2d σ模型的可积性与可重归化性：λ模型的2环RG流

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"这篇学术论文深入探讨了2维可积分sigma模型与2回路重整化群(RG)流之间的关联，特别是在λ模型的上下文中。文章由Ben Hoare、Nat Levine和Arkady A. Tseytlin合作撰写，发表于JHEP12(2019)146，并在SISSA的支持下，由Springer出版。" 在2维sigma模型的研究中，可重归一化性与可积性之间的相互作用是理论物理学家关注的重点。λ模型，作为一个与群或对称空间相关的可积模型，具有独特的特性，能够包容度量的Wess-Zumino-Witten (WZW)模型以及非阿贝尔对偶性的“插值模型”。λ模型的参数包括WZ级别k和耦合常数λ，其场变量包括在群G中取值的g和在相应代数中的2d向量A±。作者们提出了一种新的形式化方法，将λ模型表述为在扩展的G×G×G配置空间（g，h，h $$ \overline{h} $$）上的σ模型。在这个扩展的空间中，h和h $$ \overline{h} $$通过A+ = h∂+ h^(-1)和A_ = h $$ \overline{h} $$^(-1) - h $$ \overline{h} $$^(-1)定义。他们的重要观察是，在这个扩展配置空间上的λ模型可以不经过任何形变地进行重整化，只有λ参数会运行。这与标准的σ模型形成对比，后者在积分A±时，只有在添加特定的有限局部对立项后，2环可重归一化性才能实现，这会导致目标空间的几何形状在量子层面上发生变形。论文中，作者们计算了λ模型的一般群和对称空间的2环β函数，并在SU(2)/U(1)和SU(2)的实例中进行了具体分析。这些计算对于理解模型的重整化行为和量子修正至关重要，也提供了关于如何在理论中维持可积性的洞察。这项工作对理解2维量子场论的重整化过程以及如何在理论中融合可积性和可重归一化性提供了新的视角，对于进一步探索非阿贝尔对偶性、WZW模型以及sigma模型在量子物理中的应用具有重要意义。通过这样的研究，物理学家可以更深入地理解这些模型在凝聚态物理、弦理论和其他领域中的潜在作用。

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JHEP12(2019)146

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

= h ∂

−1

, A

−

h ∂

−

−1

, h,

h ∈ H , (2.1)

– 5 –

JHEP12(2019)146

and using the Polyakov-Wiegmann identity [46], we can rewrite the Lagrangian in (1.6) as

a combination of WZW models (cf. (1.5))

L = k



G/H

(g, A) − γTr[A

−

]



= k



−1

h) − L

−1

h) − γ



−1

h) − L

−1

) − L

(





(2.2)

The change of variables in (2.1) results in a Jacobian contributing to the action as [46, 47]

∆L = −2c



−1

h) + qTr[A

−

]



= −2c



−1

) + L

(

h) + (q − 1)Tr[∂

h h

−1

∂

−

−1

]



, (2.3)

where c

= c

(H) is the dual Coxeter number of H and an arbitrary coeﬃcient q

parametrizes the ambiguity of adding a local counterterm Tr[A

−

]. Combining (2.3)

and the classical Lagrangian (2.2) gives

L = kL

−1

h)−(k+2c

−1

h)−(kγ+2qc

)



−1

h)−L

−1

)−L

(



. (2.4)

In the special cases γ = 0 (gWZW model) and γ = −1 (cWZW model), we can choose q

such that (2.4) is a sum of WZW models [10, 11]

gWZW : γ = 0, q = 0, L= kL

−1

h)−(k+2c

−1

h), (2.5)

cWZW : γ = −1, q = −1, L= kL

−1

h)−(k+2c

)



−1

)+L

(



. (2.6)

Since the arguments {˜g = h

−1

h = h

−1

h} in (2.5) or {˜g = h

−1

h, h

−1

h} in (2.6) may

be treated as independent ﬁelds, it follows that these two models are conformally invariant.

For general values of γ, choosing q = −1 as in the cWZW case (2.6), we may

rewrite (2.4) as

L = kL

(˜g) + L

(

h) , ˜g ≡ h

−1

h ∈ G ,

h ≡ h

−1

∈ H , (2.7)

= −(k + 2c

)



(

h) + L

(



+ k(1 + γ)Tr[

−1

∂

h ∂

−

−1

] . (2.8)

This model is deﬁned on the extended conﬁguration space (˜g,

h) ∈ G × H × H. The

ﬁrst G WZW term (which is conformal on its own) decouples and then we are left with

the “truncated” model L

on H × H which is simply a sum of two group H WZW models

perturbed by the product of the left and right currents. Like the chiral gauged WZW

model (2.6) the Lagrangian L

in (2.8) is invariant under the chiral gauge symmetry

h →

u(σ

−

)

h →

h w(σ

), u, w ∈ H as well as the global H symmetry

h →

−1

h, v

∈ H. As we shall argue for the λ-model, these two symmetries imply that the

γ-model is also renormalizable with only one coupling γ running.

Let us note that the γ-model (1.6), (2.2) admits also a generalization similar to the

coset case of the λ-model in (1.2): with γA

−

term replaced by γA

P A

−

, where P is the

projector onto the H/F coset part of the algebra of H (with F ⊂ H ⊂ G). When H/F is a

symmetric space this model should again be renormalizable on the extended conﬁguration

space (˜g,

h) ∈ G × H × H due to chiral gauge symmetry.

– 6 –

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