WZNW项Yang-Baxterσ模型的E模型研究

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"这篇论文是关于物理领域中的可积σ模型，特别是研究了WZNW项作为E模型的Yang-Baxter σ模型。该研究指出，许多在群流形上定义的可积σ模型都属于E模型类别，这与Poisson-Lie T对偶有密切关系。作者Ctirad Klimˇcik在文章中证明了Delduc、Magro和Vicedo在先前的工作中提出的带有WZNW项的Yang-Baxter σ模型同样符合这一理论。该论文发表于《Physics Letters B》杂志，并被归类为开放访问文章，由SCOAP3资助。" 正文: 在理论物理学中，σ模型是一种描述高维膜在低维空间中运动的理论，尤其在弦理论中扮演着重要角色。非线性σ模型是这类模型的一个分支，它们通常涉及目标空间的几何结构，如度量和联络。这些模型的可积性是研究的关键属性，因为它允许通过精确解来理解物理系统的行为，而无需进行数值模拟。 Poisson-Lie T对偶是一种在广义对称性框架下的对偶性，它将一个物理系统的动力学映射到另一个等效但不同的系统。这种对偶性在理解和统一不同物理现象中起着重要作用，特别是在可积系统中。 E模型是与Poisson-Lie T对偶紧密相关的特殊类型的σ模型。它们在理论上有其独特的特性，比如具有丰富的对称性和复杂的几何结构。E模型的可积性使得它们在研究多体系统、量子场论以及广义对称性等领域有广泛的应用。杨-巴克斯特（Yang-Baxter）方程是量子群理论和可积系统的核心工具，它描述了一种特殊的交换关系，可以确保模型的可积性。WZNW（Wess-Zumino-Witten）项则是σ模型中的一个额外贡献，它来源于拓扑效应，与模型的规范对称性和拓扑性质有关。在本文中，作者Ctirad Klimˇcik展示了带有WZNW项的Yang-Baxter σ模型也属于E模型的范畴。这意味着这种模型不仅保持了可积性，还可能与Poisson-Lie T对偶有直接联系。这个发现扩展了我们对可积σ模型的理解，尤其是当它们涉及到非平凡的拓扑特性和复杂的对称结构时。进一步的研究可能包括深入探索这种模型的精确解，分析其物理性质，以及利用这些性质来推导新的对偶关系。此外，这个工作也可能对理解和构建新的可积物理模型提供启示，尤其是在弦理论和其他相关的量子场论中。

Physics Letters B 772 (2017) 725–730

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Yang–Baxter σ -model with WZNW term as E-model

Ctirad Klim

cík

Institut de Mathématiques de Luminy, 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

3 July 2017

Accepted

24 July 2017

Available

online 26 July 2017

Editor:

N. Lambert

It turns out that many integrable σ -models on group manifolds belong to the class of the so-called

E-models which are relevant in the context of the Poisson–Lie T-duality. We show that this is the case

also for the Yang–Baxter σ -model with WZNW term introduced by Delduc, Magro and Vicedo in [5].

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

1. Introduction

Integrability and T-dualisability are both rare properties of non-

linear

σ -models and it is not known what necessary conditions

must be imposed on a target space metric G and on closed 3-form

ﬁeld H in such a way that the corresponding σ -model enjoys

either one or another. It turns out, however, that those two prop-

erties

share something in common, in the sense that the majority

of integrable sigma models [1,13–15] recently constructed fall in

the class of the so-called E-models [2], which are the building-

blocks

for the construction of T-dualisable models [3,4]. It is not

yet clear why it is so, but we ﬁnd that trying to put all inte-

grable

σ -models under the common roof of the E-models is a

nice guiding principle which may eventually lead to a deeper un-

derstanding

of this phenomenon. In particular, the purpose of the

present note is to give the E -model interpretation of one of a

few integrable σ -models for which this was not yet done, namely

the Yang–Baxter σ -model with WZNW term. This model was in-

troduced

in [5] (see also [6] for previous work) and it is a two

parameter

deformation of the principal chiral model deﬁned by

the action

S[g]=−





dτ





−1

∂

−

g,(1 + η

+ AR + η

−1

∂





−1



(

−1

dg, [g

−1

∂

g, g

−1

dg])



. (1.1)

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

Since the constant K is just the overall normalisation constant and the relation

(1.2) is supposed to hold, the action (1.1) is indeed a two-parameter deformation of

the principal chiral model; the latter is recovered for the particular values η = A =

k = 0.

Here (., .) is the Killing–Cartan form on the Lie algebra G of a sim-

ple

compact Lie group G, R : G → G is the so-called Yang–Baxter

operator (see Section 3 for its deﬁnition), τ and σ stand respec-

tively

for the world-sheet time and space variables (the time and

space integrals are notationally separated as



and



expressing

the angular character of the variable σ ), g ≡ g(σ , τ ) is a dynami-

cal

ﬁeld conﬁguration with values in G, we set also ∂

:= ∂

± ∂

and, ﬁnally, the parameters of the model are constrained by the

relation

A = η



1 −

1 +η

. (1.2)

While Delduc, Magro and Vicedo showed in [5] that the model

(1.1) is integrable only if the relation (1.2) holds, we show in

the present paper that (1.1) has the structure of the E -model

also only in the case when the constraint (1.2) is imposed.

This fact illustrates again the mysterious relation between the

E-model formalism and the integrability which motivates our

work.

The

plan of this note is as follows: We ﬁrst review how the

concept of the E-model is built up on that of the Drinfeld dou-

ble,

then we introduce a particular Drinfeld double and show that

the E -model which corresponds to it is just the Delduc, Magro and

Vicedo model (1.1). We also express the Lax pair of the integrable

model (1.1) in the formalism of the E-model therefore our paper

can be viewed also as an alternative demonstration of the integra-

bility

of the model (1.1).

Note also our perhaps non-standard way of writing of the Wess–Zumino term

which is however very useful for “travelling” between the ﬁrst order and the second

order description of the σ -model dynamics. Mathematically speaking, the object



(

−1

dg, [g

−1

dg, g

−1

∂

g]) should be viewed as a two-form on the inﬁnite dimen-

sional

loop group manifold LG.

http://dx.doi.org/10.1016/j.physletb.2017.07.051

0370-2693/

SCOAP

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