A $(q,t)$-deformation of the 2d Toda
integrable hierarchy

Jean-Emile<EMAIL_ADDRESS>Alexandr
<EMAIL_ADDRESS>

School of Mathematics and Statistics
University of Melbourne
Parkville, Victoria 3010, Australia

A $(q,t)$-deformation of the 2d Toda integrable hierarchy is introduced by
enhancing the underlying symmetry algebra
$\widehat{\mathfrak{gl}}(\infty)\simeq\text{q-W}_{1+\infty}$ to the quantum
toroidal $\mathfrak{gl}(1)$ algebra. The difference-differential equations of
the hierarchy are obtained from the expansion of $(q,t)$-bilinear identities,
and two equations refining the 2d Toda equation are found in this way. The
derivation of the bilinear identities follows from the isomorphism between the
Fock representation of level $(2,0)$ of the quantum toroidal
$\mathfrak{gl}(1)$ algebra and the tensor product of the q-deformed Virasoro
algebra with a $u(1)$ Heisenberg algebra. It leads to identify the
$(q,t)$-deformed Casimir with the screening charges of the deformed Virasoro
algebra. Due to the non-trivial coproduct, equations of the hierarchy no
longer involve a single tau-function, but instead relate a set of different
tau functions. We then define the universal refined tau function using the
$L$-matrix of the quantum toroidal $\mathfrak{gl}(1)$ algebra and interpret it
as the generating function of the deformed tau functions. The equations of the
hierarchy, written in terms of the universal refined tau function, combine
into two-term quadratic equations similar to the $RLL$ equations.

##  1 Introduction

The 2d Toda hierarchy is an integrable hierarchy of difference-differential
equations introduced in [1] as an extension of the KP hierarchy. The first
equation of the hierarchy is the celebrated 2d Toda lattice equation [2]. A
great deal of knowledge on this model has been accumulated over the last forty
years [3, 4], and we refer to [5] for a short review. Following the method
developed by the Kyoto School of integrability [6], a large class of solutions
can be constructed using the fermionic representation of the algebra
$\widehat{\mathfrak{gl}}(\infty)$. These solutions are indexed by the group
elements $\widehat{GL(\infty)}$, and particular choices of elements produce
notable solutions e.g. hypergeometric functions [7], or generalizations of the
well-known solitons and polynomial solutions of the KP hierarchy. The Toda
hierarchy has been found to enter in a remarkably wide range of problems in
mathematical physics, including 2d string theory [8], Hurwitz numbers [9],
ABJM matrix model [10] and topological string theory [11, 12].

In this paper, we introduce a $(q,t)$-deformation of the 2d Toda integrable
hierarchy. One of our motivations is the correspondence between integrable
hierarchies and topological strings first noticed in [11]. This correspondence
is based on an equivalence between the algebras
$\widehat{\mathfrak{gl}}(\infty)$ and quantum $W_{1+\infty}$. The integrable
properties of the hierarchy stem from the presence of these infinite
dimensional algebras of symmetries. A refinement of topological strings theory
has been introduced in [13, 14] motivated by the correspondence with non-
perturbative partition functions of 5d ${\mathcal{N}}=1$ gauge theories in the
omega-deformed background
$\mathbb{C}_{\varepsilon_{1}}\times\mathbb{C}_{\varepsilon_{2}}\times S^{1}$.
Refined topological strings amplitudes depend on two parameters
$(q_{1},q_{2})\equiv(e^{\varepsilon_{1}},e^{\varepsilon_{2}})$, they reduce to
the original amplitudes in the self-dual limit $(q_{1},q_{2})\to(q^{-1},q)$
where $q=e^{g_{\text{str}}}$ is identified with the exponentiated strings
coupling constant. Refined topological strings also possess an infinite
dimensional symmetry algebra [15, 16, 17, 18], the quantum toroidal
$\mathfrak{gl}(1)$ algebra, which can be constructed as a deformation of the
quantum $W_{1+\infty}$ to which it reduces in the self-dual limit [19].
Following this chain of correspondences, one of the authors proposed in [20]
to introduce a deformation of integrable hierarchies with the quantum toroidal
$\mathfrak{gl}(1)$ algebra as the underlying symmetry. This paper presents a
concrete realization of this idea as a refinement of the Kyoto School
formalism.

A second motivation for investigating this deformation of the Toda hierarchy
is the well-known reduction of integrable hierarchies to Painlevé equations
[21, 22]. It was observed in [23] that solutions of the Painlevé VI equation
can be constructed from Virasoro conformal blocks with central charge $c=1$,
or 4d ${\mathcal{N}}=2$ instanton partition functions in the self-dual omega-
background $\varepsilon_{1}+\varepsilon_{2}=0$ following the celebrated AGT
correspondence [24]. Similar quantities built from the q-deformed Virasoro
algebra with parameters $q=t$ were shown to produce solutions of the
q-Painlevé VI equation in [25], these quantities being related to self-dual 5d
${\mathcal{N}}=1$ instanton partition functions through the 5d AGT
correspondence [26]. This correspondence between q-Painlevé equations and
conformal blocks / instantons partitions functions can be extended to generic
parameters $(q,t)$ by introducing a deformation of the q-Painlevé VI equation
called quantum q-Painlevé VI (or qq-Painlevé VI for short)[27]. It is then
natural to wonder if these deformed equations also follow from the periodic
reduction of a $(q,t)$-deformed integrable hierarchy. While it is not clear
yet if the formalism developed in this paper has any relation with qq-Painlevé
equations, we observe that they are both built upon the same underlying
symmetry algebra. In fact, the recent observation in [28] of a connection
between the qq-Painlevé VI equation and the non-stationary difference equation
[29], a generalization of the q-difference Toda equation, also points in this
direction. The non-stationary Ruijsenaars functions solving this equation have
deep connection to quantum toroidal symmetry [30, 31], which is the key
ingredient in our construction.

Our third motivation is the connection with Yang–Baxter integrability. The
$(q,t)$-deformation of the 2d Toda integrable hierarchy which we consider in
this paper is intimately related to the integrable model which is defined on
tensor products of Fock spaces of quantum toroidal $\mathfrak{gl}(1)$ algebra
in [32, 33].

#### Deformed hierarchy

Concretely, the deformation of the hierarchy is introduced using (a
bosonization of) the free fermion formalism developed in [34, 35, 6, 36] (see
also the review [37] and the book [38]). Refined versions of the Toda tau
functions indexed by elements of the quantum toroidal $\mathfrak{gl}(1)$
algebra are introduced. A deformation of the bilinear identity is obtained
from the vanishing commutation relation between the coproduct of the algebra
elements (evaluated in the horizontal Fock representation), and the screening
charges of the deformed Virasoro algebra. The vanishing of this commutator is
a consequence of the isomorphism found in [39] between the representation of
levels $(2,0)$ of the quantum toroidal $\mathfrak{gl}(1)$ algebra and the
deformed Virasoro algebra tensored with a Heisenberg algebra (the so-called
$u(1)$ factor). Equations of the hierarchy follow from the usual expansion of
the bilinear integral equation in the difference of time parameters.

In practice, the refinement of the hierarchy leads to two main issues. The
first one is the loss of the free fermion presentation, since the quantum
toroidal $\mathfrak{gl}(1)$ algebra has no fermionic representation. Instead,
we need to use the horizontal Fock representation defined in [40, 41] which
acts on the free boson Fock space. Thus, all fermionic formulas have to be
bosonized before introducing their $(q,t)$-deformation. For instance, the
Casimir operator involved in the basic bilinear equation has the following
fermionic expression

$\Psi=\sum_{r\in\mathbb{Z}+1/2}\bar{\psi}_{-r}\otimes\psi_{r},$ (1.1)

it is replaced by one of the q-Virasoro screening charges constructed in [42].
The latter does indeed reduce in the self-dual limit to the bosonized formula
for the Casimir $\Psi$. Fortunately, the derivation of the bilinear identity
is based on the bosonic formalism and there is no major difficulty for the
refinement.

The second issue is the deformation of the coproduct. Unlike the group
elements of $\widehat{GL(\infty)}$ which are co-commutative, i.e.
$\Delta(G)=G\otimes G$, the Drinfeld coproduct of a generic element $G$ of the
toroidal algebra takes the form of a (possibly infinite) series

$\Delta(G)=\sum_{\alpha}G_{\alpha}\otimes G^{\prime}_{\alpha}.$ (1.2)

As a consequence, in general the bilinear integral equations (and so the
difference-differential equations of the hierarchy) no longer involve a single
tau function $\tau(G)$, but instead they relate members of a set of refined
tau functions indexed by elements $G\in{\mathcal{X}}$ closed under coproduct
$\Delta({\mathcal{X}})\subset{\mathcal{X}}\otimes{\mathcal{X}}$.111Note that
${\mathcal{X}}$ does not need to be an algebra, i.e. it does not have to be
closed under the product. In the simple example where $G$ is built from
generators of the Cartan subalgebra, the coproduct is co-commutative (up to
deformation by central elements), and ${\mathcal{X}}$ contains only a single
element. However, the resulting tau function is almost trivial. More
interesting examples can be obtained as certain matrix elements of the
R-matrix. In this case, the deformed hierarchy produces a closed set of
equations for the refined tau functions which can be checked explicitly.

Our main results are the following bilinear integral equations obeyed by the
refined tau functions $\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$
defined in (3.1)

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})\tau_{uq^{-n}t^{-1}}({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{\prime})p^{k}z^{-k}}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})\tau_{uq^{-n}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\\\
\end{split}$ (1.3)

where ${\boldsymbol{t}}$, $\bar{\boldsymbol{t}}$ denotes the sets of time
parameters, and $u$ deforms the discrete parameter of the Toda lattice, and
$n\in\mathbb{Z}$. The parameters $(q,t)$ of the deformed Virasoro algebra are
identified with the quantum group parameters $(q_{1},q_{2})\equiv(t^{-1},q)$
and $\gamma=t^{1/2}q^{-1/2}$, and we have assumed that the coproduct
$\Delta(G)$ takes the generic form (1.2). The shift of time parameters
${\boldsymbol{t}}\pm[z]_{q,t}$ is defined in (3.17), it reduces to the
standard shift $t_{k}\pm z^{k}/k$ in the self-dual limit. A second equation
with the role of $q$ and $t$ exchanged is also derived in (3.16) starting from
a different screening charge.

The difference-differential equations of the refined hierarchy are obtained by
expanding the integral equation (1.3) in the differences of time parameters
$\gamma^{|k|}t_{k}-t^{\prime}_{k}$. New equations which trivialize in the
self-dual limit are found at low orders, they can be found in (3.30), (3.33)
and (3.34). More interestingly, two equations deforming the Toda lattice
equation were found

$\displaystyle\begin{split}&\sum_{\alpha}\left[\partial_{1}\tau_{u}({\boldsymbol{t}},{\boldsymbol{t}}^{\prime}|G_{\alpha})\partial_{-1}\tau_{uqt^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})-qt^{-1}\tau_{u}({\boldsymbol{t}},{\boldsymbol{t}}^{\prime}|G_{\alpha})\partial_{1}\partial_{-1}\tau_{uqt^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})\right]\\\
=&-\sum_{\alpha}\tau_{ut^{-1}}({\boldsymbol{t}},{\boldsymbol{t}}^{\prime}|G_{\alpha})\tau_{uq}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha}),\\\
&\sum_{\alpha}\left[\partial_{-1}\tau_{u}({\boldsymbol{t}},{\boldsymbol{t}}^{\prime}|G_{\alpha})\partial_{1}\tau_{uq^{-1}t}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})-qt^{-1}\tau_{u}({\boldsymbol{t}},{\boldsymbol{t}}^{\prime}|G_{\alpha})\partial_{1}\partial_{-1}\tau_{uq^{-1}t}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})\right]\\\
=&-\sum_{\alpha}\tau_{ut}({\boldsymbol{t}},{\boldsymbol{t}}^{\prime}|G_{\alpha})\tau_{uq^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha}),\end{split}$
(1.4)

with the time derivatives $\partial_{\pm 1}=\partial/\partial t_{\pm 1}$, and
$\boldsymbol{\gamma}{\boldsymbol{t}}$,
$\boldsymbol{\gamma}{\boldsymbol{t}}^{\prime}$ indicating that the times
$t_{k}$ are rescaled by a factor $\gamma^{|k|}$. The fact that we find two
inequivalent equations is related to the non-cocommutativity of the coproduct,
i.e. in general the opposite coproduct $\Delta^{op}$ differs from $\Delta$. In
the self-dual limit, both reduce to the co-commutative coproduct and the two
previous equations reproduce the usual Toda equation (3.37) for the tau
function.

