规范理论与置换：风味单峰的新视角

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"这篇学术文章探讨了规范理论中的风味单峰作为置换的概念，主要集中在U（N c）规范理论中与SO（N f）风味群相关的单线态操作员。研究利用了Gelfand对和Schur-Weyl对偶性来阐述这个思想，并特别关注了当N f = 6时，与N = 4 $$ \mathcal{N} = 4 $$超 Yang-Mills（SYM）理论的标量扇区相关的操作员。" 在规范不变算子的背景下，作者们提出了一种简单的公式，用于计算在自由场限制下gY ≥ M2 ≥ 0条件下的两点函数。他们展示了这些自由的两点函数等同于在具有特定边界和缺陷的置换拓扑场论中的分区函数。通过对置换等价类的傅立叶变换，他们能够将这些算子转化为表示基础，这些基础在有限的N c 和 N f 值下确保了两点函数的正交性。文章还涉及了规范不变算子的计数公式，这有助于理解这些操作员的数量和结构。此外，作者们在一环混合矩阵方面的工作导出了线性算子，这些算子在置换等价类上起作用。这些结果不仅深化了对规范不变性的理解，还可能对理解和计算高能物理中的散射过程以及量子场论中的其他复杂现象有所助益。文章"Flavours in singlet gauge theory as permutations"由Yusuke Kimura、Sanjaye Ramgoolam和Ryo Suzuki合作完成，分别来自日本Okayama研究所、英国伦敦大学玛丽皇后学院、南非理论物理研究所和巴西圣保罗州立大学。它是在2016年由Springer为SISSA出版的，并在JHEP期刊上发表，为开放获取，便于全球研究人员查阅和引用。这篇文章提供了一个新颖的方法来理解和描述规范不变算子，特别是在考虑风味群和单线态操作员时，其理论和数学工具的应用具有广泛的潜在影响，可能促进对基本粒子物理和量子场论更深层次的理解。

JHEP12(2016)142

where T

consists of permutations of cycle type p. This is also equal to

σ∈H

ϕ(σ), ϕ(σ) =

γ∈[2

]

(γσγ

−1

), (2.16)

where the elements of H

commute with a ﬁxed permutation of cycle type p. A derivation

of the equality of these formulae, along with counting at ﬁnite N

is in section 4.3.

When N

< 2n or N

< 2n, there exist a number of linearly dependent relations among

operators. They are called ﬁnite N

constraints or ﬁnite N

constraints. These constraints

can be expressed in terms of Young diagrams.

In section 3, we will construct a set of oper-

ators with representation labels, where the ﬁnite N

and ﬁnite N

constraints are manifest.

2.2 Two-point functions

Consider the free two-point functions of the mesonic operators (2.4). Using the Wick

contraction rule (A.1), we obtain

i= δ

k=1

(Φ

)

(k)

m=1

(Φ

)

(m)

= δ

σ∈S

k=1

(m)

(k)



m=σ

−1

(k)

= δ

σ∈S

k=1

−1

(k)

σα

−1

(k)

(2.19)

where σ represents all possible Wick contractions, and δ

i=1

2i−1

. The results can

be expressed by permutations,

i =

σ∈S

W (σ

−1

) tr

(α

σα

−1

) =

σ∈S

W (σ

−1

) N

C(α

σα

−1

)

. (2.20)

Here C(σ) is (2.6), and W (σ) is the ﬂavour factor

W (σ) = δ

k=1

σ(k)

≡ δ

(σ)

,··· ,a

,··· ,b

. (2.21)

We have W (1) = N

. The σ in (2.21) acts on V

⊗2n

, with matrix elements equal to Kro-

necker deltas. We can also write W (σ) by introducing contraction operators C

, C

···,

where C

acts on the ﬁrst and second tensor factors of V

⊗2n

⊗ e

= δ

⊗ e

(2.22)

Here is a simple example of a ﬁnite N

constraint. For a 2 × 2 matrix X, we have the identity

tr(X

) =

trX tr(X

) −

(trX)

. (2.17)

We can rewrite this identity in terms of the projection operator associated with the anti-symmetric repre-

sentation as

]

⊗3

) = 0. (2.18)

In short, we cannot anti-symmetrise more than N

= 2 indices.

(σ) = tr

(σ 1) is a special case of (2.3).

– 7 –

JHEP12(2016)142

Indeed, the function z(σ) satisﬁes

z(σ) = z(γ

σγ

), ∀γ

, γ

∈ S

], (2.28)

We will show in appendix B that it can be expressed as

z(σ) =

C(Σ

σΣ

−1

). (2.29)

Since permutations can be multiplied eﬃciently using group theory software such as GAP

or with Mathematica, this is a very useful expression for practical calculations. It does not

involve explicitly doing sums over indices ranging from 1 to N

. The quantity (2.29) also

has a nice mathematical meaning. It is equal to the number of cycles in the coset type of

σ as explained in appendix A.3.

For later purposes, let us deﬁne

Ω

(f)

σ∈S

z(σ)

−1

, Ω

σ∈S

C(σ)

−1

. (2.30)

In terms of these quantities the two-point functions (2.20) have the form of

i =

σ∈S

z(σ)

C(α

σα

−1

)

σ∈S

C(σΣ

−1

)

C(α

σα

−1

)

(Ω

(f)

σ) δ

(Ω

σα

−1

). (2.31)

From (2.28), the Ω

(f)

satisﬁes

Ω

(f)

= Ω

(f)

, (2.32)

and we recover W (σ) by

W (σ) = N

(Ω

(f)

σ). (2.33)

3 Orthogonal two-point functions

We now propose a set of operators labelled by representations,

R,Λ

,τ

= tr

R,Λ

,τ

⊗ Φ

⊗ ··· ⊗ Φ

⊗ Φ

R,Λ

,τ

α∈S

τ,

(α

−1

) [α],

(3.1)

where R, Λ

are irreducible representations of S

, τ runs over 1, . . . , C(R, R, Λ

) is the

Clebsch-Gordan multiplicity for S

tensor products (see (A.21)), and the S

τ,

is the

Clebsch-Gordan coeﬃcient (A.20). The use of this group theory data in the construction

of covariant bases was developed in [9, 12]. The B

, called a branching coeﬃcient, is

deﬁned in terms of the reduction of the irreducible representation Λ

of S

in terms

of a direct sum of irreducible representations of S

] ⊂ S

. A generic state |Λ

, ki

can be expanded in terms of irreducible representations of S

]. The one-dimensional

irreducible representation of S

] is known to appear in the decomposition of Λ

with

– 9 –

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