新方法构造：修整陪集与多参数可积分变形

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"修整陪伴和多参数可积分变形" 这篇学术文章主要探讨了在E $$ \mathcal{E} $$-模型框架下修整陪集（dressing cosets）σ模型的新构建方法。E $$ \mathcal{E} $$-模型是一种用于描述二维场论的理论，尤其在研究可积系统时具有重要意义。各向同性度量是这种模型中的关键概念，它涉及到模型在不同方向上的对称性处理。作者Ctirad Klimˇcık指出，新方法基于各向异性度量的规范化，这使得修整陪集σ模型能够被更深入地理解和处理。这一新方法的应用揭示了一个重要的发现：最近提出的主手性模型的多参数可积分变形实际上属于修整陪集类别。这意味着这些模型不仅在数学上具有结构上的优美性，而且在物理上也有着自动重整化（automatic renormalizability）的特性，即它们在高能量尺度下的行为可以被精确控制，不会出现无限大的系数需要校正。自动重整化是量子场论中的一个重要概念，它确保理论在所有能量尺度下都是有意义的，而无需引入无限级数的耦合常数修正。在这种情况下，多参数可积分变形的修整陪集模型的动力学可以通过当前代数（current algebras）完全表征，这为理解和计算模型的物理性质提供了一种有效工具。文章分为三个主要部分： 1. 概要结果：这部分简要概述了研究的主要发现，即新构造的修整陪集模型以及它们与多参数可积分变形之间的关系。 2. 引言：作者在这里介绍了问题背景，阐述了研究的动机和目标，可能包括可积系统、修整陪集模型和多参数可积分变形在量子场论中的地位和重要性。 3. 修整陪集：这部分详细讨论了修整陪集模型的构造，分为非退化E $$ \mathcal{E} $$-模型和退化 $$ \hat{E} $$-模型两个子节。非退化模型通常对应于更简单的理论结构，而退化模型则可能包含更复杂的对称性和相互作用。在非退化E $$ \mathcal{E} $$-模型部分，作者可能探讨了这些模型的基本性质和规范化的具体步骤。而在退化 $$ \hat{E} $$-模型部分，可能涉及到了模型的特殊对称性打破情况以及如何在这种情况下应用新的构造方法。这篇文章为理解二维可积场论和相关模型的复杂性提供了一个新的视角，同时强调了修整陪集模型在自动重整化和当前代数表征中的优势。通过这种方法，研究者可以更有效地探索这些理论的物理含义和潜在应用。此外，由于该研究是开放访问的，意味着全球的研究人员都可以自由地获取和利用这些成果，进一步推动相关领域的研究进展。

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JHEP07(2019)176

Here Ad

stands for the adjoint action on D of the element k ∈ K ⊂ D and J,

J are

projectors; J pr ojects to K with the kerne l

K and

J project s to

K with the kernel K.

Recall also that the operator Π(k) :

K → K encodes the so called Poisson-Lie bracket

of two functions f

, f

on the group K in the sense of the formula:

, f

}

(k) = (∇

, Π(k)∇

)

. (3.10)

Here ∇

K-valued diﬀere ntial operator acting on the fu nct i ons on K as

(∇

f, x)

(k) := (∇

f)(k) ≡

df(e



s=0

, x ∈ K. (3.11)

Of course, every element l(σ) of the loop group LD can be decomposed also i n the

dual way as

l(σ) =

k(σ)h(σ),

k ∈ L

K, h ∈ LK. (3.12)

Inserting the decomposition (

3.12) into (3.1) and into (3.2), and then repeating all the

procedure as before leads to the dual σ-model living on the target

(

k) =

dτ





E +

Π(



−1

∂

−1

, ∂

−

−1



, (3.13)

where

Π(

k) = −

JAd

−1

J. (3.14)

Of course, the linear ope r ator

E : K →

K is again such that its graph {x +

Ex, x ∈ K}

coincides with the image of the operator Id+E, which implies that the duality between the

models ( 3. 8) and (3.13) holds under the condition, that the operator

E is inverse of the

operator E.

