理性gl(2)六边形模型：解析超杨-米尔斯不变量

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本文主要探讨了在理数点gl(2)平面上的Baxter六顶点模型与杨-米尔斯理论的连接。杨-米尔斯理论是一种量子场论，特别关注于规范场的相互作用。文章提出了一种新颖的方法，即利用仰角不变量的Bethe ansatz技术来研究这些理论中的关键对象——杨简群不变量。这些不变量在物理学中扮演着重要的角色，特别是在解析散射振幅计算中，它们体现了理论的对称性和结构。 Bethe ansatz是一种强大的工具，它最初是由Bethe在研究量子Heisenberg磁模型时提出的，用于求解一维量子系统的精确本征态。在这个特定的上下文中，作者将这个方法扩展到了辅助非均匀自旋链的特征函数分析中，这些特征函数与杨简群的某些单元表示关联。这些特征函数是特定传递矩阵问题的特解，传递矩阵在量子统计力学中起到演化操作的作用。研究的关键是将这种模型与平面，但不一定是矩形网格的结构相结合。这意味着虽然模型的背景几何不是传统的，但它提供了一个独特视角来探索杨简群不变量的性质。通过这种方法，研究者得以构造出完全对称且有限维的杨简群表示的显式形式，这对于理解和计算相关物理现象具有重要意义。论文的主要成果是提出了一种从Bethe ansatz的角度来构造这些不变量的具体途径，这不仅限于gl(2)的特殊情况，而是具有普适性，可以推广到一般（超级）李代数和一般表示。这为理解更广泛的物理系统，如超级杨-米尔斯理论中的散射振幅，开辟了新的计算策略。值得注意的是，这项工作发表在《核物理学B》杂志上，是开放获取的，可以在www.sciencedirect.com和www.elsevier.com/locate/nuclphysb上获取。该论文由来自杜伦大学、柏林 Humboldt 大学、阿尔伯特-爱因斯坦研究所和韩国亚太理论物理中心的研究人员合作完成，展示了理论物理学和数学交叉领域的前沿进展。这篇论文对于杨-米尔斯理论和量子统计力学的数学框架做出了重要贡献，特别是通过Bethe ansatz方法处理复杂问题的能力，它为未来在理论物理中的进一步应用打开了新的研究路径。

R. Frassek et al. / Nuclear Physics B 883 (2014) 373–424 381

Fig. 3.1. Example of a (generalized) Baxter lattice with N = 3 lines and G = ((1, 3), (2, 5), (4, 6)), left side. Each line

carries a gl(n) representation Λ

, a spectral parameter θ

, two state labels α

and α

at the endpoints i

and

an orientation indicated by an arrow. The dotted circle and the reference point B are not part of the Baxter lattice. The

associated partition function Z(G, Λ, θ , α) is shown on the right, cf. (3.6).

3.1. Vertex models on Baxter lattices

Let us repeat the deﬁnition of the Baxter lattice spelled out in Section 2, and extend it by the

generalizations just mentioned. See also the simple example lattice in the left part of Fig. 3.1.

We start with a dotted circle on which we mark a reference point B.

Notice that the circle and

the reference point are only used for the construction and will not become part of the Baxter lat-

tice itself. N straight lines, each connecting two points on the dotted circle, are speciﬁed in such

a way that in the interior of the circle only two of the lines intersect at a single point. Starting at

the reference point, the N lines and the 2N endpoints of these lines are labeled counterclockwise.

Each line has an orientation, which for the k-th line with endpoints i

isgivenbyanarrow

pointing from j

towards i

. The choice of the reference point clearly affects the orientation of

the lines. In addition, we assign a gl(n) representation Λ

and a complex spectral parameter θ

to the k-th line. To the endpoints of this line we assign states of the representation Λ

labeled

by α

and α

. A (generalized) Baxter lattice including boundary conditions is deﬁned with this

data by the ordered sets

G =



), . . . , (i

)



Λ = (Λ

,...,Λ

), θ = (θ

,...,θ

), α = (α

,...,α

). (3.1)

In order to introduce a vertex model on such a Baxter lattice we also have to generalize the

Boltzmann weights of Section 2. We introduce them as

α, γ |R

ΛΛ





θ −θ





|β,δ=

Λ,θ



,θ



, (3.2)

which are matrix elements of the R-matrix

ΛΛ





θ −θ





Λ,θ



,θ



. (3.3)

382 R. Frassek et al. / Nuclear Physics B 883 (2014) 373–424

This R-matrix is an operator acting on the tensor product V

⊗ V



of the spaces of the two

gl(n) representations Λ and Λ



with spectral parameters θ and θ



. The Boltzmann weights are

deﬁned using orthonormal basis states of V

and V



labeled by Greek indices α, β and γ , δ,

respectively. Graphically each space is associated with one line. The orientation of (i.e. arrow

on) a line speciﬁes the order of multiple R-matrices acting on one space. R-matrices “earlier”

on the line are right of “later” ones in the corresponding formula. In this sense the arrows in

(3.3) point from the “inputs” of the R-matrix towards the “outputs” or, in component language

(3.2), from the kets towards the bras. We will switch between the operator language and the

Boltzmann weights whenever it is convenient. The R-matrix (3.3) is required to be a solution of

the Yang–Baxter equation

ΛΛ





θ −θ





ΛΛ





θ −θ













−θ













−θ





ΛΛ





θ −θ





ΛΛ





θ −θ





, (3.4)

which acts in the tensor product V

⊗V



⊗V



and reads graphically

Λ,θ



,θ





,θ



Λ,θ



,θ





,θ



. (3.5)

Now we can deﬁne the partition function of a vertex model on a (generalized) Baxter lattice,

see again the example in Fig. 3.1, employing the component language of Boltzmann weights.

To each internal edge of the lattice we assign a state of the given representation Λ, while the

states at the boundary edges are naturally ﬁxed by α. Recall that Λ is associated to the entire

line, and therefore all internal and boundary edges which make up the line carry states in this

representation. Each vertex of the lattice is then translated into a Boltzmann weight as shown in

(3.2). The partition function Z(G, Λ, θ , α) is the sum, ranging over all possible conﬁgurations

of states at the internal edges, of the product of all Boltzmann weights of the lattice. As already

in (2.14), we write symbolically

Z(G, Λ, θ , α) =



internal

state

conﬁg.



vertices

Boltzmann weight. (3.6)

If there is a line consisting of a single edge with differing states at the boundary, the partition

function vanishes.

In the operator description this partition function is a matrix element of a product of

R-matrices.

These R-matrices and their order in the product are given by the form of the Bax-

ter lattice. A line with a single edge is translated into an identity operator on the corresponding

representation space. The matrix element is speciﬁed by α, and the sum and product in (3.6)

translate into matrix multiplication.

3.2. Partition function as eigenvalue problem

As a ﬁrst step to understand the partition function of a Baxter lattice as an eigenvalue problem

within the QISM, we derive an identity satisﬁed by Z(G, Λ, θ , α ). Recall that the construction

of Baxter lattices involves a dotted circle. Here we replace this circle by an arc, which is opened

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