#### Refined tau functions and the universal R-matrix

The presence of an additional quantum group structure is the main new feature
of our $(q,t)$-deformed model. It leads to a new class of solutions defined in
section 5 using the $L$-matrix of the quantum group (i.e. the universal
$R$-matrix evaluated in specific representations). In the self-dual limit, the
coproduct becomes co-commutative, the quantum group structure trivializes, and
the model reduces to the usual Toda integrable hierarchy.

The bilinear equations (1.3) can be combined and written in a compact form in
terms of the universal refined tau function
$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$ which can be viewed as the
generator of $\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$. The
universal refined tau function is a formal series in ${\boldsymbol{t}}$ and
$\bar{\boldsymbol{t}}$ with coefficients in the quantum toroidal algebra
$\mathfrak{gl}(1)$. Using the definition of the universal $R$-matrix of the
quantum toroidal algebra $\mathfrak{gl}(1)$ we show that
$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$ can be defined via the
$L$-matrix ${\mathcal{L}}(u)$ [32] associated to the horizontal Fock
representation of $\mathfrak{gl}(1)$ with the spectral parameter $u$. The
bilinear equations in terms of ${\mathcal{L}}$ take the form

$\displaystyle(\gamma^{cn}\otimes\Phi^{+}_{n}){\mathcal{L}}_{13}(q^{-n}t^{-1}u){\mathcal{L}}_{12}(u)$
$\displaystyle={\mathcal{L}}_{13}(q^{-n}u){\mathcal{L}}_{12}(t^{-1}u)(1\otimes\Phi^{+}_{n}),$
(1.5)
$\displaystyle(\gamma^{cm}\otimes\Phi^{-}_{m}){\mathcal{L}}_{13}(t^{m}qu){\mathcal{L}}_{12}(u)$
$\displaystyle={\mathcal{L}}_{13}(t^{m}u){\mathcal{L}}_{12}(qu)(1\otimes\Phi_{m}^{-}),$
(1.6)

where $n,m\in\mathbb{Z}$, $c$ a central element of the algebra and
$\Phi^{\pm}_{n}$ are the modes of the screening operators without the zero
modes (see (LABEL:def_Spm), (2.46) and (5.25)). The modes $\Phi^{\pm}_{n}$ act
in the tensor product of two horizontal Fock representations of the quantum
toroidal algebra $\mathfrak{gl}(1)$. The $RLL$ equation of the algebra
associated to this representation in our conventions reads

$\displaystyle\check{\rm
R}_{23}(u_{2}/u_{1}){\mathcal{L}}_{13}(u_{2}){\mathcal{L}}_{12}(u_{1})={\mathcal{L}}_{13}(u_{1}){\mathcal{L}}_{12}(u_{2})\check{\rm
R}_{23}(u_{2}/u_{1}),$ (1.7)

where $\check{\rm R}(u)=P{\rm R}(u)$, $P$ being the permutation matrix and
${\rm R}(u)$ is the horizontal-horizontal R-matrix normalized such that its
vacuum matrix element is equal to $1$. By looking at the special cases $n=0$,
$m=0$ and $n=1$, $m=-1$ in (1.5) and (1.6) one finds that these equations
reproduce (1.7) at special values of $u_{1}$ and $u_{2}$ by identifying222For
a similar connection between the R-matrix and screening charges in the affine
Yangian case see [43].

$\displaystyle\check{\rm R}(t^{-1})=\Phi^{+}_{0},\qquad\check{\rm
R}(q)=\Phi^{-}_{0},\qquad\underset{x=qt^{-1}}{\text{Res}}\check{\rm
R}(x)=qt^{-1}(1-qt^{-1})\Phi^{+}_{1}\Phi^{-}_{-1}.$ (1.8)

By this consideration we establish a connection between refined tau functions
and integrability objects of quantum toroidal $\mathfrak{gl}(1)$.

#### Outline

This paper is organized as follows. In Section 2, we summarize the algebraic
results needed for our analysis. Section 3 is the core of this paper, we
define the refined tau functions, show that they obey two bilinear identities,
and expand them to obtain the differential-difference equations of the
hierarchy. In Sections 4 and 5, we present the analysis of several classes of
tau functions with Section 5 focusing on the universal refined tau function.
We conclude with a brief discussion on the deformation of the known 2d Toda
tau functions. In Appendix A we discuss the $(q,t)$-deformation of the
fermionic formalism and define the associated Baker-Akhiezer wave functions.
Appendix B provides the technical details of the derivation of the bilinear
identities. Finally, a brief reminder on the intertwining operators of the
quantum toroidal $\mathfrak{gl}(1)$ algebra used to express the vertical-
horizontal R-matrix is given in Appendix C.

Fock space | Vacuum state | Heisenberg modes | Algebra (rep.)
---|---|---|---
${\mathcal{F}}_{n}$ | $\ket{n}=e^{nQ}\ket{\emptyset}$ | $J_{k}$ | qu. $W_{1+\infty}$ ($\rho_{f}$)
${\mathcal{F}}_{u}$ | $\ket{u}=u^{{\hat{d}}}\ket{\emptyset}$ | $J_{k}$ | qu. tor. $\mathfrak{gl}(1)$ ($\rho_{u}^{(1,0)}$)
${\mathcal{F}}_{v}$ | $\ket{\emptyset}\\!\rangle$ | $-$ | qu. tor. $\mathfrak{gl}(1)$ ($\rho_{v}^{(0,1)}$)
${\mathcal{F}}_{m,n}$ | $\ket{m,n}=e^{\gamma_{m,n}\hat{q}}\ket{\emptyset}$ | $\alpha_{k}$ | q-Virasoro
Figure 1: Summary of the different Fock spaces used in this paper.

#### Notations

For reference, a brief summary of the different Fock spaces used in this paper
is provided in figure 1. We also recall here the dictionary between quantum
group parameters and q-deformed Virasoro parameters which is used throughout
the paper

$(q_{1},q_{2})=(t^{-1},q),\quad q_{3}=(q_{1}q_{2})^{-1}=\gamma^{2}=p^{-1}.$
(1.9)

##  2 Quantum toroidal $\mathfrak{gl}(1)$ and $q$-deformed Virasoro algebras

The main technical tool we use to introduce the refined tau functions is the
deformation of the quantum $W_{1+\infty}$ symmetry algebra into the quantum
toroidal $\mathfrak{gl}(1)$ algebra, and a relation between the latter and the
q-deformed Virasoro algebra. In this section, we review the definitions of
these algebras, and the roles they play in the hierarchy.

###  2.1 $\widehat{\mathfrak{gl}}(\infty)$ and q-W1+∞ symmetries

In this subsection, we recall the fermionic construction of 2d Toda tau
functions, and highlight the role of the underlying symmetry algebras.

#### Free bosons

Let $J_{k}$ and $Q$ denote the modes of the free bosonic field $\phi(z)$
satisfying the Heisenberg algebra

$\phi(z)=Q+J_{0}\log
z-\sum_{k\in\mathbb{Z}^{\times}}\dfrac{1}{k}z^{-k}J_{k},\quad[J_{k},J_{l}]=k\delta_{k+l,0},\quad[J_{k},Q]=\delta_{k,0}.$
(2.1)

Let $\ket{\emptyset}$ denote the (neutral) vacuum state annihilated by
positive modes $J_{k\geq 0}\ket{\emptyset}=0$, and
$\ket{n}=e^{nQ}\ket{\emptyset}$ the charged vacuum state of charge
$n\in\mathbb{Z}$. By construction, $\ket{0}=\ket{\emptyset}$, $J_{k}\ket{n}=0$
for $k>0$ and $J_{0}\ket{n}=n\ket{n}$. Let ${\mathcal{F}}_{n}$ and
${\mathcal{F}}$ denote respectively the Fock space of charge $n$ and the total
Fock space defined as

${\mathcal{F}}_{n}=\mathbb{C}[J_{-1},J_{-2},\cdots]\ket{n},\quad{\mathcal{F}}=\bigoplus_{n\in\mathbb{Z}}{\mathcal{F}}_{n}.$
(2.2)

The bosonic normal-ordering which consists in moving positive modes $J_{k\geq
0}$ to the right will be denoted $:\cdots:$. The adjoint action is obtained as
$J_{k}^{\dagger}=J_{-k}$, $Q^{\dagger}=-Q$, in particular the dual vacua
$\bra{n}=\bra{\emptyset}e^{-Qn}$ are annihilated by negative modes, and
$\bra{n}\\!\\!\\!\ket{m}=\delta_{n,m}$.

#### Free fermions

The Dirac fermionic fields $\bar{\psi}(z)$, $\psi(z)$ are defined as

$\psi(z)=\sum_{r\in\mathbb{Z}+1/2}z^{-r-1/2}\psi_{r},\quad\bar{\psi}(z)=\sum_{r\in\mathbb{Z}+1/2}z^{-r-1/2}\bar{\psi}_{r},\quad\\{\bar{\psi}_{r},\psi_{s}\\}=\delta_{r+s,0},$
(2.3)

and $\\{\psi_{r},\psi_{s}\\}=\\{\bar{\psi}_{r},\bar{\psi}_{s}\\}=0$. Through
the boson-fermion correspondence, these fields can be realized as vertex
operators on the bosonic Fock space

$\displaystyle\begin{split}&\bar{\psi}(z):{\mathcal{F}}_{n}\to{\mathcal{F}}_{n+1},\quad\bar{\psi}(z)=:e^{\phi(z)}:,\\\
&\psi(z):{\mathcal{F}}_{n}\to{\mathcal{F}}_{n-1},\quad\psi(z)=:e^{-\phi(z)}:.\end{split}$
(2.4)

Expanding their action on the vacuum state, we deduce that positive fermionic
modes annihilate the vacuum $\ket{\emptyset}$, and more generally
$\psi_{r}\ket{n}=0$ for $r>n$ and $\bar{\psi}_{r}\ket{n}=0$ for $r>-n$. The
normal-ordering which consists in moving the positive fermionic modes to the
right will be denoted $\vdots\cdots\vdots$.

#### Tau functions

Let ${\boldsymbol{t}}=(t_{1},t_{2},\cdots)$,
$\bar{\boldsymbol{t}}=(t_{-1},t_{-2},\cdots)$ denote two infinite sets of time
parameters, and $G\in\text{End}({\mathcal{F}})$ an operator acting on the Fock
space. The correlation function

$\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=\bra{n}e^{\sum_{k>0}t_{k}J_{k}}Ge^{\sum_{k>0}t_{-k}J_{-k}}\ket{n}$
(2.5)

defines a tau function of the Toda hierarchy if the operator $G$ satisfies the
basic bilinear condition

$[G\otimes
G,\Psi]=0,\quad\Psi=\sum_{r\in\mathbb{Z}+1/2}\bar{\psi}_{-r}\otimes\psi_{r}.$
(2.6)

In particular, the tau functions obey a bilinear integral equation, called the
bilinear identity, from which all the difference-differential equations of the
hierarchy can be deduced. We recall the derivation of the bilinear identity
from the basic bilinear condition in Appendix B.

#### Fermionic representation of $\widehat{\mathfrak{gl}}(\infty)$

The basic bilinear condition (2.6) admits for solutions operators of the form

$G=\exp\left(\sum_{r,s\in\mathbb{Z}+1/2}\alpha_{r,s}\vdots\bar{\psi}_{r}\psi_{s}\vdots\right),\quad\alpha_{r,s}\in\mathbb{C}.$
(2.7)

They define a representation of the group $\widehat{GL}(\infty)$. The
corresponding Lie algebra $\widehat{\mathfrak{gl}}(\infty)$ is generated by
the elements $E_{r,s}$ indexed by half-integer $r,s\in\mathbb{Z}+\frac{1}{2}$
with the commutation relations333As a general rule in this paper, indices
$k,l,m,n$ are integer valued while indices $r,s,t,u$ are half-integer valued.

$\left[E_{r,s},E_{t,u}\right]=\delta_{s+t}E_{r,u}-\delta_{r+u}E_{t,s}+c_{1}(\theta(r)-\theta(t))\delta_{s+t}\delta_{r+u},\quad
r,s,t,u\in\mathbb{Z}+\dfrac{1}{2}$ (2.8)

where we used the Heaviside function $\theta(r)=1$ when $r>0$ and
$\theta(r)=0$ when $r<0$, and $c_{1}$ is a central element. This algebra is a
central extension of the algebra $\mathfrak{gl}(\infty)$ of matrices with
infinite size. It has a fermionic representation of level $\rho_{f}(c_{1})=1$
defined as $\rho_{f}(E_{r,s})=\vdots\bar{\psi}_{r}\psi_{s}\vdots$ which can be
bosonized to define a map
$\rho_{f}:\widehat{\mathfrak{gl}}(\infty)\to\text{End}({\mathcal{F}})$.

