共形场论揭示一类弦幅值：N=4超 Yang-Mills 非平面关联与IIB弦理论高阶散射

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共形场理论的属一弦幅值研究是当前理论物理学领域中的一个重要课题，特别是在高能物理和弦理论的交叉点上。该研究聚焦于N $$ \mathcal{N} $$ = 4超级杨米尔斯（Super Yang-Mills）理论中的非平面关联函数与IIB型弦理论中更高阶类的闭合弦振幅之间的联系。N $$ \mathcal{N} $$ = 4超级杨米尔斯理论以其丰富的对称性和关联函数的解析性在量子色动力学和超对称性方面具有重要地位。研究者通过共形场论（Conformal Field Theory，CFT）的技术手段，成功构建了一个在AdS5×S5背景下，低能量展开下的属一（Genus-one）四点弦振幅。这种振幅在强耦合扩展中对应着't Hooft极限（'t Hooft coupling limit）下的N $$ \mathcal{N} $$ = 4超级杨米尔斯关联函数，其中次导项在1/c²的级数展开中占据重要位置。这里，'t Hooft极限是指将弦论的耦合强度与普朗克常数的倒数联系起来，它对于理解理论的非perturbative行为至关重要。在平面空间的限制下，这项工作揭示了II维闭合弦在10维空间中的属一四点散射幅度。这种转换提供了一个桥梁，使得从弦论的角度重新审视和验证了之前通过弦扰动理论得到的结果。弦论中的这些计算不仅检验了理论的自洽性，还可能揭示出新的物理现象或统一理论的线索。此外，研究者还展示了一种创新的方法，即通过在AdS振幅的平坦空间极限中固定次引线项，这在M理论（Membrane Theory）和弦论的相互作用中可能具有深远的意义。这种固定机制有助于我们更好地理解弦论与更高维引力理论之间的关联，以及它们如何在不同能量尺度上表现。这篇发表在JHEP06(2019)010上的论文，通过对共形场论的巧妙应用，不仅深化了我们对N $$ \mathcal{N} $$ = 4超级杨米尔斯与弦理论之间关系的理解，而且还展示了在特定极限下的计算技术，这对于理论物理学的未来发展具有重要的指导意义。

JHEP06(2019)010

By taking this limit, we generate predictions for the discontinuities of A

(g=1)

involving

sugra, R

and ∂

vertices. Using previous results of [25], we ﬁnd a match.

We also use our CFT methods to compute the functional form of certain discontinuities

to all orders: ﬁrst, any term in the α

expansion of A

(g=1)

involving at least one R

vertex; and second, the complete discontinuity of A

(g=1)

in the limit of forward scattering

(t → 0). We independently derive these results, and determine the actual coeﬃcients,

using the string theory techniques of [25]. (See (4.22) and (4.25).) To our knowledge, these

expressions have not appeared elsewhere.

Flat space limit: subleading order. In taking the ﬂat space limit, we run into an

interesting open question for holography. Subleading terms at large s, t represent “ﬁnite

size corrections” due to AdS curvature, and are not accessible using naive application of

the ﬂat space limit. One would like to know whether these subleading terms — indeed,

the full AdS amplitudes themselves — may be recast as certain scattering observables in

the higher-dimensional string/M-theory and, if so, which ones.

One of our main observations is that subleading terms in the s, t → ∞ limit of tree-level

AdS Mellin amplitudes may actually be ﬁxed by constructing the one-loop AdS amplitude,

and matching its ﬂat space limit to a one-loop string/M-theory amplitude. The basic point

is that since the one-loop amplitude is essentially the square of the tree-level amplitude

by AdS unitarity, the subleading terms in the tree-level amplitude feed into the one-loop

amplitude. Then by matching the latter to the string/M-theory one-loop amplitude, these

subleading terms can be at least partially ﬁxed. For the case of hO

i at O(1/c), the

ﬁrst such subleading term appears at O(λ

−5/2

), where the Mellin amplitude takes the form

β(s

+ t

+ u

) + β

+ β

s (1.9)

where (β, β

, β

) are functions of p. Only β, given in (2.24), can be ﬁxed by matching onto

the Virasoro-Shapiro amplitude at O(α

). By matching to the genus-one string amplitude

at O(α

) we ﬁx β

= 2p(p −2)(p + 1)

, and reduce β

to two constants.

