量子椭圆Calogero-Moser系统的轨距折纸与超对称规范理论

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本文主要探讨了量子椭圆Calogero-Moser系统（eCM）及其推广的深入关系，以及这些系统与超对称规范理论之间的联系。量子椭圆Calogero-Moser模型是量子力学中一种重要的非线性动力学系统，它在数学物理中有着广泛的应用，特别是在统计力学、粒子物理学和拓扑量子场论等领域。作者们首先关注的是eCM系统的特征多项式构建问题。他们通过分析对应超对称规范理论中的某些特殊“折纸”瞬时子分配函数，巧妙地构造了适用于eCM系统的特征多项式。这种折纸过程实际上是将高维理论通过特定的折叠或缩放映射到低维，引入了所谓的“共维度缺陷”，这在数学上表现为某种拓扑变换，使得系统的性质在新的框架下得以展现。在这个过程中，关键的概念是“gauge origami”，即通过折叠的方式实现理论空间的结构变换，这在物理上可以理解为从全局视角到局部视角的转换，或者说是从高维几何结构到低维几何结构的映射。这种“折纸”技术对于理解和计算复杂系统具有重要意义，因为它简化了问题的处理，揭示了隐藏的对称性和结构。接着，文章进一步探讨了如何通过“折纸折纸”技术构造出折叠瞬时子分区函数，这是一种推广的策略，它不仅适用于eCM系统，还能精确地得到被称为椭圆双Calogero-Moser（edCM）系统的双版本特征多项式。edCM系统是对eCM系统的一种增强，包含了更多的动态行为和对称性，这在数学物理模型中是重要的拓展。这篇论文的核心贡献在于提供了一种新颖的方法，即利用超对称规范理论中的折纸概念，来解析和计算量子椭圆Calogero-Moser系统及其推广的特征多项式，这对于理解这些系统的基本性质、寻找新的对称性以及探索它们在更深层次物理理论中的应用具有深远的意义。此外，文章还展示了数学物理中从高维到低维理论转换的实用工具，这对于理论物理学家和数学家来说是一篇富有洞察力的读物。

JHEP02(2020)108

2.2 Finding the commuting Hamiltonians: warm up

After demonstrating how the BAE can arise from four dimensional N = 2

∗

SU(N) gauge

theory, it is natural to consider if it is possible to similarly obtain the commuting Hamilto-

nians. To t hi s end, we ﬁrst conside r the T (x) function de ﬁne d in (

2.14) and (2.26) which

is a natural candidate for generating all commuting Hamiltonians of the eCM system by

identifying its expansi on coeﬃcients in appropriate asymptotic regime.

2.2.1 T-function from gauge theory

As a si mp l i ﬁc ati on to illustrate the procedures, let us consider pure N = 2 SU(N) gauge

theory by setting m → ∞ to integrate out the adjoint hypermultiplet, it is known that the

associated integrable system is

N−1

Toda latti ce (戸田格子) system:

Toda

= −

α=1

∂

∂x

+ Λ

α=1

−x

α−1

; x

∼ x

α+N

, q = Λ

. (2.27)

Now in the m → ∞ limit , the saddle point equation (

2.14) now be c omes BAE for

N−1

Toda system:

1 = −q

Q(x

αi

− ǫ)

Q(x

αi

+ ǫ)

, (2.28)

such that T (x) deﬁned in (

2.26) reduces to:

T (x) =

Q(x + ǫ)

Q(x)



1 + q

Q(x)

Q(x − ǫ)



= Y (x + ǫ) + q

Y (x)

, (2.29)

as can also be deduced from lim

m→∞

Y (x ± m) → 1.