The algebra $\widehat{\mathfrak{gl}}(\infty)$ is compatible with the co-
commutative coproduct $\Delta(E_{r,s})=E_{r,s}\otimes 1+1\otimes E_{r,s}$.
This coproduct takes the form $\Delta(G)=G\otimes G$ for the group elements
$G=e^{\sum_{r,s}\alpha_{r,s}E_{r,s}}$, and the basic bilinear relation (2.6)
simply expresses the fact that $\Psi$ is a Casimir for the representation
$\rho_{f}\otimes\rho_{f}$

$[(\rho_{f}\otimes\rho_{f})\Delta(E_{r,s}),\Psi]=0.$ (2.9)

This simple remark is a starting point of the Kac-Wakimoto construction [36].

#### Quantum $W_{1+\infty}$

In order to introduce the $(q,t)$-deformation, we need to reformulate the
$\widehat{\mathfrak{gl}}(\infty)$ symmetry in the language of the quantum
$W_{1+\infty}$ algebra [44]. This algebra444We refer to [45, 46] for a brief
summary of the main properties of this algebra. Note that we have flipped the
parameter $q\to q^{-1}$ with respect to [46], in agreement with the new
identification of the parameters $(q_{1},q_{2})=(q^{-1},q)$ in the self-dual
limit. is generated by elements $W_{m,n}$ indexed with two integers
$m,n\in\mathbb{Z}$, and two central elements $(c_{1},c_{2})$ with the
following relations involving the quantum parameter $q\in\mathbb{C}^{\times}$

$[W_{m,n},W_{m^{\prime},n^{\prime}}]=(q^{-m^{\prime}n}-q^{-mn^{\prime}})\left(W_{m+m^{\prime},n+n^{\prime}}+c_{1}\dfrac{\delta_{m+m^{\prime},0}}{1-q^{-n-n^{\prime}}}-c_{2}\dfrac{\delta_{n+n^{\prime},0}}{1-q^{-m-m^{\prime}}}\right).$
(2.10)

At level $c_{2}=0$, this algebra is equivalent to the algebra
$\widehat{\mathfrak{gl}}(\infty)$ upon a simple linear transformation of the
generators reminiscent of a Fourier transform

$W_{m,n}=\sum_{s\in\mathbb{Z}+1/2}q^{(s+1/2)n}E_{m-s,s}.$ (2.11)

The algebraic relations (2.10) are compatible with the co-commutative
coproducts $\Delta(W_{m,n})=W_{m,n}\otimes 1+1\otimes W_{m,n}$, and so is the
linear relation (2.11) between the two algebras. Using this relation, the
fermionic representation of $\widehat{\mathfrak{gl}}(\infty)$ can be extended
to the generators of quantum $W_{1+\infty}$

$\rho_{f}(W_{m,n})=\sum_{s\in\mathbb{Z}+1/2}q^{(s+1/2)n}\vdots\bar{\psi}_{m-s}\psi_{s}\vdots=\oint{\dfrac{dz}{2i\pi}\vdots\bar{\psi}(z)z^{m}\psi(q^{-n}z)\vdots}.$
(2.12)

We observe that the dependence in the zero-mode $Q$ drops in the expression of
$\rho_{f}(W_{m,n})$, and so
$\rho_{f}(W_{m,n})\in\text{End}({\mathcal{F}}_{k})$. The Casimir $\Psi$ also
characterizes the elements of the fermionic representation of quantum
$W_{1+\infty}$ on $\text{End}({\mathcal{F}}\otimes{\mathcal{F}})$ as

$[(\rho_{f}\otimes\rho_{f})\Delta(W_{m,n}),\Psi]=0.$ (2.13)

We would like to emphasize that the quantum $W_{1+\infty}$ algebra is not a
symmetry of the Toda hierarchy per se, and the parameter $q$ does not enter in
the equations of the hierarchy. Instead, it enters in some of the solutions of
the hierarchy which are expressed using the generators $W_{m,n}$.

###  2.2 Quantum toroidal $\mathfrak{gl}(1)$ algebra

The quantum toroidal $\mathfrak{gl}(1)$ algebra ${\mathcal{E}}$ depends on two
quantum parameters $q_{1},q_{2}\in\mathbb{C}^{\times}$, and it is customary to
introduce a third one $q_{3}$ such that $q_{1}q_{2}q_{3}=1$. We will also
denote $\gamma=q_{3}^{1/2}$. In the Drinfeld presentation, the algebra is
generated by the modes of the following currents

$x^{\pm}(z)=\sum_{k\in\mathbb{Z}}z^{-k}x^{\pm}_{k},\quad\psi^{\pm}(z)=\sum_{k\geq
0}z^{\mp k}\psi_{\pm k}^{\pm},$ (2.14)

together with the central elements $(c,{\bar{c}})$ such that
$\psi_{0}^{\pm}=\gamma^{\mp{\bar{c}}}$. These currents obey the algebraic
relations

$\displaystyle\begin{split}&[\psi^{\pm}(z),\psi^{\pm}(w)]=0,\quad\psi^{+}(z)\psi^{-}(w)=\dfrac{g(\gamma^{c}z/w)}{g(\gamma^{-c}z/w)}\psi^{-}(w)\psi^{+}(z),\\\
&\psi^{+}(z)x^{\pm}(w)=g(\gamma^{\pm c/2}z/w)^{\pm
1}x^{\pm}(w)\psi^{+}(z),\quad\psi^{-}(z)x^{\pm}(w)=g(\gamma^{\mp c/2}z/w)^{\pm
1}x^{\pm}(w)\psi^{-}(z),\\\ &\prod_{\alpha=1,2,3}(z-q_{\alpha}^{\pm 1}w)\
x^{\pm}(z)x^{\pm}(w)=\prod_{\alpha=1,2,3}(z-q_{\alpha}^{\mp 1}w)\
x^{\pm}(w)x^{\pm}(z),\\\
&[x^{+}(z),x^{-}(w)]=\kappa\left(\delta(\gamma^{-c}z/w)\psi^{+}(\gamma^{c/2}w)-\delta(\gamma^{c}z/w)\psi^{-}(\gamma^{-c/2}w)\right),\\\
&\operatorname*{\text{Sym}}_{z_{1},z_{2},z_{3}}\dfrac{z_{2}}{z_{3}}[x^{\pm}(z_{1}),[x^{\pm}(z_{2}),x^{\pm}(z_{3})]]=0\quad\text{(Serre
relations)},\end{split}$ (2.15)

where $\kappa$ is a $\mathbb{C}$-number and $g(z)$ a rational function, both
depending on the parameters $q_{1},q_{2}$

$\kappa=\dfrac{(1-q_{1})(1-q_{2})}{1-q_{1}q_{2}},\quad
g(z)=\prod_{\alpha=1,2,3}\dfrac{1-q_{\alpha}z}{1-q_{\alpha}^{-1}z}.$ (2.16)

In (LABEL:def_DIM), $\delta(z)=\sum_{k\in\mathbb{Z}}z^{k}$ denotes the formal
delta power series, we refer to the first chapter of [47] for more details
about its mathematical definition and main properties. In the $\psi\psi$ and
$\psi x$ relations, the rational functions $g(\alpha z/w)$ are expanded in
positive/negative powers of $z/w$, while the $xx$ relations lead to quadratic
relations between the modes $x_{k}^{\pm}$. We should emphasize that the
currents $\psi^{\pm}(z)$ bear no direct relation with the fermionic fields
$\psi(z)$, $\bar{\psi}(z)$.

The modes $\psi_{\pm k}^{\pm}$ together with the central elements generate the
equivalent of a Cartan subalgebra. This subalgebra can also be presented using
the modes $a_{k}$ obtained from a different expansion of the currents
$\psi^{\pm}(z)$

$\psi^{+}(z)=\psi_{0}^{+}e^{\sum_{k>0}z^{-k}a_{k}},\quad\psi^{-}(z)=\psi_{0}^{-}e^{-\sum_{k>0}z^{k}a_{-k}}.$
(2.17)

We note that the modes $a_{k}$ and $x_{k}^{\pm}$ obey the following
commutation relations

$[a_{k},a_{l}]=(\gamma^{ck}-\gamma^{-ck})c_{k}\delta_{k+l,0},\quad[a_{k},x_{l}^{\pm}]=\pm\gamma^{\mp|k|c/2}c_{k}x_{l+k}^{\pm},\quad\text{with}\quad
c_{k}=-\dfrac{1}{k}\prod_{\alpha=1,2,3}(1-q_{\alpha}^{k}).$ (2.18)

#### Gradings

The algebra ${\mathcal{E}}$ can be supplemented by two grading operators $d$
and ${\bar{d}}$ with the commutation relations

$\displaystyle\begin{split}&[d,x_{k}^{\pm}]=-kx_{k}^{\pm},\quad[d,\psi_{\pm
k}^{\pm}]=\mp k\psi_{\pm k}^{\pm},\quad[d,c]=[d,{\bar{c}}]=0,\\\
&[{\bar{d}},x_{k}^{\pm}]=\pm x_{k}^{\pm},\quad[{\bar{d}},\psi_{\pm
k}^{\pm}]=0,\quad[{\bar{d}},c]=[{\bar{d}},{\bar{c}}]=0.\end{split}$ (2.19)

This extension of ${\mathcal{E}}$ will be denoted by ${\mathcal{E}}^{\prime}$.
They can be used to define the following automorphisms indexed by
$\alpha\in\mathbb{C}^{\times}$

$\displaystyle\begin{split}&\alpha^{d}x^{\pm}(z)\alpha^{-d}=x^{\pm}(\alpha
z),\quad\alpha^{d}\psi^{\pm}(z)\alpha^{-d}=\psi^{\pm}(pz),\\\
&\alpha^{\bar{d}}x^{\pm}(z)\alpha^{-{\bar{d}}}=\alpha^{\pm
1}x^{\pm}(z),\quad\alpha^{{\bar{d}}}\psi^{\pm}(z)\alpha^{-{\bar{d}}}=\psi^{\pm}(z).\end{split}$
(2.20)

#### Coproduct

The algebra ${\mathcal{E}}$ is a Hopf algebra equipped with the Drinfeld
coproduct

$\displaystyle\begin{split}&\Delta(x_{k}^{+})=x^{+}_{k}\otimes 1+\sum_{l\geq
0}\gamma^{-(c\otimes 1)(k+l/2)}\ \psi^{-}_{-l}\otimes
x^{+}_{k+l},\quad\Delta(x_{k}^{-})=\sum_{l\geq 0}\gamma^{-(1\otimes
c)(k-l/2)}\ x^{-}_{k-l}\otimes\psi^{+}_{l}+1\otimes x_{k}^{-},\\\
&\Delta(a_{k})=a_{k}\otimes\gamma^{-|k|c/2}+\gamma^{|k|c/2}\otimes
a_{k},\quad\Delta(c)=c\otimes 1+1\otimes
c,\quad\Delta({\bar{c}})={\bar{c}}\otimes 1+1\otimes{\bar{c}}.\end{split}$
(2.21)

It can be extended to include the grading elements that are co-commutative.

#### Self-dual limit

In the self-dual limit $(q_{1},q_{2})\to(q^{-1},q)$ (i.e. $q_{3}\to 1$), the
toroidal algebra ${\mathcal{E}}$ reduces to the quantum $W_{1+\infty}$
algebra. The limit of the Drinfeld generators reads

$\displaystyle\begin{split}&x_{k}^{+}\to(1-q^{-1})q^{-k/2}W_{k,1}+c_{1}\delta_{k,0},\quad
x_{k}^{-}\to(1-q)q^{k/2}W_{k,-1}+c_{1}\delta_{k,0},\\\
&\dfrac{ka_{k}}{(q_{1}^{k/2}-q_{1}^{-k/2})(q_{3}^{k/2}-q_{3}^{-k/2})}\to
W_{k,0}-\dfrac{c_{2}}{1-q^{-k}},\quad(c,{\bar{c}})\to(c_{1},-c_{2}),\end{split}$
(2.22)

and the algebraic relations (LABEL:def_DIM) reduce to the commutation
relations (2.10). Remarkably, in this limit the Drinfeld coproduct becomes co-
commutative, in agreement with our previous observation. The quantum toroidal
algebra is a deformation of the universal enveloping algebra of the quantum
$W_{1+\infty}$ algebra, and group-like elements now correspond to quantum
group elements of co-unit one.555For an element with coproduct
$\Delta(G)=G\otimes G$, the antipode is $S(G)=G^{-1}$ and the co-unit
$\varepsilon(G)=1$.

