M理论群岛：超对称下的ABJM理论与3d N=8 CFT研究

49 浏览量更新于2024-07-16 收藏 1.17MB PDF 举报

本文主要探讨的是M理论群岛(M-Theory Archipelago)，这是一种结合了超对称定位(Supersymmetric Localization)和数值共形引导程序(Numerical Conformal Bootstrap)的方法，用于研究三维最大超对称(N=8)CFT，即在N个重合的M2膜(M2-branes)上，具体指的是U(N)_k×U(N)_{-k}的ABJM理论，其中k=1。ABJM理论是一种重要的弦理论模型，它在Chern-Simons水平上具有独特的性质。研究焦点在于压力张量的超共形原色(Tensor Multiplet)和可能存在的、由3d N=8超共形对称性所允许的下一个最低维度的半BPS多重态(Half-BPS Multiplet)。在已知的所有三维超对称N=8 SCFTs中，k=1的ABJM理论是唯一拥有这两种不同类型的多重态的理论。通过精确计算超对称定位中的短OPE系数值，作者能够确定这种理论中这类系数在半短OPE空间中的精确孤岛(islands)，即一系列特定的系数组合。有趣的是，他们发现这些孤岛的大小随着N的增大而减小，显示出理论结构中的非平凡依赖关系。文章不仅局限于ABJM理论，也扩展到了更广泛的3d N=8 SCFTs，即使它们的操作符谱不包含ABJM理论特有的特征，作者也对那些含有两种类型多重态的理论进行了半短OPE系数的上限和下限分析，以及最低未受保护的标量算子的研究。这种方法为理解这类理论的性质提供了新的洞察，并展示了超对称定位与共形对置技术的有效结合，对于揭示高维量子场论的深层次结构具有重要意义。本文的核心内容是利用超对称定位和数值共形引导程序来探索M2膜理论中的超对称CFT特性，尤其是关注不同类型多重态的混合相关器自举研究，以及这些理论中半BPS性质的定量分析。这一工作不仅深化了我们对弦理论的理解，也为未来类似理论的探索奠定了基础。

JHEP02(2020)010

Type BPS ∆ Spin so(8)

(A, 0) (long) 0 ≥ ∆

+ ` + 1 ` [a

]

(A, 1) 1/16 ∆

+ ` + 1 ` [a

]

(A, 2) 1/8 ∆

+ ` + 1 ` [0a

]

(A, 3) 3/16 ∆

+ ` + 1 ` [00a

]

(A, +) 1/4 ∆

+ ` + 1 ` [00a

(A, −) 1/4 ∆

+ ` + 1 ` [000a

]

(B, 1) 1/8 ∆

0 [a

]

(B, 2) 1/4 ∆

0 [0a

]

(B, 3) 3/8 ∆

0 [00a

]

(B, +) 1/2 ∆

0 [00a

(B, −) 1/2 ∆

0 [000a

]

conserved 5/16 ` + 1 ` [0000]

identity 1 0 0 [0000]

Table 1. Multiplets of osp (8|4) and the quantum numbers of their corresponding superconformal

primary operator. The conformal dimension ∆ is written in terms of ∆

≡ a

+(a

)/2. The

Lorentz spin can take the values ` = 0, 1/2, 1, 3/2, . . .. Representations of the so(8) R-symmetry

are given in terms of the four so(8) Dynkin labels, which are non-negative integers.

multiplet denoted by (A, 0) is a long multiplet and does not obey any shortening conditions.

The other multiplets of type A have the property that certain so(2, 1) irreps of spin ` −1/2

are absent from the product between the supercharges and the superconformal primary.

The multiplets of type B have the property that certain so(2, 1) irreps of spin ` ± 1/2 are

absent from this product, and consequently, the multiplets of type B are smaller.

The

half-BPS multiplets that we discussed above with bottom component S

are of (B, +) type

and have a

= k. The conserved current multiplet appears in the decomposition of the

long multiplet at unitarity: ∆ → ` + 1. This multiplet contains higher-spin conserved

currents, and therefore can only appear in the free theory [69]. We will sometimes denote

the superconformal multiplets by X

]

∆,`

, with (∆, `) and [a

] representing the

so(3, 2) and so(8)

quantum numbers of the superconformal primary, and the subscript X

denoting the type of shortening condition (for instance, X = (A, 2) or X = (B, +)).

