DIM表示与5d $$\mathcal{N}=1$$ Instanton分区函数和qq-字符研究

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"DIM表示的（p，q）-web，5d N = 1 $$ \mathcal{N} = 1 $$ 实例分区函数和qq-字符" 这篇研究论文探讨了5维N = 1超杨-米尔斯理论中的瞬子（Instanton）分区函数，该函数与IIB型弦理论在S 1 上的还原有密切联系。在这个理论框架中，（p，q）-brane网络图用于描述这些瞬子现象。DIM（Ding-Iohara-Miki）代数在此过程中扮演了重要角色，它作用于分区函数并对其产生影响。DIM代数是一种特殊的量子代数，由Ding、Iohara和Miki提出，其结构和性质允许对分区函数进行深入分析。在（p，q）-web中，每一段都与DIM代数的一个表示相关联，而这些代数的顶点则通过Awata、Feigin和Shiraishi构造的缠结算子来标识。这些缠结算子是拓扑字符串理论中的关键组成部分，它们在顶点处连接和重组不同的表示，从而影响分区函数的计算。研究中，作者定义了一个新的缠绕器，作用于级别（1，n）⊗（0，m）→（1，n + m）的表示空间，这扩展了原有的结构，使其能处理更高级别的m。这一创新使得计算过程可以使用更简单的（p，q）-网络图的折叠形式，简化了实际操作，使得之前由其他研究者得到的Gaiotto状态和垂直交织器的表示得以提升，转化为在水平表示的Fock空间中操作的算子关系。此外，论文还发展了一种基于DIM元素的水平作用来构建线性颤动的qq-字符的方法。qq-字符是研究量子场论和数学物理中重要的工具，尤其在理解分类和对称性方面。基本的qq特征可以通过副产物构造，但对于更复杂的qq特征，则需要引入作用于DIM生成器张量积上的量子Weyl反射。这种方法提供了一种新的理解和计算qq-字符的途径，特别是在处理高维度问题时。这篇论文是开放获取的，发表在JHEP11(2017)034，并由Springer于2017年11月8日发布。作者来自韩国高等研究所、意大利博洛尼亚大学的INFN分部和东京大学的物理系，他们通过深入研究DIM代数和（p，q）-web的相互作用，为5d N = 1超杨-米尔斯理论的瞬子分区函数和相关量子物理问题提供了新的见解和计算工具。

JHEP11(2017)034

The notation [···]

refers to an expansion in powers of z

∓1

of the quantity inside the

brackets. They will be used to deﬁne the qq-character.

The action on the bra states will be referred to as the dual vertical representation. In

this representation, the roles of x

and x

−

are exchanged:

hh~v,

λ|ˆρ

(0,m)

(z)) = −γ

−1

x∈R(

λ)

δ(z/χ

) Res

z=χ

−1

(zq

−1

)hh~v,

λ − x|,

hh~v,

λ|ˆρ

(0,m)

−

(z)) = −γ

−m+1

x∈A(

λ)

δ(z/χ

) Res

z=χ

(z)

hh~v,

λ + x|,

hh~v,

λ|ˆρ

(0,m)

(ψ

(z)) = γ

−m



(z)



hh~v,

λ|.

(2.15)

and ˆρ

(0,m)

(ˆγ) = 1. Strictly speaking, this is not a representation of the DIM algebra

because some of the q-commutation relations are no longer satisﬁed. Instead, it should be

seen as a representation on the dual states hh~v,

λ| that are orthogonal to the basis |~v,

λii,

hh~v,

λ|~v,

ii = δ

λ,

−1

, (2.16)

such that we have the property



hh~v,

λ|ˆρ

(0,m)

(e)



|~v,

ii = hh~v,

λ|



(0,m)

(e)|~v,



(2.17)

for any element e of the DIM algebra. The norm of the states involves the coeﬃcients

which will play an important role in the construction of instanton partition functions.

They are deﬁned in terms of the vector multiplet contribution to the instanton partition

function Z

vect.