The grading operators defined by the relation (LABEL:def_gradings) reduce to
the natural gradings of quantum $W_{1+\infty}$, i.e. with a slight abuse of
notation,

$[d,W_{m,n}]=-mW_{m,n},\quad[{\bar{d}},W_{m,n}]=nW_{m,n}.$ (2.23)

###  2.3 Horizontal Fock representation

The terminology Fock representations refers to a set of representations $\rho$
of levels $(\rho(c),\rho({\bar{c}}))=(m,n)\in\mathbb{Z}\times\mathbb{Z}$ of
the quantum toroidal algebra $\mathfrak{gl}(1)$ acting on the bosonic Fock
space ${\mathcal{F}}$ or its tensor products. In this paper, we are mainly
interested in the representations with $(m,n)=(1,n)$ (resp. $(m,n)=(0,1)$),
which we call horizontal (resp. vertical) Fock representations following [15].
We denote these representations by $\rho^{(m,n)}$, but the symbol will often
be omitted to lighten the expressions if no confusion ensues. In fact, our
main construction is based on the horizontal representation $\rho^{(1,0)}$
which replaces the fermionic representation $\rho_{f}$ in the original
construction. We will also consider in the next subsection the representation
$\rho^{(2,0)}=(\rho^{(1,0)}\otimes\rho^{(1,0)})\Delta$ acting on the tensor
product ${\mathcal{F}}\otimes{\mathcal{F}}$. The vertical Fock representation
will only enter in the definition of a certain class of refined tau function
in section 5.1, and so we decided to present it in Appendix C to lighten this
section.

In the horizontal Fock representation $\rho^{(1,n)}$, the Cartan modes $a_{k}$
form a deformed Heisenberg algebra and they can be identified with the modes
$J_{k}$ up to a factor depending on $q_{1},q_{2}$

$\rho^{(1,n)}(a_{k})=-\dfrac{\gamma^{-k/2}}{k}(1-q_{2}^{k})(1-q_{3}^{k})J_{k},\quad\rho^{(1,n)}(a_{-k})=-\dfrac{\gamma^{-k/2}}{k}(1-q_{1}^{k})(1-q_{3}^{k})J_{-k},\quad(k>0).$
(2.24)

On the other hand, the currents $x^{\pm}(z)$ are expressed as

$\rho^{(1,n)}(x^{\pm}(z))=z^{\mp n}\eta^{\pm}(z),$ (2.25)

using the vertex operators

$\displaystyle\begin{split}&\eta^{+}(z)=\exp\left(\sum_{k>0}\dfrac{z^{k}}{k}(1-q_{1}^{k})J_{-k}\right)\exp\left(-\sum_{k>0}\dfrac{z^{-k}}{k}(1-q_{2}^{k})J_{k}\right),\\\
&\eta^{-}(z)=\exp\left(-\sum_{k>0}\dfrac{z^{k}}{k}\gamma^{k}(1-q_{1}^{k})J_{-k}\right)\exp\left(\sum_{k>0}\dfrac{z^{-k}}{k}\gamma^{k}(1-q_{2}^{k})J_{k}\right).\end{split}$
(2.26)

These operators satisfy the normal ordering

$\displaystyle\eta^{+}(z)\eta^{+}(w)$
$\displaystyle=\zeta\left(\frac{z}{w}\right):\eta^{+}(z)\eta^{+}(w):,$
$\displaystyle\qquad\eta^{-}(z)\eta^{-}(w)$
$\displaystyle=\zeta\left(\frac{w}{z}\right):\eta^{-}(z)\eta^{-}(w):,$ (2.27)
$\displaystyle\eta^{+}(z)\eta^{-}(w)$
$\displaystyle=\zeta\left(\frac{z}{\gamma
w}\right)^{-1}:\eta^{+}(z)\eta^{-}(w):,$
$\displaystyle\qquad\eta^{-}(z)\eta^{+}(w)$
$\displaystyle=\zeta\left(\frac{w}{\gamma
z}\right)^{-1}:\eta^{-}(z)\eta^{+}(w):,$ (2.28)

where

$\displaystyle\zeta(x):=\frac{(x-1)(x-q_{1}q_{2})}{(x-q_{1})(x-q_{2})}.$
(2.29)

A complex parameter $u\in\mathbb{C}^{\times}$ can be included as the weight of
the representation as explained below.

#### Gradings and zero-modes

The representation of the grading operators $(d,{\bar{d}})$ is a little
subtle, so we first restrict ourselves to the level $n=0$. In the
representation $\rho^{(1,0)}$, the grading operator $d$ is quadratic in the
Heisenberg modes

$\rho^{(1,0)}(d)=L_{0}\quad\text{with}\quad L_{0}=\sum_{k>0}J_{-k}J_{k}.$
(2.30)

Then, the automorphism defined in (2.20) follows from the property

$p^{L_{0}}J_{k}p^{-L_{0}}=p^{-k}J_{k},\quad p\in\mathbb{C}^{\times}.$ (2.31)

An easy way to realize the second grading operator ${\bar{d}}$ in this
representation is to define it as a derivative with respect to the weight
$\log u$. However, it has been observed in [20] that the horizontal weight $u$
plays the role of charge $n$ in the definition (2.5) of the tau functions,
i.e. we have to recover $u\to q^{n}$ in the self-dual limit. Thus, in this
context it is natural to introduce the charged vacua $\ket{u}$. More
precisely, we introduce two operators $\hat{d}$ and $\hat{u}$
satisfying666Equivalently, we have the quantum torus relation
$p^{\hat{d}}\hat{u}=p\hat{u}p^{\hat{d}}$.

$\displaystyle\qquad[\hat{d},\hat{u}]=\hat{u},$

and modify the expressions (2.25) of the Drinfeld currents $x^{\pm}(z)$ by
inserting the operator $\hat{u}$

$\rho^{(1,0)}(x^{\pm}(z))=\hat{u}^{\pm 1}\eta^{\pm}(z).$ (2.32)

Then we set $\rho^{(1,0)}({\bar{d}})=\hat{d}$ which ensures the validity of
the commutation relations (LABEL:def_gradings). The charged Fock spaces are
defined on the vacua $\ket{u}$ in the usual way,
${\mathcal{F}}_{u}=\mathbb{C}[J_{-1},J_{-2},\cdots]\ket{u}$777In fact, since
we are only considering here the representation of grading elements of the
form $q^{m{\bar{d}}/2}t^{n{\bar{d}}/2}$ we can work with the total Fock space
${\mathcal{F}}=\bigoplus_{m,n\in\mathbb{Z}}{\mathcal{F}}_{uq^{n/2}t^{m/2}}.$
(2.33) with the actions

$\displaystyle\hat{u}\ket{u}=u\ket{u},\qquad\ket{u}=u^{-\hat{d}}\ket{\emptyset},$

and we identify $\ket{\emptyset}=\ket{1}$. The formulas (2.24) and (2.32) then
define the representations

$\displaystyle\rho_{u}^{(1,0)}:{\mathcal{E}}\to\text{End}({\mathcal{F}}_{u}).$
(2.34)

This construction generalizes to horizontal representations of levels $(1,n)$,
but the grading now contains a mixing term

$\rho^{(1,n)}(d)=L_{0}-n\hat{d},\quad\rho^{(1,n)}({\bar{d}})=\hat{d}.$ (2.35)

#### Self-dual limit and fermionic representation

The horizontal representation $\rho^{(1,0)}$ of ${\mathcal{E}}$ reduces in the
self-dual limit to the bosonization of the fermionic representation of quantum
$W_{1+\infty}$ defined previously. In particular, the charged Fock spaces
${\mathcal{F}}_{u}$ reduce to the ones defined previously

$\hat{u}\to q^{J_{0}},\quad\hat{d}\to-\dfrac{Q}{\log
q},\quad\ket{u}\to\ket{n},\quad{\mathcal{F}}_{u}\to{\mathcal{F}}_{n}.$ (2.36)

The Drinfeld currents $x^{\pm}(z)$ reduce to

$x^{+}(z)\to:\bar{\psi}(z)\psi(q^{-1}z):,\quad
x^{-}(z)\to:\bar{\psi}(q^{-1}z)\psi(z):,$ (2.37)

in agreement with (2.22) after taking into account the different normal-
orderings.

###  2.4 Free field representation of the deformed Virasoro algebra

From our analysis of the role of the symmetry algebra in the construction of
tau functions we deduced that the tensor product $G\otimes G$ in the basic
bilinear condition (2.6) is replaced after $(q,t)$-deformation by the
evaluation of an element $G\in{\mathcal{E}}$ in the representation

$\rho_{u_{1},u_{2}}^{(2,0)}:{\mathcal{E}}\to\text{End}({\mathcal{F}}_{u_{1}}\otimes{\mathcal{F}}_{u_{2}}),\qquad\rho_{u_{1},u_{2}}^{(2,0)}=(\rho_{u_{1}}^{(1,0)}\otimes\rho_{u_{2}}^{(1,0)})\Delta.$
(2.38)

It is helpful to keep in mind the representation $\rho^{(2,0)}$

$\displaystyle\rho^{(2,0)}:{\mathcal{E}}\to\text{End}({\mathcal{F}}\otimes{\mathcal{F}}),\qquad\rho^{(2,0)}=(\rho^{(1,0)}\otimes\rho^{(1,0)})\Delta.$
(2.39)

In this representation instead of the scalars $u_{1}$ and $u_{2}$ we encounter
invertible operators $\hat{u}^{(1)}=\hat{u}\otimes 1$ and
$\hat{u}^{(2)}=1\otimes\hat{u}$. In the view of (2.38), the Casimir $\Psi$ in
(2.13) should be replaced by operators commuting with
$\rho_{u_{1},u_{2}}^{(2,0)}({\mathcal{E}})$ on
$\text{End}({\mathcal{F}}_{u_{1}}\otimes{\mathcal{F}}_{u_{2}})$. In this
subsection, we construct these operators for specific values of the ratio
$u_{1}/u_{2}$ by exploiting the relation between the representation
$\rho_{u_{1},u_{2}}^{(2,0)}$ and the deformed Virasoro algebra [42] (see also
[48]).

#### Deformed Virasoro algebra

The deformed Virasoro algebra (or q-Virasoro algebra) is an algebra depending
on two complex parameters $(q,t)$, and generated by the modes $T_{k}$ of the
current $T(z)=\sum_{k\in\mathbb{Z}}z^{-k}T_{k}$ called the stress-energy
tensor. This current satisfies the algebraic relation

$f(w/z)T(z)T(w)-f(z/w)T(w)T(z)=-\dfrac{(1-q)(1-t^{-1})}{1-p}\left(\delta(pw/z)-\delta(p^{-1}w/z)\right),$
(2.40)

with the function

$f(z)=\exp\left(\sum_{k>0}\dfrac{1}{k}\dfrac{(1-q^{k})(1-t^{-k})}{1+p^{k}}z^{k}\right),$
(2.41)

where we denoted $p=q/t$.

The free field representation defines an action of the stress-energy tensor on
yet another set of Fock spaces. These Fock spaces ${\mathcal{F}}_{m,n}$ are
indexed by two integer charges $m,n\in\mathbb{Z}$, they are built using the
modes of a $(q,t)$-deformed Heisenberg algebra888To avoid the confusion with
the Cartan modes of the toroidal algebra, we use here $\alpha_{k}$ instead of
$a_{k}$.

