Quasi-Hopf twist and elliptic Nekrasov factor

Panupong<EMAIL_ADDRESS>and
Hiroaki<EMAIL_ADDRESS>

aGraduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan
bKMI, Nagoya University, Nagoya, 464-8602, Japan

> Abstract. We investigate the quasi-Hopf twist of the quantum toroidal
> algebra of $\mathfrak{gl}_{1}$ as an elliptic deformation. Under the quasi-
> Hopf twist the underlying algebra remains the same, but the coproduct is
> deformed, where the twist parameter $p$ is identified as the elliptic
> modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the
> relation to the elliptic lift of the Nekrasov factor for instanton counting
> of the quiver gauge theories on $\mathbb{R}^{4}\times T^{2}$. The same $R$
> matrix also appears in the commutation relation of the intertwiners, which
> implies an elliptic quantum KZ equation for the trace of intertwiners. We
> also show that it allows a solution which is factorized into the elliptic
> Nekrasov factors and the triple elliptic gamma function.

## § 1 Introduction

The quantum toroidal algebra of type $\mathfrak{gl}_{1}$ (a.k.a. Ding-Iohara-
Miki algebra [15], [46], the elliptic Hall algebra [11], [65], [66] and
others) is the fundamental symmetry which controls five dimensional (or $K$
theoretic lift of) Nekrasov partition function [54], [56], [51], [52]. Its
manifestation is the celebrated AGT relation [1], [69], [48], [10] to the
conformal block of deformed Virasoro and $W$ algebras. The fact that the
quantum toroidal algebra of type $\mathfrak{gl}_{1}$ has several names shows
its ubiquity and broad applications to many areas in mathematics and physics.
From the viewpoint of representation theory one of the advantages of the
quantum toroidal algebra is that it has a coproduct which allows us to take
the tensor product of the representations. In fact representations of the
deformed Virasoro and $W$ algebras by free bosons are derived from the tensor
product of the Fock representations of the quantum toroidal algebra [20].

On the top of the hierarchy of supersymmetric gauge theories without coupling
to the gravity is the six dimensional theory. Recall that the BPS state
counting of five dimensional theories on $\mathbb{R}^{4}\times S^{1}$ is
naturally related to the supersymmetric quantum mechanics on the instanton
moduli space, where $S^{1}$ is identified with the (periodic) time direction.
In the same manner the partition function of BPS state counting of the six
dimensional theories on $\mathbb{R}^{4}\times T^{2}$ can be identified with
the elliptic genus of the instanton moduli space [32]. On the algebraic side
we thus expect the appearance of the elliptic algebra and elliptic integrable
systems [57]. In fact an elliptic lift of Ding-Iohara-Miki (DIM) algebra
together with its connection to the six dimensional Nekrasov function and the
elliptic Virasoro algebra has been already discussed in [58], [34].

Most of the existing literatures employ a version of elliptic DIM algebra that
was first introduced by Y. Saito [64] for the purpose of describing an
elliptic version of Macdonald polynomials. This kind of elliptic algebra
introduces a second set of deformed bosons, which is the same as the Clavelli-
Shapiro method in old string theory [14] . The method allows us to rewrite the
trace of the product of vertex operators (intertwiners) that appears in the
one-loop diagrams in string theory by the vacuum expectation value with
respect to the Fock vacuum of doubled boson system [39], [29]. This also
reminds us of the method of thermo field dynamics in statistical mechanics
[67], [35], [61]. However, there is another construction of an elliptic lift
of DIM algebra, which is relatively unexplored [19]333However, see [41] and
[42].. This construction does not employ an additional boson, but makes use of
the quasi-Hopf twist [36], [37]. In this paper we investigate the quasi-Hopf
twist of DIM algebra. The original DIM algebra has two parameters $(q,t)$ and
the quasi-Hopf twist introduces a third parameter $p$. As we have emphasized
the quantum toroidal algebra has a coproduct. The quasi-Hopf twist deforms the
coproduct by what is called twistor $\mathcal{F}(p)$. The generating currents
of the algebra are also twisted accordingly. It turns out that the deformation
parameter $p$ is identified with an elliptic parameter. In the commutation
relations of the twisted currents there appears the theta function whose
elliptic norm is $p$.

As a quantum group the quantum toroidal algebra has an universal $R$ matrix
[21]. Since the quasi-Hopf twist deforms the coproduct, it also changes the
universal $R$ matrix. As has been shown in [6], [7], [8], the Cartan part of
the universal $R$ matrix is closely related to the Nekrasov factor through the
generalized Knizhnik-Zamolodchikov (KZ) equation for the correlation function
of intertwiners of DIM algebra. Just like the Wick theorem for the free
fields, the solutions to the generalized KZ equation are factorized into a
product of two point functions. In accord with the AGT correspondence the
Nekrasov factor plays the role of the two point function [7], [13]. The
elliptic Nekrasov factor is given by [58], [70], [5];

$\displaystyle\mathcal{N}_{\lambda\mu}(u|q,t,p)$ $\displaystyle=$
$\displaystyle\prod_{i,j=1}^{\infty}\frac{\Gamma(uq^{\lambda_{j}-\mu_{i}}t^{i-j};q,p)}{\Gamma(ut^{i-j};q,p)}\cdot\frac{\Gamma(ut^{i-j+1};q,p)}{\Gamma(uq^{\lambda_{j}-\mu_{i}}t^{i-j+1};q,p)}$
(1.1) $\displaystyle=$
$\displaystyle\prod_{\square\in\lambda}\theta_{p}(uq^{a_{\lambda}(\square)}t^{\ell_{\mu}(\square)+1})\prod_{\square\in\mu}\theta_{p}(uq^{-a_{\mu}(\square)-1}t^{-\ell_{\lambda}(\square)}),$

where $(\lambda,\mu)$ is a pair of partitions and
$(q,t)=(e^{\epsilon_{1}},e^{-\epsilon_{2}})$ is the $\Omega$ background of
Nekrasov [54]. $\Gamma(u;q,p)$ is the elliptic gamma function. In this paper
we will show that the universal $R$ matrix after the quasi-Hopf twist is
related to the elliptic Nekrasov factor by the relation;

$\mathfrak{q}^{|\lambda|+|\mu|}\mathcal{N}_{\lambda\mu}(\mathfrak{q}^{-2}z|q,t,p)=\overline{R}_{\lambda\mu}(z;p)\mathcal{N}_{\lambda\mu}(z|q,t,p),$
(1.2)

where $\mathfrak{q}=\sqrt{t/q}$ and $\overline{R}_{\mu\lambda}$ is the
normalized $R$ matrix of the vertical Fock representation. With appropriate
specialization of the spectral parameter $u$, the elliptic Nekrasov factor
gives the contribution of the bifundamental matter hypermultiplet to the
instanton partition function of the lift of $\mathcal{N}=2$ quiver gauge
theory to $\mathbb{R}^{4}\times T^{2}$, where the elliptic modulus of the two
dimensional torus $T^{2}$ is identified with $p$. Hence, contrary to the case
of [58], [34], we do not have to introduce an additional boson (Heisenberg
algebra) to obtain the elliptic Nekrasov factor (1.1). Only the quasi-Hopf
twist suffices. This is one of the main messages of the present paper. The
same normalized $R$ matrix also appears in the commutation relations of the
intertwiners $\Psi_{\lambda}(v;p)$ and the dual intertwiners
$\Psi^{*}_{\lambda}(v;p)$;

$\displaystyle\Psi_{\lambda}(v;p)\Psi_{\mu}(w;p)$
$\displaystyle=\frac{G_{2}(\frac{w}{v};p_{*},q,t^{-1})}{G_{2}(\mathfrak{q}^{-2}\frac{v}{w};p_{*},q,t^{-1})}\cdot\overline{R}_{\lambda\mu}\Big{(}\frac{v}{w};p_{*}\Big{)}\Psi_{\mu}(w;p)\Psi_{\lambda}(v;p),$
(1.3) $\displaystyle\Psi^{*}_{\lambda}(v;p)\Psi^{*}_{\mu}(w;p)$
$\displaystyle=\frac{G_{2}(\frac{w}{v};p,q,t^{-1})}{G_{2}(\frac{v}{w};p,q,t^{-1})}\cdot\overline{R}_{\lambda\mu}\Big{(}\frac{v}{w};p\Big{)}^{-1}\Psi^{*}_{\mu}(w;p)\Psi^{*}_{\lambda}(v;p),$
(1.4)

where $p_{*}=p\mathfrak{q}^{-2}$ and $G_{2}(u;p,q,t^{-1})$ is the double
elliptic gamma function. The ratio of the double elliptic gamma functions,
which is independent of $\lambda$ and $\mu$, comes from the vacuum
contribution. Note that the elliptic parameter in $G_{2}$ and
$\overline{R}_{\lambda\mu}$ for the commutation relation of
$\Psi_{\lambda}(v;p)$ is not $p$, but $p_{*}$.

Based on the braiding relations (1.3), (1.4) and the cyclic property of the
trace, we can derive a difference equation for the trace of the product of
intertwiners,

$\mathrm{Tr}\left[\widetilde{Q}^{d_{1}}Q^{d_{2}}\Psi^{*}_{\mu_{1}}(w_{1};p)\cdots\Psi^{*}_{\mu_{n}}(w_{n};p)\Psi_{\lambda_{1}}(z_{1};p)\cdots\Psi_{\lambda_{n}}(z_{n};p)\right],$
(1.5)

where $(d_{1},d_{2})$ is a pair of grading operaters of DIM algebra. The shift
parameter of the difference equation is $Q$ and the $Q$-shift produces a
product of the $R$ matrices $\overline{R}_{\lambda\mu}$ (see section 6 for
explicit forms). Thus, we can regard the $Q$-difference equation as a
generalization of q-KZB equation for a genus one conformal blocks [25], [23],
[22]. As in the case of [7], [13], there is a solution whose building blocks
are the Nekrasov factors. A typical example of such building blocks looks like

$\displaystyle G_{3}(z;p,Q,q,t^{-1})\cdot G_{3}(Qz^{-1};p,Q,q,t^{-1})$
$\displaystyle\quad\prod_{k=0}^{\infty}\mathcal{N}_{\lambda\mu}(Q^{k}z|q,t^{-1},p)\mathcal{N}_{\mu\lambda}(Q^{k+1}z^{-1}|q,t^{-1},p),$
(1.6)

where the triple elliptic gamma function $G_{3}$ represents the “vacuum”
contribution. In particular, when all the partitions are trivial in (1.5), the
Nekrasov factors become also trivial and only $G_{3}$ factors, which are
completely symmetric in four parameters $(p,Q,q,t^{-1})$, survive.

The fact that we keep the underlying algebra and only twist the coproduct
seems to have the following advantage. The DIM algebra has an
$SL(2,\mathbb{Z})$ automorphism, called Miki automorphism [46]. This is an
automorphism of the associative algebra, but not of the bialgebra. Hence, the
Miki automorphism survives after the quasi-Hopf twist, though it is not
clearly seen in terms of the twisted (elliptic) currents. On the other hand if
we introduce an additional boson as in [64], the existence of the Miki
automorphism is not clear at all. Incidentally, we are also led to the
following question; Since the Miki automorphism is not an automorphism of the
bialgebra, it deforms the coproduct structure. Hence, one can ask if the
change of the coproduct by the automorphism is described by a twisting of the
coproduct by an appropriate twistor.

One of the interesting aspects of the quasi-Hopf twist of DIM algebra is the
emergence of $SU(4)$ equivariant parameters. We have seen there appear two
kinds of parameters $p$ and $p_{*}$ in the commutation relations (1.3) and
(1.4). More basically, as we will see in the next section, in addition to the
theta functions with the elliptic modulus $p$, the exchange relations of the
twisted currents involve those with the elliptic modulus $p_{*}=pC^{-2}$,
where $C$ is one of the central charges of DIM algebra. This is in sharp
contrast with the elliptic DIM algebra defined in [64], where only the theta
function with parameter $p$ appears. Recall that the original DIM algebra has
parameters $(q_{1},q_{2},q_{3})$ with $q_{1}q_{2}q_{3}=1$. The standard Fock
representation in terms of a free boson has the central charge
$C=q_{3}^{1/2}$. After the quasi-Hopf twist with the twist parameter $p$, the
Fock representation has $p_{*}=pq_{3}^{-1}=pq_{1}q_{2}$. Hence defining
$q_{3}=p$ and $q_{4}=p_{*}^{-1}$, we obtain $SU(4)$ parameters with
$q_{1}q_{2}q_{3}q_{4}=1$. We should not forget that these parameters are not
associated with the algebra itself, but only arise in its Fock representation.
However, the emergence of the $SU(4)$ parameters is quite suggestive. It is
tempting to regard them as the equivariant parameters (or the $\Omega$
background) of the torus action on $\mathbb{C}^{4}$, which plays the role of
the ambient space of the spiked instanton (or the gauge origami) proposed by
Nekrasov [55]. It was introduced to provide a physical definition of the
$qq$-character of the $\mathcal{N}=2$ quiver gauge theories in terms of the
brane configuration in type IIB string theory.

#### Organization of material

The materials of this paper are structured as follows: in section 2 we provide
the definition of the quasi-Hopf twist of DIM algebra and the formula of its
coproduct in terms of the twisted currents. Since we keep the underlying
associative algebra, representations of the original DIM algebra also work as
representations after the quasi-Hopf twist. We give corresponding
representations in terms of the twisted currents in section 3. On the other
hand, since the coproduct is deformed, the intertwiners will change. In
section 4 after the intertwiner and the dual intertwiner are defined using the
coproduct, we express them explicitly as operators on the Fock space of free
bosons. We also provide formulas of the zero mode factor, which plays an
important role in deriving their commutation relations. The quasi-Hopf twist
of the $R$-matrix is derived from the universal $R$-matrix of the DIM algebra
in section 5. We double-check the computation by confirming that it agrees
with the coefficient which results from interchanging the elliptic Fock
intertwiners themselves, and dual elliptic Fock intertwiners themselves. We
also check the unitarity of the $R$ matrix and show a remarkable relation to
the elliptic Nekrasov factor. In section 6 a difference equation for the trace
of intertwiners and dual intertwiners is derived from the cyclic property of
the trace and the commutation relations among the intertwiners. Some of
technical details and auxiliary contents are relegated to Appendices.

### Elliptic functions

The complete odd theta function is defined by

$\vartheta_{1}(z;p):=\sqrt{-1}\sum_{n\in\mathbb{Z}}(-1)^{n}p^{\frac{1}{2}\left(n-\frac{1}{2}\right)^{2}}z^{n-\frac{1}{2}}.$
(1.7)

By the Jacobi triple product formula

$\sum_{m\in\mathbb{Z}}p^{\frac{m^{2}}{2}}z^{m}=\prod_{n=1}^{\infty}(1-p^{n})(1+p^{n-\frac{1}{2}}z)(1+p^{n-\frac{1}{2}}z^{-1}),$
(1.8)

we see

$\sqrt{-1}(p;p)_{\infty}\cdot\theta_{p}(z)=p^{-\frac{1}{8}}z^{\frac{1}{2}}\vartheta_{1}(z;p),$
(1.9)

where we have defined a “short” theta function by

$\theta_{p}(z):=(z;p)_{\infty}(pz^{-1};p)_{\infty}=\exp\left(-\sum_{n\neq
0}\frac{z^{n}}{n(1-p^{n})}\right).$ (1.10)

The “short” theta function enjoys the quasi-periodicity;

$\theta_{p}(p^{n}z)=(-z)^{-n}p^{-\frac{1}{2}n(n-1)}\theta_{p}(z),$ (1.11)

and the inversion formula;

$\theta_{p}(z)=(-z)\theta_{p}(z^{-1}).$ (1.12)

We also use the elliptic gamma function;

$\Gamma(z;q,p):=\frac{(qpz^{-1};q,p)_{\infty}}{(z;q,p)_{\infty}}=\exp\left(\sum_{n\neq
0}\frac{z^{n}}{n(1-q^{n})(1-p^{n})}\right).$ (1.13)

The elliptic gamma function is symmetric in $q$ and $p$. It satisfies the
$q$-difference equation;

$\Gamma(qz;q,p)=\theta_{p}(z)\Gamma(z;q,p).$ (1.14)

In general we can define an elliptic deformation of the multiple
$q$-Pochhammer symbol by

$\displaystyle(u;q_{1},q_{2},\cdots,q_{n})_{\infty}$
$\displaystyle:=\exp\left(-\sum_{k=1}^{\infty}\frac{u^{k}}{k(1-q_{1}^{k})(1-q_{2}^{k})\cdots(1-q_{n}^{k})}\right)$
$\displaystyle\longrightarrow(u;q_{1},\cdots,q_{n},p)_{\infty}\cdot(q_{1}\cdots
q_{n}pu^{-1};q_{1},\cdots,q_{n},p)_{\infty}^{(-1)^{n}}.$ (1.15)

In the literature [59] a multi-parameter generalization of the elliptic gamma
function is defined by

$G_{n}(u;q_{0},\cdots,q_{n}):=(u;q_{0},\cdots,q_{r})_{\infty}^{(-1)^{n}}\cdot(q_{0}\cdots
q_{n}u^{-1};q_{0},\cdots q_{r})_{\infty},$ (1.16)

such that
$G_{0}(u;q)=\theta_{q}(u),G_{1}(u;q_{0},q_{1})=\Gamma(u;q_{0},q_{1})$. Thus,
the multiple elliptic gamma function
$G_{n}(u;p,q_{1},\cdots,q_{n})^{(-1)^{n}}$ provides the elliptic lift of of
the multiple $q$-Pochhammer symbol (1). They satisfy the recursion relation

$G_{n}(q_{k}u;q_{0},\cdots,q_{n})=G_{n-1}(u;q_{0},\cdots,\stackrel{{\scriptstyle
k}}{{\vee}},\cdots,q_{n-1})\cdot G_{n}(u;q_{0},\cdots,q_{n}).$ (1.17)

The function $G_{2}(u;q_{0},q_{1},q_{2})$ is also called double elliptic gamma
function.

## § 2 Elliptic algebra from quasi-Hopf twist

### 2.1 Quasi-Hopf twist of Ding-Iohara-Miki algebra

Let us begin with a review of the quantum toroidal algebra of type
$\mathfrak{gl}_{1}$, which we call Ding-Iohara-Miki (DIM) algebra in the
present paper. The DIM algebra has the parameters $(q_{1},q_{2},q_{3})$ with
$q_{1}q_{2}q_{3}=1$ and enjoys the triality of the permutation of $q_{i}$. We
assume they are generic in the sense that for any $a,b,c\in\mathbb{Z}$,

$\displaystyle q^{a}_{1}q^{b}_{2}q^{c}_{3}=1\implies a=b=c.$ (2.1)

We use the notation

$\displaystyle\kappa_{n}$
$\displaystyle:=\prod_{i=1}^{3}(q_{i}^{\frac{n}{2}}-q_{i}^{-\frac{n}{2}})=\prod_{i=1}^{3}(q_{i}^{n}-1)=\prod_{i=1}^{3}(1-q_{i}^{-n})=\sum_{i=1}^{3}(q_{i}^{n}-q_{i}^{-n}),$
(2.2)

which satisfies $\kappa_{-n}=-\kappa_{n}$. By convention we often take
$q=q_{1}$ and $t=q_{2}^{-1}$ as independent parameters. It is convenient to
introduce the notation $\mathfrak{q}:=q_{3}^{\frac{1}{2}}=\sqrt{t/q}$ as the
parameter of quantum deformation.

We define the DIM algebra
$\mathcal{U}:=U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1})$ to be the
associative algebra with the generators
$E_{k},F_{k},K_{0}^{\pm},H_{r}\,\,(k\in\mathbb{Z},r\in\mathbb{Z}\backslash\\{0\\})$
and $C$. It is convenient to introduce the generating functions (currents);

$\displaystyle E(z)=\sum_{k\in\mathbb{Z}}E_{k}z^{-k},\quad
F(z)=\sum_{k\in\mathbb{Z}}F_{k}z^{-k},\quad
K^{\pm}(z)=K_{0}^{\pm}\exp\left(\pm\sum_{r=1}^{\infty}H_{\pm r}z^{\mp
r}\right).$ (2.3)

There are several conventions of the Cartan currents $K^{\pm}(z)$. The
original convention is $K^{\pm}(z)=K^{\pm,\mathrm{here}}(C^{\frac{1}{2}}z)$.
The advantage of our convention is that we can eliminate $C^{\frac{1}{2}}$
from the defining relations of the algebra. Some literatures use the
convention $K^{\pm}(z)=K^{\pm,\mathrm{here}}(Cz)$.

The DIM algebra has two-dimensional center spanned by $(C,K_{0}^{\pm})$. Note
that $K_{0}^{+}$ is the inverse of $K_{0}^{-}$ by definition. We will not
write down the defining relations among the currents, since they can be
recovered from the relations after the quasi-Hopf twist by putting the
deformation (elliptic) parameter $p=0$. But we only quote the commutation
relation

$\displaystyle[H_{r},H_{s}]$
$\displaystyle=\delta_{r+s,0}\frac{\kappa_{r}}{r}(C^{r}-C^{-r}),$ (2.4)

since it determines the normalization of the Cartan generators. Actually there
are also the Serre’s relations in the defining relations, which we do not
write down explicitly, since they are not used in this paper.

