JHEP03(2018)041
symmetries are found [3] via the expansion of the gauge transformed Lax around special
points corresponding to poles in the twist function of the Maillet algebra [37].
Much of the story for the general η-deformed model was first established for the case
of g = su(2) which corresponds to the sigma model on a squashed S
3
(the Kalb-Ramond
potential encoded by eq. (2.2) is pure gauge in this case and though it doesn’t effect the
equation of motions it corresponds to an improvement term to ensuring flatness of currents).
The integrability was established many years ago by Cherednik [38]. Somewhat later the
classical Yangian symmetry was shown in [39] and the (affine) quantum group symmetry
in [40, 41].
2
Now we turn to the case where k 6= 0 which is the main focus of this paper. Again
historically this was first well explored for the case of g = su(2). The left acting symmetry
is still a Yangian [34] but the right acting symmetry is more mysterious [35] (we review the
construction of the charges generating these generalised symmetries in appendix B). One
finds a structure similar to an affine quantum group U
q
(
[
su(2)) with,
3
q = exp
8πΘ
Θ
2
+ k
2
, (2.14)
but with a modification in how the affine tower of charges is build up. Namely, instead of
taking successively the Poisson bracket to access the next charges in the tower, the Poisson
bracket is multiplied at each step by an additional factor,
γ
−
γ
+
= −
k + iΘ
k −iΘ
. (2.15)
To move down, the Poisson bracket needs to be multiplied by its inverse (see figure 2 of [35]
for further details). Here the combination,
Θ
2
=
α
α(α − γ) − k
2
γ
, (2.16)
will play a distinguished role in what follows; it will be seen to be an RG invariant.
As mentioned above, several partial results were already obtained for the YB-WZ
model in the particular case where g = su(2). In this paper we mostly focus on the case
where g is arbitrary. As we will see the general case shows several features which are absent
when g = su(2).
4
3 Renormalisation of the YB-WZ model
Our aim is to calculate the β-functions for the couplings {α, β, γ} in the theory defined
by eq. (2.2) without first assuming that the couplings lie on the integrable locus eq. (2.8).
2
There is a small but potentially important subtlety here. In [41] the affine charges are constructed from
the expansion of a trigonometric Lax at infinity and appear in the principal gradation. When the charges
are extracted from the gauge transformation of the rational Lax evaluated around the poles in the twist
function as in [36] they appear in the homogeneous gradation; to go between the two gradations requires a
spectral parameter dependent redefinition of generators.
3
Here we restore the overall normalisations to the results in [35] and map to our conventions.
4
Mathematically all differences between the general case and the simpler case where g = su(2) arise from
the fact that su (2) is the only simple Lie algebra where all roots are simple roots.
– 6 –
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