双Yang-Baxter模型的Poisson-Lie T-对偶验证与扩展

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"这篇论文研究了Bi-Yang-Baxter模型的Poisson-Lie T-对偶，并证实了Sfetsos、Siampos和Thompson的猜想，即这种对偶性的适当解析延拓与广义λ-模型相匹配。作者进一步扩展了这一成果，通过分析Tseytlin在1993年提出的通用WZW型σ模型的解析延拓，揭示它实际上等同于由Klimčík和Ševera在1995年定义的Poisson-Lie对称σ模型的Poisson-Lie T-对偶。" 在物理学领域，Poisson-Lie T-对偶是一种重要的对称性理论，特别是在弦理论和非线性σ模型中有着深远的影响。非线性σ模型是描述二维膜在高维空间中运动的理论，其中Bi-Yang-Baxter模型是一种特殊的非线性σ模型，它的特征在于其拉格朗日量中包含了Bi-Yang-Baxter方程，这使得模型在处理对称性和相互作用时具有更丰富的结构。 Sfetsos、Siampos和Thompson的猜想表明，Bi-Yang-Baxter模型的Poisson-Lie T-对偶可以通过解析延拓得到广义λ-模型。解析延拓是一种数学方法，将函数或方程从一个定义域扩展到另一个定义域，通常是为了揭示新的物理性质或对称性。这里，这个过程揭示了两种看似不同的模型之间的深层联系。 Tseytlin在1993年提出的“通用WZW型”σ模型是Wess-Zumino-Witten (WZW)模型的一种推广，WZW模型是描述拓扑和规范对称性的基本理论。通过解析延拓，Tseytlin的模型被证明可以视为Poisson-Lie T-对偶的另一面，与Klimčík和Ševera的Poisson-Lie对称σ模型相对应。这表明，尽管这两个模型在原始形式上可能看起来不同，但它们在特定变换下实际上是等价的。 Klimčík和Ševera在1995年提出的Poisson-Lie对称σ模型是Poisson-Lie群理论在非线性σ模型中的应用，Poisson-Lie群是一类带有Poisson结构的Lie群，这种结构允许研究更广泛的对称性和相互作用。这项工作加深了我们对非线性σ模型和其对称性的理解，特别是Poisson-Lie T-对偶如何连接不同类型的模型。通过解析延拓，研究者能够揭示隐藏的对称性和模型间的等价性，这对于理解和构建更复杂的物理理论具有重要意义。同时，该论文作为开放访问资源，可供全球学者查阅和引用，进一步促进了学术交流和知识共享。

Physics Letters B 760 (2016) 345–349

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Poisson–Lie T-duals of the bi-Yang–Baxter models

Ctirad Klim

cík

Aix Marseille Université, CNRS, Centrale Marseille I2M, UMR 7373, 13453 Marseille, France

a r t i c l e i n f o a b s t r a c t

Article history:

Received

23 June 2016

Accepted

30 June 2016

Available

online 9 July 2016

Editor:

M. Cveti

Keywords:

T-duality

Nonlinear

σ -models

We prove the conjecture of Sfetsos, Siampos and Thompson that suitable analytic continuations of

the Poisson–Lie T-duals of the bi-Yang–Baxter sigma models coincide with the recently introduced

generalized λ-models. We then generalize this result by showing that the analytic continuation of a

generic σ -model of “universal WZW-type” introduced by Tseytlin in 1993 is nothing but the Poisson–Lie

T-dual of a generic Poisson–Lie symmetric σ-model introduced by Klim

cík and Ševera in 1995.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Two kinds of integrable nonlinear σ -models, the so-called

η-deformation of the principal chiral model [1,2] and the λ-defor-

mation

of the WZW model [3], have recently attracted much at-

tention

because of their relevance in string theory or in non-

commutative

geometry [4]. The integrability of those models was

proven at the level of the Lax pair in [2,3] and at the level of the

so called r/s exchange relations in [5]. Both the η-model and the

λ-model turned out to be deformable further to give rise to several

families of multi-parametric integrable σ -models

[6–8] living on

general semi-simple group targets (those families generalize some

of the integrable families of σ -models on low dimensional group

targets obtained previously in [10]).

three recent papers [11,12,8], there was suggested that

the η-deformation of the principal chiral model [1,2] and the

λ-deformation of the WZW model should be related by the

Poisson–Lie T-duality [13,14] followed by an appropriate analytic

continuation of the geometry of the λ-model target. In particular,

such suggestion was fully worked out for the simplest group tar-

get

SU(2) in [8] where it was shown that the Poisson–Lie T-dual of

the bi-Yang–Baxter model [7] coincides with the analytically con-

tinued

generalized λ-model [8]. Furthermore, Sfetsos, Siampos and

Thompson conjectured that the same result should hold for the

bi-Yang–Baxter model living on a general simple compact group

target. We have partially proved this conjecture in [16] in the fol-

lowing

sense: we did work with the general simple compact group

E-mail address: ctirad.klimcik@univ-amu.fr.

The strong integrability in the r/s-sense of the so-called bi-Yang–Baxter model

of [7] was further established in [9].

target but we have switched off one of the two deformation pa-

rameters

of the bi-Yang–Baxter model. Said in other words, we

have established in [16] for every simple compact group target

that the Poisson–Lie T-dual of the Yang–Baxter model [1,2] coin-

cides

with the analytically continued λ-deformation of the WZW

model [3]. The ﬁrst purpose of the present letter is to switch on

also the second parameter and, hence, to prove the conjecture of

Sfetsos, Siampos and Thompson in its strongest form.

The

second purpose of our work is to reveal a highly nontriv-

ial

structural relation between two classes of σ -models introduced

more than twenty years ago: the class of “universal WZW-type

conformal σ -models” introduced by Tseytlin in [18]; we shall re-

fer

to them as T-models; and the class of “Poisson–Lie T-dualizable

σ -models on a compact group target” introduced by Klim

cík and

Ševera in [14]; we shall call them KS-models. Namely, we show

that the T-models are nothing but the analytic continuations of

the Poisson–Lie T-duals of the KS-models.

the way, we ﬁnd truly remarkable that the T-models and the

KS-models were orbiting around for two decades without “know-

ing

about each other”. The reason for this is that the authors of

[14] have worked out the target space geometries of the Poisson–

Lie

T-duals of the KS-models in coordinates natural from the point

of view of Poisson-geometry but not natural for the comparison

with the T-model. The parametrization of the dual target suitable

for this comparison was introduced in [16] and here we use it to

establish the announced result. We ﬁnd also interesting that the

KS-models were originally invented as new objects, the reason of

existence of which was their T-dualizability of a new kind, and the

authors of [14] were not aware that those models were closely re-

lated

to the T-models already existing on the market which had

their independent reason of existence.

http://dx.doi.org/10.1016/j.physletb.2016.06.077

0370-2693/

SCOAP

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