全息杨米尔斯理论中的θ依赖效应研究

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"这篇学术论文深入探讨了全息杨米尔斯理论中θ参数的影响，该理论是由Witten提出的4维全息杨米尔斯模型。在该模型中，θ参数是导致CP对称性破缺的拓扑项的系数。文章首先回顾了与θ相关的引力背景，特别是对与θ参数成对的RR形式的全面反向反应的分析。接着，通过全息计算方法，作者们计算了一系列可观测物理量，如基态能量密度、弦张力、't霍夫特环、轻标胶球的质量、重子质量尺度以及解除约束的临界温度，从而构建了（T，θ）相图。此外，还讨论了纠缠熵的θ依赖性。" 在这篇研究中，作者提供了一个简单规则，用于推导模型中所有CP中立可观测值的θ修正项，这扩展了我们对全息杨米尔斯理论的理解。值得注意的是，部分研究中的可观测值已经在纯4d Yang-Mills的晶格模拟中被考察过。在晶格模拟的小θ范围内，结果显示出极低的θ平方项贡献。全息计算结果与这些晶格数据一致，并表明θ参数导致的质量比例降低。这种精确度对于理解Yang-Mills理论中的高阶校正是非常有价值的。文章发表于JHEP08(2015)090，由Springer出版，于2015年7月3日提交，7月28日接受，8月18日发布。作者来自意大利的多个研究机构，包括INFN—Sezione di Pisa、Dipartimento di Fisica e Astronomia, Università di Firenze以及Dipartimento di Fisica "E. Fermi"，Università di Pisa。研究团队通过电子邮件进行联系，进一步深化了全息杨米尔斯理论在θ参数影响下的物理现象研究。这项工作为理解全息杨米尔斯理论和θ参数如何影响高维量子场论提供了新的见解，同时为纯4d Yang-Mills理论的晶格模拟提供了重要的比较和参考。这些发现不仅有助于理论物理的发展，也为实验物理学家提供了预测和解释数据的新工具。

JHEP08(2015)090

The relevant type IIA gravity action, in string frame, reads

S =

√

−g



−2φ



R + 4(∂φ)



−



. (2.2)

Here 2k

= (2π)

where l

≡

√

is the string length, F

= dC

is the RR four-form

which is magnetically sourced by the N

D4-branes, F

= dC

, and φ is the dilaton.

The θ-backreacted background [4, 5] (see appendix A for a detailed re-derivation based

on the 1d reduced action) is a solution of the equations of motion derived from the action

above. The string frame metric reads





3/2



√







3/2



+ u

dΩ



, (2.3)

where

f = 1 −

, H

= 1 −

1 + Θ

. (2.4)

The background also includes a running dilaton, a RR one-form and a four-form ﬁeld

strength given by

= g





3/4

, C

, F

= 3R

, (2.5)

with the ﬂux quantization condition ﬁxing the value of R as

= 8π

, R = (πg

)

1/3

. (2.6)

Here Θ is a parameter which, as it will be clear in a moment, is proportional to θ. The

original background at zero θ angle, found in [1], can be readily obtained from the one

above setting Θ = 0. Let us recall that in the expression above µ = 0, 1, 2, 3 are the

1 + 3 Minkowski directions where the Yang-Mills theory is deﬁned, dΩ

is the metric of

a S

of radius one, u is the transverse radial coordinate u ∈ [u

, ∞) - holographically

mapped to the RG scale of the dual ﬁeld theory - x

is the compact coordinate of length

2πM

−1

and R is a curvature radius. The isometry group of S

is mapped into a global

SO(5) symmetry group in the dual ﬁeld theory, which acts non-trivially on the Kaluza-

Klein massive modes (signaling that these are, in fact, not decoupled in the limit we are

considering). Finally g

is the string coupling and ω

is the volume form of the transverse

, of volume V

= 8π

/3.

Notice also that, crucially, the S

circle shrinks to zero size when u = u

. Absence of

conical singularities at u = u

is guaranteed if

1 + Θ

, (2.7)

which thus implies that the (x

, u) subspace has the topology of a disk.

Reliability of the background requires e

to be small: when this condition is violated

we should better make use of the 11d (“M-theory”) completion of the model. As it was

– 5 –

JHEP08(2015)090

shown in [1], in fact, the 10d solution can be obtained starting from an AdS

× S

planar

black hole solution of 11d supergravity, reducing it on the M-theory circle and performing

a double Wick rotation. We will make use of this picture in the following.

As it will be reviewed below, the integral of C

over S

at u → ∞, is ﬁxed to be

proportional to θ by the holographic dictionary. Then, since the S

circle shrinks, a

trivial solution like C

∼ θdx

is not allowed. Regularity requires in fact that C

goes to

zero at u = u

, a condition which is precisely satisﬁed by the solution above.

The UV ’t Hooft coupling and the θ angle of the gauge theory, can be related to the

gravity parameters by considering the low energy limit of the D4-brane action

= −τ

x dx

−φ

−det(G + F) + τ

∧ TrF ∧ F , (2.8)

where F

αβ

≡ 2πα

αβ

is proportional to the gauge ﬁeld strength, τ

= (2π)

−4

−5

, C

the electric ﬁve-form sourced by the branes (its ﬁeld strength F

is the Hodge dual to

) and G

αβ

is the induced metric on the world-volume. Expanding the action to second

order in derivatives, considering the UV asymptotics u → ∞, integrating over the compact

direction and comparing the resulting 4d action with the Minkowski version of (2.1),

one gets

λ = g

≡ 2λ

≡ 2g

Y M

= 4πg

, θ + 2πk =

, (2.9)

where k is an integer. In the above expression we have introduced the parameters g

Y M

and λ

which are often referred to as the (UV) 4d gauge and ’t Hooft coupling of the

holographic model in the literature, despite the fact that they diﬀer for a factor of 2 from

the standard bare ones appearing in eq. (2.1). As we have observed above, λ

∼ T

parameterizes how far the holographic model is from pure Yang-Mills.

The second relation in (2.9) deﬁnes θ mod 2π integer shifts (since the integral of C

is gauge invariant only modulo 2πZ) realizing the expected multi-branched feature of the

dual gauge theory vacuum. Moreover, together with (2.5), it implies that the bare θ angle

is related to the Θ parameter of the background by

Θ ≡

4π



θ + 2kπ



8π



θ + 2kπ



. (2.10)

As it is also suggested by eq. (2.1), in the ’t Hooft limit, the corrections to the physics

due to the θ parameter, w.r.t. the θ = 0 case, are actually weighed by the combination

appearing in Θ. Moreover, since this parameter depends on k, what we actually get on the

gravity side is an inﬁnite family of solutions corresponding to possible ﬁeld theory vacua.

As it was shown in [5] the curvature invariants of the background remain small if

|Θ|  λ

1/4

. A similar constraint is obtained by studying the tension of the domain walls

(identiﬁed with wrapped D6-branes) separating the various k-vacua in the dual ﬁeld theory:

if |Θ| is too large, the (metastable) vacua become unstable. Our results will thus be reliable

provided Θ satisﬁes these bounds. Actually, as we will see in the following section, on the

Remember that

TrF ∧ F ≡ (1/2)

xTrF

F .

– 6 –

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