Organization. In section 2 we set up the problem, introduce the tree-level amplitudes

in the 1/λ expansion, and explain where the subleading terms at large s, t come from in

terms of the AdS

× S

reduction. In section 3 we construct the dDisc of the one-loop

amplitudes to the ﬁrst several orders in 1/λ, reveal their transcendental structure, and

use this to parameterize the coeﬃcients of subleading terms in the tree-level correlator at

O(λ

−5/2

). In section 4 we make contact with the type II genus-one string amplitudes.

We relate their analytic parts to the structure of one-loop anomalous dimensions and UV

divergences. Taking the ﬂat space limit of our dDiscs, we reproduce the discontinuities of

various terms in the genus-one string amplitudes and constrain the subleading coeﬃcients

(p) and β

(p) given above. We end with a handful of open problems, while various

appendices supplement the main text.

At O(λ

−3/2

), the leading stringy correction, the amplitude is just a constant which can be matched

using the ﬂat space limit and the known 10d R

vertex, see [26] and appendix B.

– 6 –

JHEP06(2019)010

2 Generalities and tree-level solutions

Our object of study is the four-point function of O

, the superconformal primary in the

stress tensor multiplet of N = 4 SYM. O

is a rank-two symmetric traceless tensor

of SO(6)

≈ SU(4)

. Contracting its R-symmetry indices with polarization vectors y

obeying the null condition y

· y

= 0, we introduce the index-free four-point function

, y

)i =

· y

)

· y

)

(σ, τ)G

(z, ¯z)

(2.1)

where the sum runs over SU(4)

representations R ∈ [0, 2, 0]×[0, 2, 0]. We have introduced

cross-ratios in position space

z¯z =

, (1 −z)(1 − ¯z) =

, x

= x

− x

(2.2)

and polarization space,

σ ≡

· y

)(y

· y

)

· y

)(y

· y

)

, τ ≡

· y

)(y

· y

)

· y

)(y

· y

)

. (2.3)

The Y

(σ, τ) are SO(6) harmonics which may be found in [27]. We will work in the

Lorentzian regime, where z, ¯z are independent complex variables. (For the physical cor-

relator, they are real variables.) Superconformal Ward identities [27, 28] allow to write

all G

(z, ¯z) in terms of a single function G(z, ¯z) ≡ G

105

(z, ¯z)/(z¯z)

, where the irrep

105 ≡ [0, 4, 0]. Under the crossing transformation z ↔ 1 − ¯z, this satisﬁes the relation

((1 −z)(1 − ¯z))

G(z, ¯z) − (z¯z)

G(1 − ¯z, 1 − z)

+ ((z¯z)

− ((1 − z)(1 − ¯z))

) +

z¯z −(1 − z)(1 − ¯z)

= 0 (2.4)

where the central charge c = (N

−1)/4. See [29] for a detailed discussion. The contribution

to G(z, ¯z) from protected intermediate operators, belonging to short multiplets, can be

computed exactly and is denoted by G

short

(z, ¯z). We then split

G(z, ¯z) = G

short

(z, ¯z) + H(z, ¯z) (2.5)

where H(z, ¯z) carries the dynamically non-trivial information and admits a decomposition

in superconformal blocks,

H(z, ¯z) =

∆,`

(z, ¯z) , (2.6)

with squared three-point coeﬃcients a

∆,`

. The sum runs over superconformal primaries in

long multiplets, of dimension ∆ and (traceless symmetric) Lorentz spin `. The supercon-

formal blocks are given by

∆,`

(z, ¯z) = (z¯z)

∆−`

`+1

∆+`+4

(z)F

∆−`+2

(¯z) − ¯z

`+1

∆+`+4

(¯z)F

∆−`+2

(z)

z − ¯z

(2.7)

– 7 –

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