility with
this duality may be a useful way to further explore the conjectured relationship between
integrability and renormalizability.
The rest of the paper is organized as follows. In section 2 we consider the λ-model on
the extended configuration space (g, A
+
, A
−
). We argue that it should be renormalizable
with only the parameter λ running and compute the 2-loop β-function of the λ-model based
on general groups and symmetric spaces. As a consequence, the same renormalizability
conclusion holds for the model (1.10) interpolating between the PCM and its NAD, with
the same 2-loop β-function for h as in the PCM (and the same in the symmetric space case).
In section 3 we study the renormalization of the λ-model defined by the standard σ-
model action, after integrating out A
±
. In this case its invariance under the 2-loop RG
flow requires a specific deformation of the classical geometry. While in the SU(2)/U(1)
case the required counterterm is simple [7], in the SU(2) case, the corresponding quantum-
corrected σ-model action has a rather intricate structure. We also consider a particular
limit of the SU(2) λ-model where it becomes T-dual to a σ-model for a squashed 3-sphere,
explaining the consistency of the quantum deformation of the original λ-model with the
known quantum correction to the T-duality transformation rule. We also discuss the 2-loop
β-function for the NAD of the SU(2) PCM.
Some concluding remarks are made in section 4. In appendix A we summarize our no-
tation and group-theory conventions. In appendix B we compute the 2-loop β-function
for a two-coupling “squashed” principal chiral model that interpolates between the G
group space PCM and the G/H coset σ-model, determining also the 2-loop β-function
for the latter.
2 Renormalizability of λ-model: extended configuration space
In this section we shall study renormalization of the λ-model on the extended configuration
space. It is first useful to draw an analogy with the γ-model [10, 11] defined in (1.6) (where
G and H ⊂ G are simple Lie groups) that interpolates between the gauged WZW (gWZW)
and chiral gauged WZW (cWZW) theories. Changing variables (A
+
, A
−
) → (h,
¯
h) as
A
+
= h ∂
+
h
−1
, A
−
=
¯
h ∂
−
¯
h
−1
, h,
¯
h ∈ H , (2.1)
– 5 –