If the Drinfeld double is not perfect, there exists a generalization of the T-duality

between the models (

3.8) and (3.13), where the two σ-models live, respectively, on the

spaces of cosets D/

K and D/K [

55, 56, 65, 66]. If we parametrize (possibl y patch by

patch) the coset space D/

K by a section m ∈ D of the bundle D → D/

K and the coset

space D/K by a section ˜m ∈ D of the bundle D → D/K, then the decompositions l = m

and l = ˜mh ge ne r al i ze those (

3.4) and (3. 12) and lead respectively to the following dual

pair of σ-models wi t h the WZW terms:

(m) =

dτ



(1−2P

(E)) m

−1

∂

m, m

−1

∂

−



−1



−1

dm, [m

−1

∂

m, m

−1

dm]



;

(3.15)

( ˜m) =

dτ



1−2

˜m

(E)



˜m

−1

∂

˜m, ˜m

−1

∂

−

˜m



−1



˜m

−1

d ˜m, [ ˜m

−1

∂

˜m, ˜m

−1

d ˜m]



(3.16)

Here the projectors P

(E) : D → D and

˜m

(E) : D → D have the res pective images

and K and their respective kernels are given by the linear spaces (Id+Ad

−1

EAd

)

K and

(Id+Ad

˜m

−1

EAd

˜m

)K.

Of course, the Poisson-Li e T-duality relating the models (

3.8) and (3. 13) , or, more

generally, relating the models (

3.15) and (3.16), is the main result that we review in this

– 7 –

JHEP07(2019)176

section 3.1, but we add few more formulas about the E-models t hat will be useful in what

follows. First of all, the ﬁrst order Hamiltonian equations of motions derived from the

formulas (

3.1) and (3.2) can be written in two useful ways, either as

∂

−1

= E∂

−1

(3.17)

or as

∂

j = ∂

(Ej) + [Ej, j], (3.18)

where

j(σ) := ∂

l(σ)l(σ)

−1

. (3.19)

Moreover, the inversion of th e s y mpl e ct i c for m ω

gives th e standard current algebra

Poisson brackets for the LD-valued variable j(σ):

(σ), j

(σ

′

)} = F

(σ)δ(σ − σ

′

) + D

∂

δ(σ − σ

′

). (3.20)

Here

(σ) := (j(σ), T

)

, [T

, T

] = F

, D

:= (T

, T

)

(3.21)

and T

∈ D is some basis of D.

3.2 Degenerate

E-models

We start the exposi t i on of the degenerate

E-mode l s from the end, that is, we ﬁrst write to

what kin d of T-duality they give rise to. To grasp the idea, it is suﬃcient to consider the

perfect Dri nf el d doubles and the resulting dual pair of the “dressing cosets” σ-models is

then given by th e actions

(k) =

dτ





Id +

EΠ(k)



−1

E∂

−1

, ∂

−

−1



, (3.22)

(

k) =

dτ





E +

Π(



−1

∂

−1

, ∂

−

−1



. (3.23)

Well, if

E : K →

K is an invertible operator then the dressing cosets models (

3.22)

and (

3.23) look identical to the models (3.8) and (3.13), so what is then new in the dressing

coset story? Is it just the circumstance that we release the condition that the operator

be invertible? No, this is not the cas e. In fact, even if

E is invertible, t he dressing cosets

pair (

3.22) and (3.23) may c ontain substantially diﬀerent physics as the standard Poisson-

Lie T-dual pair (

3.8) and (3.13). What happens is that in the const r uct i on of the standard

pair (

3.8) and (3.13) from the E-model it follows automatically that the symmetric part

(

E +

∗

) of the operator

E : K →

K is also invertible (the symbol

∗

stands for the

adjoint of the operator

E with respect to the bilinear form (., .)

.) In the dressing cosets

case, the operator

(

E +

∗

) need not be invertible and the T-duality of the models (

3.22)

and (

3.23) still holds (if some further invariance conditions on

E are fulﬁlled). This is not

a trivial generalization of the standar d Poisson-Lie T-duality, however, because the lack of

invertibility of the operator

E or of i t s symmetric part

(

E +

∗

) has dras ti c consequences

– 8 –

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新方法构造：修整陪集与多参数可积分变形

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