η-变形AdS5×S5的σ模型与超弦理论谜题

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本文主要探讨了η变形AdS 5×S 5空间中的弦理论模型。AdS 5×S 5是一个重要的双曲空间，在弦理论中扮演着关键角色，尤其是在AdS/CFT（阿贝尔-杨弦理论/共形场论）对应中，它与四维N=4超 Yang-Mills理论有深刻的关联。η变形是一种非平凡的量子群对称性，它扩展了原始的对称性结构，为理论带来了新的动力学性质。研究者们首先推导出了η变形AdS 5×S 5上的一种σ模型的Lagrangian，这个模型的特点是仅依赖于玻色子，同时具有费米子的二次项。这表明了理论中费米子与玻色子之间的非平凡相互作用。随后，他们发现了关键的场重定义技巧，通过这种方法，他们将这个Lagrangian转换到了IIB Green-Schwarz类型超弦的标准形式，这是超弦理论中常见的框架，有助于理解和计算弦论物理量。然而，当考察相关的RR耦合时，他们发现这些耦合并不满足运动的超引力方程，这可能意味着理论在某些方面存在不一致性或有待进一步修正。尽管如此，通过特定的缩放比例，他们的解能复现Maldacena和Russo之前发现的超重力背景，这为理论提供了重要的验证点。文章还深入到世界sheet散射矩阵的研究，利用费米拉格朗日函数计算了树级散射矩阵的新矩阵元素。值得注意的是，当对两个不具有单粒子分解的粒子态进行统一变换后，T矩阵可以被分解为两个等效的部分，每个部分都独立满足经典的Yang-Baxter方程。这个结果与q变形S矩阵的极限行为相一致，展示了η变形AdS 5×S 5模型在散射过程中的量子群对称性的体现。这篇文章揭示了η变形AdS 5×S 5空间上的弦理论模型的一些新颖特性，包括其特殊的Lagrangian形式、场重定义、以及与经典对称性和量子群结构的联系。这些发现不仅深化了我们对量子重力和量子群对称性在高维理论中的理解，也为未来的研究提供了新的方向，特别是在弦理论的背景选择和动力学行为的探索上。

JHEP12(2015)049

where

is a vielbein for the deformed metric which we present in appendix A.4 and η is

the Minkowski metric (A.20).

In the expansion of the Lagrangian (2.1) in powers of fermions, contributions to a given

power come from three sources: from the current A

, from the operator O

−1

and from the

current A

. Thus, the quadratic fermionic Lagrangian is a sum of six terms

(2)

= L

{002}

+ L

{200}

+ L

{101}

+ L

{011}

+ L

{110}

+ L

{020}

(2.23)

where three numbers in the brackets indicate powers of fermions coming from A

, O

−1

and

, respectively. For the ﬁrst two contributions we ﬁnd

{002}

= −

˜g

(γ

αβ

− 

αβ

)

{200}

= −

˜g

(γ

αβ

− 

αβ

)

(2.24)

where k

= η

. Note that the sum of L

{002}

+ L

{200}

gives a non-trivial con-

tribution also to the Wess-Zumino term, since the matrix k

has a non-vanishing anti-

symmetric part.

Concerning the contribution {101}, in appendix B.2 we manipulate the initial re-

sult (B.24) to bring it to the form most close to the canonical one

{101}

= −

˜g



αβ

i e



−

1 +

√

1 + κ





, (2.25)

which holds up to a total derivative.

Now we spell out the contributions stemming from the inverse operator taken at ﬁrst

order in the θ expansion. The two contributions {011}, {110} can be naturally considered

together

{011}+{110}

= −

˜g

(γ

αβ

− 

αβ

)



− (κσ

− (−1 +

1 + κ

)δ

)



iγ



+(κσ

− (−1 +

1 + κ

)

) iγ



+ k

)θ

. (2.26)

Finally, the last contribution to the Lagrangian is delivered by the term where the inverse

The result can be put in this form thanks to the properties (B.7).

– 9 –

JHEP12(2015)049

operator is taken at order θ

. We ﬁnd

{020}

= −

˜g

(γ

αβ

− 

αβ

)

− 2δ





−



− 



− γ



−(−1 +

1 + κ

)δ



−





+ γ



−(−1 +

1 + κ

)



− γ







−



+κσ



−





− γ



−κσ







−



. (2.27)

The last two expressions involve the coeﬃcients λ

, λ

which are collected in

appendix B.1.

Summing up all the above contributions, we discover that the result is not the standard

Green-Schwarz Lagrangian, see the Introduction. Yet, in the undeformed limit it reduces

to that one. Indeed, when κ → 0, the contributions L

{011}+{110}

and L

{020}

vanish, while

in eq. (2.24) becomes η

, so that (2.24) transforms into the standard kinetic term,

while (2.25) provides its Wess-Zumino completion.