To define the quasi-Hopf twist of the DIM algebra, let us introduce the
operators $b_{\pm n}$ defined via

$\displaystyle
K^{+}(z)=K^{+}_{0}\exp\bigg{(}\sum_{n=1}^{\infty}b_{n}C^{n}z^{-n}\bigg{)},\quad
K^{-}(z)=K^{-}_{0}\exp\bigg{(}-\sum_{n=1}^{\infty}b_{-n}z^{n}\bigg{)}.$ (2.5)

The coproduct of $b_{\pm n}$ is given by

$\displaystyle\Delta(b_{\pm n})=b_{\pm n}\otimes C_{2}^{-n}+1\otimes b_{\pm
n}.$ (2.6)

Then in term of the twistor

$\displaystyle\mathcal{F}(p)=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{np^{n}C^{-n}_{2}}{\kappa_{n}(1-p^{n}C^{-2n}_{2})}b_{n}\otimes
b_{-n}\bigg{)}\in\mathcal{U}\otimes\mathcal{U},\qquad C_{2}:=1\otimes C,$
(2.7)

we define the twisted coproduct by [19]

$\Delta_{p}(a)=\mathcal{F}(p)\Delta(a)\mathcal{F}(p)^{-1},\qquad
a\in\mathcal{U}.$ (2.8)

Note that the twistor $\mathcal{F}(p)$ is invertible444$\epsilon$ denotes the
counit, which is non-vanishing only on the central elements.;
$(\epsilon\otimes\mathrm{id})\mathcal{F}(p)=(\mathrm{id}\otimes\epsilon)\mathcal{F}(p)=1$.
By (2.6) one can check that it satisfies the shifted cocycle condition [19];

$\displaystyle\mathcal{F}^{(23)}(p)(\mathrm{id}\otimes\Delta)\mathcal{F}(p)=\mathcal{F}^{(12)}(pC_{3}^{-2})(\Delta\otimes\mathrm{id})\mathcal{F}(p)$
(2.9)

on $\mathcal{U}^{\otimes 3}$. We would like to emphasize that by the quasi-
Hopf twist the underlying algebra $\mathcal{U}$ itself remains the same. But
the coproduct has been deformed and hence the definition of tensor product
representations will change. Originally the DIM algebra $\mathcal{U}$ is a
(quasi-triangular) Hopf algebra. But due to the deformation of the coproduct
it is no longer true and it becomes a quasi-Hopf algebra (hence the name
“quasi-Hopf twist”) with the Drinfeld associator;

$\displaystyle\Phi^{(D)}=\mathcal{F}^{(23)}(p)(\mathrm{id}\otimes\Delta)\mathcal{F}(p)\cdot(\mathcal{F}^{(12)}(p)(\Delta\otimes\mathrm{id})\mathcal{F}(p))^{-1}.$
(2.10)

For a quasi-Hopf algebra the coassociativity is modified by $\Phi^{(D)}$555For
a quantum group the cocommutativity is broken, but the universal $R$ matrix
compensates it.. When $\Phi^{(D)}=1$ the coassociativity holds and it is a
Hopf algebra. Note that if there was no shift in the cocycle condition (2.9),
we had $\Phi^{(D)}=1$. Hence the shift of parameters in (2.9) causes the
violation of the coassociativity. On the other hand in the case of the
elliptic DIM algebra introduced in [64], the algebra is extended by an extra
Heisenberg algebra, keeping the coproduct intact.

### 2.2 Coproduct and exchange relations among elliptic currents

The new coproduct $\Delta_{p}$ is neatly expressed in terms of the elliptic
currents to be defined shortly. Introducing the twisting currents by

$\displaystyle U^{+}(z;p)$
$\displaystyle=\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{p^{n}C^{-n}}{1-p^{n}C^{-2n}}b_{-n}z^{n}\bigg{)},$
(2.11) $\displaystyle U^{-}(z;p)$
$\displaystyle=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{p^{n}}{1-p^{n}}b_{n}z^{-n}\bigg{)},$
(2.12)

we define the elliptic generating currents by

$\displaystyle E(z;p)=U^{+}(z;p)E(z),\qquad F(z;p)=F(z)U^{-}(z;p),$ (2.13)
$\displaystyle K^{+}(z;p)=U^{+}(z;p)K^{+}(z)U^{-}(C^{-1}z;p),$ (2.14)
$\displaystyle K^{-}(z;p)=U^{+}(C^{-1}z;p)K^{-}(z)U^{-}(z;p).$ (2.15)

Explicitly the twisted Cartan currents are

$\displaystyle K^{+}(z;p)$
$\displaystyle=K^{+}_{0}\exp\left(-\sum_{n=1}^{\infty}\frac{p^{n}C^{-n}}{1-p^{n}C^{-2n}}b_{-n}z^{n}\right)\exp\left(\sum_{n=1}^{\infty}\frac{C^{n}}{1-p^{n}}b_{n}z^{-n}\right),$
(2.16) $\displaystyle K^{-}(z;p)$
$\displaystyle=K^{-}_{0}\exp\left(-\sum_{n=1}^{\infty}\frac{1}{1-p^{n}C^{-2n}}b_{-n}z^{n}\right)\exp\left(\sum_{n=1}^{\infty}\frac{p^{n}}{1-p^{n}}b_{n}z^{-n}\right).$
(2.17)

Hence, they are related by the scaling of the spectral parameters;

$K^{+}(p^{-1}Cz;p)=K^{+}_{0}(K^{-}_{0})^{-1}\cdot K^{-}(z;p).$ (2.18)

Note that this relation ceases to hold for $p=0$, since it involves $p^{-1}$.

In terms of the elliptic currents the coproduct is given by

$\displaystyle\Delta_{p}\big{(}E(z;p)\big{)}$ $\displaystyle=E(z;p_{*})\otimes
1+K^{-}(C_{1}z;p_{*})\otimes E(C_{1}z;p),$ (2.19)
$\displaystyle\Delta_{p}\big{(}F(z;p)\big{)}$
$\displaystyle=F(C_{2}z;p_{*})\otimes K^{+}(C_{2}z;p)+1\otimes F(z;p),$ (2.20)
$\displaystyle\Delta_{p}\big{(}K^{+}(z;p)\big{)}$
$\displaystyle=K^{+}(z;p_{*})\otimes K^{+}(C_{1}^{-1}z;p),$ (2.21)
$\displaystyle\Delta_{p}\big{(}K^{-}(z;p)\big{)}$
$\displaystyle=K^{-}(C_{2}^{-1}z;p_{*})\otimes K^{-}(z;p),$ (2.22)

where $C_{1}:=C\otimes 1,C_{2}:=1\otimes C$ and $p_{*}=pC_{2}^{-2}$. The
deformed coproduct $\Delta_{p}$ becomes complicated in terms of the original
currents. But the twisted currents makes it quite similar to the original
copruduct $\Delta$. In fact $\Delta_{p}$ takes the same form as $\Delta$
except the shift of the elliptic parameter $p\to p_{*}$ in the first factor of
the tensor product. Since the coproduct is a homomorphism;
$\Delta_{p}(ab)=\Delta_{p}(a)\Delta_{p}(b)$, we can define the tensor product
representation of $\rho_{1}$ and $\rho_{2}$ by
$(\rho_{1}\otimes\rho_{2})(a)=\rho_{1}\otimes\rho_{2}(\Delta_{p}(a))$. We
would like to give a remark that the above coproduct is not a coproduct in the
strict sense. The reason is that for each order $k\in\mathbb{Z}$ of $z^{k}$,
the expression of the coproduct contains an infinite summation of generators,
which is not well-defined in general.

Let us define an elliptic lift of the structure function of the DIM algebra
by666The second equality is due to the condition $q_{1}q_{2}q_{3}=1$.

$\displaystyle\mathcal{G}(x;p):=\frac{\theta_{p}(q_{1}^{-1}x)\theta_{p}(q_{2}^{-1}x)\theta_{p}(q_{3}^{-1}x)}{\theta_{p}(q_{1}x)\theta_{p}(q_{2}x)\theta_{p}(q_{3}x)}=\frac{\vartheta_{1}(q_{1}^{-1}x;p)\vartheta_{1}(q_{2}^{-1}x;p)\vartheta_{1}(q_{3}^{-1}x;p)}{\vartheta_{1}(q_{1}x;p)\vartheta_{1}(q_{2}x;p)\vartheta_{1}(q_{3}x;p)}.$
(2.23)

Using the relation (1.11) and the property $q_{1}q_{2}q_{3}=1$, we can check
that $\mathcal{G}(x;p)$ is periodic; $\mathcal{G}(px;p)=\mathcal{G}(x;p)$.
Similarly (1.12) implies $\mathcal{G}(x^{-1};p)=\mathcal{G}(x;p)^{-1}$. Then
the exchange relations of among elliptic currents can be stated as follows;

$\displaystyle
K^{\pm}(z;p)K^{\pm}(w;p)=\frac{\mathcal{G}(w/z;p_{*})}{\mathcal{G}(w/z;p)}K^{\pm}(w;p)K^{\pm}(z;p),$
(2.24) $\displaystyle
K^{+}(z;p)K^{-}(w;p)=\frac{\mathcal{G}(w/\mathfrak{q}z;p_{*})}{\mathcal{G}(\mathfrak{q}w/z;p)}K^{-}(w;p)K^{+}(z;p),$
(2.25) $\displaystyle
K^{+}(z;p)E(w;p)=\mathcal{G}(w/z;p_{*})E(w;p)K^{+}(z;p),$ (2.26)
$\displaystyle
K^{-}(\mathfrak{q}z;p)E(w;p)=\mathcal{G}(w/z;p_{*})E(w;p)K^{-}(\mathfrak{q}z;p),$
(2.27) $\displaystyle
K^{+}(\mathfrak{q}z;p)F(w;p)=\mathcal{G}(w/z;p)^{-1}F(w;p)K^{+}(\mathfrak{q}z;p),$
(2.28) $\displaystyle
K^{-}(z;p)F(w;p)=\mathcal{G}(w/z;p)^{-1}F(w;p)K^{-}(z;p),$ (2.29)
$\displaystyle E(z;p)E(w;p)=\mathcal{G}(w/z;p_{*})E(w;p)E(z;p),$ (2.30)
$\displaystyle F(z;p)F(w;p)=\mathcal{G}(w/z;p)^{-1}F(w;p)F(z;p),$ (2.31)
$\displaystyle[E(z;p),F(w;p)]=\tilde{g}\bigg{(}\delta\Big{(}\frac{Cw}{z}\Big{)}K^{+}(z;p)-\delta\Big{(}\frac{Cz}{w}\Big{)}K^{-}(w;p)\bigg{)},$
(2.32)

where the normalization factor $\tilde{g}$ of the commutation relation of
$E(z;p)$ and $F(z;p)$ does not change under the quasi-Hopf twist. Hence we can
keep the same normalization $\tilde{g}=\kappa_{1}^{-1}$ as [13]. On the other
hand, the elliptic DIM algebra of [64] choose a different normalization; the
factor in $\tilde{g}$ is lifted to the theta functions. Since we can change
$\tilde{g}$ by the rescaling of $E(z;p)$ and $F(z;p)$ without affecting other
exchange relations, the rescaling

$\displaystyle
E(z,p)\longrightarrow\frac{(1-q_{1})(q_{1}^{-1}p;p)_{\infty}(q_{2}^{-1};p)_{\infty}}{(p;p)_{\infty}(q_{3}p;p)_{\infty}}E(z,p),$
(2.33) $\displaystyle
F(z,p)\longrightarrow\frac{(1-q_{1}^{-1})(q_{1}p;p)_{\infty}(q_{2};p)_{\infty}}{(p;p)_{\infty}(q_{3}^{-1}p;p)_{\infty}}F(z,p).$
(2.34)

is allowed for the matching of the normalization. Namely, by the rescaling
(2.33) and (2.34), we have

$\frac{1}{\kappa_{1}}\longrightarrow\frac{(q_{1};p)_{\infty}(q_{1}^{-1}p;p)_{\infty}(q_{2};p)_{\infty}(q_{2}^{-1}p;p)_{\infty}}{(p;p)_{\infty}^{2}(q_{3}p;p)_{\infty}(q_{3}^{-1};p)_{\infty}}=\frac{\theta_{p}(q_{1})\theta_{p}(q_{2})}{(p;p)_{\infty}^{2}\theta_{p}(q_{1}q_{2})},$
(2.35)

which exactly matches with the coefficient of the commutation relation (2) in
[68]777The convention of the theta function in [68] is different from ours..

Since $\mathcal{G}(x;p)$ has infinitely many poles, it is mathematically
precise to write the exchange relation (2.30) in the following way;

$\displaystyle-\left(\frac{w}{z}\right)^{3}\theta_{p_{*}}(q_{1}^{-1}\frac{z}{w})\theta_{p_{*}}(q_{2}^{-1}\frac{z}{w})\theta_{p_{*}}(q_{3}^{-1}\frac{z}{w})\cdot
E(z;p)E(w;p)$
$\displaystyle=\theta_{p_{*}}(q_{1}^{-1}\frac{w}{z})\theta_{p_{*}}(q_{2}^{-1}\frac{w}{z})\theta_{p_{*}}(q_{3}^{-1}\frac{w}{z})\cdot
E(w;p)E(z;p).$ (2.36)

The same remark applies to other exchange relations.

The elliptic parameter appearing in the exchange relations involving $F(z;p)$
is $p$, while it is the shifted parameter $p_{*}$ for $E(z;p)$888Since the
power of $p_{*}$ appears frequently, we have changed the original notation
$p^{*}$ to $p_{*}$.. Note also that the relations (2.24) – (2.32) are
consistent with the scaling relation (2.18) of $K^{\pm}(z;p)$. In other words
the relations involving $K^{-}(z;p)$ follow from those of $K^{+}(z;p)$. When
$C=1$ and hence $p_{*}=p$, these exchange relations agree with those of the
elliptic DIM algebra introduced [64] up to the normalization factor
$\tilde{g}$ of the commutation relation $[E(z;p),F(w;p)]$. This in particular
implies that the vertical representations with $C=1$ of Saito’s elliptic
algebra are also the vertical representations of the quasi-Hopf twist of the
DIM algebra.

## § 3 Representations of the elliptic currents

Since the underlying algebra does not change as an associative algebra, the
representations of the original DIM algebra provide also those of the quasi-
Hopf twisted algebra as representations of the associative algebra. In
particular there are central elements $(C,K_{0}^{-})$ which are constant, if
the representation is irreducible. Under the quasi-Hopf twist these values do
not change. Since only integer powers of $\mathfrak{q}$ appear as the values
of the central elements in the present paper, we take the additive convention
and define a representation has level $(n,m)$, if
$(C,K_{0}^{-})=(\mathfrak{q}^{n},\mathfrak{q}^{m})$. On the other hand, the
tensor product representations will change, since the coproduct is twisted. As
we will see this leads to an issue on the construction of the vertical Fock
representation. In this section we will express known representations of the
original DIM algebra in terms of the elliptic currents. The advantage of using
the twisted currents is that the coproduct $\Delta_{p}$ takes a simple form.

### 3.1 Vector Representation

To obtain the vector representation of the elliptic currents, we first start
with the vector representation of the DIM algebra [18], and then perform the
twisting procedure. For each $v\in\mathbb{C}$ called spectral parameter, let
$V(v)$ be the vector space over $\mathbb{C}$ with a basis
$\\{[v]_{i}|~{}i\in\mathbb{Z}\\}$. Recall that in the vector representation of
the DIM algebra, we have

$K^{+}(z)[v]_{i}=\widetilde{\psi}(q_{1}^{i}v/z)[v]_{i},\qquad
K^{-}(z)[v]_{i}=\widetilde{\psi}(q_{1}^{-i-1}z/v)[v]_{i},$ (3.1)

where

$\widetilde{\psi}(z)=\exp\left(\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{z^{n}}{1-q_{1}^{-n}}\right).$
(3.2)

Recall also that we define the operators $b_{\pm n}$ by (2.5). Since the
vector representation has level $(0,0)$, we see

$b_{\pm n}[v]_{i}=\frac{\kappa_{n}}{n}\frac{1}{1-q_{1}^{\mp
n}}(q_{1}^{i}v)^{\pm n}[v]_{i}.$ (3.3)

Hence, the twisting currents are given by

$U^{+}(z;p)[v]_{i}=\prod_{k=1}^{\infty}\widetilde{\psi}(p^{k}q_{1}^{-i-1}z/v)[v]_{i},\qquad
U^{-}(z;p)[v]_{i}=\prod_{k=1}^{\infty}\widetilde{\psi}(p^{k}q_{1}^{i}v/z)[v]_{i}.$
(3.4)

It is straightforward to check that in terms of the elliptic currents the
vector representation $\rho_{v}^{V}$ is described by

$\displaystyle K^{+}(z;p)[v]_{i}$
$\displaystyle=\frac{\theta_{p}(q_{2}^{-1}q_{1}^{i}v/z)\theta_{p}(q_{3}^{-1}q_{1}^{i}v/z)}{\theta_{p}(q_{1}^{i}v/z)\theta_{p}(q_{1}^{i+1}v/z)}[v]_{i}$
(3.5) $\displaystyle K^{-}(z;p)[v]_{i}$
$\displaystyle=\frac{\theta_{p}(q_{3}q_{1}^{-i}z/v)\theta_{p}(q_{2}q_{1}^{-i}z/v)}{\theta_{p}(q_{1}^{-i-1}z/v)\theta_{p}(q_{1}^{-i}z/v)}[v]_{i},$
(3.6) $\displaystyle E(z;p)[v]_{i}$
$\displaystyle=\frac{(pq_{2};p)_{\infty}(pq_{3};p)_{\infty}}{(1-q_{1})(p;p)_{\infty}(pq_{1}^{-1};p)_{\infty}}\delta\Big{(}q_{1}^{i+1}\frac{v}{z}\Big{)}[v]_{i+1},$
(3.7) $\displaystyle F(z;p)[v]_{i}$
$\displaystyle=\frac{(pq_{2}^{-1};p)_{\infty}(pq_{3}^{-1};p)_{\infty}}{(1-q_{1}^{-1})(p;p)_{\infty}(pq_{1};p)_{\infty}}\delta\Big{(}q_{1}^{i}\frac{v}{z}\Big{)}[v]_{i-1}.$
(3.8)

### 3.2 Vertical Fock Representation by Tensor Product

As in the case of DIM algebra, we can construct the so-called vertical Fock
representation from the vector representation via the inductive limit [18].
The first step of this procedure is to perform the tensor product of vector
representations with appropriate shift of spectral parameters;

$\displaystyle V^{n}(v)\stackrel{{\scriptstyle\text{def}}}{{=}}V(v)\otimes
V(q_{2}v)\otimes\cdots\otimes V(q_{2}^{n-1}v).$ (3.9)

For each $\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathbb{Z}^{n}$, we define
$|\lambda\rangle\in V^{n}(v)$ to be

$\displaystyle|\lambda\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}[v]_{\lambda_{1}-1}\otimes[q_{2}v]_{\lambda_{2}-1}\otimes\cdots\otimes[q^{n-1}_{2}v]_{\lambda_{n}-1}.$
(3.10)

It is clear that $\\{|\lambda\rangle~{}|\lambda\in\mathbb{Z}^{n}\\}$ forms a
basis of $V^{n}(v)$. We can endow the structure of $\mathcal{U}$-module
$\rho^{(n)}:\mathcal{U}\rightarrow\operatorname{End}(V^{n}(v))$ to the vector
space $V^{n}(v)$ by999Since the coassociativity does not hold for
$\Delta_{p}$, there is an ambiguity in the definition of $\Delta_{p}^{n}$ in
general. However, since $C=1$ for the vertical representations, such a problem
can be evaded. We use the definition in [9] and [13].

$\displaystyle\rho^{(n)}(a)|\lambda\rangle=\Big{[}\rho^{V}_{v}\otimes\rho^{V}_{q_{2}v}\otimes\cdots\otimes\rho^{V}_{q_{2}^{n-1}v}\Big{]}\Delta^{n-1}_{p}(a)|\lambda\rangle,\qquad
a\in\mathcal{U}.$ (3.11)

Note that here we can set $C_{1}=C_{2}=1$ in the formulas of the coproduct
(2.19) – (2.22), since we are focusing on the vector representations. For the
elliptic currents the $n$-fold tensor product of vector representations is
given by

$\displaystyle\rho^{(n)}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\bigg{[}\prod_{i=1}^{n}\frac{\theta_{p}(q_{1}^{\lambda_{i}}q_{2}^{i}v/z)\theta_{p}(q_{1}^{\lambda_{i}-1}q_{2}^{i-2}v/z)}{\theta_{p}(q_{1}^{\lambda_{i}}q_{2}^{i-1}v/z)\theta_{p}(q_{1}^{\lambda_{i}-1}q_{2}^{i-1}v/z)}\bigg{]}|\lambda\rangle,$
(3.12) $\displaystyle\rho^{(n)}\big{(}K^{-}(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\bigg{[}\prod_{i=1}^{n}\frac{\theta_{p}(q_{1}^{-\lambda_{i}}q_{2}^{-i}z/v)\theta_{p}(q_{1}^{-\lambda_{i}+1}q_{2}^{2-i}z/v)}{\theta_{p}(q_{1}^{-\lambda_{i}}q_{2}^{-i+1}z/v)\theta_{p}(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+1}z/v)}\bigg{]}|\lambda\rangle,$
(3.13)

and

$\displaystyle\rho^{(n)}\big{(}E(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\sum_{k=1}^{n}\frac{(pq_{2};p)_{\infty}(pq_{3};p)_{\infty}}{(1-q_{1})(p;p)_{\infty}(pq_{1}^{-1};p)_{\infty}}\delta\Big{(}q_{1}^{\lambda_{k}}q_{2}^{k-1}\frac{v}{z}\Big{)}$
$\displaystyle~{}~{}\,\,\cdot\bigg{[}\prod_{i=1}^{k-1}\frac{\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i-1})\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i+1})}{\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i})\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i})}\bigg{]}|\lambda+1_{k}\rangle,$
(3.14) $\displaystyle\rho^{(n)}\big{(}F(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\sum_{k=1}^{n}\frac{(pq_{2}^{-1};p)_{\infty}(pq_{3}^{-1};p)_{\infty}}{(1-q_{1}^{-1})(p;p)_{\infty}(pq_{1};p)_{\infty}}\delta\Big{(}q_{1}^{\lambda_{k}-1}q_{2}^{k-1}\frac{v}{z}\Big{)}$
$\displaystyle~{}~{}\,\,\cdot\bigg{[}\prod_{i=k+1}^{n}\frac{\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}+1}q_{2}^{i-k+1})\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k-1})}{\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}+1}q_{2}^{i-k})\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k})}\bigg{]}|\lambda-1_{k}\rangle,$
(3.15)

where $\lambda\pm 1_{k}$ means the shift of the $k$-th component
$\lambda_{k}\to\lambda_{k}\pm 1$. Let us denote the set of partitions with
length at most $n$ by

$\mathcal{P}_{n}:=\left\\{\begin{array}[]{@{}l@{}}\lambda=(\lambda_{1},\cdots,\lambda_{n})\in\mathbb{Z}^{n}\text{
s.t. }\lambda_{1}\geq\cdots\geq\lambda_{n}\geq 0\end{array}\right\\}.$ (3.16)

For later convenience we also introduce the set of all partitions
$\mathcal{P}$. By the judicious choice of the $q_{2}$-shift of spectral
parameters for a sequence of the vector representations, there is an invariant
subspace

$\displaystyle
W^{n,+}(v)\stackrel{{\scriptstyle\text{def}}}{{=}}\text{span}\Big{\\{}|\lambda\rangle\in
V^{n}(v)|~{}\lambda\in\mathcal{P}_{n}\Big{\\}}.$ (3.17)

This can be confirmed by investigating the positions of zeros appearing in the
action of the creation operator $E(z)$ and the annihilation operator $F(z)$
[18].

