三角Ruijsenaars-Schneider系统的新紧凑形式与参数y的影响

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"三角Ruijsenaars-Schneider系统的新紧凑形式" 这篇研究论文深入探讨了鲁伊斯纳尔斯-施耐德（Ruijsenaars-Schneider）系统的三角紧凑形式，这是一种在量子力学和非经典力学中的重要模型。文章特别关注SU(n)的拟哈密顿双态的还原，这一过程构成了Ruijsenaars的紧致三角n体系统的基础。研究的核心是参数y，其在早期的研究中被限制在0 < y < π/n的范围内，这是因为Ruijsenaars原始压缩方法依赖于特定的等价条件。作者发现，当放宽这个限制，允许参数y取更广的范围0 < y < π/2时，系统出现了两种全新的自对偶紧凑形式。这两种形式在性质上有显著差异，分别标记为类型（i）和类型（ii）。类型（i）的情况与标准情况相似，其特点是减少了的相空间具有全局平滑的动作和位置变量，并且可以证明是辛同构于汉密尔顿复曲面流形的CP^n-1。这表明在类型（i）中，系统保持了经典力学中的某些结构特性。另一方面，类型（ii）的情况则更为复杂。在这种情况下，位置变量和动作变量在相空间的某些无处不稠密的子集上表现出奇异性。这种奇异性的存在可能会影响系统动力学的行为和解的性质，使得分析和理解变得更加挑战。论文进一步提供了关于参数y的完整分类，基于类型（i）和类型（ii）的二分法。为了具体说明这些新发现，作者详细描述了π/n < y < π/(n-1)区间内的一个最简单的新型（i）系统。这是一个实际应用和进一步研究的实例，有助于读者更好地理解和掌握这些新紧凑形式的特性。该研究的重要性在于它扩展了我们对Ruijsenaars-Schneider系统的理解，特别是在参数y的广义选取下系统的多样性。这种新的理解和分类对于理论物理学家、数学家以及那些在非经典力学领域工作的研究人员来说，将提供新的研究方向和潜在的应用场景。此外，由于文章是开放访问的，全球的研究社群都能无障碍地获取这些成果，从而促进相关领域的交流与合作。

102 L. Fehér, T.J. Kluck / Nuclear Physics B 882 (2014) 97–127



δ(ξ)



:=ξ and Ξ



ηgη

−1



=Ξ(g), ∀η, g ∈ G. (2.2)

Note that Ξ is G-invariant, its real component functions are globally continuous on G, and their

restrictions to G

reg

belong to C

∞

reg

). It is also important to know that Ξ is not differentiable

at G

sing

=G \G

reg

consisting of matrices with multiple eigenvalues (see Appendix A). It follows

that the functions

α: (A, B) →Ξ(A) and β:(A, B) → Ξ(B) (2.3)

engender continuous maps

ˆα: P



(y)



→A and

β:P



(y)



→A. (2.4)

These maps are globally smooth if (1.16) is valid, in which case they take their values in the

interior of the alcove, denoted

reg



ξ ∈A



=0 (∀k =1,...,n)



. (2.5)

Throughout this section, we restrict our attention to the open submanifold

−1



reg



⊂P



(y)



, (2.6)

where the components of

β are C

∞

functions. This submanifold equals P(μ

(y)) if (1.16) holds,

and it will be shown to be an open dense subset for any y satisfying (2.1).

The linearly independent smooth functions

:=λ

,j=1,...,n−1, (2.7)

induce Hamiltonian ﬂows on the submanifold (2.6). These are 2π -periodic and thus generate a

n−1

-action [19]. To describe this torus action, let us take a representative (A, B) ∈μ

−1

(μ

(y))

of the point [(A, B)]∈P(μ

(y)). Then diagonalize B, that is, introduce ξ ∈A

reg

and g ∈G by

gBg

−1

=δ(ξ). (2.8)

The action of

τ =(τ

,...,τ

n−1

) ∈T

n−1

(2.9)

is provided by the following formula:



(A, B)



→



−1

(τ)g, B



(2.10)

with

(τ) :=diag(1/τ

,τ

/τ

,τ

/τ

,...,τ

n−2

/τ

n−1

,τ

n−1

). (2.11)

It can be shown (see below) that this Hamiltonian T

n−1

-action on

−1

reg

), which we call the

β-generated torus action,

is an effective action.

Since P(μ

(y)) is compact and connected [20], we see that P(μ

(y)) is a Hamiltonian toric

manifold under the β-generated torus action whenever (1.16) is valid. Then we can invoke the

powerful Atiyah–Guillemin–Sternberg and Delzant theorems of symplectic geometry [22,23]

that determine the structure of a Hamiltonian toric manifold in terms of the moment map. In par-

ticular, under (1.16), we know that the image of the map

β is a closed convex polytope in A

reg

Of course one also has an analogously operating α-generated torus action on ˆα

−1

reg

) [19].

L. Fehér, T.J. Kluck / Nuclear Physics B 882 (2014) 97–127 103

The polytope is the convex hull of its vertices, which are the images of the ﬁxed points of the

β-generated torus action. The correspondence between the vertices and the ﬁxed points is one-

to-one. Moreover, the polytope completely characterizes the Hamiltonian toric manifold.

On account of the above, at least in the presence of (1.16),

we may establish the structure of

(y)) if we can ﬁnd its image under the map

β. Next, we shall present a characterization of

the image β(μ

−1

(μ

(y)) ∩(G ×G

reg

)), and study the equations that determine ﬁxed points of

the β-generated torus action.

2.2. The β-regular part of the reduced phase space

The open submanifold

−1

reg

) (2.6) will be called the β-regular part of the reduced phase

space. Here, we are interested in the

β-image of this submanifold, given by

reg



(y)



∩A

reg

=β



−1



(y)



∩(G ×G

reg

)



. (2.12)

Our description of this image relies on the functions z



(ξ, y) deﬁned on A

reg

by the formula



(ξ, y) :=

2iy

−1

2niy

−1



j=1

j=

(ξ) −e

2iy



(ξ)

(ξ) −δ



(ξ)

, ∀ =1,...,n. (2.13)

By using formula (1.9) and the periodicity convention (1.12) we can spell out this function as



(ξ, y) =

sin(y)

sin(ny)



j=1

j=



−e

−iy



−δ

sin(y)

sin(ny)

+n−1



j=+1



sin(



j−1

m=

−y)

sin(



j−1

m=

)



. (2.14)

The proof of the following result can be extracted from Section 3.2 of [19]. Nevertheless we

sketch it here since it is required for our later arguments.

Lemma 1. The

element ξ ∈ A

reg

belongs to the

β-image (2.12) if and only if z



(ξ, y) is non-

negative for all  =1,...,n.

Proof. Suppose that we have

ABA

−1

=μ

(y). (2.15)

Since B is conjugate to δ(ξ) with some ξ ∈A, (2.15) is equivalent to

δ(ξ)





−1



gμ

(y)g

−1



δ(ξ), (2.16)

where g is a unitary matrix for which

δ(ξ) =gBg

−1

and A

=gAg

−1

. (2.17)

Denoting by v ∈C

the last column of the matrix g,



:=g

n

, (2.18)

it is easily checked that

gμ

(y)g

−1

2iy



2i(1−n)y

−e

2iy



†

. (2.19)

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