$[\alpha_{k},\alpha_{l}]=k\dfrac{1-q^{|k|}}{1-t^{|k|}}\delta_{k+l,0},\quad[\alpha_{0},\hat{q}]=\beta^{-1},$
(2.42)

where $\beta=\log t/\log q$ (i.e. $t=q^{\beta}$). From the neutral vacuum
state $\ket{\emptyset}$ annihilated by $\alpha_{k}$ with $k\geq 0$, we define
the charged vacua $\ket{m,n}=e^{\gamma_{m,n}\hat{q}}\ket{\emptyset}$ with
$\gamma_{m,n}=(m+1)\beta/2-(n+1)/2$, $m,n\in\mathbb{Z}$, and the corresponding
bosonic Fock spaces
${\mathcal{F}}_{m,n}=\mathbb{C}[\alpha_{-1},\alpha_{-2},\cdots]\ket{m,n}$. In
this representation, the stress-energy tensor has the form

$T(z)=\Lambda(z)+:\Lambda(pz)^{-1}:,$ (2.43)

with the vertex operator

$\Lambda(z)=p^{1/2}\exp\left(-\sum_{k>0}\dfrac{1}{k}\dfrac{1-t^{k}}{1+p^{k}}t^{-k}p^{-k/2}z^{k}\alpha_{-k}\right)\exp\left(-\sum_{k>0}\dfrac{1}{k}(1-t^{k})p^{k/2}z^{-k}\alpha_{k}\right)q^{\beta\alpha_{0}}.$
(2.44)

#### Screening currents

The representation of the q-Virasoro algebra on the Fock spaces
${\mathcal{F}}_{m,n}$ can be characterized by the commutation of the operators
with certain screening charges built from the screening currents

$\displaystyle\begin{split}&S_{+}(z)=e^{\sum_{k>0}\frac{z^{k}}{k}\frac{1-t^{k}}{1-q^{k}}\alpha_{-k}}e^{-\sum_{k>0}\frac{z^{-k}}{k}\frac{1-t^{k}}{1-q^{k}}(1+p^{k})\alpha_{k}}e^{\beta\hat{q}}z^{2\beta\alpha_{0}},\\\
&S_{-}(z)=e^{-\sum_{k>0}\frac{z^{k}}{k}\alpha_{-k}}e^{\sum_{k>0}\frac{z^{-k}}{k}(1+p^{-k})\alpha_{k}}e^{-\hat{q}}z^{-2\alpha_{0}}.\end{split}$
(2.45)

Since $z^{2\beta\alpha_{0}}\ket{m,n}=z^{(m+1)\beta-(n+1)}\ket{m,n}$, the
screening current $S_{+}(z)$ is expanded in integer powers of $z$ only when it
acts on Fock spaces ${\mathcal{F}}_{m,n}$ with $m=-1$. Similarly, since
$z^{-2\alpha_{0}}\ket{m,n}=z^{-(m+1)+(n+1)/\beta}\ket{m,n}$, the current
$S_{-}(z)$ has an expansion in integer powers of $z$ only when it acts on Fock
spaces ${\mathcal{F}}_{m,n}$ with $n=-1$. When the currents have an expansion
in integer powers, we can define the screening charges

$\Psi_{\pm}=\oint_{0}\dfrac{dz}{2i\pi}S_{\pm}(z).$ (2.46)

Thus, our previous requirement forces us to restrict the action of these
operators to the corresponding Fock spaces. It was shown in [42] that the
screening charges commute with the deformed Virasoro algebra on these spaces,
i.e. $[T(z),\Psi_{+}]=0$ (resp. $[T(z),\Psi_{-}]=0$) when acting on states
belonging to the Fock space ${\mathcal{F}}_{-1,n}$ (resp.
${\mathcal{F}}_{m,-1}$). In the next paragraph, we will argue that these
screening charges play the role of the Casimir $\Psi$ for the refined
hierarchy.

Note that it is possible to consider Fock spaces with more general charges if
we introduce the higher screening charges

$\Psi_{\pm}^{(n)}=\oint\prod_{a=1}^{n}\dfrac{dz_{a}}{2i\pi}S_{\pm}(z_{1})S_{\pm}(z_{2})\cdots
S_{\pm}(z_{n}).$ (2.47)

The derivation of the bilinear identity presented in the next section is
expected to extend to these higher screening charges, and should produce
bilinear integral equations involving several nested contours. However, we
will not address this problem here and simply focus on the simpler case of the
Fock spaces ${\mathcal{F}}_{-1,n}$ and ${\mathcal{F}}_{m,-1}$.

#### Relation with the quantum toroidal $\mathfrak{gl}(1)$ algebra

A remarkable relation between the representations of levels $(2,n)$ of the
quantum toroidal $\mathfrak{gl}(1)$ algebra and the deformed Virasoro algebra
has been observed in [39]. With the application to integrable hierarchies in
mind, we restrict ourselves to the representation $\rho_{u_{1},u_{2}}^{(2,0)}$
defined in (2.38). In this representation, the Cartan modes $a_{k}$ form a
(twisted) Heisenberg algebra $\beta_{k}=k\rho_{u_{1},u_{2}}^{(2,0)}(a_{k})$
that is sometimes called the $u(1)$-factor

$\beta_{k}=-\gamma^{-k}(1-q_{2}^{k})(1-q_{3}^{k})(J_{k}^{(1)}+\gamma^{k}J_{k}^{(2)}),\quad\beta_{-k}=\gamma^{-k}(1-q_{1}^{k})(1-q_{3}^{k})(J_{-k}^{(1)}+\gamma^{k}J_{-k}^{(2)}),$
(2.48)

with the shortcut notation $J_{k}^{(1)}=J_{k}\otimes 1$, $J_{k}^{(2)}=1\otimes
J_{k}$. To build the isomorphism, the key observation is that this Heisenberg
algebra can be factorized from the representation of the Drinfeld currents
$x^{\pm}(z)$, and the remaining factor identified with the q-Virasoro stress-
energy tensor $T(z)$

$\rho^{(2,0)}(x^{\pm}(z))=V_{b}^{\pm}(z)T(z),\quad
V_{b}^{\pm}(z)=(u^{(1)}u^{(2)})^{\pm 1/2}:e^{\mp\sum_{k\neq
0}\frac{\gamma^{\mp|k|}}{k(q_{3}^{k}-q_{3}^{-k})}z^{-k}\beta_{k}}.$ (2.49)

This relation holds upon the identification of parameters (1.9) and the
relations

$\displaystyle\alpha_{k}=-\dfrac{1-q^{k}}{1-t^{k}}p^{k}\dfrac{\gamma^{k}J_{k}^{(1)}-J_{k}^{(2)}}{1+p^{k}},$
$\displaystyle\alpha_{-k}=-(\gamma^{k}J_{-k}^{(1)}-J_{-k}^{(2)}),$ (2.50)
$\displaystyle q^{1-\beta+2\beta\alpha_{0}}=\hat{u}^{(2)}/\hat{u}^{(1)},$
$\displaystyle e^{\hat{q}}=q^{{\hat{d}}^{(1)}-{\hat{d}}^{(2)}}.$ (2.51)

where we suppressed the algebra homomorphism notation. The $u(1)$-factor
decouples from the deformed Virasoro algebra since $[\alpha_{k},\beta_{l}]=0$,
and we can write formally

$\rho^{(2,0)}({\mathcal{E}})=\left(\text{Heisenberg}\right)\times\left(\text{q-Virasoro}\right).$
(2.52)

As a consequence of this relation, the screening charges (2.46) of the
deformed Virasoro algebra define the Casimir of the representation
$\rho_{u_{1},u_{2}}^{(2,0)}$ for specific values of the ratio $u_{2}/u_{1}$
($u_{2}/u_{1}=q^{m\beta-n}$ in ${\mathcal{F}}_{m,n}$ due to (2.51)). It means
that the operators of $\rho_{u_{1},u_{2}}^{(2,0)}({\mathcal{E}})$ in
$\text{End}({\mathcal{F}}_{u_{1}}\otimes{\mathcal{F}}_{u_{2}})$ are fully
characterized by the fact that they commute with these screening charges (up
to the adjustment of the zero modes as explained below). Using the relations
(2.50) between the Heisenberg modes, we can rewrite the screening currents as

$\displaystyle\begin{split}&S_{+}(z)=\exp\left(-\sum_{k>0}\dfrac{z^{k}}{k}\dfrac{1-t^{k}}{1-q^{k}}(\gamma^{k}J_{-k}^{(1)}-J_{-k}^{(2)})\right)\exp\left(\sum_{k>0}\dfrac{z^{-k}}{k}p^{k}(\gamma^{k}J_{k}^{(1)}-J_{k}^{(2)})\right)e^{\beta\hat{q}}z^{2\beta\alpha_{0}},\\\
&S_{-}(z)=\exp\left(\sum_{k>0}\dfrac{z^{k}}{k}(\gamma^{k}J_{-k}^{(1)}-J_{-k}^{(2)})\right)\exp\left(-\sum_{k>0}\dfrac{z^{-k}}{k}\dfrac{1-q^{k}}{1-t^{k}}(\gamma^{k}J_{k}^{(1)}-J_{k}^{(2)})\right)e^{-\hat{q}}z^{-2\alpha_{0}},\end{split}$
(2.53)

and for the zero modes the relations (2.51) are implied.

#### Zero modes

From the correspondence between the representation
$\rho_{u_{1},u_{2}}^{(2,0)}$ and the free field representation of the deformed
Virasoro algebra, we deduced the following identification of the vacuum states

$\ket{u_{1}}\otimes\ket{u_{2}}=\ket{v_{0}}_{\text{u(1)}}\otimes\ket{m,n}_{\text{qVir}},\quad
v_{0}=(u_{1}u_{2})^{-1/2},\quad u_{2}=t^{m}q^{-n}u_{1}.$ (2.54)

The screening charges $\Psi_{\pm}$ are operators mapping Fock spaces of
different charges

$\Psi_{+}:{\mathcal{F}}_{-1,n}\to{\mathcal{F}}_{1,n},\quad\Psi_{-}:{\mathcal{F}}_{m,-1}\to{\mathcal{F}}_{m,1}.$
(2.55)

As a result, they intertwine representations of the toroidal algebra with
different weights

$\rho_{u_{1}^{\prime},u_{2}^{\prime}}^{(2,0)}({\mathcal{E}})\Psi_{\pm}=\Psi_{\pm}\rho_{u_{1},u_{2}}^{(2,0)}({\mathcal{E}}),$
(2.56)

with $u_{2}=t^{-1}q^{-n}u_{1}$ and $u^{\prime}_{2}=tq^{-n}u^{\prime}_{1}$ in
the case of $\Psi_{+}$. Similarly, $u_{2}=qt^{m}u_{1}$ and
$u^{\prime}_{2}=q^{-1}t^{m}u^{\prime}_{1}$ in the case of $\Psi_{-}$. On the
other hand, the products of weights $u_{1}u_{2}$ and
$u_{1}^{\prime}u^{\prime}_{2}$ are equal since $\Psi_{\pm}$ do not contain any
dependence in the algebra associated to the $u(1)$-factor. These relations fix
all the weights but one, and we take

$\displaystyle\begin{split}&\Psi_{+}:\quad u_{1}=u,\quad
u_{2}=t^{-1}q^{-n}u,\quad u^{\prime}_{1}=t^{-1}u,\quad
u^{\prime}_{2}=q^{-n}u,\\\ &\Psi_{-}:\quad u_{1}=u,\quad u_{2}=qt^{m}u,\quad
u^{\prime}_{1}=qu,\quad u^{\prime}_{2}=t^{m}u.\end{split}$ (2.57)

The relation (2.56) is the refined version of the basic bilinear condition
(2.6). In the self-dual limit, $S_{+}(z)$ (resp. $S_{-}(z)$) reduces to
$\psi(z)\otimes\bar{\psi}(z)$ (resp. $\bar{\psi}(z)\otimes\psi(z)$).999In
particular, for the zero modes $\hat{q}\to Q^{(2)}-Q^{(1)}$ and
$2\alpha_{0}\to J_{0}^{(2)}-J_{0}^{(1)}$. Thus $\Psi_{-}$ reduces to $\Psi$
and $\Psi_{+}$ to $\bar{\Psi}=\oint\psi(z)\otimes\bar{\psi}(z)$. Since the
coproduct $\Delta$ becomes co-commutative in this limit, the relation (2.56)
does indeed reduce to (2.6). In fact, a $(q,t)$-analogue of the Dirac fermions
can also be introduced using the expression of the screening currents, this
idea is explored in the Appendix A.

##  3 Refined Toda hierarchy

In this section, we introduce a $(q,t)$-deformation of the Toda tau functions,
and show that they obey a set of bilinear identities. Expanding these
identities, we derive the first difference-differential equations of a refined
Toda hierarchy.