(~v,

λ) as follows,

= Z

vect.

(~v,

λ)

l=1

(−γv

)

−|

λ|

x∈

. (2.18)

The vector contribution Z

vect.

(~v,

λ) will be deﬁned in the section 2.5 below, it is expressed

in terms of the Nekrasov factor (2.33) as a result of localization. As shown in [11, 40], the

Nekrasov factor obeys a set of discrete Ward identities. Consequently, the coeﬃcients a

also obey similar identities. They can be written in terms of the function Y

(z) as

λ+x

1 − q

(1 − q

)(1 − q

)

−m

Res

z=χ

(z)Y

(zq

−1

)

λ−x

= −

1 − q

(1 − q

)(1 − q

)

−m

m−2

Res

z=χ

(z)Y

(zq

−1

(2.19)

Due to the change of states normalization performed in appendix A, and since the original states were

orthonormal, the coeﬃcient a

is expected to be proportional to N (

λ)

−2

. The additional factors are chosen

to simplify the formulation of the AFS lemmas below.

– 8 –

JHEP11(2017)034

By deﬁnition, the vacuum state |∅i

(1,n)

is annihilated by the positive modes a

, and

(z)|∅i

(1,n)

= |∅i

(1,n)

. The dual vacuum state

(1,n)

h∅| is annihilated by negative modes,

and

(1,n)

h∅|ϕ

−

(z) =

(1,n)

h∅|. The normal ordering, denoted : ··· :, corresponds to write

all the positive modes on the right, and all the negative modes on the left. Correlators of

operators O

) acting in the Fock space are deﬁned as the vacuum expectation values

) ···O

)i =

(1,n)

h∅|O

) ···O

)|∅i

(1,n)

, (2.27)

with the radial ordering |z

| > |z

| > ··· > |z

2.5 Reminder on 5d N = 1 instanton partition functions

The quiver Super Yang-Mills gauge theories with N = 1 in 5d reduced on S

are charac-

terized by a simply laced Dynkin diagram Γ. Each node i ∈ Γ is associated to a vector

multiplet with gauge group U(m

), and an exponentiated gauge coupling q

. To each edge

< ij >∈ Γ corresponds a bifundamental matter multiplet of mass µ

that transforms un-

der the gauge group U(m

) ×U(m

). In addition, a Chern-Simons term of level κ

coupled

to the gauge group U(m

) can be introduced at each node i. Thus, each node bears two

integer labels (m

, κ

) with m

> 0 that will later be related to the levels (l

, l

) of DIM

representations. Extra matter ﬁelds in the fundamental/antifundamental representation of

the gauge group U(m

) can also be attached to each node, and the corresponding masses

will be denoted µ

(f)

i,j

with j = 1 ···f

, and µ

(af)

i,j

with j = 1 ···

respectively.

The expression of the instanton contribution to the (K-theoretic) partition function

reﬂects the particle content of the theory. It is written as a sum over m

-tuple Young

diagrams

, and each term is factorized into vector, Chern-Simons, (anti)fundamental

and bifundamental contributions:

inst.

[Γ] =

{

}

i∈Γ

vect.

(~v

(κ

fund.

(~µ

(f)

a.f.

(~µ

(af)

)

<ij>∈Γ

bfd.

(~v

, ~v

|µ

), (2.28)

where the (exponentiated) Coulomb branch vevs ~v

are related to the vacuum expectation

value of the scalar ﬁeld in the gauge multiplets. From this expression, it is possible to

deﬁne a normalized trace of functions depending on the realization of the set of (tuple)

Young diagrams {

} as follows,

F [{

}]

gauge

inst.

[Γ]

{

}

F [{

}]

i∈Γ

vect.

(~v

(κ

fund.

(~µ

(f)

)

× Z

a.f.

(~µ

(af)

)

<ij>∈Γ

bfd.

(~v

, ~v

|µ

). (2.29)

This operation will be very useful in order to express the qq-characters of the gauge theory.

– 10 –

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