AdS4/CFT3强弱弦耦合研究

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"AdS4 / CFT3从弱到强的字符串耦合——探究了U（N）k×U（N）-k ABJM理论中应力张量多重态的算子四点函数在N趋向无穷大，N/k保持固定条件下的性质。在此极限下，ABJM理论与AdS4×ℂℙ3上的IIA型弦理论全息对应，弦耦合g s与(N/k)^1/4成正比。研究中关注的第一个超前校正是由R 4相互作用顶点引起的树级维滕图修正。利用超共形 Ward 恒等式、整体局部性以及之前通过超对称局部化计算得到的所有阶1/N的球面自由能，确定了R 4校正值与N/k的关系。结果验证了IIA型S矩阵的已知R 4贡献，仅从零模和一模弦世界片中获得gs微扰。这是首次在AdS / CFT背景下检查强耦合下的g s效应，具体针对四点相关函数。结果在大N、大霍夫特耦合和有限k极限之间进行插值，同时也为小弦耦合下的AdS4×ℂℙ3上的IIA型弦理论与AdS4×S 7 /ℤk上的M理论之间的关系提供了见解。" 这篇研究详细探讨了AdS/CFT对应中的一个重要方面，即从弱到强的字符串耦合转换。AdS/CFT对应是一种强大的工具，它将量子场论与弦理论在特定条件下联系起来，允许在一种理论中计算的物理量在另一种理论中找到对应。在本研究中，特定的背景是U(N)k×U(N)−k的ABJM理论，这是一个描述M2-branes的理论，N/k的值固定，N趋于无穷大，这意味着弦耦合g s随着N/k的增加而增强。研究的核心是四点函数，特别是应力张量多重态中的算子。在1/N的领先阶，可以使用IIA型超引力来计算这些函数。然而，为了深入理解强耦合效应，作者们关注了第一个次主导的修正，这源于R 4交互项的树级维滕图。这种修正对于理解弦理论在强耦合下的行为至关重要，因为它揭示了如何从微扰系列中获取信息。作者们利用超共形 Ward 恒等式，这是一种在共形场论中非常有用的工具，可以用来关联不同算子的关联函数。结合整体局部性和之前通过超对称局部化方法得到的结果，他们能够确定R 4校正值，即它如何依赖于N/k的大小。这种分析不仅验证了已知的IIA型弦理论S矩阵中的R 4贡献，还显示了这些贡献仅来源于零模和一模弦的世界片。此外，研究的结果对于四点相关函数在不同的理论极限之间的插值有重要意义。这意味着它们不仅适用于大N和大霍夫特耦合的极限，也适用于N固定但k较大的情况。这为我们提供了从弱耦合到强耦合转变的连续视角，同时为AdS4×ℂℙ3上的IIA型弦理论与AdS4×S 7 /ℤk上的M理论之间的插值提供了新见解，这在小弦耦合下特别有趣。这项工作展示了AdS/CFT对应在处理强耦合问题时的潜力，并提供了对弦理论非微扰特性的深入理解。通过研究高阶修正和在不同理论参数空间中的插值，我们可以更好地洞察这些复杂理论的内在结构。

JHEP01(2020)034

Up to crossing there are ﬁve possible 4 particle superamplitudes we can construct from

Φ and Ψ. However, under CPT the two supermultiplets Φ and Ψ are conjugates, and

their scattering amplitudes are related by complex conjugation (see appendix B.2 for a

description of how discrete space-time symmetries act on the scattering amplitudes):

A[ΨΨΨΨ] = (A[ΦΦΦΦ])

∗

, A[ΨΨΨΦ] = (A[ΦΦΦΨ])

∗

. (2.17)

This leaves us only three independent superamplitudes, A[ΦΦΨΨ], A[ΦΦΦΨ], and

A[ΦΦΦΦ]. Our task now is to constrain the forms of these superamplitudes, beginning

with invariance under supersymmetry.

As explained in [77], for a given particle i the supermomentum is deﬁned to be

= |iiη

, ˜q

= |i]

∂

∂η

(2.18)

and it satisﬁes the on-shell SUSY algebra by construction. For a given amplitude the total

supermomentum is thus:

˜q

(2.19)

Superamplitudes must be annihiliated by these supercharges. For a four-point amplitude

such as A[ΦΦΨΨ] this implies that

A[ΦΦΨΨ] = 0 ,

A[ΦΦΨΨ] = 0 . (2.20)

Imposing these conditions uniquely ﬁxes any four-point superamplitudes up to an arbitrary

function of s and t:

A[ΦΦΨΨ] = δ

(Q)

[12]

h34i

(s, t) ,

A[ΦΦΦΨ] = δ

(Q)

[12]

h14ih24i

h34i

(s, t) ,

A[ΦΦΦΦ] = δ

(Q)

[12]

h34i

(s, t) ,

(2.21)

where the ﬁrst factor is the Grassmann delta function

(Q) =

I=1

i,j=1

hijiη

, (2.22)

which is annihilated by both Q

and

, and f

(s, t) are arbitrary functions

of s and t.

The delta function δ

(Q) is automatically invariant under SU(6)

, even if the full theory

does not preserve SU(6)

[72].

Note that every term in each superamplitude contains

Note that [ij]

∗

= hjii in terms of the spinor-helicity angle and square brackets.

Since A[ΦΦΨΨ] is self-conjugate under CPT we ﬁnd that f

(s, t) is real, as we show in appendix B.

We furthermore show that for CT -invariant theories f

2,3

(s, t) are also real.

