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首页缠结块与链接不变式理论的新探索
"链接不变式理论中的缠结块" 在量子场论和数学物理领域,链接不变式理论是一个关键的研究方向,特别是在二维共形场论中。这篇由Mironov、Morozov和An.Morozov合作的Open Access文章探讨了缠结块在链接不变式理论中的作用和重要性。文章发表于JHEP09(2018)128,并由Springer于2018年9月发布。 文章的核心在于全纯因式分解,这是二维共形理论的一项基本发现。这一概念将相关函数表述为共形块的双线性组合,这些共形块可以方便地进行切割和连接,无需对整个庞大的希尔伯特状态空间求和。这一特性大大简化了计算过程,使得复杂问题可以通过更简单的方式处理。 缠结块的概念与链接多项式紧密相关,它们是通过多线性组合缠结块并只对中间状态的有限子集求和来定义的。这种方法的威力在于,同一个缠结可以出现在不同类型的结中,使得我们可以从简单的构造中推导出复杂的结构。尽管技术上目前只在树状结的框架内发展,但作者指出这一方法实际上具有更广泛的适用性。 为了深入理解缠结块,文章通过研究一系列原型示例来系统展开讨论,这些示例不仅揭示了非平凡的结果,而且超越了传统技术的局限。缠结运算则关注如何粘合缠结以及Hom(缠结)中的函数映射。这种运算方式直观地实现了乘法合成结构,这是在传统打结理论中较为隐晦的一部分。 缠结块的理论不仅有助于我们理解链接不变量,还有可能在量子信息、编码理论以及统计物理等领域找到应用。通过这种方式,研究者可以构建更有效的计算策略,解决那些传统方法难以处理的复杂问题。这项工作为链接不变式理论提供了一个新的视角,推动了理论的发展,可能开启新的研究方向。
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JHEP09(2018)128
Notation, framing, normalization.
Notation. Throughout the text, we use the following notation: the quantum dimension
of the representation R is given by the celebrated hook formula
H
unknot
R
= D
R
=
Y
(i,j)∈R
{Aq
i−j
}
{q
h
i,j
}
(1.10)
where h
i,j
is the hook length for the element (i, j) of the Young diagram R, A = q
N
and
{x} := x − x
−1
, so that the quantum number [n] =
{q
n
}
{q}
. In the main body of the paper,
we give examples of this formula in the case of two-line Young diagrams, (4.20) and in
the case of representations from the adjoint tower, (4.17). Since we consider only finite-
dimensional representations of SU(N), each of them is associated with a Young diagram.
Throughout the text, we do not differ between the representation R and the corresponding
Young diagram. We also use the notation µ
Y
for the eigenvalues of the R-matrix in the
channel R ⊗
¯
R = ⊕Y and λ
Q
, in the channel R ⊗ R = ⊕Q.
Normalization. We denote by calligraphic letters the unreduced knot/link invariants: the
HOMFLY-PT invariant is denoted through H. The reduced HOMFLY-PT invariant, i.e.
that divided by the product of quantum dimensions of the representations coloring the link
components is denoted through just H. Note that the normalization of the link invariants
in [44] is different: as soon as they consider uncolored links, they normalize the invariants
dividing them just by one quantum dimension of the fundamental representation D
=
{A}
{q}
.
We also often need a mirror image of the knot/link, which, in terms of the invariant,
is described by the transformation (A, q) → (1/A, 1/q). We denote the corresponding
invariants with the bar:
H(A, q) := H
1
A
,
1
q
. (1.11)
Framing. Typically, all the invariants are given in the topological framing for knots and
in the standard (or canonical) framing for links [45–48]. This framing contains a trivial
U(1)-factor
U = q
2
P
i>j
|R
i
||R
j
|L
ij
/N
(1.12)
where the sum goes over all components of the link, the i-th component being colored
with the representation R
i
, and L
ij
is the linking number of the i-th and j-th components.
This factor is absent for knots, and we omit it for links unless the inverse is stated.
1
It is
essential only when one reduces representations, for the concrete SU(N), removing from
the Young diagrams the rows of the length N.
We sometimes need to consider various relations between knot/link invariants inspired
by group theory identities. Hence, these relations are most naturally realized in the framing
consistent with the group theory structure of the invariants, and, in the case of knots, this
is nothing but the vertical framing. In the case of links, this is the so called differential
framing, which is also consistent with the differential expansion of invariants, hence the
name [48]. We denote the invariants in this framing with a check-superscript:
∨
H.
1
It is often omitted, see [49, 50].
– 6 –
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JHEP09(2018)128
Important note. In this paper, for the sake of simplicity, we deal only with the cases of
trivial multiplicities. Otherwise, there would appear additional matrices in the space of
multiplicities, and the formulas become too overload.
2 Simple building blocks
2.1 The vacuum block: the simplest among cut-and-join
Let us demonstrate how our procedure works in the simplest example of the unreduced
colored HOMFLY polynomial for the unknot, which is equal to the quantum dimension of
the coloring representation R,
H
unknot
R
= D
R
. (2.1)
If the simplest unknot diagram (a circle) is cut into two semicircles, we get a decomposition
over all representations Y from the product of
R
=
P
Y ∈R⊗
¯
R
Y
R
¯
R
However, actually contributing to the sum is only the singlet representation Y = ∅ with
dimension D
∅
= 1. Thus the formula which we should associate with this picture is
D
R
=
X
Y ∈R⊗
¯
R
D
Y
V
Y
R
V
Y
R
= V
∅
R
V
∅
R
(2.2)
where V =
¯
V are the two blocks in the left and right parts of the second picture, which we
can naturally call vacuum. It follows that the vacuum block
R
¯
R
Y
V
Y
R
= = δ
Y
∅
·
√
D
R
As mentioned in the Introduction, already in this example we see that the tangle block
depends not only on the apparent argument Y , but also on the “hidden” R. Thus the
vacuum block is the simplest example of the cut-and-join blocks.
2.2 The lock blo ck
The central role in (1.6) is played by the lock element τ
Y
R
, which was actually calculated
long ago in the study of twist knots in [43]. From this block τ , one can construct a few
– 7 –
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