Next, we shall take the inductive limit of the tensor product of vector
representations constructed above. The reason why we are interested in taking
the inductive limit is that we would like to remove the restriction on the
length of the partitions $\lambda$. Thus, we consider the vector space
$\mathcal{F}_{v}$ which is defined by

$\displaystyle\mathcal{F}_{v}\stackrel{{\scriptstyle\text{def}}}{{=}}\mathop{\lim_{\longrightarrow}}W^{n,+}(v),$
(3.18)

where the inductive limit is taken in the category of vector spaces. We would
like to endow the structure of left $\mathcal{U}$-module on $\mathcal{F}_{v}$.
At first glance, it is natural to define this representation
$\rho^{\mathcal{F}}:\mathcal{U}\rightarrow\text{End}(\mathcal{F}_{v})$ as
follows ; for each $\lambda=(\lambda_{1},\dots,\lambda_{n},0,\dots)$ where
$\lambda_{n}\neq 0$,

$\displaystyle\rho^{\mathcal{F}}\big{(}X(z;p)\big{)}|\lambda\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}\rho^{(n+1)}\big{(}X(z;p)\big{)}|\lambda\rangle=\rho^{(n+2)}\big{(}X(z;p)\big{)}|\lambda\rangle=\cdots,$
(3.19)

where $X=K^{+},K^{-},E$ and $F$. That is, we expect that for all $k\geq 1$,
$\rho^{(n+k)}\big{(}X(z;p)\big{)}|\lambda\rangle$ are equal, since the
partitions
$(\lambda_{1},\dots,\lambda_{n}),(\lambda_{1},\dots,\lambda_{n},0),(\lambda_{1},\dots,\lambda_{n},0,0),\dots$
are identified in the inductive limit. However, this is not the case, since
from (3.12), we see that for $\lambda=(\lambda_{1},\dots,\lambda_{n},0,\dots)$
with $\lambda_{n}\neq 0$,

$\displaystyle\frac{\langle\lambda|\rho^{(n+2)}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle}{\langle\lambda|\rho^{(n+1)}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle}=\frac{\theta_{p}(q_{1}^{-1}q_{2}^{n}v/z)\theta_{p}(q_{2}^{n+2}v/z)}{\theta_{p}(q_{1}^{-1}q_{2}^{n+1}v/z)\theta_{p}(q_{2}^{n+1}v/z)}\neq
1.$ (3.20)

This means that we need certain modification factors in (3.19) for $X=K^{+}$.
The same situation also occurs in the case $X=K^{-}$. Hence, let us define

$\displaystyle\rho^{\mathcal{F}}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}\beta_{n+1}\rho^{(n+1)}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle=\beta_{n+2}\rho^{(n+2)}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle=\cdots.$
(3.21)

Then we obtain a consistency condition for the modification factors
$\beta_{n}$;

$\displaystyle\frac{\beta_{n+2}}{\beta_{n+1}}\frac{\theta_{p}(q_{1}^{-1}q_{2}^{n}v/z)\theta_{p}(q_{2}^{n+2}v/z)}{\theta_{p}(q_{1}^{-1}q_{2}^{n+1}v/z)\theta_{p}(q_{2}^{n+1}v/z)}=1.$
(3.22)

Therefore, we conclude that $\beta_{n}$ takes the following form;

$\displaystyle\beta_{n}=f(v/z)\frac{\theta_{p}(q_{1}^{-1}q_{2}^{n-1}v/z)}{\theta_{p}(q_{2}^{n}v/z)},$
(3.23)

where $f(v/z)$ is a proportional factor which is independent of $n$.

By the same line of arguments, if we define for each
$\lambda=(\lambda_{1},\dots,\lambda_{n},0,\dots)$ with $\lambda_{n}\neq 0$,

$\displaystyle\rho^{\mathcal{F}}\big{(}K^{-}(z;p)\big{)}|\lambda\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{n+1}\rho^{(n+1)}\big{(}K^{-}(z;p)\big{)}|\lambda\rangle=\gamma_{n+2}\rho^{(n+2)}\big{(}K^{-}(z;p)\big{)}|\lambda\rangle=\cdots.$
(3.24)

We then find that

$\displaystyle\gamma_{n}=g(z/v)\cdot
q_{3}\frac{\theta_{p}(z/q_{1}^{-1}q_{2}^{n-1}v)}{\theta_{p}(z/q_{2}^{n}v)}.$
(3.25)

From [9], we expect that the modification factors $\beta_{n}$ for $K^{+}(z;p)$
and $\gamma_{n}$ for $K^{-}(z;p)$ are the same. From this and the inversion
formula (1.12) we obtain that $f(v/z)=g(z/v)$.

On the other hand, in the case $X=E$, the problem does not arise. Namely, if
$\lambda=(\lambda_{1},\dots,\lambda_{n},0,\dots)$ with $\lambda_{n}\neq 0$,
then the action

$\displaystyle\rho^{\mathcal{F}}\big{(}E(z;p)\big{)}|\lambda\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}\rho^{(n+1)}\big{(}E(z;p)\big{)}|\lambda\rangle=\rho^{(n+2)}\big{(}E(z;p)\big{)}|\lambda\rangle=\cdots$
(3.26)

is well-defined. From this result, we immediately see that we also have to
introduce a modification factor to the action of $F(z;p)$, and, moreover, it
has to be equal to $\gamma_{n}$. That is,

$\displaystyle\rho^{\mathcal{F}}\big{(}F(z;p)\big{)}|\lambda\rangle\stackrel{{\scriptstyle\text{def}}}{{=}}\gamma_{n+1}\rho^{(n+1)}\big{(}F(z;p)\big{)}|\lambda\rangle=\gamma_{n+2}\rho^{(n+2)}\big{(}F(z;p)\big{)}|\lambda\rangle=\cdots.$
(3.27)

The reason is that the defining relation (2.32) must be satisfied.

Finally, to accomplish the task, we have to determine an explicit expression
of $f(v/z)$. We require that the original Fock representation is recovered in
the limit $p\rightarrow 0$. For simplicity, we also assume that $f(v/z)$ does
not depend on $p$. Thus, we conclude that $f(v/z)=g(z/v)=\mathfrak{q}^{-1}$.

As already mentioned, we can find the invariant subspace $\mathcal{F}_{v}$,
which is spanned by the set of partitions $\mathcal{P}$. This is an
irreducible subrepresentation generated by the empty partition $\varnothing$.
We call it vertical Fock representation. It is a highest weight representation
with the empty partition $\varnothing$ being the highest weight state. In
summary, we have constructed a representation
$\rho^{\mathcal{F}}:\mathcal{U}\rightarrow\text{End}(\mathcal{F}_{v})$ with
the spectral parameter $v$;

$\displaystyle\rho^{\mathcal{F}}\big{(}K^{+}(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\mathfrak{q}^{-1}\prod_{i=1}^{\infty}\frac{\theta_{p}(q_{1}^{\lambda_{i}}q_{2}^{i}v/z)\theta_{p}(q_{1}^{\lambda_{i}-1}q_{2}^{i-2}v/z)}{\theta_{p}(q_{1}^{\lambda_{i}}q_{2}^{i-1}v/z)\theta_{p}(q_{1}^{\lambda_{i}-1}q_{2}^{i-1}v/z)}|\lambda\rangle,$
(3.28)
$\displaystyle\rho^{\mathcal{F}}\big{(}K^{-}(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\mathfrak{q}\prod_{i=1}^{\infty}\frac{\theta_{p}(q_{1}^{-\lambda_{i}}q_{2}^{-i}z/v)\theta_{p}(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+2}z/v)}{\theta_{p}(q_{1}^{-\lambda_{i}}q_{2}^{-i+1}z/v)\theta_{p}(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+1}z/v)}|\lambda\rangle,$
(3.29) $\displaystyle\rho^{\mathcal{F}}\big{(}E(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(pq_{2};p)_{\infty}(pq_{3};p)_{\infty}}{(1-q_{1})(p;p)_{\infty}(pq_{1}^{-1};p)_{\infty}}\delta\Big{(}q_{1}^{\lambda_{k}}q_{2}^{k-1}\frac{v}{z}\Big{)}$
$\displaystyle~{}~{}\,\,\cdot\bigg{[}\prod_{i=1}^{k-1}\frac{\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i-1})\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i+1})}{\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i})\theta_{p}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i})}\bigg{]}|\lambda+1_{k}\rangle,$
(3.30) $\displaystyle\rho^{\mathcal{F}}\big{(}F(z;p)\big{)}|\lambda\rangle$
$\displaystyle=\mathfrak{q}^{-1}\sum_{k=1}^{\infty}\frac{(pq_{2}^{-1};p)_{\infty}(pq_{3}^{-1};p)_{\infty}}{(1-q_{1}^{-1})(p;p)_{\infty}(pq_{1};p)_{\infty}}\delta\Big{(}q_{1}^{\lambda_{k}-1}q_{2}^{k-1}\frac{v}{z}\Big{)}$
$\displaystyle~{}~{}\,\,\cdot\bigg{[}\prod_{i=k+1}^{\infty}\frac{\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}+1}q_{2}^{i-k+1})\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k-1})}{\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}+1}q_{2}^{i-k})\theta_{p}(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k})}\bigg{]}|\lambda-1_{k}\rangle.$
(3.31)

These are universal formulas which do not depend on the length of partition
$\ell(\lambda)$. Our prescription for the infinite product appearing in
$K^{\pm}(z;p)$ and $F(z;p)$ is as follows; we make a successive cancellation
of the factors in the denominator and the numerator for
$\lambda_{n}=0~{}(\ell(\lambda)<n)$, which reduces the infinite product to a
finite product once the partition $\lambda$ is fixed. The factor
$\mathfrak{q}^{\pm 1}$ is regarded as a result of the regularization of the
infinite product by this prescription. Note that it does not appear for
$E(z;p)$ which does not require the infinite product. Similarly the infinite
sum for $E(z;p)$ and $F(z;p)$ reduces to a finite sum up to $\ell(\lambda)+1$,
because when the adjacent lengths of the partition agree;
$\lambda_{j}=\lambda_{j+1}$, it is possible to have a factor
$\theta_{p}(1)=0$. This also implies that when $\lambda+1_{k}$ or
$\lambda-1_{k}$ is no longer a partition, the corresponding coefficient
automatically vanishes. From (3.29) we see that the vertical Fock
representation has level $(0,1)$. After the scaling (2.33) and (2.34) of
$E(z;p)$ and $F(z;p)$, our result agrees with the vertical representation in
[68].

### 3.3 Vertical Fock Representation by Twisting

In the last subsection, we have constructed the vertical Fock representation
by using the inductive limit of the tensor product of vector representations.
On the other hand, since the original DIM algebra has the vertical Fock
representation [18], we may construct a vertical representation directly from
the quasi-Hopf twist. In the vertical Fock representation, the operator
$b_{n}$ acts as follows:

$\displaystyle b_{n}|\lambda\rangle$
$\displaystyle=\frac{v^{n}}{n}\frac{\kappa_{n}}{1-q_{1}^{-n}}\bigg{[}\sum_{s=1}^{\ell(\lambda)}x^{n}_{s}+\frac{1}{1-q_{2}^{n}}x^{n}_{\ell(\lambda)+1}\bigg{]}|\lambda\rangle$
$\displaystyle=\frac{v^{n}}{n}\frac{\kappa_{n}}{1-q_{1}^{-n}}\bigg{[}\sum_{s=1}^{\infty}x^{n}_{s}\bigg{]}|\lambda\rangle,\qquad
x_{s}:=q_{1}^{\lambda_{s}-1}q_{2}^{s-1}.$ (3.32)

We note that the eigenvalues of $b_{n}$ are proportional to
$A_{\lambda}(q_{1}^{n},q_{2}^{n})$ to be defined below (see (4.12)). Thus from
(2.13), (2.14) and (2.15), we can check that the action of the elliptic Cartan
currents $K^{\pm}(z;p)$ is the same as (3.28) and (3.29). On the other hand,
we find some discrepancy in the action of the elliptic currents $E(z;p)$ and
$F(z;p)$. Namely there appear the following remainder factors against the
formulas (3.30) and (3.2);

$\displaystyle R_{\lambda}^{(k)}(q_{1},q_{2};p)$
$\displaystyle:=\prod_{s=1}^{k-1}\frac{(pq_{1}^{\lambda_{s}-\lambda_{k}}q_{2}^{s-k};p)_{\infty}(pq_{1}^{\lambda_{s}-\lambda_{k}-1}q_{2}^{s-k};p)_{\infty}}{(pq_{1}^{\lambda_{s}-\lambda_{k}}q_{2}^{s-k+1};p)_{\infty}(pq_{1}^{\lambda_{s}-\lambda_{k}-1}q_{2}^{s-k-1};p)_{\infty}}$
$\displaystyle\cdot\prod_{s=k+1}^{\infty}\frac{(pq_{1}^{\lambda_{k}-\lambda_{s}+1}q_{2}^{k-s+1};p)_{\infty}(pq_{1}^{\lambda_{k}-\lambda_{s}}q_{2}^{k-s-1};p)_{\infty}}{(pq_{1}^{\lambda_{k}-\lambda_{s}+1}q_{2}^{k-s};p)_{\infty}(pq_{1}^{\lambda_{k}-\lambda_{s}}q_{2}^{k-s};p)_{\infty}},$
(3.33)

for $E(z;p)$ and

$\displaystyle\widetilde{R}_{\lambda}^{(k)}(q_{1},q_{2};p)$
$\displaystyle=R_{\lambda-1_{k}}^{(k)}(q_{1},q_{2};p)^{-1}$ (3.34)

for $F(z;p)$. The relation (3.34) allows us to understand the remainders
$R_{\lambda}^{(k)}$ and $\widetilde{R}_{\lambda}^{(k)}$ in the following
manner; In the vertical representations the Cartan currents $K^{\pm}(z;p)$ are
commuting and we employ a basis consisting of simultaneous eigenstates
$|\lambda\rangle$ of $K^{\pm}(z;p)$. Since the eigenvalues are non-degenerate,
they are orthogonal. But there is an ambiguity of the (relative) normalization
of $|\lambda\rangle$, in particular it may depend on $\lambda$ and the
elliptic parameter $p$. The change of the normalization does not affect the
matrix elements of $K^{\pm}(z;p)$, but the matrix elements of $E(z;p)$ and
$F(z;p)$ will change, since they are off-diagonal. In fact let us consider the
change of the normalization;
$|\lambda\rangle\longrightarrow\mathcal{C}_{\lambda}(q_{1},q_{2};p)|\lambda\rangle$,
where $\mathcal{C}_{\lambda}(q_{1},q_{2};p)$ is determined by the recursion
relation

$\frac{\mathcal{C}_{\lambda+1_{k}}(q_{1},q_{2};p)}{\mathcal{C}_{\lambda}(q_{1},q_{2};p)}=R_{\lambda}^{(k)}(q_{1},q_{2};p).$
(3.35)

Then one can see this change of the normalization eliminates both
$R_{\lambda}^{(k)}$ and $\widetilde{R}_{\lambda}^{(k)}$. Finally with the
initial condition $\mathcal{C}_{\varnothing}(q_{1},q_{2};p)=1$, the recursion
relation is solved by

$\mathcal{C}_{\lambda}(q_{1},q_{2};p)=\prod_{s\in\lambda}(pq_{1}^{a(s)}q_{2}^{-\ell(s)-1};p)_{\infty}.$
(3.36)

Here $a(s):=\lambda_{i}-j$ and $\ell(s):=\lambda^{\vee}_{j}-i$ are the arm-
length and the leg-length of the box $s=(i,j)$ in the partition $\lambda$.
Note that

$C_{\lambda}(q_{1},q_{2}):=\prod_{s\in\lambda}(1-q_{1}^{a(s)}q_{2}^{-\ell(s)-1})$
(3.37)

is the normalization factor which appears in the integral form of the
Macdonald polynomials [45].

### 3.4 Horizontal Fock Representation

In subsection 2.2, we have seen the relation between the elliptic currents and
the original DIM algebra, which has a Fock representation in terms of free
bosons $\widetilde{a}_{n}$ [19]. The free boson operators obey the commutation
relations of the deformed Heisenberg algebra;

$\displaystyle[\widetilde{a}_{n},\widetilde{a}_{m}]=\frac{n}{\kappa_{n}}(\mathfrak{q}^{n}-\mathfrak{q}^{-n})\delta_{n+m,0}=\frac{n\mathfrak{q}^{-n}}{(1-q_{1}^{n})(1-q_{2}^{n})}\delta_{n+m,0}.$
(3.38)

By using the relations (2.13)–(2.15), we can obtain the horizontal Fock
representation of the elliptic currents. Let $\mathcal{H}_{u}$ be a vector
space over $\mathbb{C}$, which has

$\displaystyle\left\\{\widetilde{a}_{-\lambda_{1}}\cdots\widetilde{a}_{-\lambda_{n}}|0;u\rangle|\;\middle|\;\begin{array}[]{@{}l@{}}n\in\mathbb{Z}^{\geq
0},\\\ \lambda_{1},\cdots,\lambda_{n}\in\mathbb{Z}^{>0}\text{ s.t.
}\lambda_{1}\geq\cdots\geq\lambda_{n}\end{array}\right\\}$ (3.41)

as a basis. Here we have introduced the horizontal spectral parameter $u$ and
the state $|0;u\rangle$ is defined to be annihilated by the positive mode
operators $\\{\widetilde{a}_{n}|~{}n\in\mathbb{Z}^{>0}\\}$.

If we define
$\rho^{(0)}_{H}:\mathcal{U}\rightarrow\text{End}(\mathcal{H}_{u})$ by

$\displaystyle\rho^{(0)}_{H}\Big{(}K^{+}(z;p)\Big{)}=$
$\displaystyle\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{p^{n}\mathfrak{q}^{-3n/2}}{1-p^{n}_{*}}\widetilde{a}_{-n}z^{n}\bigg{)}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{n/2}}{1-p^{n}}\widetilde{a}_{n}z^{-n}\bigg{)},$
(3.42) $\displaystyle\rho^{(0)}_{H}\Big{(}K^{-}(z;p)\Big{)}=$
$\displaystyle\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{-n/2}}{1-p^{n}_{*}}\widetilde{a}_{-n}z^{n}\bigg{)}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{p^{n}\mathfrak{q}^{-n/2}}{1-p^{n}}\widetilde{a}_{n}z^{-n}\bigg{)},$
(3.43) $\displaystyle\rho^{(0)}_{H}\Big{(}E(z;p)\Big{)}=$
$\displaystyle\frac{u}{(1-q_{1})(1-q_{2})}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{-n/2}(1-p^{n})}{(1-p^{n}_{*})(\mathfrak{q}^{n}-\mathfrak{q}^{-n})}\widetilde{a}_{-n}z^{n}\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{-n/2}}{\mathfrak{q}^{n}-\mathfrak{q}^{-n}}\widetilde{a}_{n}z^{-n}\bigg{)},$
(3.44) $\displaystyle\rho^{(0)}_{H}\Big{(}F(z;p)\Big{)}=$
$\displaystyle\frac{u^{-1}}{(1-q_{1}^{-1})(1-q_{2}^{-1})}\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{n/2}}{\mathfrak{q}^{n}-\mathfrak{q}^{-n}}\widetilde{a}_{-n}z^{n}\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{n/2}(1-p^{n}_{*})}{(1-p^{n})(\mathfrak{q}^{n}-\mathfrak{q}^{-n})}\widetilde{a}_{n}z^{-n}\bigg{)},$
(3.45)

where $p_{*}=p\mathfrak{q}^{-2}$. This is a level $(1,0)$ representation of
$\mathcal{U}$. We can also obtain a level $(1,N)$ representation for any
integer $N$. By definition, the zero modes of $K^{\pm}(z;p)$ are
$\mathfrak{q}^{\mp N}$. The zero modes of $E(z;p)$ and $F(z;p)$ are fixed by
consistency;

$\mathbf{e}(z)\mathbf{f}(\mathfrak{q}^{-1}z)=\mathfrak{q}^{-N},\qquad\mathbf{e}(\mathfrak{q}^{-1}z)\mathbf{f}(z)=\mathfrak{q}^{N}.$
(3.46)

Using a canonical solution

$\mathbf{e}(z)=\left(\frac{\mathfrak{q}}{z}\right)^{N},\qquad\quad\quad\mathbf{f}(z)=\left(\frac{\mathfrak{q}}{z}\right)^{-N},$
(3.47)

we define the homomorphism $\rho^{(N)}_{H}$;

$\displaystyle\rho^{(N)}_{H}\Big{(}K^{+}(z;p)\Big{)}=\mathfrak{q}^{-N}\rho^{(0)}_{H}\Big{(}K^{+}(z;p)\Big{)},\quad\quad\rho^{(N)}_{H}\Big{(}K^{-}(z;p)\Big{)}=\mathfrak{q}^{N}\rho^{(0)}_{H}\Big{(}K^{-}(z;p)\Big{)},$
$\displaystyle\rho^{(N)}_{H}\Big{(}E(z;p)\Big{)}=\mathbf{e}(z)\rho^{(0)}_{H}\Big{(}E(z;p)\Big{)},\quad\quad\rho^{(N)}_{H}\Big{(}F(z;p)\Big{)}=\mathbf{f}(z)\rho^{(0)}_{H}\Big{(}F(z;p)\Big{)}.$
(3.48)

In summary in the same way as the original DIM algebra, the horizontal Fock
representations are characterized by the level $N$ and the spectral parameter
$u$. We will denote the free boson Fock space for the representation
$\rho^{(N)}_{H}$ by $\mathcal{H}_{u}^{(N)}$.

