可积Yang-Baxter-Wess-Zumino模型：经典与量子的联系

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"Yang-Baxter Wess-Zumino模型的古典和量子方面" 这篇学术论文探讨了2d主手性模型的可积Yang-Baxter变形，特别关注了在模型中引入Wess-Zumino项后的物理行为。Wess-Zumino项是一种在量子场论中常见的拓扑项，它在某些情况下能够影响理论的性质，如对称性和相互作用。文章中，研究人员计算了一环β函数，这是量子场论中描述耦合常数随能量尺度变化的关键量。他们发现，对于任意的群结构，经典可积性条件对于阻止在重正化过程中产生新的耦合是至关重要的。这意味着在经典层面上的可积性不仅影响到理论的解，而且在量子层面上也起到决定性作用，防止了理论结构的复杂化。论文还揭示了一个令人惊讶的联系，即经典物理学与量子物理学之间的紧密关系。这种联系体现在，如果一个理论在经典意义上是可积的，那么它在量子重正化过程中更可能保持其结构的简单性，不引入额外的耦合。这为理解量子场论中的可积性提供了新的视角。作者进一步展示了这些理论如何优雅地实现泊松-李T对偶，这是一种耦合常数的简单反演。这种对偶性在物理学中具有重要意义，因为它允许我们从不同角度理解同一个理论，甚至可能揭示理论的隐藏对称性。在这些模型中，自对偶点对应于Wess-Zumino-Witten（WZW）模型，这些模型是Renormalization Group（RG）流的红外（IR）不动点，意味着在低能量极限下，它们保持不变。论文还讨论了对这些Yang-Baxter Wess-Zumino模型进行超对称扩展的可能性。然而，研究发现通常情况下，这样的扩展是困难的或不可能的，这暗示了在保持可积性的同时增加超对称性的严格限制。这为探索超对称性和可积性之间的平衡点提供了新的挑战。这项工作深入探讨了可积Yang-Baxter变形与Wess-Zumino项相结合的2d主手性模型，揭示了经典和量子物理之间的深刻联系，以及这些理论在重正化、对偶性和超对称性方面的特性。这些发现对于理解量子场论的基本原理，特别是在可积系统和对称性保护的结构方面，有着重要的理论价值。

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

(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].

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

(

[

su(2)) with,

q = exp



8πΘ

+ k



, (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,

−

= −

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,



α(α − γ) − k



, (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).

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).

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.

Here we restore the overall normalisations to the results in [35] and map to our conventions.

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 –

JHEP03(2018)041

The coupling k being integer quantised evidently does not run. To do so we will proceed

geometrically; for a general two-dimensional non-linear sigma model the β-function for the

metric G

µν

and Kalb-Ramond two-form potential B

µν

in local coordinates x

are given by,

at one-loop,

dµ

µν

= α



µν

−

µν



+ O(α

)

dµ

µν

= α



−

∇

λµν



+ O(α

)

(3.1)

where H = dB is the torsion 3-form and the connections and curvatures are to be calculated

using G. However, the diﬀeomorphism and gauge covariance of G and B means that these

β-functions are ambiguous (even at one-loop order) [43, 44] allowing us to modify them by,

µν

+ ∇

(µ

ν)

µν

+ (ι

µν

+ (dΛ)

µν

(3.2)

with W and Λ arbitrary target space one-forms. For the sigma model deﬁned in eq. (2.2),

of which the left acting G symmetry is unaltered by the deformation, the target space data

is most naturally expressed in a non-orthonormal frame formalism with frames deﬁned by

the left-invariant one-forms u = g

−1

dg = −iu

as,

= α κ

+ γ R

, (3.3)

and torsion,

ABC

= 3β F

[AB

C]D

− k F

ABC

. (3.4)

Here κ

= hT

, T

i, R(T

) = R

and algebra indices out of position are lowered

with κ

. To completely ﬁx things one should set α

= 2 so that the standard normalisation

of the WZW models is recovered in the case α = |k|, β = γ = 0.

After a long battle making use of the properties listed in appendix C and the expres-

sions of the geometry in the non-orthonormal frame listed in appendix D one ﬁnds that

the β-functions are given by,

= −c



(α − 2γ)

2α(γ − α)

−

(γ − α)

−

2(γ − α)

−



−



1 −

+ β

(γ − α)



−



(γ − α)



)

, (3.5)

2β

α − γ

+ c

α − γ

. (3.6)

The terms in blue involve tensor structures that are not present in the metric ansatz. If

these terms are not removed it would mean that under the RG ﬂow the metric would ﬂow

out of the ansatz speciﬁed by eq. (3.3). Let us exploit the diﬀeomorphism symmetry to

Note that the Lie derivative acts on L

B = ι

H + dι

B and the latter term being a total derivative

can be discarded within the action.

– 7 –

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