Only certain M can appear in the OPE of half-BPS multiplets. Comparing the so(8)

irreps (2.4) that can appear in S

×S

to table 1, we see that S

×S

can contain only

the following series of multiplets

(A, 0)

[0 n−m 2m 0]

∆≥n+`+1,`

, (A, 2)

[0 n−m 2m 0]

n+`+1,`

, (A, +)

[0 0 2n 0]

n+`+1,`

(B, 2)

[0 n−m 2n 0]

n,0

, (B, +)

[0 0 2n 0]

n,0

(2.12)

Furthermore, it was shown in [70] that for n =

+ k

) − 1 we have the

This description is correct only when ` > 0. When ` = 0, the deﬁnition of the multiplets also requires

various conditions when acting on the primary with two supercharges.

– 9 –

JHEP02(2020)010

following extra constraints:

n =

+ k

) : (A, 0), (A, 2), and (A, +) do not appear ,

n =

+ k

) − 1 : (A, 0) does not appear .

(2.13)

One ﬁnal constraint is that the irreps that appear in the (anti)-symmetric product of the

OPE of identical operators can contain only (odd) even spins. If the spins set in (2.12)

and (2.13) conﬂict with this constraint, then the multiplet is forbidden.

2.2 Four point functions of S

and S

We will now restrict our attention to the following four-point functions:

h2222i, h3333i, h2233i, h2323i, h3223i,

(2.14)

which we consider in the s-channel as discussed above, and we use the notation hijkli ≡

i. The so(8)

tensor products that we must consider for the S

× S

, S

× S

and S

× S

OPEs are

⊗35

= 1⊕28⊕35

⊕300⊕567

⊕294

⊗112

= 8

⊕112

⊕160

⊕672

⊕1400

⊕1568

112

⊗112

= 1⊕28⊕35

⊕300⊕567

⊕294

⊕1386

⊕1925⊕3696

⊕4312

(2.15)

The supermultiplets M that can then appear in these OPEs according to the rules de-

scribed above are listed in tables 2 and 3. Note that h2233i and h2323i contain the same

supermultiplets as h2222i and h3223i, respectively.

We can compute the superblocks for each of these M in two steps, following the com-

putation for h2222i in [39]. First, we must determine which conformal primaries appear

in each supermultiplet. This can be done by decomposing the character of each osp(8|4)

supermultiplet under the bosonic subgroup so(5) ⊕so(8)

, using the formulae in appendix

B of [39], based on [68]. The results are written in appendix B. Second, we plug the

conformal block expansion (2.11) of each superblock into the Ward identities (2.8), using

the explicit expressions for the relevant Y

(σ, τ ) that we give in appendix A, and

expand these constraints to high enough order in r until we have ﬁxed all the pairs of

OPE coeﬃcients λ

∆,`,nm

of the superconformal descendents in terms of

the superconformal primary OPE coeﬃcients λ

. For instance, the stress ten-

sor multiplet (B, +)

[0020]

1,0

has a superblock that can be written in the A

(U, V ) basis

of (2.5) in terms of conformal blocks as

0,0

(U,V ) = G

0,0

1,0

(U,V ), A

0,0

(U,V ) = −G

0,0

2,1

(U,V ), A

0,0

(U,V ) =

0,0

3,2

(U,V ), (2.16)

where note that the relative sign of the odd spin conformal primary is opposite that of

the even spins, as expected in general for a four-point function of scalars. The analogous

expressions for the other superconformal blocks will have linear combinations of many

– 10 –

剩余56页未读，继续阅读

weixin_38544978

粉丝: 1
资源: 916

M理论群岛：超对称下的ABJM理论与3d N=8 CFT研究

multiplepassage:发生大（M7ish）地震后，请从宽带站获取地震图，绘制频谱图，并可选地覆盖理论瑞利波到达时间以演示表面波的多次通过

2016高考地理二轮复习专题五地壳运动规律练习

echarts地图隐藏南沙群岛

echarts 地图 隐藏南海群岛部分

伽马群岛由若干小岛构成，开发者在某些小岛间修建了水上通路，使得群岛大部分连通，但也不排除部分小岛仍为孤岛，创建伽马群岛的地图，例如： image.png 编写函数数一数该群岛共有多少个小岛。

echarts 海南地图将海南群岛弄成缩略图

请介绍一下三沙市的基本情况。

1923年至2021年，每年的4月2日发生过那些大事

求海南省行政区划图！

请讲述一些有趣的未解之谜

最新资源

echarts 地图隐藏南海群岛部分