This is of course expected because the

canonical form of the undeformed Lagrangian is intrinsically built in our construction based

on global symmetries and the choice (2.8) of the coset representative. On the other hand in

the deformed model the AdS

×S

coset plays an auxiliary role because only six commuting

isometries remain unbroken. It is thus clear that the Lagrangian we got describes couplings

of bosons with fermion bilinears written with a more or less arbitrary choice of coordinates

and that ﬁeld redeﬁnitions will in general modify its form. Our next task is therefore to

ﬁnd a ﬁeld redeﬁnition that will cast (2.23) in the desired canonical form.

To search for necessary ﬁeld redeﬁnitions we need a guidance principle. All terms in

(2)

can be split into two parts: the kinetic part L

∂

, which contains all couplings of the

form θ∂θ, and the mass part θθ, which constitutes the rest of the Lagrangian. The idea

is to concentrate just on the kinetic part and ﬁnd ﬁeld redeﬁnitions which bring it to the

canonical form. In the process new mass terms will be generated and we look at all of them

at the very end. Clearly, two types of ﬁeld redeﬁnitions are possible: rotations of fermions

→ U

with coeﬃcients U

depending on bosons and shifts of bosons by fermion

bilinears. In the second case, the bosonic Lagrangian L

(0)

will generate contributions to

(2)

and if we do not want to create higher derivatives of θ, the corresponding shifts should

be of the form

→ X

(X) θ

(2.28)

with boson-dependent coeﬃcients f

(X).

In particular, the self-dual ﬁve-form of the AdS

×S

background arises from the term with 

in the

deﬁnition (2.14) of the operator D

[26].

– 10 –

JHEP12(2015)049

Next, all the terms in L

∂

are naturally divided according to their symmetry properties

into two categories L

∂

and L

∂

−

. Given an expression of the form θ

∂θ

, we classify

it according to

∂θ

= +∂θ

=⇒ L

∂

∂θ

= −∂θ

=⇒ L

∂

−

(2.29)

The symmetry properties are manifested through purely algebraic manipulations, not by

integrating by parts. They are inherited from symmetries of gamma matrices contained

in M

and from the behaviour of M

under the exchange of I, J. We then show in

appendix B.3 that there exists a choice of the coeﬃcients in (2.28) such that the corre-

sponding shift completely removes L

∂

, leaving behind a bunch of new mass terms. As to

∂

−

, it remains untouched under this shift because the symmetry properties of the deriva-

tive couplings generated by (2.28) are opposite to that of L

∂

−

. The only manipulations we

are left with at this point are boson-dependent rotations of fermions. Since L

∂

−

and the

canonical kinetic term share the same symmetry (2.29), a rotation which transforms one

into the other always exists and we ﬁnd its explicit form in appendix B.3.

Through the shift of bosons and the rotation of fermions we generated quite a lot of new

mass terms. It is now time to sum them up and group together according to their tensorial

structures. Quite remarkably, after this is done, the mass part turns out to automatically

ﬁt the canonical arrangement. In terms of a 32-dimensional Majorana fermion Θ of positive

chirality (A.46), our Lagrangian is therefore

(2)

= −

˜g

(γ

αβ

+ 

αβ

)

(2.30)

where the operator

enjoys the canonical form

= δ



∂

−



mnp

−





(1)

pqr

(3)

pqr

2 · 5!



pqrst

(5)

pqrst



(2.31)

In the last equation

is the same vielbein of the η-deformed metric, cf. appendix A.4,

that features in the bosonic Lagrangian, while

is the spin connection that is related

by the standard formula (A.54). Finally, for the 3-form H

mnp

we ﬁnd the following

two non-vanishing components

234

= −4κρ

1 + ρ

1 − κ

sin ζ

1 + κ

sin

, H

789

= +4κr

√

1 − r

√

1 + κ

sin ξ

1 + κ

sin

. (2.32)

These are precisely the ﬁeld strength components of the B-ﬁeld (2.6) written with the

ﬂat space indices. Thus, we completely restore the NSNS background of the η-deformed

theory at the level of the quadratic fermionic action, which is rather non-trivial by itself

and provides a strong validity check of our computation.

Postponing the discussion of the RR couplings till the next section, we conclude by

pointing out that the ﬁeld redeﬁnitions of (X, θ) we used do not involve world-sheet deriva-

tives and, as such, they can be viewed as a certain κ-dependent reparametrisation of the

original coset representative (2.8) of AdS

× S

The 10-dimensional Γ-matrices are given in (A.42).

– 11 –

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η-变形AdS5×S5的σ模型与超弦理论谜题

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