S对偶与DIM代数：纤维基对偶的代数视角

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"这篇学术文章探讨了代数观点下的基于光纤的对偶，特别是如何应用于5D N=1规范理论和IIB型弦理论中的(p, q)骨架网。S对偶在此理论中的作用是交换NS5和D5粗线，并将(p, q)粗线映射到(-q, p)粗线，从而在5D规范理论之间产生对偶关系。文章还提到了这些理论可以通过Calabi-Yau流形上拓扑字符串的压缩得到，S对偶在这种情况下表现为基于纤维的对偶。作者进一步引入了Ding-Iohara-Miki (DIM)代数，利用其代数对象来构建5D规范理论，特别是通过操作符T $$ \mathcal{T} $$的真空期望值来计算实例分割函数。T $$ \mathcal{T} $$-运算符与DIM代数的共代数结构密切相关，而Miki的自同构被证明可用于实现S对偶的扭曲。该研究发表于JHEP03(2019)003，并由Springer为SISSA出版。" 这篇论文深入研究了弦理论和5维量子场论中的对称性与对偶性，特别是在代数框架下的理解和应用。首先，5D N=1规范理论是描述IIB型弦理论中(p, q)骨架网低能动态的有效工具。S对偶是一种重要的对称性，它不仅交换了NS5和D5粗线，还在5D规范理论之间建立了对偶关系，扩大了理论的可探索范围。其次，理论的另一种构建方法是通过Calabi-Yau流形上的拓扑字符串理论的压缩。在这一背景下，S对偶表现为一种基于纤维的对偶性，这为理解和研究这些理论提供了新的视角。最近，DIM代数在构建5D规范理论中的作用引起了关注。DIM代数的代数对象，特别是互缠子和T $$ \mathcal{T} $$-运算符，被用于计算与拓扑顶点相关的实例分割函数。这种代数方法揭示了S对偶的内在结构，并且通过Miki的自同构，可以将S对偶的效应体现在代数结构的扭曲上。这篇论文通过代数手段深入剖析了S对偶在5D规范理论中的实现，以及它如何在不同的理论框架下起作用，这为理解和统一不同对偶关系提供了新的工具和见解。这项工作对于深化我们对弦理论和量子场论的理解具有重要意义。

JHEP03(2019)003

We can deﬁne in a similar way a twisted coproduct ∆

for any power S

of Miki’s

automorphism. The action of S

on the Drinfeld currents (2.1) takes a simple form:

· x

(z) = −x

∓

−1

), S

· ψ

(z) = ψ

∓

−1

), S

· (c, ¯c) = (−c, −¯c), S

· a

= −a

−k

(2.19)

As a result, it is possible to write down explicitly the action of the twisted coproduct

∆

on these generators. Remarkably, this twisted coproduct coincides with the opposite

coproduct ∆

involved in the deﬁnition of the universal R-matrix. This leads to interpret

the automorphism S as a sort of square root of the R-matrix. This result should have

tremendous consequences for integrable systems built upon DIM algebra [48–50]. We hope

to come back to this important issue in a future publication.

2.3.2 Twist by a two-tensor

The so-called Drinfeld twist provides another way to deform a coalgebraic structure [47, 51].

In this case, the coproduct is twisted by an invertible two-tensor F that should satisfy the

following properties:

(∆ ⊗ 1)F = F

(1 ⊗ ∆)F, (ε ⊗ 1)F = (1 ⊗ ε)F = 1. (2.20)

Here we used the standard notation F

for the action of F in ith and jth tensor space

and ε denotes the co-unit. The twisted coproduct F∆F

−1

deﬁnes a new quasi-triangular

Hopf algebra with universal R-matrix F

∗

−1

, F

∗

= F

denoting the permutation of the

two-tensor, i.e. PFP. Although a general statement does not seem to exist, it has been

observed in some cases that the twisting by an automorphism can also be realized as a

two-tensor twist. For instance, this is the case for Lusztig’s automorphism of U

(g) where

g is a simple, ﬁnite-dimensional, Lie algebra [47]. This is also the case for the twist by the

automorphism S

, since ∆

identiﬁes with the opposite coproduct, and the corresponding

two-tensor F

is simply the universal R-matrix R. In fact, the existence of the two tensor

has been prooved in [49] (lemma A.5) in the restricted case c = 0.

In this paper, it will be convenient, although not essential, to assume that the

S-twisted coproduct ∆

deﬁned in (2.18) can also be obtained by a Drinfeld twist with the

two-tensor F

∆

= F

∆F

−1

, R

= F

∗

−1

, (2.21)

where F

∗

= F

S21

is the permutation of F

. This assumption will provide us a general

intuition on the transformation of algebraic objects. However, strictly speaking, we do not

need the existence of the universal object F

, but only its realization in speciﬁc represen-

tations. The latter will be constructed explicitly below, up to a normalization factor, using

a product of intertwiners.