## § 4 Intertwiner and Dual Intertwiner

In this section we construct the intertwining operator and the dual
intertwining operator. Historically they appeared in the theory of solvable
lattice models associated with the quantum affine algebra
$U_{q}(\hat{\mathfrak{g}})$, where they were called vertex operators of type
II and of type I, respectively [47]. The vertical Fock representation
$\mathcal{F}_{v}$ corresponds to the evaluation module in the case of the
solvable lattice models and the horizontal Fock space $\mathcal{H}_{u}^{(N)}$
is a generalization of the level one highest weight module of the quantum
affine algebra. The intertwining operator and the dual intertwining operator
for the elliptic DIM algebra introduced by Y.Saito [64] are constructed in
[70], [24].

The intertwining operator
$\Psi(v;p):\mathcal{F}_{v}\otimes\mathcal{H}^{(N)}_{u}\rightarrow\mathcal{H}^{(N+1)}_{w}$
is determined by the following intertwining condition [2];

$\displaystyle a\Psi(v;p)=\Psi(v;p)\Delta_{p}(a),\quad a\in\mathcal{U}.$ (4.1)

Here $\mathcal{F}_{v}$ denotes the vertical Fock representation that has level
$(0,1)$, while $\mathcal{H}^{(N)}_{u}$ and $\mathcal{H}^{(N+1)}_{w}$ are
horizontal representations of level $(1,N)$ and $(1,N+1)$, respectively. Let
$\Psi(v)$ be the intertwiner defined by the original coproduct $\Delta$, which
is given in [2]. Since the twisted coproduct $\Delta_{p}$ is defined by (2.8),
we see that schematically

$\Psi(v;p)=\Psi(v)\cdot(\rho_{\mathcal{F}}\otimes\rho_{H})(\mathcal{F}(p)^{-1})$
(4.2)

satisfies the condition (4.1). Recall that
$\\{|\lambda\rangle\\}_{\lambda\in\mathcal{P}}$ forms a basis of
$\mathcal{F}_{v}$. We define the $\lambda$-component of the elliptic
intertwiner
$\Psi_{\lambda}:\mathcal{H}^{(N)}_{u}\rightarrow\mathcal{H}^{(N+1)}_{w}$ by

$\displaystyle\Psi_{\lambda}(v;p)\big{(}\bullet\big{)}=\Psi(v;p)\Big{(}|\lambda\rangle\otimes\bullet\Big{)}.$
(4.3)

Similarly the dual intertwiner
$\Psi^{*}(v;p):\mathcal{H}^{(N)}_{u}\rightarrow\mathcal{H}^{(N-1)}_{w}\otimes\mathcal{F}_{v}$
is determined by the dual intertwining relation;

$\displaystyle\Psi^{*}(v;p)a=\Delta_{p}(a)\Psi^{*}(v;p),\quad
a\in\mathcal{U}.$ (4.4)

Again, from (2.8), if $\Psi^{*}(v)$ is the dual intertwiner before the quasi-
Hopf twist, then

$\Psi^{*}(v;p)=(\rho_{H}\otimes\rho_{\mathcal{F}})(\mathcal{F}(p))\cdot\Psi^{*}(v)$
(4.5)

gives a formal solution to the condition (4.4). We define the
$\lambda$-component of the elliptic dual intertwiner
$\Psi^{*}_{\lambda}(v;p):\mathcal{H}^{(N)}_{u}\rightarrow\mathcal{H}^{(N-1)}_{w}$
by

$\displaystyle\Psi^{*}(v;p)\Big{(}\bullet\Big{)}=\sum_{\lambda}\Psi^{*}_{\lambda}(v;p)\Big{(}\bullet\Big{)}\otimes|\lambda\rangle.$
(4.6)

### 4.1 Elliptic Fock Intertwiner

For the intertwining relation (4.1), we have $C_{1}=1$ and
$C_{2}=\mathfrak{q}$ in (2.19) – (2.22) and the intertwining relations are
explicitly;

$\displaystyle K^{+}(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\langle\lambda|K^{+}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)K^{+}(z;p),$
(4.7) $\displaystyle K^{-}(\mathfrak{q}z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\langle\lambda|K^{-}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)K^{-}(\mathfrak{q}z;p),$
(4.8) $\displaystyle E(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\sum_{k=1}^{\ell(\lambda)+1}\langle\lambda+1_{k}|E(z;p_{*})|\lambda\rangle\Psi_{\lambda+1_{k}}(v;p)+\langle\lambda|K^{-}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)E(z;p),$
(4.9) $\displaystyle F(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\sum_{k=1}^{\ell(\lambda)}\langle\lambda-1_{k}|F(\mathfrak{q}z;p_{*})|\lambda\rangle\Psi_{\lambda-1_{k}}(v;p)K^{+}(\mathfrak{q}z;p)+\Psi_{\lambda}(v;p)F(z;p),$
(4.10)

where $p_{*}=p\mathfrak{q}^{-2}$. Note that here the inner product is
calculated in the vertical representation $\mathcal{F}_{v}$. For the existence
of the intertwiner the horizontal spectral parameters of the source and the
target Fock spaces have to be related by $w=-uv$ [2] (see also Appendix B). We
can express the elliptic Fock intertwiner $\Psi_{\lambda}(v;p)$ by the
trivalent diagram in Figure 1 below, where the change of the level and the
spectral parameter of the horizontal Fock space is indicated.

$u$$(1,N)$$\lambda$$(0,1)$$v$$(1,N+1)$$-uv$ Figure 1. Trivalent diagram of the
elliptic Fock intertwiner $\Psi_{\lambda}(v;p)$

The intertwining relations (4.7) and (4.8) mean $\Psi_{\lambda}(v;p)$ is an
eigenstate of the adjoint action of $K^{+}(z;p)$ and $K^{-}(\mathfrak{q}z;p)$
with eigenvalues $\langle\lambda|K^{\pm}(z;p_{*})|\lambda\rangle$. These two
conditions fix $\Psi_{\lambda}(v;p)$ up to the overall factor
$z_{\lambda}(v)\mathcal{G}^{-1}_{\lambda}(p_{*})$ as follows;

$\displaystyle\Psi_{\lambda}(v;p)=z_{\lambda}(v)\mathcal{G}^{-1}_{\lambda}(p_{*})$
$\displaystyle\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}\mathfrak{q}^{n/2}(q_{2}^{n}-1)(q_{1}^{n}-1)\widetilde{a}_{-n}v^{n}A_{\lambda}(q_{1}^{n},q_{2}^{n})\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{n}\mathfrak{q}^{n/2}(q_{2}^{n}-1)(q_{1}^{n}-1)\widetilde{a}_{n}v^{-n}A_{\lambda}(q_{1}^{-n},q_{2}^{-n})\bigg{)},$
(4.11)

where

$A_{\lambda}(q_{1},q_{2}):=\sum_{(i,j)\in\lambda}x_{ij}-\frac{1}{(q_{1}-1)(q_{2}-1)},\qquad
x_{ij}:=q_{1}^{j-1}q_{2}^{i-1}$ (4.12)

is fixed by the eigenvalues $\langle\lambda|K^{\pm}(z;p_{*})|\lambda\rangle$.
The remaining two conditions (4.9) and (4.10) determine $z_{\lambda}(v)$ and
$\mathcal{G}^{-1}_{\lambda}(p_{*})$ as follows;

$z_{\lambda}(v)=\prod_{(i,j)\in\lambda}(-\mathfrak{q}q_{2}^{i-1}x^{-1}_{ij})u\cdot\mathbf{e}(x_{ij}v)=q_{2}^{n(\lambda)}(-v)^{-N|\lambda|}u^{|\lambda|}f_{\lambda}(q_{1},q_{2})^{-N-1},$
(4.13)

and

$\mathcal{G}_{\lambda}(p_{*})=\prod_{s\in\lambda}\frac{(q_{1}^{-a(s)}q_{2}^{\ell(s)+1};p_{*})_{\infty}}{(pq_{1}^{-a(s)}q_{2}^{\ell(s)+1};p_{*})_{\infty}}.$
(4.14)

Here the framing factor $f_{\lambda}(q_{1},q_{2})$ is defined by

$f_{\lambda}(q_{1},q_{2})\stackrel{{\scriptstyle\text{def}}}{{=}}\prod_{(i,j)\in\lambda}(-1)q_{1}^{j-1}q_{2}^{i-1}\mathfrak{q}^{-1}=\prod_{s\in\lambda}(-1)q_{1}^{a(s)+\frac{1}{2}}q_{2}^{\ell(s)+\frac{1}{2}}.$
(4.15)

The intertwining relations of $E(z;p)$ and $F(z;p)$ are responsible for the
formula of the zero mode factor $z_{\lambda}(v)$ the normalization
$\mathcal{G}_{\lambda}(p_{*})$. More precisely, $z_{\lambda}(v)$ comes form
the choice of the zero modes of the horizontal representation and
$\mathcal{G}_{\lambda}(p_{*})$ depends on the normalization of the basis of
the vertical Fock representation (See Appendices A and B for computations of
$\mathcal{G}_{\lambda}(p_{*})$ and $z_{\lambda}(v)$). The appearance of
$\mathcal{G}_{\lambda}(p_{*})$ and $z_{\lambda}(v)$ can be described in the
following manner. Since the vertical Fock representation is constructed as the
semi-infinite product of the vector representations (see subsec.3.2), we can
express the elliptic Fock intertwiner (4.11) as a composition of elliptic
intertwiners for the vector representations [9], [13]. The factor
$\mathcal{G}_{\lambda}(p_{*})$ is related to the normal ordering of this
composition. By the relation (4.2) the zero factor $z_{\lambda}(v)$ is the
same as the intertwiners for the original coproduct $\Delta$, which are e.g.
given in [9].

We note that $\Psi_{\lambda}(v;p)$ is expressed as a normal ordered product of
the oscillator part $\eta(z;p)$ of the elliptic current $E(z;p)$;

$\displaystyle\Psi_{\lambda}(v;p)$
$\displaystyle=z_{\lambda}(v)\mathcal{G}^{-1}_{\lambda}(p):\Psi_{\varnothing}(v;p)\prod_{(i,j)\in\lambda}\eta(q_{1}^{j-1}q_{2}^{i-1}v;p):,$
(4.16)
$\displaystyle\Psi_{\varnothing}(v;p):=:\prod_{i.j=1}^{\infty}\eta(q_{1}^{j-1}q_{2}^{i-1}v;p)^{-1}:.$
(4.17)

The zero mode factor $z_{\lambda}(v)$, which depends on the level $N$ and the
spectral parameters takes care of the zero mode factor of $E(z;p)$. This
structure is exactly the the same as the intertwiner of the original DIM
algebra, which is recovered by $p\to 0$, and explains the appearance of
$A_{\lambda}(q_{1},q_{2})$ which geometrically is the evaluation of the
equivariant character of the tautological sheaf on the universal bundle of
instantons at the fixed point labeled by $\lambda$.

It is remarkable that the shift of the spectral parameter
$q_{1}^{j-1}q_{2}^{i-1}v$ in (4.16) and (4.17) comes from the way of
constructing the vertical Fock representation. In fact we have used the matrix
elements $\langle\lambda|K^{\pm}(z;p_{*})|\lambda\rangle$ to fix this part. It
is the vertical Fock representation that arises naturally from the geometry of
the Hilbert scheme of points on $\mathbb{C}^{2}$ [50]. In this way the Fock
intertwiner incorporates the geometry of $U(1)$ instanton moduli space into
the vertex operators on the Fock space of free bosons. To describe the moduli
space of $U(N)$ instantons we have to take the $N$-fold tensor product of the
boson Fock spaces.

### 4.2 Elliptic Dual Fock Intertwiner

In the case of the dual intertwining relation (4.4),
$C_{1}=\mathfrak{q},C_{2}=1$ in (2.19) – (2.22). Hence, the intertwining
relations of the dual Fock intertwiner are explicitly;

$\displaystyle\Psi^{*}_{\lambda}(v;p)K^{+}(z;p)$
$\displaystyle=\langle\lambda|K^{+}(\mathfrak{q}^{-1}z;p)|\lambda\rangle
K^{+}(z;p)\Psi^{*}_{\lambda}(v;p),$ (4.18)
$\displaystyle\Psi^{*}_{\lambda}(v;p)K^{-}(z;p)$
$\displaystyle=\langle\lambda|K^{-}(z;p)|\lambda\rangle
K^{-}(z;p)\Psi^{*}_{\lambda}(v;p),$ (4.19)
$\displaystyle\Psi^{*}_{\lambda}(v;p)E(z;p)$
$\displaystyle=E(z;p)\Psi^{*}_{\lambda}(v;p)+\sum_{k=1}^{\ell(\lambda)}\langle\lambda|E(\mathfrak{q}z;p)|\lambda-1_{k}\rangle
K^{-}(\mathfrak{q}z;p)\Psi^{*}_{\lambda-1_{k}}(v;p),$ (4.20)
$\displaystyle\Psi^{*}_{\lambda}(v;p)F(z;p)$
$\displaystyle=\langle\lambda|K^{+}(z;p)|\lambda\rangle
F(z;p)\Psi^{*}_{\lambda}(v;p)+\sum_{k=1}^{\ell(\lambda)+1}\langle\lambda|F(z;p)|\lambda+1_{k}\rangle\Psi^{*}_{\lambda+1_{k}}(v;p).$
(4.21)

We can show that the solution of the intertwining relations (4.18) - (4.21) is

$\displaystyle\Psi^{*}_{\lambda}(v;p)=$ $\displaystyle
z^{*}_{\lambda}(v)\mathcal{G}^{*-1}_{\lambda}(p)\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{n}\mathfrak{q}^{3n/2}(q_{1}^{n}-1)(q_{2}^{n}-1)\widetilde{a}_{-n}v^{n}A_{\lambda}(q_{1}^{n},q_{2}^{n})\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1-p^{n}_{*}}{n(1-p^{n})}(q_{1}^{n}-1)(q_{2}^{n}-1)\mathfrak{q}^{3n/2}\widetilde{a}_{n}v^{-n}A_{\lambda}(q_{1}^{-n},q_{2}^{-n})\bigg{)},$
(4.22)

where

$\displaystyle z^{*}_{\lambda}(v)=$
$\displaystyle\mathfrak{q}^{|\lambda|}\prod_{(i,j)\in\lambda}(-\mathfrak{q}q_{2}^{i-1}x^{-1}_{ij})u^{-1}\cdot\mathbf{f}(x_{ij}v)=\mathfrak{q}^{|\lambda|}q_{2}^{n(\lambda)}(-v)^{N|\lambda|}u^{-|\lambda|}f_{\lambda}(q_{1},q_{2})^{N-1},$
(4.23)

and

$\displaystyle\mathcal{G}^{*}_{\lambda}(p)=$
$\displaystyle\prod_{s\in\lambda}\frac{\Big{(}q_{1}^{a(s))}q_{2}^{-\ell(s)-1};p\Big{)}_{\infty}}{\Big{(}p_{*}q_{1}^{a(s)}q_{2}^{-\ell(s)-1};p\Big{)}_{\infty}}.$
(4.24)

For the existence of the dual intertwiner the horizontal spectral parameters
of the source and the target Fock spaces have to be related by $w=-u/v$ [2].
The elliptic dual Fock intertwiner $\Psi^{*}_{\lambda}(v;p)$ is expressed by
the trivalent diagram in Figure 2 below.

$u$$(1,N)$$\lambda$$v$$(0,1)$$(1,N-1)$$-u/v$ Figure 2. Trivalent diagram of
the elliptic dual Fock intertwiner $\Psi^{*}_{\lambda}(v;p)$

We can observe a similarity to the case of the intertwiner. In fact we have

$\displaystyle\Psi^{*}_{\lambda}(v;p)$
$\displaystyle=z^{*}_{\lambda}(v)\mathcal{G}^{*-1}_{\lambda}(p):\Psi^{*}_{\varnothing}(v;p)\prod_{(i,j)\in\lambda}\xi(q_{1}^{j-1}q_{2}^{i-1}v;p):,$
(4.25)
$\displaystyle\Psi^{*}_{\varnothing}(v;p):=:\prod_{i.j=1}^{\infty}\xi(q_{1}^{j-1}q_{2}^{i-1}v;p)^{-1}:,$
(4.26)

where $\xi(z;p)$ is the oscillator part of $F(z;p)$. Namely, $E(z;p)$ for
$\Psi_{\lambda}(v;p)$ is simply replaced by $F(z;p)$ for the dual intertwiner
$\Psi^{*}_{\lambda}(v;p)$. Again, the factor $\mathcal{G}^{*-1}_{\lambda}(p)$
appears by removing the normal ordered product, when we express the dual
intertwiner as a composition of those for the vector representation. Note also
that the factor $z^{*}_{\lambda}(v)$ depends on the level $N$ of the
horizontal Fock representation.

## § 5 Vertical $R$-matrix and Elliptic Nekrasov Factor

In this section, we determine the $R$-matrix corresponding to the vertical
Fock representation. According to [36], the quasi-Hopf twist of the universal
$R$-matrix $\widetilde{\mathcal{R}}$ is

$\displaystyle\widetilde{\mathcal{R}}=\mathcal{F}^{(21)}(p)\cdot\mathcal{R}\cdot\mathcal{F}^{-1}(p),$
(5.1)

where $\mathcal{R}$ is the universal $R$-matrix of the original DIM algebra
and $\mathcal{F}(p)$ is the twistor given by (2.7). Here the notation
$\mathcal{F}^{(21)}(p)$ means that we interchange the order of the elements in
the tensor product of the expression of $\mathcal{F}(p)$;

$\displaystyle\mathcal{F}^{(21)}(p)=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{np^{n}C^{-n}_{1}}{\kappa_{n}(1-p^{n}C^{-2n}_{1})}b_{-n}\otimes
b_{n}\bigg{)}.$ (5.2)

According to [21], the universal $R$-matrix $\mathcal{R}$ of DIM algebra
factorizes as follows;

$\mathcal{R}=\mathfrak{q}^{-(c^{\perp}\otimes d^{\perp}+d^{\perp}\otimes
c^{\perp})}\mathcal{R}_{+}\mathcal{R}_{0}\mathcal{R}_{-},$ (5.3)

where $\mathfrak{q}^{c^{\perp}}=K_{0}^{-}$ and $d^{\perp}=d_{1}$ (the grading
operator for the principal degree). See also [28] for computations of the $R$
matrix for the horizontal Fock representation. What is most relevant in the
present paper is the Cartan factor $\mathcal{R}_{0}$ with the contribution of
the centers101010The definition of $\kappa_{n}$ in this paper is $-\kappa_{n}$
in [21]. We have changed the convention of the $R$ matrix from [13].;

$\displaystyle\mathcal{R}_{0}^{\prime}=\mathfrak{q}^{-(c^{\perp}\otimes
d^{\perp}+d^{\perp}\otimes
c^{\perp})}\exp\bigg{(}-\sum_{n=1}^{\infty}n\kappa_{n}h_{-n}\otimes
h_{n}\bigg{)},$ (5.4)

where $h_{\pm n}$ is defined via $\kappa_{n}h_{\pm n}=\pm H_{\pm n}$. Recall
that any explicit formula of the universal $R$ matrix depends on the choice of
the Borel subalgebra from which the quantum group is reconstructed as the
Drinfeld double. It is interesting to find that the $R$-matrix
$[\rho^{\mathcal{F}}_{v_{1}}\otimes\rho^{\mathcal{F}}_{v_{2}}](\mathcal{R}_{0})$,
which appears shortly below, coincides with the infinite slope $R$-matrix
$R_{\infty}$ which is ubiquitous in the Khoroshkin-Tolstoy factorization of
the slope $s$ $R$-matrix introduced in [60]. From the viewpoint of the
elliptic Hall algebra, $\mathcal{R}_{0}^{\prime}$ is a universal $R$ matrix of
the vertical (or slope infinity) Heisenberg subalgebra [53]. As noticed in
[60], $R_{\infty}$ corresponds to multiplication by a class of normal bundles
in $K$-theory and is diagonal in the fixed point basis of the torus action.