###  3.1 Refined tau functions

Let ${\boldsymbol{t}}=(t_{1},t_{2},\cdots)$ and
$\bar{\boldsymbol{t}}=(t_{-1},t_{-2},\cdots)$ denote two sets of time
parameters, $u\in\mathbb{C}^{\times}$ a complex parameter, and
$G\in{\mathcal{E}}$ an element of the quantum toroidal $\mathfrak{gl}(1)$
algebra. We define the refined 2d Toda tau functions
$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$ as the following
correlation functions in the bosonic Fock space ${\mathcal{F}}_{u}$ introduced
in Section 2.3

$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=\bra{u}e^{\sum_{k>0}t_{k}J_{k}}\rho_{u}^{(1,0)}(G)e^{\sum_{k>0}t_{-k}J_{-k}}\ket{u},\quad
G\in{\mathcal{E}}.$ (3.1)

These tau functions also depend on the quantum group parameters
$(q_{1},q_{2})\equiv(t^{-1},q)$ but we will omit them in the notation since
they are spectators in our discussion. In the self-dual limit $t\to q$,
$\rho_{u}^{(1,0)}(G)$ reduces to an element of the quantum $W_{1+\infty}$
algebra in the (bosonized) fermionic representation, and the expression (3.1)
reproduces the usual definition (2.5) of the 2d Toda tau function
$\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$, with the complex
parameter $u$ reducing to the integer parameter $n$ of the as $u\to
q^{n}$.101010In the self-dual limit, $q^{n}$ coincides with one of the two
weights of the fermionic representation $\rho_{f}$ of $W_{1+\infty}$. To avoid
introducing too many technicalities, we decided to omit these weights in the
definition 2.12 of the representation, but they can be found in the section
2.1.2 of [20].

Toda tau functions have a natural expansion in Schur functions, and their
$(q,t)$-deformations (3.1) can be expanded in a similar way in Macdonald
functions. Indeed, Macdonald functions are closely related to the horizontal
Fock representation of the quantum toroidal $\mathfrak{gl}(1)$ algebra [39].
This is better understood by introducing the Macdonald states
$\ket{P_{\lambda},u}\in{\mathcal{F}}_{u}$ obtained as the inverse image
$\iota^{-1}(P_{\lambda}({\boldsymbol{x}}))$ of Macdonald function under the
isomorphism $\iota$ sending the Fock space ${\mathcal{F}}_{u}$ to the ring of
symmetric functions

$J_{-k}\to p_{k}({\boldsymbol{x}}),\quad J_{k}\to k\dfrac{\partial}{\partial
p_{k}},\quad\ket{u}\to 1,$ (3.2)

where $p_{k}({\boldsymbol{x}})$ are the power sum symmetric functions
$\sum_{\alpha=1}^{\infty}x_{\alpha}^{k}$. With this definition, the action of
a certain commutative subalgebra $b_{k}\in{\mathcal{E}}$ is diagonal on these
states, with eigenvalues111111This subalgebra can be defined as the image of
the Cartan modes $a_{k}$ under Miki’s automorphism, and in particular $b_{\pm
1}=x_{0}^{\pm}$[19, 49].

$\rho_{u}^{(1,n)}(b_{k})\ket{P_{\lambda},u}=(-\gamma)^{-k}u^{k}c_{k}\left(\sum_{i,j\in\lambda}q_{1}^{(i-1)k}q_{2}^{(j-1)k}-\dfrac{1}{(1-q_{1}^{k})(1-q_{2}^{k})}\right)\ket{P_{\lambda},u}.$
(3.3)

The dual states $\bra{P_{\lambda},u}$ can be introduced by identifying the
inner product with Macdonald’s scalar product

$\bra{P_{\lambda},u}\\!\\!\\!\ket{P_{\mu},u}=\delta_{\lambda,\mu}\left\langle
P_{\lambda},P_{\lambda}\right\rangle_{q,t},\quad\left\langle
P_{\lambda},P_{\lambda}\right\rangle_{q,t}=\prod_{(i,j)\in\lambda}\dfrac{1-q^{\lambda_{i}-j+1}t^{\lambda^{\prime}_{j}-i}}{1-q^{\lambda_{i}-j}t^{\lambda^{\prime}_{j}-i+1}}.$
(3.4)

where $\lambda^{\prime}$ is the transposed partition. Inserting the Cauchy
identity

$1=\sum_{\lambda}\dfrac{\ket{P_{\lambda},u}\bra{P_{\lambda},u}}{\left\langle
P_{\lambda},P_{\lambda}\right\rangle_{q,t}}$ (3.5)

in the definition (3.1) of the refined tau functions, and using the properties

$P_{\lambda}({\boldsymbol{x}})=\bra{u}e^{\sum_{k>0}\frac{p_{k}({\boldsymbol{x}})}{k}J_{k}}\ket{P_{\lambda},u}=\bra{P_{\lambda},u}e^{\sum_{k>0}\frac{1-t^{k}}{1-q^{k}}\frac{p_{k}({\boldsymbol{x}})}{k}J_{-k}}\ket{u},$
(3.6)

we end up with a double expansion over partitions

$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=\sum_{\lambda,\mu}\dfrac{P_{\lambda}({\boldsymbol{x}})P_{\mu}({\boldsymbol{y}})}{\left\langle
P_{\lambda},P_{\lambda}\right\rangle_{q,t}\left\langle
P_{\mu},P_{\mu}\right\rangle_{q,t}}\bra{P_{\lambda},u}\rho_{u}^{(1,0)}(G)\ket{P_{\mu},u}.$
(3.7)

where the time variables are related to the Miwa variables in the following
way

$t_{k}=\dfrac{1}{k}p_{k}({\boldsymbol{x}}),\quad
t_{-k}=\dfrac{1}{k}\dfrac{1-t^{k}}{1-q^{k}}p_{k}({\boldsymbol{y}}).$ (3.8)

#### Refined hypergeometric tau functions

When the operator $\rho^{(1,0)}(G)$ is diagonal on the Macdonald basis, the
refined tau function can be expanded in a single sum over partitions121212It
was pointed to us by a referee that the tau function in (3.9) can be
interpreted as the expectation value of the operator $G$ with respect to the
Macdonald measure [50, 51].

$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=\sum_{\lambda}\dfrac{P_{\lambda}({\boldsymbol{x}})P_{\lambda}({\boldsymbol{y}})}{\left\langle
P_{\lambda},P_{\lambda}\right\rangle_{q,t}}G_{\lambda},\quad\text{with}\quad\rho^{(1,0)}(G)\ket{P_{\lambda}}=G_{\lambda}\ket{P_{\lambda}}.$
(3.9)

This is a natural $(q,t)$-deformation of a class of hypergeometric tau
functions. In particular, this class contains the tau functions introduced by
Okounkov in the context of Hurwitz numbers [9]

$\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|\mathfrak{q},\beta)=\bra{n}e^{\sum_{k>0}t_{k}J_{k}}\mathfrak{q}^{L_{0}}e^{\beta
W_{0}}e^{\sum_{k>0}t_{-k}J_{-k}}\ket{n},\quad
W_{0}=\sum_{r\in\mathbb{Z}+1/2}r^{2}\vdots\bar{\psi}_{-r}\psi_{r}\vdots,$
(3.10)

for which the $(q,t)$-deformation reads

$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|\mathfrak{q},\beta)=\bra{u}e^{\sum_{k>0}t_{k}J_{k}}\mathfrak{q}^{L_{0}}F^{\beta}e^{\sum_{k>0}t_{-k}J_{-k}}\ket{u}=\sum_{\lambda}\dfrac{P_{\lambda}({\boldsymbol{x}})P_{\lambda}({\boldsymbol{y}})}{\left\langle
P_{\lambda},P_{\lambda}\right\rangle_{q,t}}\mathfrak{q}^{|\lambda|}\prod_{(i,j)\in\lambda}q_{1}^{(i-1)\beta}q_{2}^{(j-1)\beta}$
(3.11)

where the operator $W_{0}$ has been replaced with the framing operator $\log
F$ introduced in [20]131313Strictly speaking, this operator $F$ does not
belong to the representation of the algebra ${\mathcal{E}}$, but it can be
obtained in a certain limit of operators in ${\mathcal{E}}$ from which we hope
to be able to define its coproduct.

$F\ket{P_{\lambda}}=\prod_{(i,j)\in\lambda}q_{1}^{i-1}q_{2}^{j-1}\ket{P_{\lambda}}\operatorname*{\longrightarrow}_{s.d.}q^{W_{0}}\ket{s_{\lambda}}=\prod_{(i,j)\in\lambda}q^{j-i}\ket{s_{\lambda}}.$
(3.12)

This type of tau functions are closely related to the $(q,t)$-deformed matrix
models studied in [52, 53]. For instance, choosing $G=e^{\alpha b_{1}}$ with
$b_{1}=x_{0}^{+}=\oint(2i\pi z)^{-1}x^{+}(z)dz$, we have the associated
eigenvalues

$G_{\lambda}=\exp\left(-\alpha
u\gamma^{-1}(1-q_{1})(1-q_{3})\sum_{i=1}^{\infty}q_{1}^{i-1}q_{2}^{\lambda_{i}}\right).$
(3.13)

But, it is also possible to expand this $(q,t)$-tau function in the parameter
$\alpha$ and exploit the normal-ordering properties of the vertex operator
$\eta^{+}(z)$ to rewrite the expansion as a sum of integrals reminiscent of
the $(q,t)$-deformed matrix model,

$\displaystyle\begin{split}\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)&=\sum_{n=0}^{\infty}\dfrac{\alpha^{n}u^{n}}{n!}\oint\dfrac{dz_{1}}{2i\pi
z_{1}}\cdots\dfrac{dz_{n}}{2i\pi
z_{n}}\bra{\emptyset}e^{\sum_{k>0}t_{k}J_{k}}\eta^{+}(z_{1})\cdots\eta^{+}(z_{n})e^{\sum_{k>0}t_{-k}J_{-k}}\ket{\emptyset}\\\
&=\sum_{n=0}^{\infty}\dfrac{\alpha^{n}u^{n}}{n!}\oint\dfrac{dz_{1}}{2i\pi
z_{1}}\cdots\dfrac{dz_{n}}{2i\pi
z_{n}}\prod_{\genfrac{}{}{0.0pt}{}{i,j=1}{i<j}}^{n}\zeta(z_{i}/z_{j})\times
e^{\sum_{i=1}^{n}\sum_{k>0}(1-q_{1}^{k})t_{k}z_{i}^{k}}e^{-\sum_{i=1}^{n}\sum_{k>0}(1-q_{2}^{k})t_{-k}z_{i}^{-k}},\\\
&=\sum_{n=0}^{\infty}\dfrac{\alpha^{n}u^{n}}{n!}\oint\dfrac{dz_{1}}{2i\pi
z_{1}}\cdots\dfrac{dz_{n}}{2i\pi
z_{n}}\prod_{\genfrac{}{}{0.0pt}{}{i,j=1}{i<j}}^{n}\zeta(z_{i}/z_{j})\times\prod_{i=1}^{n}\prod_{\alpha}\dfrac{(1-t^{-1}z_{i}x_{\alpha})(1-z_{i}^{-1}y_{\alpha})}{(1-z_{i}x_{\alpha})(1-tz_{i}^{-1}y_{\alpha})},\end{split}$
(3.14)

where we introduced the Miwa variables (3.8) to write the last line. Similar
integral formulas can be derived for operators of the form $G=e^{\alpha
b_{k}}$, albeit involving more integrations.

Furthermore, for a certain operator $G$, the tau functions (3.9) reproduce the
$\beta$-Hurwitz tau functions studied in [54] in the Jack limit
$q=t^{\beta}\to 1$. Ultimately, we would like to derive the hierarchy of
difference-differential equations satisfied by this type of tau functions.
Unfortunately, because of the complexity of the coproduct for these operators,
we will not be able to provide an explicit answer in this paper. We hope to
come back to this particular problem in a future work.

###  3.2 Refined bilinear identities

Refined tau functions (3.1) are related through a set of bilinear integral
equations which stems from the commutation relation (2.56) involving the
screening charges $\Psi_{\pm}$ of the q-Virasoro algebra and the coproduct of
the element $G$141414Note that the summation over $\alpha$ may contain
infinitely many terms.