Several interesting properties follow from our assumption (2.21). First, the opposite

coproduct ∆

can be twisted in the same manner,

∆

(e) = (S

−1

⊗ S

−1

)∆

(S · e) = F

∗

∆

(e)F

∗−1

. (2.22)

Moreover, since S

= 1, we notice that ∆

= ∆

−1

, which implies the existence of

the two tensor F

−1

= F

∗

R associated with the twist by the automorphism S

−1

, and the

– 8 –

JHEP03(2019)003

corresponding twisted R-matrix is R

−1

= R

−1

. Similarly, the identiﬁcation ∆

−1

= ∆

provides the permuted relation F

∗

−1

= F

−1

. Finally, inverting these relations, it is

possible to express the universal R-matrices solely in terms of F

and F

−1

(and their

permutation), and re-interpret them as factors in a certain decomposition,

R = F

∗−1

−1

= F

∗−1

−1

, R

= F

∗

∗−1

−1

= F

−1

. (2.23)

2.4 Representations

In the algebraic engineering, vertical representations are associated to (multiple) D5-branes,

and horizontal representations to NS5-branes (possibly dressed by extra D5-branes). These

representations have already been presented in several papers, we reproduce them here for

consistency and follow the conventions employed in [21]. Then, we investigate the S-dual

representations and derive the expression of several matrices M

(`,

2.4.1 Vertical representations

The modules of vertical representations, denoted here (0, m)

were introduced in [52].

They are inﬁnite dimensional vector spaces with a basis of states |~v,

λii parameterized by

an m-tuple Young diagram

λ = (λ

(1)

, · · · , λ

(m)

), and on which the Drinfeld currents act

as follows:

(0,m)

(z)) |~v,

λii =

x∈A(

λ)

δ(z/χ

) Res

z=χ

(z)

|~v,

λ + xii,

(0,m)

−

(z)) |~v,

λii = γ

−m

x∈R(

λ)

δ(z/χ

) Res

z=χ

−1

z) |~v,

λ − xii,

(0,m)

(ψ

(z)) |~v,

λii = γ

−m



(z)



|~v,

λii.

(2.24)

These representations have levels (`,

`) = (0, m), they are lowest weight representations,

with the vacuum |~v, ∅ii corresponding to empty Young diagrams, and the lowest weights

given by the vector ~v = (v

, · · · , v

). They are associated to m parallel D5-branes, with

(exponentiated) positions v

, · · · , v

along the axis ‘5’ of the 56-plane. In the gauge the-

ory, the level m coincides with the rank of the gauge group U(m), and the weight with the

exponentiated Coulomb branch v.e.v. .

The modes of the currents x

(z) behave as instantons creation/annihilation operators,

and their action involves a summation over the sets of boxes A(

λ) (resp. R(

λ)) that can

be added to

λ (or removed from

λ). To each box x = (l, i, j) in

λ with coordinates (i, j) in

the lth Young diagram, has been associated the complex parameter χ

= v

i−1

j−1

∈ C.

The vertical action (2.24) has been written using the functions

(z) =

−1

(z)

, Y

(z) =

x∈A(

λ)

1 − χ

x∈R(

λ)

1 − χ

/(q

. (2.25)

– 9 –

JHEP03(2019)003

These two functions possess an alternative expression as a product over the box content of

each Young diagram:

(z) =

l=1

1 − q

1 − v

x∈

g(z/χ

), Y

(z) =

l=1



1 −



x∈

S(χ

/z),

S(z) =

(1 − q

z)(1 − q

(1 − z)(1 − q

(2.26)

The function S(z) is related to the function g(z) deﬁned in (2.3), and involved in the

deﬁnition of the algebra, through the relation g(z) = S(z)/S(z

−1

). It obeys the functional

identity S(q

z) = S(z

−1

By expanding the functions Ψ

(z) both around z = ∞ and z = 0, it is possible to

deduce the representation of the modes a

describing the Cartan sector:

(0,m)

)|~v,

λii = c





x∈

−

(1 − q

)(1 − q

)

l=1





|~v,

λii, k ∈ Z \ {0}. (2.27)

This expression involves the coeﬃcients c

deﬁned in (2.7) from the two expansions of the

function g(z).

It is also necessary to introduce a dual basis hh~v,

λ| for the vertical modules, which is

orthogonal (but not orthonormal) to the basis of states |~v,

λii:

hh~v,

λ|~v,

ii = δ

λ,

−1

, a

= Z

vect.

(~v,

λ)

l=1

(−γv

)

−|

λ|

x∈

. (2.28)

The scalar product involves the quantity Z

vect.

(~v,

λ) that coincides with the vector multiplet

contribution to the Nekrasov instanton partition function. A brief reminder on the various

building blocks for these partition functions can be found in appendix B. Note that in

the case of m = 1, the (inverse) norms a

are in fact independent of the weight v of the

representation. The non-trivial norm is introduced here in order to simplify the expression

of the contragredient representation ˆρ that acts on the dual basis in such a way that



hh~v,

λ|ˆρ

(0,m)

(e)



|~v,

ii = hh~v,

λ|



(0,m)

(e)|~v,



, ∀e ∈ A

DIM

. (2.29)

With our choice of normalization, the action of ˆρ reads just like (2.24), but with x

(z)

replaced by −x

∓

(z).

Vertical representations of levels (`,

`) = (0, −m) can be deﬁned using the automor-

phism S

. The corresponding modules are isomorphic to (0, m)

, and the action of the

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