Since $c^{\perp}$ and $d^{\perp}$ commute with $b_{\pm}$, we obtain

$\displaystyle\widetilde{\mathcal{R}}_{0}^{\prime}$
$\displaystyle=\mathfrak{q}^{-(c^{\perp}\otimes d^{\perp}+d^{\perp}\otimes
c^{\perp})}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{np^{n}C^{-n}_{1}}{\kappa_{n}(1-p^{n}C^{-2n}_{1})}b_{-n}\otimes
b_{n}\bigg{)}$
$\displaystyle\exp\bigg{(}-\sum_{n=1}^{\infty}n\kappa_{n}h_{-n}\otimes
h_{n}\bigg{)}\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{np^{n}C^{-n}_{2}}{\kappa_{n}(1-p^{n}C^{-2n}_{2})}b_{n}\otimes
b_{-n}\bigg{)}.$ (5.5)

The $R$-matrix of the vertical Fock representation is evaluated as follows;

$\displaystyle\biggl{\\{}[\rho^{\mathcal{F}}_{v_{1}}\otimes\rho^{\mathcal{F}}_{v_{2}}]\Big{(}\widetilde{\mathcal{R}}_{0}^{\prime}\Big{)}\biggr{\\}}\Big{(}|\lambda,v_{1}\rangle\otimes|\mu,v_{2}\rangle\Big{)}=R_{\lambda\mu}\Big{(}\frac{v_{1}}{v_{2}};p\Big{)}\Big{(}|\lambda,v_{1}\rangle\otimes|\mu,v_{2}\rangle\Big{)},$
(5.6)

where $|\lambda,v_{1}\rangle\in\mathcal{F}_{v_{1}}$ and
$|\mu,v_{2}\rangle\in\mathcal{F}_{v_{2}}$. Since $c^{\perp}=1$ for the Fock
representation and $d^{\perp}=d_{1}$ counts the degree of the horizontal
spectral parameter, we obtain

$\displaystyle R_{\lambda\mu}\Big{(}\frac{v_{1}}{v_{2}};p\Big{)}=$
$\displaystyle\mathfrak{q}^{-(|\lambda|+|\mu|)}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{1-p^{n}}\frac{v^{-n}_{1}}{1-q_{1}^{n}}\bigg{[}\sum_{s=1}^{\infty}x^{-n}_{s,\lambda_{s}}\bigg{]}\frac{v^{n}_{2}}{n}\frac{\kappa_{n}}{1-q_{1}^{-n}}\bigg{[}\sum_{r=1}^{\infty}x^{n}_{r,\mu_{r}}\bigg{]}\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{p^{n}}{1-p^{n}}\frac{v^{n}_{1}}{1-q_{1}^{-n}}\bigg{[}\sum_{s=1}^{\infty}x^{n}_{s,\lambda_{s}}\bigg{]}\frac{v^{-n}_{2}}{n}\frac{\kappa_{n}}{1-q_{1}^{n}}\bigg{[}\sum_{r=1}^{\infty}x^{-n}_{r,\mu_{r}}\bigg{]}\bigg{)}.$
(5.7)

To simplify the expression (5.7) the following formula can be used;

$\displaystyle\frac{1}{1-q_{1}}\bigg{(}\sum_{i=1}^{\infty}q_{1}^{\lambda_{i}}q_{2}^{i-1}\bigg{)}=-A_{\lambda}(q_{1},q_{2}),$
(5.8)

where $A_{\lambda}(q_{1},q_{2})$ is defined by (4.12). Then, we see that

$\displaystyle R_{\lambda\mu}\Big{(}\frac{v_{1}}{v_{2}};p\Big{)}=$
$\displaystyle\mathfrak{q}^{-(|\lambda|+|\mu|)}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{1-p^{n}}\frac{\kappa_{n}}{n}\Big{(}\frac{v_{2}}{v_{1}}\Big{)}^{n}A_{\lambda}(q_{1}^{-n},q_{2}^{-n})A_{\mu}(q_{1}^{n},q_{2}^{n})\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{p^{n}}{1-p^{n}}\frac{\kappa_{n}}{n}\Big{(}\frac{v_{1}}{v_{2}}\Big{)}^{n}A_{\mu}(q_{1}^{-n},q_{2}^{-n})A_{\lambda}(q_{1}^{n},q_{2}^{n})\bigg{)}.$
(5.9)

Let us define the normalized $R$-matrix by

$\overline{R}_{\lambda\mu}(z;p):=\frac{R_{\lambda\mu}(z;p)}{R_{\varnothing\varnothing}(z;p)},$
(5.10)

so that $\overline{R}_{\varnothing\varnothing}(z;p)=1$. Then
$\overline{R}_{\lambda\mu}(z;p)$ may also be expressed in terms of the theta
function $\theta_{p}(x)$ by the following lemma:

######  Lemma 5.1.

$\displaystyle\exp$
$\displaystyle\bigg{(}\sum_{n=1}^{\infty}\frac{1}{1-p^{n}}\frac{\kappa_{n}}{n}z^{n}A_{\lambda}(q_{1}^{-n},q_{2}^{-n})A_{\mu}(q_{1}^{n},q_{2}^{n})\bigg{)}$
$\displaystyle=$
$\displaystyle\prod_{(i,j)\in\lambda}\prod_{(k,l)\in\mu}\frac{\Big{(}zq_{3}^{-1}\frac{x_{kl}}{x_{ij}};p\Big{)}_{\infty}\Big{(}zq_{2}^{-1}\frac{x_{kl}}{x_{ij}};p\Big{)}_{\infty}\Big{(}zq_{1}^{-1}\frac{x_{kl}}{x_{ij}};p\Big{)}_{\infty}}{\Big{(}zq_{3}\frac{x_{kl}}{x_{ij}};p\Big{)}_{\infty}\Big{(}zq_{2}\frac{x_{kl}}{x_{ij}};p\Big{)}_{\infty}\Big{(}zq_{1}\frac{x_{kl}}{x_{ij}};p\Big{)}_{\infty}}\cdot\prod_{(i,j)\in\lambda}\frac{\Big{(}\frac{q_{3}z}{x_{ij}};p\Big{)}_{\infty}}{\Big{(}\frac{z}{x_{ij}};p\Big{)}_{\infty}}$
$\displaystyle\cdot\prod_{(i,j)\in\mu}\frac{(zx_{ij};p)_{\infty}}{(zq_{3}^{-1}x_{ij};p)_{\infty}}\cdot\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{1-p^{n}}\frac{z^{n}}{n}\frac{1-q_{3}^{-n}}{(q_{1}^{n}-1)(q_{2}^{n}-1)}\bigg{)}.$
(5.11)

The result is

$\displaystyle\overline{R}_{\lambda\mu}\Big{(}\frac{v_{1}}{v_{2}};p\Big{)}=$
$\displaystyle\mathfrak{q}^{-(|\lambda|+|\mu|)}\prod_{(i,j)\in\lambda}\prod_{(k,l)\in\mu}\frac{\theta_{p}\Big{(}q_{1}^{-1}\frac{v_{2}}{v_{1}}\frac{x_{kl}}{x_{ij}}\Big{)}\theta_{p}\Big{(}q_{2}^{-1}\frac{v_{2}}{v_{1}}\frac{x_{kl}}{x_{ij}}\Big{)}\theta_{p}\Big{(}q_{3}^{-1}\frac{v_{2}}{v_{1}}\frac{x_{kl}}{x_{ij}}\Big{)}}{\theta_{p}\Big{(}q_{1}\frac{v_{2}}{v_{1}}\frac{x_{kl}}{x_{ij}}\Big{)}\theta_{p}\Big{(}q_{2}\frac{v_{2}}{v_{1}}\frac{x_{kl}}{x_{ij}}\Big{)}\theta_{p}\Big{(}q_{3}\frac{v_{2}}{v_{1}}\frac{x_{kl}}{x_{ij}}\Big{)}}$
$\displaystyle\cdot\prod_{(i,j)\in\lambda}\frac{\theta_{p}(\frac{v_{2}}{v_{1}}\frac{q_{3}}{x_{ij}})}{\theta_{p}(\frac{v_{2}}{v_{1}}\frac{1}{x_{ij}})}\cdot\prod_{(i,j)\in\mu}\frac{\theta_{p}(x_{ij}\frac{v_{2}}{v_{1}})}{\theta_{p}(q_{1}q_{2}x_{ij}\frac{v_{2}}{v_{1}})}.$
(5.12)

By the inversion formula of the theta function (1.12), we can check the
unitarity of the normalized $R$ matrix;

$\overline{R}_{\mu\lambda}(z^{-1};p)=\overline{R}_{\lambda\mu}(z;p)^{-1}.$
(5.13)

We can derive the relation of the $R$ matrix for the vertical Fock
representation and the elliptic Nekrasov factor. Recall that we have obtained

$\displaystyle R_{\lambda\mu}(z;p)=$
$\displaystyle\mathfrak{q}^{-(|\lambda|+|\mu|)}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{(1-q_{2}^{-n})(1-\mathfrak{q}^{-2n})}{(1-p^{n})(1-q_{1}^{n})}\sum_{i,j=1}^{\infty}\Big{(}x_{i,\lambda_{i}}x_{j,\mu_{j}}^{-1}z\Big{)}^{-n}\bigg{)}$
$\displaystyle\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{n}\frac{q_{1}^{n}p^{n}(1-q_{2}^{n})(1-\mathfrak{q}^{2n})}{(1-p^{n})(1-q_{1}^{n})}\sum_{i,j=1}^{\infty}\Big{(}x_{i,\lambda_{i}}x_{j,\mu_{j}}^{-1}z\Big{)}^{n}\bigg{)}$
$\displaystyle=$
$\displaystyle\mathfrak{q}^{-(|\lambda|+|\mu|)}\prod_{i,j=1}^{\infty}\frac{\Gamma(q_{1}^{\mu_{j}-\lambda_{i}}q_{2}^{j-i}z^{-1};q_{1},p)\Gamma(\mathfrak{q}^{-2n}q_{1}^{\mu_{j}-\lambda_{i}}q_{2}^{j-i-1}z^{-1};q_{1},p)}{\Gamma(q_{1}^{\mu_{j}-\lambda_{i}}q_{2}^{j-i-1}z^{-1};q_{1},p)\Gamma(\mathfrak{q}^{-2n}q_{1}^{\mu_{j}-\lambda_{i}}q_{2}^{j-i}z^{-1};q_{1},p)}.$
(5.14)

On the other hand the elliptic Nekrasov factor is (cf. (1.1));

$\mathcal{N}_{\lambda\mu}(u|q,t,p)=\prod_{i,j=1}^{\infty}\frac{\Gamma(uq^{\lambda_{j}-\mu_{i}}t^{i-j};q,p)}{\Gamma(uq^{\lambda_{j}-\mu_{i}}t^{i-j+1};q,p)}\frac{\Gamma(ut^{i-j+1};q,p)}{\Gamma(ut^{i-j};q,p)}.$
(5.15)

Hence with $t=q_{2}^{-1}$ we see

$\frac{\mathcal{N}_{\lambda\mu}(z|q,t,p)}{\mathcal{N}_{\lambda\mu}(\mathfrak{q}^{-2}z|q,t,p)}=\mathfrak{q}^{|\lambda|+|\mu|}\frac{R_{\mu\lambda}(z^{-1};p)}{R_{\varnothing\varnothing}(z^{-1};p)}=\mathfrak{q}^{|\lambda|+|\mu|}\overline{R}_{\mu\lambda}(z^{-1};p).$
(5.16)

By using the combinatorial identity (see Appendix E of [3])

$\prod_{(i,j)\in\lambda}q^{\mu_{i}-j}\prod_{(i,j)\in\mu}q^{-\lambda_{i}+j-1}=\prod_{(i,j)\in\mu}q^{\mu_{i}-j}\prod_{(i,j)\in\lambda}q^{-\lambda_{i}+j-1}$
(5.17)

and the inversion formula (1.12), we can prove

$\frac{\mathcal{N}_{\lambda\mu}(z|q_{1},q_{2},p)}{\mathcal{N}_{\mu\lambda}(\mathfrak{q}^{-2}z^{-1}|q_{1},q_{2},p)}=z^{|\mu|+|\lambda|}\mathfrak{q}^{|\mu|+|\lambda|}\frac{f_{\lambda}(q_{1},q_{2})}{f_{\mu}(q_{1},q_{2})},$
(5.18)

where the framing factor $f_{\lambda}(q_{1},q_{2})$ is defined by (4.15). The
formula (5.18) also confirms the unitarity of the (normalized) $R$ matrix;

$\displaystyle\overline{R}_{\mu\lambda}(z^{-1};p)=\mathfrak{q}^{-|\mu|-|\lambda|}\frac{\mathcal{N}_{\lambda\mu}(z|q_{1},q_{2},p)}{\mathcal{N}_{\lambda\mu}(\mathfrak{q}^{-2}z|q_{1},q_{2},p)}=\mathfrak{q}^{|\mu|+|\lambda|}\frac{\mathcal{N}_{\mu\lambda}(\mathfrak{q}^{-2}z^{-1}|q_{1},q_{2},p)}{\mathcal{N}_{\mu\lambda}(z^{-1}|q_{1},q_{2},p)}=\overline{R}_{\lambda\mu}(z;p)^{-1}.$
(5.19)

We can also show that the Fock $R$-matrix constructed above is obtained as the
coefficient resulting from interchanging the intertwiners and the dual
intertwiners themselves. The normal ordering of the vertex operators produces
the elliptic Nekrasov factors. Taking the difference of the zero mode factors
into account and using (5.18), we arrive at;

$\displaystyle\Psi^{*}_{\mu}(w;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\Upsilon\Big{(}\mathfrak{q}^{-1}\Big{|}\frac{v}{w};0\Big{)}\Psi_{\lambda}(v;p)\Psi^{*}_{\mu}(w;p),$
(5.20) $\displaystyle\Psi_{\lambda}(v;p)\Psi_{\mu}(w;p)$
$\displaystyle=\overline{R}_{\lambda\mu}\Big{(}\frac{v}{w};p_{*}\Big{)}\cdot\Upsilon(\mathfrak{q}^{-2}|\frac{v}{w};p_{*})\cdot\Psi_{\mu}(w;p)\Psi_{\lambda}(v;p),$
(5.21) $\displaystyle\Psi^{*}_{\lambda}(v;p)\Psi^{*}_{\mu}(w;p)$
$\displaystyle=\overline{R}_{\lambda\mu}\Big{(}\frac{v}{w};p\Big{)}^{-1}\Upsilon\left(1\Big{|}\frac{v}{w};p\right)\Psi^{*}_{\mu}(w;p)\Psi^{*}_{\lambda}(v;p),$
(5.22)

where

$\displaystyle\Upsilon(\alpha|z;p)$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{\alpha^{n}}{(1-p^{n})(1-q_{1}^{n})(1-q_{2}^{n})}(z^{n}-z^{-n})\bigg{)}$
$\displaystyle\qquad\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{n}\frac{p_{*}^{n}\alpha^{-n}}{(1-p^{n})(1-q_{1}^{n})(1-q_{2}^{n})}(z^{n}-z^{-n})\bigg{)}$
$\displaystyle=\frac{G_{2}(\alpha z^{-1};p,q_{1},q_{2})}{G_{2}(\alpha
z;p,q_{1},q_{2})}.$ (5.23)

The equations (5.20) – (5.22) are elliptic generalization of eqs.(40) – (42)
in [7]. It is remarkable that the exchange relation between
$\Psi_{\lambda}(v;p)$ and $\Psi^{*}_{\mu}(w;p)$ is undeformed. The elliptic
parameter for the exchange relation of the intertwiner is shifted $p\to
p_{*}$. Note also that

$\Upsilon(\alpha|z^{-1};p)=\Upsilon(\alpha|z;p)^{-1}.$ (5.24)

The emergence of the double elliptic gamma function $G_{2}$ is quite amusing,
since it also appears some of computations in six dimensional supersymmetric
gauge theory (note that it is symmetric in $(p,q_{1},q_{2})$) and topological
strings [30], [31], [43], [44], [49].

## § 6 Elliptic quantum Knizhnik-Zamolodchikov Equation

In the elliptic case the quantum Knizhnik-Zamolodchikov (q-KZ) equation is a
difference equation for the trace of intertwining operators [16], [17], [40],
which is an analogue of the genus one conformal block of two dimensional
conformal field theory;

$\mathrm{Tr}\left[q^{L_{0}}\phi_{1}(z_{1})\cdots\phi_{n}(z_{n})\right].$ (6.1)

Here $q=e^{2\pi i\tau}$ (not to be confused with the parameter of the DIM
algebra), $\tau$ is the modulus of the torus and $L_{0}$ is the zero mode of
the Virasoro algebra. Contrary to the case of the vacuum expectation values
(matrix elements) of the product of intertwining operators, which corresponds
to the genus zero conformal block on $\mathbb{P}^{1}$, the shift parameter is
not fixed for the trace.

Let us consider the trace of the intertwining operators111111 For simplicity
in this section we will suppress the $p$-dependence of the intertwiners. ;

$\displaystyle\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu}):=\mathrm{Tr}_{\mathcal{H}_{u}^{(N)}}\left[\widetilde{Q}^{d_{1}}Q^{d_{2}}\Psi^{*}_{\mu_{1}}(w_{1})\cdots\Psi^{*}_{\mu_{n}}(w_{n})\Psi_{\lambda_{1}}(z_{1})\cdots\Psi_{\lambda_{n}}(z_{n})\right],$
(6.2)

where $d_{1}$ and $d_{2}$ are the grading operators that count the degree of
the horizontal and the vertical parameters (see section 2.3 of [13]). In
particular

$\Psi_{\lambda}(Qz)=Q^{d_{2}}\Psi_{\lambda}(z)Q^{-d_{2}},\qquad\Psi^{*}_{\mu}(Qw)=Q^{d_{2}}\Psi^{*}_{\mu}(w)Q^{-d_{2}}.$
(6.3)

Note that we can express the trace of the intertwining operators (6.2) by the
diagram in Figure 3 below.

$u$$(1,N)$$z_{n}$$(0,1)$$\lambda_{n}$$\dots$$z_{1}$$(0,1)$$\lambda_{1}$$\mu_{n}$$w_{n}$$(0,1)$$\dots$$\mu_{1}$$w_{1}$$(0,1)$$\widetilde{Q}^{d_{1}}Q^{d_{2}}$$(1,N)$$u$
Figure 3. Diagram representing the trace
$\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu})$. The left and the
right ends of the horizontal line are identified.

When we take the trace, the initial and the final Fock spaces must have the
same level $N$ and the same spectral parameter $u$. Hence the number of the
intetwiners and the dual intertwiners should agree. Since we can shift the
horizontal spectral parameter by $\widetilde{Q}^{d_{1}}$, we make the spectral
parameters coincide by tuning the parameter $\widetilde{Q}$ as

$\widetilde{Q}=\prod_{i=1}^{n}\frac{w_{i}}{z_{i}}.$ (6.4)

Hence there remains a free parameter $Q$. We can derive a difference equation
with the shift parameter $Q$ as follows; let us first consider the case
$z_{k}\to Qz_{k}$. By using (5.21) we can move $\Psi_{\lambda_{k}}(Qz_{k})$ to
the rightmost position in the trace, then by the cyclic property of the trace
it is moved to the leftmost position. Here it is important that this operation
causes the change of the level and the spectral parameter of each intertwiner
and dual intertwiner. We need the compensating factor $\mathcal{C}$ associated
with it. Then the commutation with $\widetilde{Q}^{d_{1}}$ adjusts the zero
mode factor of $\Psi_{\lambda_{k}}(Qz_{k})$ appropriately by the scaling of
the horizontal spectral parameter $u$ and the commutation with $Q^{d_{2}}$
cancels the $Q$-shift of $z_{k}$. Finally by using (5.20) and (5.21) again we
can move $\Psi_{\lambda_{k}}(z_{k})$ to the original position. Note that
(5.20) and (5.21) already take care of the change of the level and the
spectral parameter associated with the exchange of the intertwiners.
Consequently no additional factors arise when we use them. After all these
operations we arrive at

$\displaystyle Q^{z_{k}\frac{\partial}{\partial
z_{k}}}\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu})=$
$\displaystyle~{}\mathcal{C}\cdot\prod_{j=1}^{n}\Upsilon\left(\mathfrak{q}^{-1}|\frac{z_{k}}{w_{j}};0\right)^{-1}\prod_{i<k}\overline{R}_{\lambda_{i}\lambda_{k}}\left(\frac{z_{i}}{z_{k}};p_{*}\right)^{-1}\Upsilon\left(\mathfrak{q}^{-2}|\frac{z_{k}}{z_{i}};p_{*}\right)$
$\displaystyle\quad\prod_{k<j}\overline{R}_{\lambda_{k}\lambda_{j}}\left(\frac{Qz_{k}}{z_{j}};p_{*}\right)\Upsilon\left(\mathfrak{q}^{-2}|\frac{Qz_{k}}{z_{j}};p_{*}\right)\cdot\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu}).$
(6.5)

Similarly in the case $w_{k}\to Qw_{k}$, we can use (5.20) and (5.22) and
obtain

$\displaystyle Q^{w_{k}\frac{\partial}{\partial
w_{k}}}\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu})=$
$\displaystyle~{}\mathcal{C}^{*}\cdot\prod_{i=1}^{n}\Upsilon\left(\mathfrak{q}^{-1}|\frac{z_{i}}{Qw_{k}};0\right)\prod_{k<\ell}\overline{R}_{\mu_{\ell}\mu_{k}}\left(\frac{w_{\ell}}{Qw_{k}};p\right)\Upsilon\left(1|\frac{Qw_{k}}{w_{\ell}};p\right)$
$\displaystyle\quad\prod_{\ell<k}\overline{R}_{\mu_{k}\mu_{\ell}}\left(\frac{w_{k}}{w_{\ell}};p\right)^{-1}\Upsilon\left(1|\frac{w_{k}}{w_{\ell}};p\right)\cdot\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu}),$
(6.6)

with the factor $\mathcal{C}^{*}$ for the move of $\Psi^{*}_{\mu_{k}}(Qw_{k})$
from the rightmost to the leftmost in the trace. We can regard these
$Q$-difference equations as a generalization of elliptic $q$-KZB equation to
DIM algebra.

