AdS3×S3×T4背景中混合通量刚性纺丝弦的一环量子化研究

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"这篇研究论文详细探讨了在AdS 3×S 3×T 4背景中，具有混合R-R（Ramond-Ramond）和NS-NS（Neveu-Schwarz-Neveu-Schwarz）三形式通量的刚性闭环纺丝细线的一环量子化。作者Juan Miguel Nieto和Roberto Ruiz来自马德里康普顿斯大学的理论物理部门。文章指出，他们计算了具有两个相等角动量的刚性闭合旋转弦的经典色散关系的一环校正，并将这一分析扩展到了纯NS-NS通量极限下的任意两个角动量情况。研究过程中，作者采用了两种不同的技术进行计算。第一种方法基于欧拉-拉格朗日方程，用于解决围绕经典解的二次波动；第二种方法则利用问题的潜在可积性，通过有限间隙方程进行处理。结果表明，在纯NS-NS极限中，一环校正消失了。这是一项关于AdS/CFT对应（Anti-de Sitter/Conformal Field Theory Correspondence）领域的重要工作，涉及可积场论和sigma模型的理论。关键词包括：AdS-CFT对应、可积场论、sigma模型。该文于2018年4月10日作为预印本发布，并在同年7月23日由Springer出版，收录在JHEP（Journal of High Energy Physics）07期，编号141。" 这篇学术论文深入研究了AdS/CFT对称性的具体应用，特别是在带有混合R-R和NS-NS通量的复杂背景下的量子弦理论。AdS/CFT对应是一种强大的工具，它将引力理论（在AdS空间中）与非引力理论（通常为共形场论）联系起来。在这种背景下，刚性旋转弦的行为提供了对强耦合系统的洞察，因为它们在AdS空间中的动力学可以通过CFT中的操作来理解。作者的计算展示了在特定极限条件下（纯NS-NS极限）量子修正的消失，这是对基本物理性质的重要洞察，可能有助于更好地理解弦理论的结构和其在凝聚态物理、量子信息等领域可能的应用。同时，利用有限间隙方程揭示问题的可积性是理论物理学中一个强大的技术，它可以帮助简化复杂的数学计算，并提供更深刻的物理见解。

JHEP07(2018)141

deformation of the Neumann-Rosochatius system. The equations of motion must be sup-

plemente d with the Virasoro constraints, which read

i=1



′2

+ r

(α

′2

+ ω

)



= w

, (2.13)

i=1

′

i=1

= 0 . (2.14)

The most straightforward solutions to the equations of motion are those of con-

stant radii

(σ) = a

. (2.15)

On these solutions α‘

= m

and (2.10) and (2.11) simplify to

Λ = m

− ω

= m

− ω

+ 2q(ω

− ω

) . (2.16)

Furthermore, the restriction (2.2) and the Virasoro constraint (2.14) determine

−ω

− ω

, a

− ω

, (2.17)

or, using the deﬁnitions of the angular momenta,

+ 4πhqm

4πh(ω

+ qm

)

, a

4πh(ω

+ qm

)

. (2.18)

The dispersion relation for general values of q only can be written down as a series in

negative powers of the total angular momentum. Despite so, ther e exist two regimes where

we can ﬁn d a closed analyti c expression for the dispersion relation. The ﬁrst one is the

pure NS-NS limit we commented above, while the second one corresponds to a restriction

to a su(2) subsector of the theory, which amounts to set the J

= J

. Note that these two

regimes are not mutually exclusive.

Let us examine in more depth both cases. In the ﬁrst limit the equations of moti on

can be sol ved by

4πh

, ω

4πh

− (m

− m

) , (2.19)

where J = J

+ J

. Using the ﬁrst Virasoro constraint (

2.13) and (2.17) we can obtain the

dispersion relation

E = J + 4πhm

. (2.20)

On the other hand, if we restrict ourselves to the su(2) subsector, the second Virasoro

constraint (

2.14) implie s m

= −m

≡ m, while t he equations of motion and the deﬁnition

of the angular momenta imposes

4πh

= Υ + qm , ω

4πh

− 2qm = Υ − qm , (2.21)

where we have deﬁned

Υ =

4πh

− qm =

+ J

4πh

− qm , (2.22)

– 5 –

JHEP07(2018)141

for later convenience. The expression for the dis persion relation r e ads

E =

− 8πhqmJ + 16π

= 4πh

+ κ

, (2.23)

with κ

= 1 − q

3 Quadratic ﬂuctuations around the classical solution

In thi s section we derive the Lagragian for the quadratic ﬂuctuations around the rigid

spinning string-type solutions on R × S

. We obtain the equation for the characteristic

frequencies for them, which greatly simpliﬁes in both t he case of two equal angular mome nta

and the case of pure NS-NS ﬂux. The signed sum of the characteristic frequencies provide

the one-loop correction to t he classical energy.

We derive this eﬀective Lagrangian by splitting the target-space ﬁelds into th e classical

background and ﬂuctuation ﬁelds and trun cat i ng the latter up to second order in the action.

We treat the bosonic ﬂuctuations on the sphere, on the anti-de Si t te r space, on the torus

and the fermionic ﬂuctuations separ ate l y in order to make the section more readable.

3.1 Bosonic ﬂuctuations on S

Regarding the ﬂuctuations on the sphere, we take advantage of the spherical symmetry

and perform the following substitution

cos ϕ

→ a

cos(α

+ ω

τ) + ˜r

cos(α

+ ω

τ) − ρ

sin(α

+ ω

τ) ,

sin ϕ

→ a

sin(α

+ ω

τ) + ˜r

sin(α

+ ω

τ) + ρ

cos(α

+ ω

τ) . (3.1)

where ˜r

, ρ

denote the perturbation ﬁelds.

Introducing (3.1) into the Polyakov action with the B-ﬁeld in the c onfor m al gauge, the

Lagrangian for the quadratic ﬂuctuations of the ﬁelds turns out to be

= h

(

i=1

−

˜r

− ˙ρ

+ ˜r

′2

+ ρ

′2

+ (˜r

+ ρ

)(−ω

+ α

′2

) + 2˜r

(−ω

˙ρ

+ α

′

ρ‘

)

− 2ρ

(−ω

˜r

+ α‘

˜r

′

) − Λ (˜r

+ ρ

)

+ 2q



(˜r

+ ρ

)(ω

α‘

− α

′

)

+ ˜r

(ω

′

− α

′

˙ρ

) − ρ

(ω

˜r

′

− α

′

˜r

) +



2 ˜r

−

˜r



( ˙ρ

′

− ρ

′

)

−

(

˜r

′

− ˜r

′

) +

( ˙ρ

′

− ρ

′

˙ρ

)



)

, (3.2)

where the dot stands for derivatives with respect to τ .

In this section we do not label the origin of the frequencies (S, AdS, T or F ) as no ambiguity ar ises .

It is also possible to incorporate the ﬂuctuations as r

→ a

+ ˜r

and ϕ

→ α

+ ω

τ + ˜ϕ

instead. Both

choices are equivalent since ρ

≈ a

˜ϕ

at ﬁrst order.

– 6 –

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