$\Delta(G)=\sum_{\alpha}G_{\alpha}\otimes G^{\prime}_{\alpha}.$ (3.15)

We call these relations refined bilinear identities, they take the form

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})\tau_{uq^{-n}t^{-1}}({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{\prime})p^{k}z^{-k}}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})\tau_{uq^{-n}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\\\
&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-m-1}e^{\sum_{k>0}(\gamma^{k}t_{k}-t^{\prime}_{k})z^{k}}\tau_{u}({\boldsymbol{t}}-[\gamma
z^{-1}]_{t,q},\bar{\boldsymbol{t}}|G_{\alpha})\tau_{uqt^{m}}({\boldsymbol{t}}^{\prime}+[z^{-1}]_{t,q},\bar{\boldsymbol{t}}^{\prime}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-m-1}e^{-\sum_{k>0}(\gamma^{k}t_{-k}-t^{\prime}_{-k})\frac{1-q^{k}}{1-t^{k}}z^{-k}}\tau_{uq}({\boldsymbol{t}},\bar{\boldsymbol{t}}+[\gamma
z]|G_{\alpha})\tau_{ut^{m}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}-[z]|G^{\prime}_{\alpha})=0,\end{split}$
(3.16)

where the first equation is obtained from the commutation relation involving
$\Psi_{+}$ and the second one using $\Psi_{-}$. Each equation depends on a
free integer parameter, $n\in\mathbb{Z}$ in the first equation, and
$m\in\mathbb{Z}$ in the second one, which enters as a power of the integration
variable, and in the shift of the parameter $u$. These equations involve the
usual shift of time parameters ${\boldsymbol{t}}\pm[z]=(t_{1}\pm z,t_{2}\pm
z^{2}/2,\cdots)$, but also one of two $(q,t)$-deformations of this shift

${\boldsymbol{t}}\pm[z]_{q,t}=\left(t_{1}\pm\dfrac{1-t}{1-q}z,t_{2}\pm\dfrac{1}{2}\dfrac{1-t^{2}}{1-q^{2}}z^{2},\cdots\right),\quad{\boldsymbol{t}}\pm[z]_{t,q}=\left(t_{1}\pm\dfrac{1-q}{1-t}z,t_{2}\pm\dfrac{1}{2}\dfrac{1-q^{2}}{1-t^{2}}z^{2},\cdots\right).$
(3.17)

The shifts of the negative times $\bar{\boldsymbol{t}}$ have the same
expression with $t_{k}$ replaced with $t_{-k}$. The equations (3.16) are one
of the main results of this paper, their derivation is given in Appendix B.

In the self-dual limit, the coproduct (3.15) becomes co-commutative, i.e.
$\Delta(G)=G\otimes G$, and the summation over $\alpha$ disappears. The
$(q,t)$-deformed shifts of the time parameters ${\boldsymbol{t}}\pm[z]_{q,t}$
and ${\boldsymbol{t}}\pm[z]_{t,q}$ both reduce to the usual one
${\boldsymbol{t}}\pm[z]$. Thus, the two refined bilinear identities (3.16)
reduce to the bilinear identities of the 2d Toda hierarchy151515It is obtained
directly for the second equation, and follows from the change of variable
$z\to 1/z$ in the first equation.

$\displaystyle\begin{split}&\oint_{\infty}\dfrac{dz}{2i\pi}z^{n-n^{\prime}}e^{\sum_{k}(t_{k}-t^{\prime}_{k})z^{k}}\tau_{n}({\boldsymbol{t}}-[z^{-1}],\bar{\boldsymbol{t}}|G)\tau_{n^{\prime}}({\boldsymbol{t}}^{\prime}+[z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G)\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{n-n^{\prime}}e^{-\sum_{k}(t_{-k}-t^{\prime}_{-k})z^{-k}}\tau_{n+1}({\boldsymbol{t}},\bar{\boldsymbol{t}}+[z]|G)\tau_{n^{\prime}-1}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}-[z]|G)=0.\end{split}$
(3.18)

The derivation of these identities from the basic bilinear condition (2.6) is
also reproduced in Appendix B to allow for comparison. We note that this
derivation only employs the bosonic expression of fermionic fields, which is
why it extends so easily to the $(q,t)$-deformed framework.

#### Reduction to the mKP hierarchy

The reduction of the 2d Toda hierarchy to the mKP hierarchy is obtained by
assuming that the operator $G$ obeys the property $J_{k}G=GJ_{-k}$ for $k>0$.
In this case, the tau function only depends on a single set of time
parameters, e.g.

$\tau_{n}({\boldsymbol{t}}|G)=\bra{n}e^{\sum_{k>0}t_{k}J_{k}}G\ket{n}.$ (3.19)

The derivation of the bilinear identity is similar to the case of 2d Toda, but
the treatment of the r.h.s. is simplified by the absence of the exponential of
negative times.161616Indeed, in this case we can use the property
$\Psi(\ket{n}\otimes\ket{n^{\prime}})=0$ for $n\geq n^{\prime}$ which follows
from the action of the fermionic modes on the vacua. This property is used to
simplify the equation (LABEL:equ_H_step1) into
$\displaystyle\begin{split}&\oint_{\infty}\dfrac{dz}{2i\pi}\bra{n+1,n^{\prime}-1}\left(e^{J_{+}({\boldsymbol{t}})}\otimes
e^{J_{+}({\boldsymbol{t}}^{\prime})}\right)(\bar{\psi}(z)\otimes\psi(z))(G\otimes
G)\left(e^{J_{-}(\bar{\boldsymbol{t}})}\otimes
e^{J_{-}(\bar{\boldsymbol{t}}^{\prime})}\right)\ket{n,n^{\prime}}=0\end{split}$
(3.20) for $n\geq n^{\prime}$. In this case, the bilinear identity contains
only a single integral

$\oint_{\infty}dzz^{n-n^{\prime}}e^{\sum_{k>0}z^{k}(t_{k}-t^{\prime}_{k})}\tau_{n}({\boldsymbol{t}}-[z^{-1}]|G)\tau_{n^{\prime}}({\boldsymbol{t}}^{\prime}+[z^{-1}]|G)=0,\quad
n\geq n^{\prime}.$ (3.21)

A similar reduction can be introduced in the refined setting, provided that
all the operators $G_{\alpha}$, $G^{\prime}_{\alpha}$ entering the coproduct
of $G$ obey the property $J_{k}G_{\alpha}=\Lambda_{k}G_{\alpha}J_{-k}$ for
some fixed $\Lambda_{k}\in\mathbb{C}(q,t)$. If so, we can remove the
exponential of negative times in the definition (3.1) of the tau functions.
Using the fact that $\Psi_{+}\ket{-1,n}=0$ for $n<0$ and
$\Psi_{-}\ket{m,-1}=0$ for $m<0$, we find the following $(q,t)$-deformation of
the KP bilinear identity

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}]|G_{\alpha})\tau_{uq^{-n}t^{-1}}({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}]|G^{\prime}_{\alpha})=0,\quad
n<0.\\\
&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-m-1}e^{\sum_{k>0}(\gamma^{k}t_{k}-t^{\prime}_{k})z^{k}}\tau_{u}({\boldsymbol{t}}-[\gamma
z^{-1}]_{t,q}|G_{\alpha})\tau_{uqt^{m}}({\boldsymbol{t}}^{\prime}+[z^{-1}]_{t,q}|G^{\prime}_{\alpha})=0,\quad
m<0.\end{split}$ (3.22)

###  3.3 Difference-differential equations of the refined hierarchy

The difference-differential equations of the refined hierarchy are obtained
from the bilinear identities (LABEL:Hirota_2dToda) by setting

$\displaystyle\varepsilon_{k}=\gamma^{|k|}t_{k}-t^{\prime}_{k}$ (3.23)

and expanding at small $\varepsilon_{k}$, with $k\in\mathbb{Z}$. We focus here
on the first bilinear identity, i.e. the one derived from $\Psi_{+}$, the
treatment of the second identity being parallel with the role of $q$ and $t$
simply being flipped and the coproduct replaced by its opposite. The
difference-differential equations obtained in this way will be labeled
$\text{Eq}_{n}^{(\mu,\bar{\mu})}$ with two partitions $\mu$, $\bar{\mu}$ and
the parameter $n\in\mathbb{Z}$ coinciding with the integer parameter in the
bilinear identity. The two partitions label the order
$O(\varepsilon_{\mu}\varepsilon_{-\bar{\mu}})$ at which the equation appears,
with
$\varepsilon_{\mu}=\varepsilon_{\mu_{1}}\varepsilon_{\mu_{2}}\cdots\varepsilon_{\mu_{\ell}}$
and
$\varepsilon_{-\bar{\mu}}=\varepsilon_{-\bar{\mu}_{1}}\varepsilon_{-\bar{\mu}_{2}}\cdots\varepsilon_{-\bar{\mu}_{\bar{\ell}}}$,
where $\ell,\bar{\ell}$ denote the lengths of the corresponding partitions.

To shorten a bit our expressions, which tend to be rather lengthy, we
introduce the notation $\partial_{k}=\partial/\partial t_{k}$, and often drop
the parentheses after derivation operators, taking the unusual convention
$\partial AB$ meaning $(\partial A)B$. We also denote for short
$\boldsymbol{\gamma}{\boldsymbol{t}}=(\gamma t_{1},\gamma^{2}t_{2},\cdots)$,
$\boldsymbol{\gamma}\bar{\boldsymbol{t}}=(\gamma
t_{-1},\gamma^{2}t_{-2},\cdots)$, and

$\tau_{u}(G_{\alpha})=\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G_{\alpha}),\quad\tau_{u}^{\prime}(G_{\alpha}^{\prime})=\tau_{u}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|G^{\prime}_{\alpha}).$
(3.24)

The bilinear identities can be rewritten in terms of time differences
$\varepsilon_{k}$ as

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}r_{k}\varepsilon_{k}z^{k}}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})e^{-\sum_{k>0}\gamma^{-k}(\varepsilon_{k}\partial_{k}+\varepsilon_{-k}\partial_{-k})}\tau_{uq^{-n}t^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}}-[\gamma^{-2}z^{-1}],\boldsymbol{\gamma}\bar{\boldsymbol{t}}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}\varepsilon_{-k}p^{k}z^{-k}}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})e^{-\sum_{k>0}\gamma^{-k}(\varepsilon_{k}\partial_{k}+\varepsilon_{-k}\partial_{-k})}\tau_{uq^{-n}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\end{split}$
(3.25)

with

$r_{k}=\dfrac{1-t^{k}}{1-q^{k}}.$ (3.26)

From the combinatorial relation171717We denote
$m_{k}(\lambda)=\sharp\\{i\in\mathbb{Z}^{>0}\diagup\lambda_{i}=k\\}$ the
multiplicity of columns with $k$ boxes in $\lambda$.

$e^{\sum_{k>0}z^{k}p_{k}}=\sum_{\lambda}\tilde{z}_{\lambda}^{-1}p_{\lambda}z^{|\lambda|},\quad
p_{\lambda}=p_{\lambda_{1}}p_{\lambda_{2}}\cdots
p_{\lambda_{\ell}},\quad\tilde{z}_{\lambda}=\prod_{k\geq 1}m_{k}(\lambda)!,$
(3.27)

for any set of parameters $p_{k}$ indexed by positive integers $k$, we deduce
the integral equation $\text{Eq}_{n}^{(\mu,\bar{\mu})}$ at order
$O(\varepsilon_{\mu},\varepsilon_{-\bar{\mu}})$

$\displaystyle\begin{split}&\sum_{\alpha}\sum_{\genfrac{}{}{0.0pt}{}{\lambda,\nu}{\lambda\cup\nu=\mu}}\frac{r_{\lambda}\gamma^{|\lambda|}}{\tilde{z}_{\lambda}\tilde{z}_{\nu}\tilde{z}_{\bar{\mu}}}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1+|\lambda|}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})\partial_{\nu}\partial_{-\bar{\mu}}\tau_{uq^{-n}t^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}}-[\gamma^{-2}z^{-1}],\boldsymbol{\gamma}\bar{\boldsymbol{t}}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\sum_{\genfrac{}{}{0.0pt}{}{\bar{\lambda},\bar{\nu}}{\bar{\lambda}\cup\bar{\nu}=\bar{\mu}}}\frac{(-1)^{|\bar{\lambda}|}\gamma^{-|\bar{\lambda}|}}{\tilde{z}_{\bar{\lambda}}\tilde{z}_{\bar{\nu}}\tilde{z}_{\mu}}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1-|\bar{\lambda}|}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})\partial_{\mu}\partial_{-\bar{\nu}}\tau_{uq^{-n}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\end{split}$
(3.28)

In this expression, $\lambda\cup\nu$ denotes the partition combining the
columns of both $\lambda$ and $\nu$, and so
$\varepsilon_{\lambda}\varepsilon_{\nu}=\varepsilon_{\lambda\cup\nu}$. We also
used the notation $r_{\lambda}=r_{\lambda_{1}}r_{\lambda_{2}}\cdots
r_{\lambda_{\ell}}$,
$\partial_{\lambda}=\partial_{\lambda_{1}}\partial_{\lambda_{2}}\cdots\partial_{\lambda_{\ell}}$,
$\partial_{-\lambda}=\partial_{-\lambda_{1}}\partial_{-\lambda_{2}}\cdots\partial_{-\lambda_{\ell}}$.
Note that the first line is vanishing if $n>|\mu|$, and the second one if
$n<-|\bar{\mu}|$, and so the equation is always non-trivial. The difference-
differential equations of the hierarchy are obtained by expanding the
integrands in powers of $z^{\mp 1}$. In the remaining of this section, we
examine in details the first few orders.