### 6.1 Computation of $\mathcal{C}$

We define $\mathcal{C}$ as follows :

$\displaystyle\mathrm{Tr}\bigg{[}$
$\displaystyle\widetilde{Q}^{d_{1}}Q^{-d_{1}}Q^{d_{2}}\Psi^{*}_{\mu_{1}}(w_{1})\cdots\Psi^{*}_{\mu_{n}}(w_{n})\Psi_{\lambda_{1}}(z_{1})\cdots\stackrel{{\scriptstyle
k}}{{\vee}}\cdots\Psi_{\lambda_{n}}(z_{n})\Psi_{\lambda_{k}}(Qz_{k})\bigg{]}$
$\displaystyle=\mathcal{C}\cdot\mathrm{Tr}\bigg{[}\widetilde{Q}^{d_{1}}Q^{d_{2}}\Psi_{\lambda_{k}}(z_{k})\Psi^{*}_{\mu_{1}}(w_{1})\cdots\Psi^{*}_{\mu_{n}}(w_{n})\Psi_{\lambda_{1}}(z_{1})\cdots\stackrel{{\scriptstyle
k}}{{\vee}}\cdots\Psi_{\lambda_{n}}(z_{n})\bigg{]}.$ (6.7)

Recall that when $\Psi_{\lambda}(z)$ and $\Psi^{*}_{\mu}(w)$ act on the Fock
space of level $N$ with the spectral parameter $u$, their zero mode factors
are

$\Psi_{\lambda}(z)\sim(-z)^{-N|\lambda|}u^{|\lambda|}f_{\lambda}^{-N-1},\qquad\Psi^{*}_{\mu}(w)\sim(-w)^{N|\mu|}u^{-|\mu|}f_{\mu}^{N-1}.$
(6.8)

Chasing the change of the level and the spectral parameter, we find the total
zero mode factor on the left hand side is

$\displaystyle\mathcal{A}=$
$\displaystyle\bigg{[}(-Qz_{k})^{-N|\lambda_{k}|}u^{|\lambda_{k}|}f^{-N-1}_{\lambda_{k}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=k+1}^{n}(-z_{i})^{-(N+n-i+1)|\lambda_{i}|}\Big{(}u\cdot(-Qz_{k})\prod_{j=i+1}^{n}(-z_{j})\Big{)}^{|\lambda_{i}|}f^{-N-2-n+i}_{\lambda_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{k-1}(-z_{i})^{-(N+n-i)|\lambda_{i}|}\Big{(}u\cdot(-Qz_{k})\prod_{j=k+1}^{n}(-z_{j})\prod_{l=i+1}^{k-1}(-z_{l})\Big{)}^{|\lambda_{i}|}f^{-N-n+i-1}_{\lambda_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{n}(-w_{i})^{(N+i)|\mu_{i}|}\Big{(}\frac{u\cdot(-Qz_{k})\prod_{j\neq
k}(-z_{j})}{\prod_{j=i+1}^{n}(-w_{j})}\Big{)}^{-|\mu_{i}|}f^{N+i-1}_{\mu_{i}}\bigg{]},$
(6.9)

where the initial level and spectral parameter of the horizontal Fock space
are $N$ and $u$, respectively.

Next we are going to investigate the right hand side of (6.1). Similar
consideration gives the total zero mode factor;

$\displaystyle\mathcal{B}=$
$\displaystyle\bigg{[}(-z_{k})^{-(N-1)|\lambda_{k}|}\Big{(}\frac{u\prod_{j\neq
k}(-z_{j})}{\prod_{l=1}^{n}(-w_{l})}\Big{)}^{|\lambda_{k}|}f^{-N}_{\lambda_{k}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=k+1}^{n}(-z_{i})^{-(N+n-i)|\lambda_{i}|}\Big{(}u\prod_{j=i+1}^{n}(-z_{j})\Big{)}^{|\lambda_{i}|}f^{-N-n+i-1}_{\lambda_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{k-1}(-z_{i})^{-(N+n-i-1)|\lambda_{i}|}\Big{(}u\prod_{i=k+1}^{n}(-z_{i})\prod_{j=i+1}^{k-1}(-z_{j})\Big{)}^{|\lambda_{i}|}f^{-N-n+i}_{\lambda_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{n}(-w_{i})^{(N+i-1)|\mu_{i}|}\Big{(}\frac{u\prod_{j\neq
k}(-z_{j})}{\prod_{l=i+1}^{n}(-w_{l})}\Big{)}^{-|\mu_{i}|}f^{N+i-2}_{\mu_{i}}\bigg{]}.$
(6.10)

After making the exchange with $Q^{d_{2}}$ which shifts all the vertical
spectral parameters in $\mathcal{A}$ and $\mathcal{B}$;
$Q^{d_{2}}\mathcal{A}=\mathcal{A}^{\prime}Q^{d_{2}}$and
$Q^{d_{2}}\mathcal{B}=\mathcal{B}^{\prime}Q^{d_{2}}$, we obtain

$\displaystyle\mathcal{C}=\frac{\mathcal{A}^{\prime}}{\mathcal{B}^{\prime}}=\frac{\prod_{i=1}^{n}w_{i}^{|\mu_{i}|}f_{\mu_{i}}}{\prod_{i=1}^{n}z^{|\lambda_{i}|}_{i}f_{\lambda_{i}}}\cdot
z_{k}^{\sum_{i=1}^{n}|\lambda_{i}|-\sum_{i=1}^{n}|\mu_{i}|}\widetilde{Q}^{|\lambda_{k}|}\cdot
Q^{\sum_{i=1}^{n}|\lambda_{i}|-\sum_{i=1}^{n}|\mu_{i}|}Q^{-(N+1)|\lambda_{k}|}.$
(6.11)

### 6.2 Computation of $\mathcal{C}^{*}$

Now, we are going to compute the coefficient $\mathcal{C}^{*}$ defined by

$\displaystyle\mathrm{Tr}$
$\displaystyle\bigg{[}\widetilde{Q}^{d_{1}}Q^{d_{1}}Q^{d_{2}}\Psi^{*}_{\mu_{1}}(w_{1})\cdots\stackrel{{\scriptstyle
k}}{{\vee}}\cdots\Psi^{*}_{\mu_{n}}(w_{n})\Psi_{\lambda_{1}}(z_{1})\cdots\Psi_{\lambda_{n}}(z_{n})\Psi^{*}_{\mu_{k}}(Qw_{k})\bigg{]}$
$\displaystyle=$
$\displaystyle\mathcal{C}^{*}\cdot\mathrm{Tr}\bigg{[}\widetilde{Q}^{d_{1}}Q^{d_{2}}\Psi^{*}_{\mu_{k}}(w_{k})\Psi^{*}_{\mu_{1}}(w_{1})\cdots\stackrel{{\scriptstyle
k}}{{\vee}}\cdots\Psi^{*}_{\mu_{n}}(w_{n})\Psi_{\lambda_{1}}(z_{1})\cdots\Psi_{\lambda_{n}}(z_{n})\bigg{]}.$
(6.12)

The total zero mode factor on the left hand side is

$\displaystyle\mathcal{M}=$
$\displaystyle\bigg{[}(-Qw_{k})^{N|\mu_{k}|}u^{-|\mu_{k}|}f^{N-1}_{\mu_{k}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{n}(-z_{i})^{-(N-1+n-i)|\lambda_{i}|}\Big{(}-\frac{u}{Qw_{k}}\prod_{j=i+1}^{n}(-z_{j})\Big{)}^{|\lambda_{i}|}f^{-(N+n-i)}_{\lambda_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=k+1}^{n}(-w_{i})^{(N+i-1)|\mu_{i}|}\Big{(}\frac{u\prod_{j=1}^{n}(-z_{j})}{(-Qw_{k})\prod_{l=i+1}^{n}(-w_{l})}\Big{)}^{-|\mu_{i}|}f^{N+i-2}_{\mu_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{k-1}(-w_{i})^{(N+i)|\mu_{i}|}\Big{(}\frac{u\prod_{j=1}^{n}(-z_{j})}{Q\prod_{l=i+1}^{n}(-w_{l})}\Big{)}^{-|\mu_{i}|}f^{N+i-1}_{\mu_{i}}\bigg{]}.$
(6.13)

On the other hand the zero mode factor on the right hand side of (6.2) is

$\displaystyle\mathcal{L}=$
$\displaystyle\bigg{[}(-w_{k})^{(N+1)|\mu_{k}|}\Big{(}\frac{u\prod_{j=1}^{n}(-z_{j})}{\prod_{l\neq
k}(-w_{l})}\Big{)}^{-|\mu_{k}|}f^{N}_{\mu_{k}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{n}(-z_{i})^{-(N+n-i)|\lambda_{i}|}\Big{(}u\prod_{j=i+1}^{n}(-z_{j})\Big{)}^{|\lambda_{i}|}f^{-(N+n-i)-1}_{\lambda_{i}}D_{\lambda_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=k+1}^{n}(-w_{i})^{(N+i)|\mu_{i}|}\Big{(}\frac{u\prod_{j=1}^{n}(-z_{j})}{\prod_{l=i+1}^{n}(-w_{l})}\Big{)}^{-|\mu_{i}|}f^{N+i-1}_{\mu_{i}}\bigg{]}$
$\displaystyle\cdot\bigg{[}\prod_{i=1}^{k-1}(-w_{i})^{(N+i+1)|\mu_{i}|}\Big{(}\frac{u\prod_{j=1}^{n}(-z_{j})}{\prod_{j=i+1}^{k-1}(-w_{j})\prod_{l=k+1}^{n}(-w_{l})}\Big{)}^{-|\mu_{i}|}f^{N+i}_{\mu_{i}}\bigg{]}.$
(6.14)

As before defining $\mathcal{M}^{\prime}=Q^{d_{2}}\mathcal{M}Q^{-d_{2}}$ and
$\mathcal{L}^{\prime}=Q^{d_{2}}\mathcal{L}Q^{-d_{2}}$, we obtain

$\displaystyle\mathcal{C}^{*}=\frac{\mathcal{M}^{\prime}}{\mathcal{L}^{\prime}}=\frac{\prod_{i=1}^{n}z_{i}^{|\lambda_{i}|}f_{\lambda_{i}}}{\prod_{i=1}^{n}w^{|\mu_{i}|}_{i}f_{\mu_{i}}}w_{k}^{\sum_{i=1}^{n}|\mu_{i}|-\sum_{i=1}^{n}|\lambda_{i}|}\widetilde{Q}^{-|\mu_{k}|}Q^{(N-1)|\mu_{k}|}Q^{\sum_{i=1}^{n}|\mu_{i}|-\sum_{i=1}^{n}|\lambda_{i}|}.$
(6.15)

In [7] a pair of the q-KZ equations for the trace of intertwiners is derived
and a solution is given rather explicitly (see eq.(84)). From the results we
have derived in section 5, it is easy to guess a generalization of the
solution in [7]. For example we can replace the Nekrasov factor (denoted as
$G_{\alpha\beta}$ in [7]) by the elliptic one. To define a generalization of
the remaining factor related to $\Upsilon(\alpha|z;p)$, let us introduce

$\displaystyle\widetilde{\Upsilon}(\alpha|z;p,Q)$ $\displaystyle:=$
$\displaystyle G_{3}(\alpha z;p,Q,q_{1},q_{2})\cdot G_{3}(\alpha
Qz^{-1};p,Q,q_{1},q_{2})$ (6.16) $\displaystyle=$
$\displaystyle\exp\left(\sum_{n=1}^{\infty}\frac{\alpha^{n}(z^{n}+Q^{n}z^{-n})}{n(1-p^{n})(1-Q^{n})(1-q_{1}^{n})(1-q_{2}^{n})}\right)$
$\displaystyle~{}~{}\exp\left(-\sum_{n=1}^{\infty}\frac{\alpha^{-n}p_{*}^{n}(z^{n}+Q^{n}z^{-n})}{n(1-p^{n})(1-Q^{n})(1-q_{1}^{n})(1-q_{2}^{n})}\right).$

Then the recursion relation (1.17) among multiple elliptic gamma functions
implies

$\widetilde{\Upsilon}(\alpha|Qz;p,Q)=\Upsilon(\alpha|z;p)^{-1}\cdot\widetilde{\Upsilon}(\alpha|z;p,Q).$
(6.17)

We define the building blocks

$\displaystyle\Theta_{\lambda\mu}(z|p,Q):=$
$\displaystyle\left(\prod_{k=0}^{\infty}\mathcal{N}_{\lambda\mu}(Q^{k}z|q_{1},q_{2},p)\mathcal{N}_{\mu\lambda}(\mathfrak{q}^{-2}Q^{k+1}z^{-1}|q_{1},q_{2},p)\right)\widetilde{\Upsilon}(\mathfrak{q}^{-1}|\mathfrak{q}z;p,Q),$
$\displaystyle\Phi_{\lambda\mu}(z|p,Q):=$
$\displaystyle\left(\prod_{k=0}^{\infty}\mathcal{N}_{\lambda\mu}(\mathfrak{q}^{-2}Q^{k}z|q_{1},q_{2},p)\mathcal{N}_{\mu\lambda}(\mathfrak{q}^{-2}Q^{k+1}z^{-1}|q_{1},q_{2},p)\right)\widetilde{\Upsilon}(\mathfrak{q}^{-2}|z;p,Q),$
(6.18) $\displaystyle\overline{\Phi}_{\lambda\mu}(z|p,Q):=$
$\displaystyle\left(\prod_{k=0}^{\infty}\mathcal{N}_{\lambda\mu}(Q^{k}z|q_{1},q_{2},p)\mathcal{N}_{\mu\lambda}(Q^{k+1}z^{-1}|q_{1},q_{2},p)\right)\widetilde{\Upsilon}(1|z;p,Q).$

It should be not an accident that, if we rewrite
$\widetilde{\Upsilon}(\alpha|z;p,Q)$ in terms of the triple elliptic gamma
function $G_{3}(z;p,Q,q_{1},q_{2})$, the argument $z$ agrees with those of
$\mathcal{N}_{\lambda\mu}$ and $\mathcal{N}_{\mu\lambda}$ with $k=0$. Then our
claim is that

$\mathfrak{G}_{n}(\vec{z},\vec{w}|\vec{\lambda},\vec{\mu})=\mathcal{P}\cdot\frac{\displaystyle{\prod_{i,j=1}^{n}}\Theta_{\lambda_{i}\mu_{j}}(\mathfrak{q}^{-1}z_{i}/w_{j}|0,Q)}{\displaystyle{\prod_{i<j}}\Phi_{\lambda_{j}\lambda_{i}}(z_{j}/z_{i}|p_{*},Q)\displaystyle{\prod_{k<\ell}}\overline{\Phi}_{\mu_{\ell}\mu_{k}}(w_{\ell}/w_{k}|p,Q)},$
(6.19)

where the monomial prefactor

$\mathcal{P}:=\prod_{i=1}^{n}z_{i}^{-(N+n-i)|\lambda_{i}|-\sum_{i=1}^{n}|\mu_{i}|+\sum_{j<i}|\lambda_{j}|}\cdot
w_{i}^{(N+i)|\mu_{i}|+\sum_{j<i}|\mu_{j}|}$ (6.20)

is introduced for the matching of the $Q$ dependence of the factor
$\mathcal{C}$ and $\mathcal{C}^{*}$ evaluated above. The difference equation
for (6.19) is derived from those for the building blocks
$\Theta_{\lambda\mu}$, $\Phi_{\lambda\mu}$ and $\overline{\Phi}_{\lambda\mu}$.
When the shift parameter is $Q$, by (6.17), a direct computation shows

$\displaystyle\frac{\Theta_{\lambda\mu}(Qz|p,Q)}{\Theta_{\lambda\mu}(z|p,Q)}$
$\displaystyle=$
$\displaystyle\mathcal{N}_{\lambda\mu}(z|q_{1},q_{2},p)^{-1}\mathcal{N}_{\mu\lambda}(\mathfrak{q}^{-2}z^{-1}|q_{1},q_{2},p)\Upsilon(\mathfrak{q}^{-1}|\mathfrak{q}z;p)^{-1}$
(6.21) $\displaystyle=$
$\displaystyle(\mathfrak{q}z)^{-|\lambda|-|\mu|}(f_{\mu}/f_{\lambda})\Upsilon(\mathfrak{q}^{-1}|\mathfrak{q}z;p)^{-1},$
$\displaystyle\frac{\Phi_{\lambda\mu}(Qz|p,Q)}{\Phi_{\lambda\mu}(z|p,Q)}$
$\displaystyle=$
$\displaystyle\mathcal{N}_{\lambda\mu}(\mathfrak{q}^{-2}z|q_{1},q_{2},p)^{-1}\mathcal{N}_{\mu\lambda}(\mathfrak{q}^{-2}z^{-1}|q_{1},q_{2},p)\Upsilon(\mathfrak{q}^{-2}|z;p)^{-1}$
(6.22) $\displaystyle=$ $\displaystyle
z^{-|\lambda|-|\mu|}(f_{\mu}/f_{\lambda})\overline{R}_{\lambda\mu}(z;p)^{-1}\Upsilon(\mathfrak{q}^{-2}|z;p)^{-1},$
$\displaystyle\frac{\overline{\Phi}_{\lambda\mu}(Qz|p,Q)}{\overline{\Phi}_{\lambda\mu}(z|p,Q)}$
$\displaystyle=$
$\displaystyle\mathcal{N}_{\lambda\mu}(z|q_{1},q_{2},p)^{-1}\mathcal{N}_{\mu\lambda}(z^{-1}|q_{1},q_{2},p)\Upsilon(1|z;p)^{-1}$
(6.23) $\displaystyle=$ $\displaystyle
z^{-|\lambda|-|\mu|}(f_{\mu}/f_{\lambda})\overline{R}_{\lambda\mu}(z;p)\Upsilon(1|z;p)^{-1},$

where we have also used (5.16) and (5.18). Hence, taking the power of $Q$
coming from (6.20) into account, we can see (6.19) satisfies the difference
equations (6) and (6) with exactly the same factor $\mathcal{C}$ and
$\mathcal{C}^{*}$. Let us explain the origin of these factors. When we use the
cyclic property of the trace and move $\Psi_{\lambda_{k}}(z_{k})$ or
$\Psi^{*}_{\mu_{k}}(w_{k})$ from right to left, it should be accompanied by
the change of the level and the spectral parameter of the horizontal Fock
space in the definition of the trace (6.2). When we move
$\Psi_{\lambda_{k}}(z_{k})$, the change is
$\mathrm{Tr}_{\mathcal{H}_{u}^{(N)}}\longrightarrow\mathrm{Tr}_{\mathcal{H}_{-z_{k}u}^{(N+1)}}$.
On the other hand, when we move $\Psi^{*}_{\mu_{k}}(w_{k})$ it is
$\mathrm{Tr}_{\mathcal{H}_{u}^{(N)}}\longrightarrow\mathrm{Tr}_{\mathcal{H}_{-u/w_{k}}^{(N-1)}}$.
We can check the total changes of the zero modes are exactly given by
$\mathcal{C}$ and $\mathcal{C}^{*}$ up to the additional powers of $Q$.
Finally the power of $\widetilde{Q}$ comes from the exchange with
$\widetilde{Q}^{d_{1}}$.

Since the coupled equations (6) and (6) are $Q$-difference equations, there is
an ambiguity of “integration constant” or $Q$-periodic function in general.
Let us illustrate this point in the simplest example of $n=1$ with empty
partitions;

$\mathfrak{G}_{1}(z,w|\varnothing,\varnothing)=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\frac{(z/w)^{n}+Q^{n}\mathfrak{q}^{-2n}(w/z)^{n}}{(1-Q^{n})(1-q_{1}^{n})(1-q_{2}^{n})}\right).$
(6.24)

Unfortunately the solution is $p$ independent in this case. The parameter $Q$
is identified with the gauge coupling (the parameter of instanton expansion),
which is consistent with AGT dictionary [1]. Hence the substitution of $Q=0$
gives the perturbative part;

$\mathfrak{G}_{1}^{\mathrm{pert}}=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\frac{(z/w)^{n}}{(1-q_{1}^{n})(1-q_{2}^{n})}\right).$
(6.25)

Then the instanton part is

$\mathfrak{G}_{1}^{\mathrm{inst}}=\mathfrak{G}_{1}/\mathfrak{G}_{1}^{\mathrm{pert}}=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\frac{Q^{n}((z/w)^{n}+\mathfrak{q}^{-2n}(w/z)^{n})}{(1-Q^{n})(1-q_{1}^{n})(1-q_{2}^{n})}\right).$
(6.26)

On the other hand in this case we can compute the trace directly [33], [62],
[4], [63], [27]. For example, we quote the formula from [12];

$Z^{\mathrm{inst}}(m,Q;q_{1},q_{2})=\exp\left(\sum_{n=1}^{n}\frac{Q^{n}}{nm^{n}}\frac{(m^{n}-q_{1}^{n})(m^{n}-q_{2}^{n})}{(1-Q^{n})(1-q_{1}^{n})(1-q_{2}^{n})}\right),$
(6.27)

where $m$ is the (exponentiated) mass of the adjoint matter hypermultiplet of
$\mathcal{N}=2^{*}$ $U(1)$ gauge theory. Identifying $m=z/w$, we find a
complete agreement of (6.27) and (6.26) up to $m$ independent factor.

Acknowledgements

We would like to thank H. Awata, T. Kimura, A. Mironov, A. Morozov, H.
Nakajima, Y. Ohkubo, Y. Yoshida and Y. Zenkevich for useful discussions. In
particular, we are grateful to H. Konno and J. Shiraishi for sharing their
insights on the elliptic algebra. Our work is supported in part by Grants-in-
Aid for Scientific Research (# 18K03274) (H.K.). The work of P.C. is supported
by the MEXT Scholarship.