We will now show that T (x) is a degr e e N polynomial in spectral parameter x. Us-

ing (

2.28), we c an see that the apparent poles of T (x) coming from poles of Y ( x + ǫ) are

canceled by the corresponding zeros in the bracket. This proves that T (x) is analytic in

the compl ex x-plane (excluding x = ∞). To prove T (x) has the correct degree, we ﬁrst

consider large x behavior of Y (x). When x is large, we may approximate x

αi

≈ x

(0)

αi

. Thus

the asymptotic behavior of Y (x) behaves as:

Y (x) ∼

α=1

∞

i=1

x − a

− (i − 1)ǫ

x − a

− (i − 1)ǫ − ǫ

α=1

(x − a

) ∼ x

at x → ∞. (2.30)

We conclude that (

2.29) c onst r uc te d based on saddle point equation is a degree N polyno-

mial. Next we would like to see if it can be directly related to the characteristic polynom i al

of Toda lattice by checking if we can recover Hamiltonian given in (

2.27).

2.2.2 Surface defect via orbifolding

Here we would like to introduce a co-dimension two surface defect on C

⊂ R

= C

× C

with Z

orbifolding on the coordinates of C

by (z

, z

) → (z

, ζz

) where ζ

= 1. This

orbifolding procedure commutes with NS limit.

We will res tor e ǫ

dependence for a mo-

ment, and take NS limit after orbifolding. Such orbifolding m aps the original gauge theory

The introduction of full surface defects into pure N = 2 SYM and N = 2

∗

theories, and their connections

with quantum integrable systems was also considered earlier in [

29].

– 7 –

JHEP02(2020)108

(which can be considered as A

quiver) into so called handsaw quiver structure [30]. Shown

in [

10], the instanton partition function with surface defe c t inserted is an eigenfunction of

Hamiltonian for both Calogero-Moser and Tod a cases (eq. (

2.1) for Calogero-Moser and

eq. (

2.27) for Toda). On the other hand, N = 2 with fundamental hypermultiplet does not

require such a orbifold i ng procedur e, as discussed in [

15]. It should be noted however that

it is possible to rel at e these two diﬀerent types of surface defects via the brane creation

process similar to Hanany-Witten transition in M-t he ory, as discus se d in [

31, 32].

A Z

type surface defect in a U(N) gauge theory [

21] can be characterized by a coloring

function: c : {α = 1, 2, . . . , N } → Z

, which assigns a color α labeling Coulomb parameter

to an irreducible representation R

of Z

, ω = 0, 1, . . . , N − 1. Here we choose the

simplest form of c:

c(α) = α − 1, (2.31)

this implies we can assi gn the Coulomb parameter a

to the representation R

α−1

of Z

One may take other form of coloring function c if needed. In principle, one can consider the

lower degree orbifolding as the quotient by Z

n<N

. The defect corresponding to Z

is cal l e d

the full-type surface defect, which is relevant to our purpose. More detailed di sc us si ons can

be found in [

18–21]. Under Z

orbifolding, the c omp l ex coupling q splits into N copi es :

q = q

· · · q

N−1

; q

ω+N

= q

, (2.32)

each q

is assigned to the representation R

of Z

for complex gauge coupling. Under

Orbifolding, counting in the instanton partition function becomes

λ|

(2.33)

with the deﬁniti on

:= {(α, (i, j)) | α = 1, . . . , N ; (i, j) ∈ λ

(α)

; α + j − 1 ≡ ω mod N} (2.34)

and the following deﬁnition of notations:

= |K

|, ν

= k

− k

ω+1

(2.35)

We will show how the introduction of full surface defect aﬀects T (x). Starting from

its building block Y (x), under orbifolding, Y (x) becomes Y (x) =

(x) where each

orbifold copy is:

(x) = (x − a

)

(α,(i,j))∈K



(x − a

− (i − 1)ǫ

− (j − 1)ǫ

− ǫ

)

(x − a

− (i − 1)ǫ

− (j − 1)ǫ

)



(α,(i,j))∈K

ω+1



(x − a

− (i − 1)ǫ

− (j − 1)ǫ

− ǫ

)

(x − a

− (i − 1)ǫ

− (j − 1)ǫ

− ǫ

)



, (2.36)

For more general coloring function, the condition in (

2.34) should be c(α) + j ≡ ω. K

is a colle ct i on of Young diagram boxes which are assigned to the representation R

un-

der orbifolding. Reme mbe r that orbifolding is impose d in C

, with ǫ

charged. Adding

– 8 –

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