In ﬂat space N = 6 the supersymmetry algebra does not require there to be an R-symmetry; it is an

accidental symmetry of the supergravity action. On the other hand, the superconformal algebra does require

that at least an SO(6)

symmetry be present in order for an AdS solution to preserve all supersymmetries

of the theory.

– 10 –

JHEP01(2020)034

| A[ΦΦΨΨ] A[ΦΦΦΨ] A[ΦΦΦΦ]

0 A[φφφφ] None None

1 A[F

−

φφ] A[φφφa

] None

A[φφχ

−

] A[φφF

]

A[φχ

−

φ] A[φφg

φ]

A[φF

−

φ]

2 A[F

−

] A[F

−

] A[F

]

A[χ

−

] A[F

−

] A[g

φ]

A[φg

−

φ] A[φφψ

−

] A[g

φφ]

A[φa

−

φ] A[ψ

−

φφ] A[ψ

φφ]

A[φg

−

] A[g

−

φ] A[h

φφφ]

. . . . . .

. . . . . . . . . . . .

6 . . . . . . A[h

−

]

7 . . . A[h

−

] None

8 A[h

−

] None None

Table 2. Component amplitudes of each superamplitude, organised by total helicity

|. Here

is the helicity of the i

particle. We have not included amplitudes equivalent to the ones listed

here under crossing.

exactly 12 Grassmannian variables, and, as a result, many component amplitudes vanish,

including A[h

] = A[φφφφ] = 0. See table 2 for a list of component amplitudes

that do not vanish. The angle and square brackets in (2.21) are required so that the Φ and

Ψ components have the correct helicity, which for instance can be ﬁxed by considering

A[h

−

] = [12]

h34i

(s, t) ,

A[h

−

] = [12]

h34i

h14ih24if

(s, t) ,

A[h

−

] = [12]

h34i

(s, t) .

(2.23)

We are now left to constrain the forms of f

(s, t) using locality and crossing symmetry.

A tree-level scattering amplitude is local if and only if it can be written as a polynomial

in the spinor helicity variables [ij] and hiji; note that

s = [12]h12i = [34]h34i, t = [13]h13i = [24]h24i, u = [14]h14i = [23]h23i. (2.24)

From (2.23) we immediately see that it is not possible for f

(s, t) to contain poles in s, t

or u, or else the amplitudes in (2.23) would lead to non-polynomial expressions. Hence

(s, t) are necessarily polynomials for tree-level amplitudes. This is also suﬃcient, as when

(s, t) = 1 one can check that all amplitudes in the superamplitude are local.

Crossing symmetry imposes a series of further constraints. For instance, in (2.23) the

amplitudes must be invariant under interchanging the ﬁrst and second particles. This gives

– 11 –

JHEP01(2020)034

Mellin deg. f

(s, t) f

(s, t) Counterterms # sols.

3 — — 1 F

4 1 — — R

5 s — s

+ t

+ u

, D

6 s

, t

+ u

— stu D

, D

7 s

, s(t

+ u

) — (s

+ t

+ u

)

, D

7.5 — (s − t)(t − u)(u − s) — D

F R

Table 3. Counterterms in N = 6 supergravity, up to 15 derivatives.

us the relations

1,3

(s, t) = f

1,3

(s, u) , f

(s, t) = −f

(s, u) , (2.25)

where u = −s − t is the third Mandelstam variable. The superamplitudes A[ΦΦΦΦ] and

A[ΦΦΦΨ] are furthermore invariant under crossing which exchange the ﬁrst and third

particles, giving rise to the further conditions:

(s, t) = −f

(u, t) , f

(s, t) = f

(u, t) . (2.26)

Together, eqs. (2.25) and (2.26) suﬃce to guarantee crossing under all possible permutations.

Having determined the allowed forms of f

(s, t), we can now determine the number

of derivatives in each interaction vertex. To this count each angle and square bracket

contribute 1, δ

(Q) contributes 6, and each power of s, t, u contributes 2. For instance, if

we set f

(s, t) = s

and consider the amplitude A[ΦΦΨΨ] = s

(Q)

[12]

h34i

, it follows that

this amplitude comes from an interaction vertex with 8 + 2k derivatives, namely from an

term.

With this in mind, we can now systematically ﬁnd all local counterterms up to a certain

number of derivatives. In table 3 we list all local counterterms up to 15 derivatives, corre-

sponding to Mellin amplitudes up to degree 7.5.

In particular, the ﬁrst local counterterm

has 6 derivatives, is unique, and contributes only to A[ΦΦΦΦ]. The next local counterterm

has 8 derivatives and is also unique and contributes only to A[ΦΦΨΨ]. There are two 10

derivative counterterms, one contributing to A[ΦΦΦΦ] and one to A[ΦΦΦΦ], and so on.

The counterterm with the lowest number of derivatives that contributes to A[ΦΦΦΨ] has

15 derivatives and will not be important in this work.

2.4 Implications for N = 6 SCFts

Having systematically computed the local amplitudes in N = 6 supergravity, we will now

discuss the implications for holographic N = 6 SCFTs. First, we can deduce that there are

ﬁve independent superconformal invariants in the four point function of four stress tensor

multiplets. This counting follows from the number of unknown real functions needed to

A Mellin amplitude of degree 7.5 would seem to require non-polynomial contributions to M

(s, t). In

appendix B however we show that due to discrete symmetries the Mellin amplitudes corresponding to

A[ΦΦΦΨ] never contribute to hSSSSi, so M

(s, t) remains a polynomial in s and t.

– 12 –

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