##  Appendix A  Normalization of the intertwiner

In this appendix, we provide a detailed computation of the normalization
factor $\mathcal{G}_{\lambda}$ of the intertwiner. To set the stage, we first
fix notations and provide some definitions. For a non-negative integer $m$ we
define

$\displaystyle
B_{m}(v):=\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}q_{2}^{nm}\mathfrak{q}^{n/2}\widetilde{a}_{-n}v^{n}\bigg{)}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\widetilde{a}_{n}v^{-n}q_{2}^{-nm}\mathfrak{q}^{-3n/2}\bigg{)}.$
(A.1)

For a partition $\lambda$, and for $n\geq\ell(\lambda)$ we define

$\displaystyle\widetilde{\mathbb{I}}^{[n]}_{\lambda}(v):=\widetilde{\mathbb{I}}_{\lambda_{1}-1}(v;p)\widetilde{\mathbb{I}}_{\lambda_{2}-1}(q_{2}v;p)\cdots\widetilde{\mathbb{I}}_{\lambda_{n}-1}(q_{2}^{n-1}v;p),$
(A.2)

where

$\displaystyle\widetilde{\mathbb{I}}_{m}(z;p)=~{}:\mathrel{\prod_{l=1}^{\infty}\eta(q_{1}^{m+l}z;p)}:$
(A.3)

and $\eta(z;p)$ is the oscillator part of the horizontal representation of
$E(z;p)$ (see (3.44)). Up to the zero mode factor,
$\widetilde{\mathbb{I}}_{m}(z;p)$ gives a component of the intertwiner for the
vector representation [9], [13]. From the definition it is straightforward to
show that for $n\geq\ell(\lambda)$

$\displaystyle:\mathrel{\widetilde{\mathbb{I}}^{[n]}_{\lambda}(v)B_{n}(v)}:=:\mathrel{\widetilde{\mathbb{I}}^{[n+1]}_{\lambda}(v)B_{n+1}(v)}:.$
(A.4)

######  Remark A.1.

We can rewrite the intertwiner (4.11) as

$\displaystyle\Psi_{\lambda}(v;p)=$ $\displaystyle
z_{\lambda}\mathcal{G}^{-1}_{\lambda}:\mathrel{\widetilde{\mathbb{I}}^{[\ell(\lambda)]}_{\lambda}(v)B_{\ell(\lambda)}(v)}:$
$\displaystyle=$ $\displaystyle
z_{\lambda}\mathcal{G}^{-1}_{\lambda}:\mathrel{\,\widetilde{\mathbb{I}}_{\lambda_{1}-1}(v;p)\widetilde{\mathbb{I}}_{\lambda_{2}-1}(q_{2}v;p)\cdots\widetilde{\mathbb{I}}_{\lambda_{\ell(\lambda)}-1}(q_{2}^{\ell(\lambda)-1}v;p)B_{\ell(\lambda)}(v)\,}:.$
(A.5)

Then, from (A.4) we get that

$\displaystyle\Psi_{\lambda}(v;p)=z_{\lambda}\mathcal{G}^{-1}_{\lambda}:\mathrel{\widetilde{\mathbb{I}}^{[n]}_{\lambda}(v)B_{n}(v)}:\quad(n\geq\ell(\lambda)).$
(A.6)

Now following [9], we define $\mathcal{G}_{\lambda}$ to be the coefficient
appearing by removing the normal ordering of
$:\mathrel{\widetilde{\mathbb{I}}^{[n]}_{\lambda}(v)B_{n}(v)}:$. More
precisely, for $n>\ell(\lambda)$,

$\displaystyle:\mathrel{\widetilde{\mathbb{I}}^{[n]}_{\lambda}(v)B_{n}(v)}:$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{G}^{[n]}\mathcal{G}_{\lambda}\widetilde{\mathbb{I}}^{[n]}_{\lambda}(v)B_{n}(v),$
(A.7)

where $\mathcal{G}^{[n]}$ is defined by

$\displaystyle:\mathrel{\widetilde{\mathbb{I}}^{[n]}_{\emptyset}(v)B_{n}(v)}:$
$\displaystyle\stackrel{{\scriptstyle\text{def}}}{{=}}\mathcal{G}^{[n]}\widetilde{\mathbb{I}}^{[n]}_{\emptyset}(v)B_{n}(v).$
(A.8)

Namely we factorize the coefficient into the $\lambda$ dependent part
$\mathcal{G}_{\lambda}$ and the $n$ dependent part $\mathcal{G}^{[n]}$. Hence,
$\mathcal{G}_{\lambda}$ should be independent of $n$ as long as
$n>\ell(\lambda)$.

From (A.1) and (A.2), it is straightforward to show that for
$m>\ell(\lambda)$,

$\widetilde{\mathbb{I}}_{\emptyset}^{[m]}(v;p)B_{m}(v)=\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{1-q_{1}^{-n}}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}q_{2}^{n}m\bigg{)}:\mathrel{\widetilde{\mathbb{I}}_{\emptyset}^{[m]}(v;p)B_{m}(v)}:.$
(A.9)

Thus, we obtain that

$\displaystyle\mathcal{G}^{[m]}=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{1-q_{1}^{-n}}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}q_{2}^{n}m\bigg{)}.$
(A.10)

Similarly, replacing $\widetilde{\mathbb{I}}_{\emptyset}^{[m]}(v;p)$ by
$\widetilde{\mathbb{I}}_{\lambda}^{[m]}(v;p)$ we can show

$\displaystyle\mathcal{G}^{[m]}\mathcal{G}_{\lambda}=$
$\displaystyle\prod_{k=1}^{m-1}\exp\bigg{(}-\sum_{n=1}^{\infty}\sum_{l=0}^{k-1}\frac{1}{n}\frac{q_{2}^{n}-1}{1-q_{1}^{-n}}\frac{1-p^{n}}{1-p^{n}_{*}}q_{2}^{n(k-l)}q_{1}^{n(\lambda_{k+1}-\lambda_{l+1})}\bigg{)}$
$\displaystyle\cdot\prod_{k=1}^{m}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}q^{nm}_{2}\frac{1}{1-q_{1}^{-n}}(q_{1}^{\lambda_{k}}q_{2}^{k-1})^{-n}\bigg{)}.$
(A.11)

According to (2) in page 338 of [45], we have the following identity:

$\displaystyle(1-q)\sum_{s\in\lambda}q^{a(s)}t^{\ell(s)+1}=\sum_{i=1}^{n}(t-q^{\lambda_{i}}t^{n+1-i})-(1-t)\sum_{1\leq
i<j\leq n}q^{\lambda_{i}-\lambda_{j}}t^{j-i},$ (A.12)

where $a(s)$ and $\ell(s)$ are the arm-length and the leg-length of the box
$s$ in the partition $\lambda$, respectively. Equivalently,

$\displaystyle\sum_{k=1}^{m-1}\sum_{l=0}^{k-1}q_{2}^{n(k-l)}q_{1}^{n(\lambda_{k+1}-\lambda_{l+1})}$
$\displaystyle=\frac{1}{1-q_{2}^{n}}\bigg{[}\sum_{i=1}^{m}(q_{2}^{n}-q_{1}^{-n\lambda_{i}}q_{2}^{n(m+1-i)})-(1-q_{1}^{-n})\sum_{s\in\lambda}q_{1}^{-na(s)}q_{2}^{n(\ell(s)+1)}\bigg{]}.$
(A.13)

Applying (A.13) to (A), we obtain that

$\displaystyle\mathcal{G}^{[m]}\mathcal{G}_{\lambda}$
$\displaystyle=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1}{1-q_{1}^{-n}}\frac{1-p^{n}}{1-p^{n}_{*}}\bigg{[}\sum_{i=1}^{m}(q_{2}^{n}-q_{1}^{-n\lambda_{i}}q_{2}^{n(m+1-i)})-(1-q_{1}^{-n})\sum_{s\in\lambda}q_{1}^{-na(s)}q_{2}^{n(\ell(s)+1)}\bigg{]}\bigg{)}$
$\displaystyle\quad\cdot\prod_{k=1}^{m}\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}q^{nm}_{2}\frac{1}{1-q_{1}^{-n}}(q_{1}^{\lambda_{k}}q_{2}^{k-1})^{-n}\bigg{)}.$
$\displaystyle=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1}{1-q_{1}^{-n}}\frac{1-p^{n}}{1-p^{n}_{*}}\bigg{[}\sum_{i=1}^{m}(q_{2}^{n}-q_{1}^{-n\lambda_{i}}q_{2}^{n(m+1-i)})-(1-q_{1}^{-n})\sum_{s\in\lambda}q_{1}^{-na(s)}q_{2}^{n(\ell(s)+1)}\bigg{]}\bigg{)}$
$\displaystyle\quad\cdot\exp\bigg{(}\sum_{n=1}^{\infty}\sum_{k=1}^{m}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}\frac{1}{1-q_{1}^{-n}}q_{2}^{n(m-k+1)}q_{1}^{-n\lambda_{k}}\bigg{)}$
$\displaystyle=\exp\bigg{(}\sum_{n=1}^{\infty}\frac{1}{n}\frac{1}{1-q_{1}^{-n}}\frac{1-p^{n}}{1-p^{n}_{*}}\bigg{[}q_{2}^{n}m-(1-q_{1}^{-n})\sum_{s\in\lambda}q_{1}^{-na(s)}q_{2}^{n(\ell(s)+1)}\bigg{]}\bigg{)}.$
(A.14)

So we get that

$\displaystyle\mathcal{G}_{\lambda}=\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{1}{n}\frac{1-p^{n}}{1-p^{n}_{*}}\bigg{[}\sum_{s\in\lambda}q_{1}^{-na(s)}q_{2}^{n(\ell(s)+1)}\bigg{]}\bigg{)}.$
(A.15)

From this we see that $\mathcal{G}_{\lambda}$ does not depend on $m$, as
should be.

##  Appendix B Intertwining relations for $E$ and $F$

The free boson oscillator part of the horizontal representation is

$\displaystyle\eta(z;p)=$
$\displaystyle\exp\bigg{(}\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{-n/2}(1-p^{n})}{(1-p^{n}_{*})(\mathfrak{q}^{n}-\mathfrak{q}^{-n})}\widetilde{a}_{-n}z^{n}\bigg{)}\cdot\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{-n/2}}{\mathfrak{q}^{n}-\mathfrak{q}^{-n}}\widetilde{a}_{n}z^{-n}\bigg{)},$
(B.1) $\displaystyle\xi(z;p)=$
$\displaystyle\exp\bigg{(}-\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{n/2}}{\mathfrak{q}^{n}-\mathfrak{q}^{-n}}\widetilde{a}_{-n}z^{n}\bigg{)}\cdot\exp\bigg{(}\sum_{n=1}^{\infty}\frac{\kappa_{n}}{n}\frac{\mathfrak{q}^{n/2}(1-p^{n}_{*})}{(1-p^{n})(\mathfrak{q}^{n}-\mathfrak{q}^{-n})}\widetilde{a}_{n}z^{-n}\bigg{)}.$
(B.2)

With the standard choice of the zero modes the elliptic currents are given by

$E(z;p)=\frac{(\mathfrak{q}/z)^{N}u}{(1-q_{1})(1-q_{2})}\eta(z;p),\qquad
F(z;p)=\frac{(\mathfrak{q}/z)^{-N}u^{-1}}{(1-q_{1}^{-1})(1-q_{2}^{-1})}\xi(z;p),$
(B.3)

where they act on the horizontal Fock space with level $N$ and the spectral
parameter $u$. Note that the zero modes are independent of the elliptic
parameter $p$. We have the following OPE relations with the intertwiner;

$\displaystyle\eta(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{v}{z}\right)^{n}\frac{(1-p^{n})(1-q_{2}^{n})}{1-p_{*}^{n}}\sum_{i=1}^{\infty}q_{1}^{n\lambda_{i}}q_{2}^{n(i-1)}\right)$
$\displaystyle~{}~{}~{}:\eta(z;p)\Psi_{\lambda}(v;p):,$ (B.4)
$\displaystyle\Psi_{\lambda}(v;p)\eta(z;p)$
$\displaystyle=\exp\left(-\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{z}{v}\right)^{n}\frac{(1-p^{n})(1-q_{2}^{n})}{1-p_{*}^{n}}\sum_{i=1}^{\infty}q_{1}^{-n(\lambda_{i}-1)}q_{2}^{-n(i-1)}\right)$
$\displaystyle~{}~{}~{}:\eta(z;p)\Psi_{\lambda}(v;p):,$ (B.5)
$\displaystyle\xi(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\exp\left(-\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{v}{z}\right)^{n}\mathfrak{q}^{n}(1-q_{2}^{n})\sum_{i=1}^{\infty}q_{1}^{n\lambda_{i}}q_{2}^{n(i-1)}\right)$
$\displaystyle~{}~{}~{}:\xi(z;p)\Psi_{\lambda}(v;p):,$ (B.6)
$\displaystyle\Psi_{\lambda}(v;p)\xi(z;p)$
$\displaystyle=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}\left(\frac{z}{v}\right)^{n}\mathfrak{q}^{n}(1-q_{2}^{n})\sum_{i=1}^{\infty}q_{1}^{-n(\lambda_{i}-1)}q_{2}^{-n(i-1)}\right)$
$\displaystyle~{}~{}~{}:\xi(z;p)\Psi_{\lambda}(v;p):.$ (B.7)

Assume that $\Psi_{\lambda}(v;p)$ maps level $N$ and the spectral parameter
$u$ to $N+1$ and $w$. The intertwining relation for the vacuum component
$F(z;p)\Psi_{\varnothing}(v;p)=\Psi_{\varnothing}(v;p)F(z;p)$ implies

$(\mathfrak{q}/z)^{-N-1}w^{-1}\left(1-\frac{\mathfrak{q}v}{z}\right)=(\mathfrak{q}/z)^{-N}u^{-1}\left(1-\frac{z}{\mathfrak{q}v}\right).$
(B.8)

Hence we obtain the condition $w=-uv$ for the existence of
$\Psi_{\lambda}(v;p)$.

Now let us turn to the intertwining relation for $E(z;p)$;

$\displaystyle E(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\sum_{k=1}^{\ell(\lambda)+1}\langle\lambda+1_{k}|E(z;p_{*})|\lambda\rangle\Psi_{\lambda+1_{k}}(v;p)$
$\displaystyle\qquad+\langle\lambda|K^{-}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)E(z;p),$
(B.9)

Up to the zero mode factors we have

$\displaystyle\eta(z;p)\Psi_{\lambda}(v;p)$
$\displaystyle=\prod_{i=1}^{\infty}\frac{(p_{*}q_{1}^{\lambda_{i}-1}q_{2}^{i-2}(v/z);p_{*})_{\infty}(q_{1}^{\lambda_{i}}q_{2}^{i}(v/z);p_{*})_{\infty}}{(q_{1}^{\lambda_{i}}q_{2}^{i-1}(v/z);p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-1}q_{2}^{i-1}(v/z);p_{*})_{\infty}}:\eta(z;p)\Psi_{\lambda}(v;p):,$
(B.10)

and

$\displaystyle\langle\lambda|K^{-}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)\eta(z;p)$
$\displaystyle=\mathfrak{q}\prod_{i=1}^{\infty}\frac{\theta_{p_{*}}(q_{1}^{-\lambda_{i}}q_{2}^{-i}(z/v))\theta_{p_{*}}(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+2}(z/v))}{\theta_{p_{*}}(q_{1}^{-\lambda_{i}}q_{2}^{-i+1}(z/v))\theta_{p_{*}}(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+1}(z/v))}$
$\displaystyle\prod_{i=1}^{\infty}\frac{(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+1}(z/v);p_{*})_{\infty}(p_{*}q_{1}^{-\lambda_{i}}q_{2}^{-i+1}(z/v);p_{*})_{\infty}}{(q_{1}^{-\lambda_{i}+1}q_{2}^{-i+2}(z/v);p_{*})_{\infty}(p_{*}q_{1}^{-\lambda_{i}}q_{2}^{-i}(z/v);p_{*})_{\infty}}:\eta(z;p)\Psi_{\lambda}(v;p):$
$\displaystyle=\mathfrak{q}\prod_{i=1}^{\infty}\frac{(1-q_{1}^{-\lambda_{i}}q_{2}^{-i}(z/v))}{(1-q_{1}^{-\lambda_{i}}q_{2}^{-i+1}(z/v))}\frac{(p_{*}q_{1}^{\lambda_{i}}q_{2}^{i}(v/z);p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-1}q_{2}^{i-2}(v/z);p_{*})_{\infty}}{(p_{*}q_{1}^{\lambda_{i}}q_{2}^{i-1}(v/z);p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-1}q_{2}^{i-1}(v/z);p_{*})_{\infty}}$
$\displaystyle\qquad:\eta(z;p)\Psi_{\lambda}(v;p):.$ (B.11)

Hence, incorporating the zero mode factor with $w=-uv$, we find

$\displaystyle(1-q_{1})(1-q_{2})\left(E(z;p)\Psi_{\lambda}(v;p)-\langle\lambda|K^{-}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)E(z;p)\right)$
$\displaystyle=\left(\frac{\mathfrak{q}}{z}\right)^{N+1}(-uv)\prod_{i=1}^{\infty}\frac{(p_{*}q_{1}^{\lambda_{i}}q_{2}^{i}(v/z);p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-1}q_{2}^{i-2}(v/z);p_{*})_{\infty}}{(p_{*}q_{1}^{\lambda_{i}}q_{2}^{i-1}(v/z);p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-1}q_{2}^{i-1}(v/z);p_{*})_{\infty}}:\eta(z;p)\Psi_{\lambda}(v;p):$
$\displaystyle~{}\left(\frac{1}{1-q_{2}^{\ell(\lambda)}(v/z)}\prod_{i=1}^{\ell(\lambda)}\frac{1-q_{1}^{\lambda_{i}}q_{2}^{i}(v/z)}{1-q_{1}^{\lambda_{i}}q_{2}^{i-1}(v/z)}-\frac{-z/v}{1-q_{2}^{-\ell(\lambda)}(z/v)}\prod_{i=1}^{\ell(\lambda)}\frac{1-q_{1}^{-\lambda_{i}}q_{2}^{-i}(z/v)}{1-q_{1}^{-\lambda_{i}}q_{2}^{-i+1}(z/v)}\right)~{}.$
(B.12)

Note that the last factor in the big parentheses is independent of the
elliptic parameter $p$ and $p$ dependence appears only through the factor
$(p_{*}X;p_{*})_{\infty}$, which becomes trivial when $p=0$. The overall
monomial factor is nothing but the zero mode (3.47) for the target Fock space.

Now to evaluate the last factor we use the following lemma; for a rational
function of the form

$f_{+}(z)=\prod_{i=1}^{n}\frac{1-\alpha_{i}z}{1-\beta_{i}z},$ (B.13)

we assume that $\beta_{j}$ are all distinct and

$\prod_{i=1}^{n}\alpha_{i}=\prod_{i=1}^{n}\beta_{i}.$ (B.14)

Namely all the poles of $f_{+}(z)$ are simple and

$\lim_{z\to\infty}f_{+}(z)=1.$ (B.15)

If a given rational function is

$\tilde{f}_{+}(z)=\frac{\prod_{i=1}^{m}(1-\alpha_{i}z)}{\prod_{j=1}^{n}(1-\beta_{j}z)},\qquad\prod_{i=1}^{m}\alpha_{i}=\prod_{j=1}^{n}\beta_{j},$
(B.16)

with $m<n$ like (B). We can multiply $(1-z)^{n-m}$ and consider

$f_{+}(z)=(z-1)^{n-m}\tilde{f}_{+}(z).$ (B.17)

By using the condition (B.14), we can see

$f_{-}(z)=\prod_{i=1}^{n}\frac{1-\alpha_{i}^{-1}z^{-1}}{1-\beta_{i}^{-1}z^{-1}},$
(B.18)

satisfies

$f_{+}(z)=f_{-}(z)$ (B.19)

for $z\neq\beta_{j}^{-1}$. Then the following formula holds;

$f_{+}(z)-f_{-}(z)=\sum_{k=1}^{n}\delta(\beta_{k}z)\frac{\prod_{i=1}^{n}(1-\beta_{k}^{-1}\alpha_{i})}{\prod_{j\neq
k}(1-\beta_{k}^{-1}\beta_{j})}.$ (B.20)

Note that the coefficients of the delta function are the residues of
$f_{+}(z)$ at the corresponding poles.

We can prove (B.20) as follows; since $f_{+}(z)$ is holomorphic on
$\mathbb{P}^{1}$ with only simple poles at $z=\beta_{j}^{-1}$, the partial
fraction decomposition of $f_{+}(z)$ is121212Note (B.15).

$f_{+}(z)=1+\sum_{k=1}^{n}\frac{c_{k}}{1-\beta_{k}z},\qquad\sum_{k=1}^{n}c_{k}=0,$
(B.21)

where $c_{k}$ is the residue of $f_{+}(z)$ at $z=\beta_{k}^{-1}$. Then by
(B.19), we must have

$f_{-}(z)=1-\sum_{i=1}^{n}\frac{c_{k}\beta_{k}^{-1}z^{-1}}{1-\beta_{k}^{-1}z^{-1}},$
(B.22)

and hence

$f_{+}(z)-f_{-}(z)=\sum_{k=1}^{n}c_{k}\left(\frac{1}{1-\beta_{k}z}+\frac{\beta_{k}^{-1}z^{-1}}{1-\beta_{k}^{-1}z^{-1}}\right)=\sum_{k=1}^{n}c_{k}\delta(\beta_{k}z).$
(B.23)

We can check that the last factor in (B) satisfies the above assumptions.
Applying the lemma we obtain

$\displaystyle(1-q_{1})(1-q_{2})\left(E(z;p)\Psi_{\lambda}(v;p)-\langle\lambda|K^{-}(z;p_{*})|\lambda\rangle\Psi_{\lambda}(v;p)E(z;p)\right)$
$\displaystyle=\left(\frac{\mathfrak{q}}{q_{1}^{\lambda_{k}}q_{2}^{k-1}v}\right)^{N+1}(-uv)\prod_{i=1}^{\infty}\frac{(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k+1};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k-1};p_{*})_{\infty}}{(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k};p_{*})_{\infty}}$
$\displaystyle\times(1-q_{2})\sum_{k=1}^{\ell(\lambda)+1}\delta\left(q_{1}^{\lambda_{k}}q_{2}^{k-1}\frac{v}{z}\right)\prod_{i\neq
k}\frac{1-q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k+1}}{1-q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k}}:\eta(q_{1}^{\lambda_{k}}q_{2}^{k-1}v;p)\Psi_{\lambda}(v;p):~{}.$
(B.24)

Since

$:\eta(q_{1}^{\lambda_{k}}q_{2}^{k-1}v;p)\Psi_{\lambda}(v;p):=\frac{z_{\lambda}}{\mathcal{G}_{\lambda}}\frac{\mathcal{G}_{\lambda+1_{k}}}{z_{\lambda+1_{k}}}\Psi_{\lambda+1_{k}}(v;p),$
(B.25)

using (3.30) and assuming $\langle\lambda+1_{k}|\lambda+1_{k}\rangle=1$, we
obtain recursion relations for the normalization factor
$\mathcal{G}_{\lambda}$ and the zero mode factor $z_{\lambda}$;

$\frac{z_{\lambda+1_{k}}}{z_{\lambda}}=q_{2}^{k-1}(-uv)\left(\frac{\mathfrak{q}}{q_{1}^{\lambda_{k}}q_{2}^{k-1}v}\right)^{N+1},$
(B.26)

and

$\displaystyle\frac{\mathcal{G}_{\lambda+1_{k}}}{\mathcal{G}_{\lambda}}\prod_{i\neq
k}\frac{(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k+1};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k-1};p_{*})_{\infty}}{(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k};p_{*})_{\infty}}$
$\displaystyle\qquad=q_{2}^{k-1}\prod_{i=1}^{k-1}\frac{\theta_{p_{*}}(q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i-1})\theta_{p_{*}}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i+1})}{\theta_{p_{*}}(q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i})\theta_{p_{*}}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i})}.$
(B.27)

Note that we have adjusted the monomial factor $q_{2}^{k-1}$ between
$z_{\lambda}$ and $\mathcal{G}_{\lambda}$ to simplify the recursion relation
for $\mathcal{G}_{\lambda}$ as follows;

$\displaystyle\frac{\mathcal{G}_{\lambda+1_{k}}}{\mathcal{G}_{\lambda}}$
$\displaystyle=\prod_{i=1}^{k-1}\frac{(p_{*}q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i-1};p_{*})_{\infty}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i+1};p_{*})_{\infty}}{(p_{*}q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i};p_{*})_{\infty}(q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i};p_{*})_{\infty}}$
$\displaystyle\qquad\prod_{i=k+1}^{\infty}\frac{(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k};p_{*})_{\infty}}{(q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k+1};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k-1};p_{*})_{\infty}}.$
(B.28)

We note the $p$ dependent factor in the recursion relation is

$\displaystyle\prod_{i=1}^{k-1}\frac{(p_{*}q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i-1};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i+1};p_{*})_{\infty}}{(p_{*}q_{1}^{\lambda_{k}-\lambda_{i}}q_{2}^{k-i};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i};p_{*})_{\infty}}$
$\displaystyle\prod_{i=k+1}^{\infty}\frac{(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k};p_{*})_{\infty}}{(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k+1};p_{*})_{\infty}(p_{*}q_{1}^{\lambda_{i}-\lambda_{k}-1}q_{2}^{i-k-1};p_{*})_{\infty}}.$
(B.29)

It is striking that this is the same as the remainder factor
$R_{\lambda}^{(k)}$ appearing in section 3.3 by the change of variables
$(q_{1},q_{2},p)\to(q_{1}^{-1},q_{2}^{-1},p_{*})$. This means the base change
discussed there eliminates the above $p$ dependence by employing the "second"
formula for the vertical Fock representation derived in section 3.3. When
$p\to 0$ or after the base change, the recursion reduces to

$\displaystyle\frac{\mathcal{G}_{\lambda+1_{k}}}{\mathcal{G}_{\lambda}}=\prod_{i=1}^{k-1}\frac{1-q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i+1}}{1-q_{1}^{\lambda_{k}-\lambda_{i}+1}q_{2}^{k-i}}\prod_{i=k+1}^{\infty}\frac{1-q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k}}{1-q_{1}^{\lambda_{i}-\lambda_{k}}q_{2}^{i-k+1}}.$
(B.30)

With the initial condition $\mathcal{G}_{\varnothing}=1$ the recursion
relation is solved by

$\mathcal{G}_{\lambda}(q_{1},q_{2})=\prod_{s\in\lambda}(1-q_{1}^{-a(s)}q_{2}^{\ell(s)+1}).$
(B.31)

After $(q_{1},q_{2})\to(q_{1}^{-1},q_{2}^{-1})$, it agrees with the standard
normalization factor for the integral form of the Macdonald function. In
general for $p\neq 0$ the solution of the recursion relation is

$\mathcal{G}_{\lambda}(q_{1},q_{2};p)=\frac{\prod_{s\in\lambda}(q_{1}^{-a(s)}q_{2}^{\ell(s)+1};p_{*})_{\infty}}{\prod_{s\in\lambda}(pq_{1}^{-a(s)}q_{2}^{\ell(s)+1};p_{*})_{\infty}},$
(B.32)

which agrees with the result in Appendix A. Finally the recursion relation for
$z_{\lambda}$ is solved by

$z_{\lambda}=q_{2}^{n(\lambda)}(-v)^{-N|\lambda|}u^{|\lambda|}f_{\lambda}(q_{1},q_{2})^{-N-1},$
(B.33)

where

$n(\lambda):=\sum_{k=1}^{\infty}\lambda_{k}(k-1).$ (B.34)

##  Appendix C Free field representation and $SU(4)$ Omega background

In this appendix we show an interesting connection of the free field
representation employed in this paper and $SU(4)$ Omega background
$(q_{1},q_{2},q_{3},q_{4})$ or the equivariant parameters of the toric action
on $\mathbb{C}^{4}$, which opens a way of interpreting our computation from
the viewpoint of eight dimensional gauge theory, or the spiked instantons of
Nekrasov [55]. To motivate the $SU(4)$ Omega background, let us look at the
affine quiver $\hat{A}_{0}$ with a single node and a single loop (a.k.a. the
Jordan quiver). Since it has a single node, the $\mathfrak{q}$-deformed Cartan
matrix has a single component $C=(1-\mu^{-1})(1-\mathfrak{q}^{-1}\mu)$, where
$\mu$ is a parameter associated with the loop of $\hat{A}_{0}$ quiver. The
quiver gauge theory for $\hat{A}_{0}$ is the supersymmetric gauge theory with
adjoint hypermultiplet, usually called $\mathcal{N}=2^{*}$ theory and the
parameter $\mu$ is physically the exponentiated mass parameter. If we follow
the prescription of [26] and [38], the commutation relation of the so-called
“root boson” takes the form;

$\displaystyle\left[\lambda_{n}.\lambda_{m}\right]$ $\displaystyle=$
$\displaystyle-n(1-q_{1})(1-q_{2})C\delta_{n+m,0}$ (C.1) $\displaystyle=$
$\displaystyle-n(1-q_{1})(1-q_{2})(1-q_{3})(1-q_{4})\delta_{n+m,0},$

where we have defined $q_{3}=\mu^{-1}$ and $q_{4}=\mathfrak{q}^{-1}\mu$ with
$\mathfrak{q}=q_{1}q_{2}$. We are going to show that for each pair $(ij)$ with
$1\leq i<j\leq 4$, there exists a Fock representation of the quasi-Hopf twist
of DIM algebra with the central charge $C=\sqrt{q_{i}q_{j}}$. Thus, we obtain
a family of six Fock representations with various central charges, which seems
to match with the six stacks of $D3$ configuration for the spiked instanton,
where the pair $(ij)$ specifies a choice of codimension four subspace of
$\mathbb{C}^{4}$ which are fixed by the toric action

$(z_{1},z_{2},z_{3},z_{4})\longrightarrow(q_{1}z_{1},q_{2}z_{2},q_{3}z_{3},q_{4}z_{4}),\qquad
q_{1}q_{2}q_{3}q_{4}=1.$ (C.2)

One can define a completely $S_{4}$ symmetric deformed Heisenberg algebra

$\left[a_{n},a_{m}\right]=-n(1-q_{1}^{\pm n})(1-q_{2}^{\pm n})(1-q_{3}^{\pm
n})(1-q_{4}^{\pm n})\delta_{n+m,0},$ (C.3)

and an $S_{4}$ symmetric vertex operator

$\Phi(z)=\exp\left(\sum_{n=1}^{\infty}\frac{a_{-n}}{n}z^{n}\right)\exp\left(\sum_{n=1}^{\infty}\frac{a_{n}}{n}z^{-n}\right).$
(C.4)

Then we define a quartet of the screening operators by

$S^{(k)}(z)=\exp\left(\sum_{n=1}^{\infty}\frac{a_{-n}}{n(1-q_{k}^{n})}z^{n}\right)\exp\left(\sum_{n=1}^{\infty}\frac{a_{n}}{n(1-q_{k}^{-n})}z^{-n}\right),\qquad(1\leq
k\leq 4)$ (C.5)

which satisfies

$\Phi(z)=:S^{(k)}(z)S^{(k)}(q_{k}z)^{-1}:.$ (C.6)

Writing the screening operator as

$S^{(k)}(z)=\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}s^{(k)}_{-n}z^{n}\right)\exp\left(\sum_{n=1}^{\infty}\frac{1}{n}s^{(k)}_{n}z^{-n}\right),$
(C.7)

we have the commutation relation

$\left[s^{(k)}_{n},s^{(k)}_{m}\right]=-n\frac{\prod_{i\neq k}(1-q_{i}^{\mp
n})}{1-q_{k}^{\pm n}}\delta_{n+m,0}.$ (C.8)

Computing OPE coefficients we obtain

$S^{(k)}(z)S^{(k)}(w)=\frac{\theta_{q_{k}}(w/z)\theta_{q_{k}}(q_{i}^{-1}q_{j}^{-1}w/z)\theta_{q_{k}}(q_{i}^{-1}z/w)\theta_{q_{k}}(q_{j}^{-1}z/w)}{\theta_{q_{k}}(z/w)\theta_{q_{k}}(q_{i}^{-1}q_{j}^{-1}z/w)\theta_{q_{k}}(q_{i}^{-1}w/z)\theta_{q_{k}}(q_{j}^{-1}w/z)}S^{(k)}(w)S^{(k)}(z),$
(C.9)

where $\\{i,j,k,\ell\\}=\\{1,2,3,4\\}$. Using the inversion formula (1.12), we
can rewrite the relation (C.9) as

$S^{(k)}(z)S^{(k)}(w)=G^{(ij)}(w/z;q_{k})S^{(k)}(w)S^{(k)}(z),$ (C.10)

where

$G^{(ij)}(u;q_{k}):=\frac{\theta_{q_{k}}(q_{i}u)\theta_{q_{k}}(q_{j}u)\theta_{q_{k}}(q_{i}^{-1}q_{j}^{-1}u)}{\theta_{q_{k}}(q_{i}^{-1}u)\theta_{q_{k}}(q_{j}^{-1}u)\theta_{q_{k}}(q_{i}q_{j}u)}.$
(C.11)

Note that since $\theta_{p^{-1}}(u)=\theta_{p}(u^{-1})^{-1}$, we have

$G^{(ij)}(u;q_{k}^{-1})=G^{(ij)}(u;q_{k})^{-1}.$ (C.12)

$G^{(ij)}$ is related to the structure function used in this paper by
$G^{(12)}=\mathcal{G}^{-1}$. Later we will take $q_{3}=p_{*}^{-1}$ and (C.12)
shows the consistency of this choice.

The commutation relation of the screening operators of different kind
generates a new operator, which we identify with the Cartan current. For
convenience let us denotes the OPE factors of two operators $A(z)$ and $B(w)$
by $c(A(z),B(w))$, namely

$A(z)B(w)=c(A(z),B(w)):A(z)B(w):,\qquad|z|>|w|.$ (C.13)

We have

$\displaystyle c(S^{(k)}(z),S^{(\ell)}(w))$ $\displaystyle=$
$\displaystyle\frac{\left(1-q_{k}q_{i}\frac{w}{z}\right)\left(1-q_{k}q_{j}\frac{w}{z}\right)}{\left(1-q_{k}\frac{w}{z}\right)\left(1-q_{\ell}^{-1}\frac{w}{z}\right)},$
$\displaystyle c(S^{(\ell)}(w),S^{(k)}(z))$ $\displaystyle=$
$\displaystyle\frac{\left(1-q_{\ell}q_{i}\frac{z}{w}\right)\left(1-q_{\ell}q_{j}\frac{z}{w}\right)}{\left(1-q_{k}^{-1}\frac{z}{w}\right)\left(1-q_{\ell}\frac{z}{w}\right)}.$
(C.14)

Hence the support of the commutation relation $[S^{(k)}(z),S^{(\ell)}(w)]$ is
only at the simple poles; $q_{i}w/z=1$ and $q_{j}^{-1}w/z=1$. Computing the
residues there, we find

$\displaystyle\left[S^{(k)}(z),S^{(\ell)}(w)\right]$ (C.15) $\displaystyle=$
$\displaystyle\frac{(1-q_{i})(1-q_{j})}{1-q_{i}q_{j}}\left(\frac{1}{1-q_{k}\frac{w}{z}}-\frac{1}{1-q_{\ell}^{-1}\frac{w}{z}}+\frac{1}{1-q_{k}^{-1}\frac{z}{w}}-\frac{1}{1-q_{\ell}\frac{z}{w}}\right):S^{(k)}(z)S^{(\ell)}(w):$
$\displaystyle=$
$\displaystyle\frac{(1-q_{i})(1-q_{j})}{1-q_{i}q_{j}}\left(\delta\left(q_{k}\frac{w}{z}\right)-\delta\left(q_{\ell}\frac{z}{w}\right)\right):S^{(k)}(z)S^{(\ell)}(w):.$

Hence introducing

$\psi^{(k\ell)}(z):=:S^{(k)}(q_{k}z)S^{(\ell)}(z):=:S^{(k)}(z)S^{(\ell)}(q_{\ell}z):,$
(C.16)

we can express the commutation relation as

$\left[S^{(k)}(z),S^{(\ell)}(w)\right]=\frac{(1-q_{i})(1-q_{j})}{1-q_{i}q_{j}}\left(\delta\left(q_{k}\frac{w}{z}\right)\psi^{(k\ell)}(w)-\delta\left(q_{\ell}\frac{z}{w}\right)\psi^{(k\ell)}(z)\right).$
(C.17)

More explicitly the Cartan current is131313After the quasi-Hopf twist, we can
treat $\psi^{\pm}$ on an equal footing.

$\psi^{(k\ell)}(z)=\exp\left(\sum_{n=1}^{\infty}\frac{1-q_{k}^{n}q_{\ell}^{n}}{n(1-q_{k}^{n})(1-q_{\ell}^{n})}a_{-n}z^{n}\right)\exp\left(\sum_{n=1}^{\infty}\frac{1-q_{k}^{-n}q_{\ell}^{-n}}{n(1-q_{k}^{-n})(1-q_{\ell}^{-n})}a_{n}z^{-n}\right).$
(C.18)

To express the OPE factor of the Cartan current, it is convenient to
temporally use the notation

$\mathfrak{q}_{1}=q_{i},\quad\mathfrak{q}_{2}=q_{j},\quad\mathfrak{q}_{3}=q_{i}^{-1}q_{j}^{-1}.$
(C.19)

Then we have

$c(\psi^{(k\ell)}(z),\psi^{(k\ell)}(w))=\prod_{m=1}^{3}\frac{\left(\mathfrak{q}_{m}\frac{q_{k}w}{z};q_{k},q_{\ell}^{-1}\right)_{\infty}\left(\mathfrak{q}_{m}^{-1}\frac{w}{q_{\ell}z};q_{k},q_{\ell}^{-1}\right)_{\infty}}{\left(\mathfrak{q}_{m}\frac{w}{q_{\ell}z};q_{k},q_{\ell}^{-1}\right)_{\infty}\left(\mathfrak{q}_{m}^{-1}\frac{q_{k}w}{z};q_{k},q_{\ell}^{-1}\right)_{\infty}}.$
(C.20)

Using the elliptic gamma function, we can simplify the exchange relation

$\psi^{(k\ell)}(z)\psi^{(k\ell)}(w)=\prod_{m=1}^{3}\frac{\Gamma_{q_{k},q_{\ell}^{-1}}(\mathfrak{q}_{m}\frac{w}{q_{\ell}z})\Gamma_{q_{k},q_{\ell}^{-1}}(\mathfrak{q}_{m}^{-1}\frac{q_{k}w}{z})}{\Gamma_{q_{k},q_{\ell}^{-1}}(\mathfrak{q}_{m}\frac{q_{k}w}{z})\Gamma_{q_{k},q_{\ell}^{-1}}(\mathfrak{q}_{m}^{-1}\frac{w}{q_{\ell}z})}\psi^{(k\ell)}(w)\psi^{(k\ell)}(z).$
(C.21)

Finally by the difference relation,

$\Gamma_{p_{1},p_{2}}(p_{1}u)=\theta_{p_{2}}(u)\Gamma_{p_{1},p_{2}}(u),\qquad\Gamma_{p_{1},p_{2}}(p_{2}u)=\theta_{p_{1}}(u)\Gamma_{p_{1},p_{2}}(u)$
(C.22)

we obtain

$\displaystyle\psi^{(k\ell)}(z)\psi^{(k\ell)}(w)$ $\displaystyle=$
$\displaystyle\prod_{m=1}^{3}\frac{G^{(ij)}(\frac{w}{z};q_{k})}{G^{(ij)}(\frac{w}{z},q_{\ell}^{-1})}\psi^{(k\ell)}(w)\psi^{(k\ell)}(z)$
(C.23) $\displaystyle=$ $\displaystyle
G^{(ij)}(\frac{w}{z};q_{k})G^{(ij)}(\frac{w}{z},q_{\ell})\psi^{(k\ell)}(w)\psi^{(k\ell)}(z),$

which is manifestly symmetric in $i\leftrightarrow j$ and
$k\leftrightarrow\ell$. In the present $S_{4}$ symmetric formulation the
exchange relations of $S^{(k)}(z),S^{(\ell)}(z)$ and $\psi^{(k\ell)}(w)$
are141414Note that $\psi^{(k\ell)}=\psi^{(\ell k)}$.

$\displaystyle S^{(k)}(z)\psi^{(k\ell)}(w)$
$\displaystyle=G^{(ij)}(q_{k}w/z;q_{k})\psi^{(k\ell)}(w)S^{(k)}(z)=G^{(ij)}(w/z;q_{k})\psi^{(k\ell)}(w)S^{(k)}(z),$
$\displaystyle S^{(\ell)}(z)\psi^{(k\ell)}(w)$
$\displaystyle=G^{(ij)}(q_{\ell}w/z;q_{\ell})\psi^{(k\ell)}(w)S^{(\ell)}(z)=G^{(ij)}(w/z;q_{\ell})\psi^{(k\ell)}(w)S^{(\ell)}(z),$
(C.24)

where we have used $G^{(ij)}(pu;p)=G^{(ij)}(u;p)$ which follows from
$\theta_{p}(px)=-x^{-1}\theta_{p}(x)$.

In summary, we have obtained a sextet of the Fock representations of the
quasi-Hopf twist of DIM algebra; $\mathcal{F}^{(k\ell)}=\mathcal{F}^{(\ell
k)}~{}(1\leq k<\ell\leq 4)$ generated by $S^{(k)}(z),S^{(\ell)}(z)$ and
$\psi^{(k\ell)}(z)$. Their commutation relations are;

$\displaystyle S^{(k)}(z)S^{(k)}(w)$
$\displaystyle=G^{(ij)}(\frac{w}{z};q_{k})S^{(k)}(w)S^{(k)}(z),$ (C.25)
$\displaystyle\psi^{(k\ell)}(z)\psi^{(k\ell)}(w)$
$\displaystyle=G^{(ij)}(\frac{w}{z};q_{k})G^{(ij)}(\frac{w}{z},q_{\ell})\psi^{(k\ell)}(w)\psi^{(k\ell)}(z),$
(C.26) $\displaystyle S^{(k)}(z)\psi^{(k\ell)}(w)$
$\displaystyle=G^{(ij)}(\frac{w}{z};q_{k})\psi^{(k\ell)}(w)S^{(k)}(z),$ (C.27)
$\displaystyle\left[S^{(k)}(z),S^{(\ell)}(w)\right]$
$\displaystyle=\frac{(1-q_{i})(1-q_{j})}{1-q_{i}q_{j}}\left(\delta\left(q_{k}\frac{w}{z}\right)\psi^{(k\ell)}(w)-\delta\left(q_{\ell}\frac{z}{w}\right)\psi^{(k\ell)}(z)\right).$
(C.28)

where $\\{i,j,k,\ell\\}=\\{1,2,3,4\\}$. In fact one can check that

$\displaystyle E(z)=S^{(k)}(\sqrt{q_{k}}z),\qquad
F(z)=S^{(\ell)}(\sqrt{q_{\ell}}z),$ (C.29) $\displaystyle
K^{+}(z)=\psi^{(k\ell)}(z/\sqrt{q_{k}}),\qquad
K^{-}(z)=\psi^{(k\ell)}(z/\sqrt{q_{\ell}}),$ (C.30)

gives a representation of the quasi-Hopf twist of DIM algebra with the central
charge $C=\sqrt{q_{k}q_{\ell}}$ and the following correspondence of the
structure functions;

$G^{(ij)}(u;q_{k})\longleftrightarrow\mathcal{G}(u;p_{*}),\qquad
G^{(ij)}(u;q_{\ell})\longleftrightarrow\mathcal{G}(u;p)^{-1}.$ (C.31)

The normalization of the commutation relation $[E(z),F(z)]$ is

$\tilde{g}=\frac{(1-q_{i})(1-q_{j})}{1-q_{i}q_{j}}.$ (C.32)

For example, take $SU(4)$ parameters;

$q_{1}=q,\qquad q_{2}=t^{-1},\qquad q_{3}=p_{*}^{-1},\qquad q_{4}=p,$ (C.33)

then $C=\sqrt{p/p_{*}}=\mathfrak{q}$ and for $p\neq 0$ we have

$\displaystyle E(z;p)=S^{(3)}(z/\sqrt{p_{*}}),\qquad
F(z;p)=S^{(4)}(\sqrt{p}z),$ (C.34) $\displaystyle
K^{+}(z;p)=\psi^{(34)}(\sqrt{p_{*}}z),\qquad
K^{-}(z;p)=\psi^{(34)}(z/\sqrt{p}).$ (C.35)

Note that after the quasi-Hopf twist $K^{\pm}(z)$ are related by the shift of
the spectral parameter. Note also that (C.35) is valid only after the quasi-
Hopf twist, since it involves $p\neq 0$.

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