#### Order $O(1)$

At this order, the expansion of the bilinear identity for the Toda hierarchy
brings only a trivial equation, but it is not the case for the refined
hierarchy. Indeed, the integral equation at this order is

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})\tau_{uq^{-n}t^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}}-[\gamma^{-2}z^{-1}],\boldsymbol{\gamma}\bar{\boldsymbol{t}}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})\tau_{uq^{-n}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\\\
\end{split}$ (3.29)

For definiteness, we restrict ourselves to the values $n=0$ and $n=\pm 1$.
Expanding the integrands, the contour integrals can be computed easily and we
end up with the following three equations

$\displaystyle\begin{split}\text{Eq}_{-1}^{(\emptyset,\emptyset)}:\quad&\sum_{\alpha}\left[\partial_{1}\tau_{u}(G_{\alpha})\tau_{uqt^{-1}}^{\prime}(G^{\prime}_{\alpha})-p\tau_{u}(G_{\alpha})\partial_{1}\tau_{uqt^{-1}}^{\prime}(G^{\prime}_{\alpha})\right]=0,\\\
\text{Eq}_{0}^{(\emptyset,\emptyset)}:\quad&\sum_{\alpha}\left[\tau_{u}(G_{\alpha})\tau_{ut^{-1}}^{\prime}(G^{\prime}_{\alpha})-\tau_{ut^{-1}}(G_{\alpha})\tau_{u}^{\prime}(G^{\prime}_{\alpha})\right]=0,\\\
\text{Eq}_{1}^{(\emptyset,\emptyset)}:\quad&\sum_{\alpha}\left[\partial_{-1}\tau_{u}(G_{\alpha})\tau_{uq^{-1}t}^{\prime}(G^{\prime}_{\alpha})-p\tau_{u}(G_{\alpha})\partial_{-1}\tau_{uq^{-1}t}^{\prime}(G^{\prime}_{\alpha})\right]=0.\end{split}$
(3.30)

In the self-dual limit, the summation disappears, both $\tau_{u}(G_{\alpha})$
and $\tau_{u}^{\prime}(G_{\alpha}^{\prime})$ tend to
$\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$, and these three equations
become trivial.

#### Order $O(\varepsilon_{1})$ and $O(\varepsilon_{-1})$

At this order, the expansion (3.28) of the bilinear identity simplifies into
respectively

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})\left(\gamma
r_{1}z+\partial_{1}\right)\tau_{uq^{-n}t^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}}-[\gamma^{-2}z^{-1}],\boldsymbol{\gamma}\bar{\boldsymbol{t}}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})\partial_{1}\tau_{uq^{-n}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\\\
\end{split}$ (3.31)

and

$\displaystyle\begin{split}&\sum_{\alpha}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G_{\alpha})\partial_{-1}\tau_{uq^{-n}t^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}}-[\gamma^{-2}z^{-1}],\boldsymbol{\gamma}\bar{\boldsymbol{t}}|G^{\prime}_{\alpha})\\\
+&\sum_{\alpha}\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G_{\alpha})\left(-\gamma^{-1}z^{-1}+\partial_{-1}\right)\tau_{uq^{-n}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}+[z]_{q,t}|G^{\prime}_{\alpha})=0,\end{split}$
(3.32)

Specializing to $n=0$, we can evaluate the integrals and write down the
corresponding equations of the refined hierarchy

$\displaystyle\begin{split}\text{Eq}_{0}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}:\quad&\sum_{\alpha}\left[r_{1}\partial_{1}\tau_{u}(G_{\alpha})\tau_{ut^{-1}}^{\prime}(G^{\prime}_{\alpha})+(1-pr_{1})\tau_{u}(G_{\alpha})\partial_{1}\tau_{ut^{-1}}^{\prime}(G^{\prime}_{\alpha})-\tau_{ut^{-1}}(G_{\alpha})\partial_{1}\tau_{u}^{\prime}(G^{\prime}_{\alpha})\right]=0,\\\
\text{Eq}_{0}^{(\emptyset,{\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&\sum_{\alpha}\left[r_{1}\partial_{-1}\tau_{ut^{-1}}(G_{\alpha})\tau_{u}^{\prime}(G^{\prime}_{\alpha})+(1-pr_{1})\tau_{ut^{-1}}(G_{\alpha})\partial_{-1}\tau_{u}^{\prime}(G^{\prime}_{\alpha})-\tau_{u}(G_{\alpha})\partial_{-1}\tau_{ut^{-1}}^{\prime}(G^{\prime}_{\alpha})\right]=0.\end{split}$
(3.33)

In the self-dual limit, these two equations also trivialize. On the other
hand, choosing $n=\pm 1$, we find four equations. Two of them, namely

$\displaystyle\begin{split}&\text{Eq}_{-1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}:\\\ &\
\sum_{\alpha}\left[p\left(1-p\dfrac{r_{1}}{2}\right)\tau_{u}(G_{\alpha})\partial_{1}^{2}\tau_{up}^{\prime}(G^{\prime}_{\alpha})-(1-pr_{1})\partial_{1}\tau_{u}(G_{\alpha})\partial_{1}\tau_{up}^{\prime}(G^{\prime}_{\alpha})-\dfrac{r_{1}}{2}\partial_{1}^{2}\tau_{u}(G_{\alpha})\tau_{up}^{\prime}(G^{\prime}_{\alpha})\right]\\\
&=\dfrac{1}{2}r_{1}\gamma^{-1}\sum_{\alpha}\left[\partial_{2}\tau_{u}(G_{\alpha})\tau_{up}^{\prime}(G^{\prime}_{\alpha})-\gamma^{-4}\tau_{u}(G_{\alpha})\partial_{2}\tau_{up}^{\prime}(G^{\prime}_{\alpha})\right],\\\
&\text{Eq}_{1}^{(\emptyset,{\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\\\ &\
\sum_{\alpha}\left[p\left(1-p\dfrac{r_{1}}{2}\right)\tau_{u}(G_{\alpha})\partial_{-1}^{2}\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})-(1-pr_{1})\partial_{-1}\tau_{u}(G_{\alpha})\partial_{-1}\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})-\dfrac{r_{1}}{2}\partial_{-1}^{2}\tau_{u}(G_{\alpha})\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})\right]\\\
&=-\dfrac{r_{2}}{2r_{1}}\sum_{\alpha}\left[\partial_{-2}\tau_{u}(G_{\alpha})\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})-\gamma^{-4}\tau_{u}(G_{\alpha})\partial_{-2}\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})\right],\end{split}$
(3.34)

trivialize in the self-dual limit, but the other two

$\displaystyle\begin{split}\text{Eq}_{-1}^{(\emptyset,{\leavevmode\hbox
to4.45pt{\vbox to4.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&\sum_{\alpha}\left[\partial_{1}\tau_{u}(G_{\alpha})\partial_{-1}\tau_{up}^{\prime}(G^{\prime}_{\alpha})-p\tau_{u}(G_{\alpha})\partial_{1}\partial_{-1}\tau_{up}^{\prime}(G^{\prime}_{\alpha})+\tau_{ut^{-1}}(G_{\alpha})\tau_{uq}^{\prime}(G^{\prime}_{\alpha})\right]=0,\\\
\text{Eq}_{1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}:\quad&\sum_{\alpha}\left[\partial_{-1}\tau_{u}(G_{\alpha})\partial_{1}\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})-p\tau_{u}(G_{\alpha})\partial_{1}\partial_{-1}\tau_{up^{-1}}^{\prime}(G^{\prime}_{\alpha})+\tau_{ut}(G_{\alpha})\tau_{uq^{-1}}^{\prime}(G^{\prime}_{\alpha})\right]=0,\end{split}$
(3.35)

do not. Instead, they reduce to the 2d Toda equation for tau function
$\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$. Indeed, noticing that in
the limit $t\to q$, we have $p=\gamma=1$, and the sum over $\alpha$ reduces to
a single term corresponding to $G_{\alpha}=G_{\alpha}^{\prime}=G$, we end up
with the equation

$\partial_{1}\tau_{u}(G)\partial_{-1}\tau_{u}(G)-\tau_{u}(G)\partial_{1}\partial_{-1}\tau_{u}(G)+\tau_{uq^{-1}}(G)\tau_{uq}(G)=0.$
(3.36)

The sum of the first two terms is equal to
$-\tau_{u}(G)^{2}\partial_{1}\partial_{-1}\log(\tau_{u}(G))$, and specializing
to $u=q^{n}$, we find for
$\tau_{q^{n}}(G)=\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)$181818The
unusual sign of the r.h.s. comes from the fact that we have flipped the sign
of the negative time $t_{-k}\to-t_{-k}$ with respect to the more traditional
conventions used for instance in [5].

$\partial_{1}\partial_{-1}\log\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=\dfrac{\tau_{n-1}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)\tau_{n+1}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)}{\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)^{2}}.$
(3.37)

This equation implies the Toda equation

$\partial_{1}\partial_{-1}\phi(n)=e^{\phi(n+1)-\phi(n)}-e^{\phi(n)-\phi(n-1)},\quad\text{for}\quad
e^{\phi(n)}=\dfrac{\tau_{n+1}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)}{\tau_{n}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)}.$
(3.38)

Thus, the two equations (3.35) are a $(q,t)$-deformation of the Toda equation,
and in consistence with our terminology we will call them refined 2d Toda
equations. The derivation of these equations is the main result of this
section.

##  4 Analysis of refined Toda $\tau$-functions

In this section, we specialize the operator $G$ to various elements of the
algebra ${\mathcal{E}}$ and analyze the corresponding refined tau functions.
Our examples in this section are obtained with $G$ given by either an
exponential of Cartan generators or by grading elements.

###  4.1 Cartan generators

In the simplest example, $G$ is defined as an exponential of the Cartan
generators $a_{k}$ for which the coproduct (LABEL:coproduct) is co-commutative
(up to factors involving the central element $\gamma^{c}$). For definiteness,
consider the case of a family of elements $G_{\boldsymbol{\xi}}$ parameterized
by a set of complex numbers $\xi_{k}\in\mathbb{C}$ and defined as follows

$G_{\boldsymbol{\xi}}=e^{\sum_{k>0}\xi_{k}a_{-k}}\in{\mathcal{E}}.$ (4.1)

The coproduct of these elements can be written in the compact form

$\Delta(G_{\boldsymbol{\xi}})=G_{\boldsymbol{\gamma}^{-c^{(2)}/2}{\boldsymbol{\xi}}}\otimes
G_{\boldsymbol{\gamma}^{c^{(1)}/2}{\boldsymbol{\xi}}},$ (4.2)

where we introduced the notation
$\boldsymbol{\gamma}^{\alpha}{\boldsymbol{\xi}}=\\{\gamma^{\alpha
k}\xi_{k}\\}$ to absorb the central elements. For these simple elements, the
correlation functions (3.1) defining the refined tau functions are easy to
compute, we find

$\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|{\boldsymbol{\xi}})=e^{\sum_{k>0}kt_{k}t_{-k}}e^{-\sum_{k>0}(1-q_{1}^{k})(1-q_{3}^{k})\gamma^{-k/2}t_{k}\xi_{k}}.$
(4.3)

Note that in this case the refined tau functions are actually independent of
the weight $u$, and we have simplified the previous notation
$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G_{{\boldsymbol{\xi}}})$ into
$\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|{\boldsymbol{\xi}})$.

#### Bilinear identities

Once again, we will focus on the first of the two bilinear identities (3.16),
and the same arguments apply to the second one as well. For this particular
choice of elements $G_{{\boldsymbol{\xi}}}$, the bilinear identity simplifies
into

$\displaystyle\begin{split}&\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})\tau({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{\prime})p^{k}z^{-k}}\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})\tau({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})=0.\end{split}$
(4.4)

Using the expression of the refined tau function (4.3) derived previously, we
can compute the products

$\displaystyle\begin{split}&\tau({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})\tau({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})=e^{\sum_{k>0}(\gamma^{k}t_{-k}-t^{\prime}_{-k})p^{k}z^{-k}},\\\
&\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})\tau({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})=e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}},\end{split}$
(4.5)

and we observe that they no longer depend on the parameters $\xi_{k}$.
Inserting these expressions in the bilinear identity, we find that the two
integrals have the same integrand, and the identity takes the form

$\displaystyle\begin{split}&\left(\oint_{\infty}+\oint_{0}\right)\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t^{\prime}_{-k})p^{k}z^{-k}}=0.\end{split}$
(4.6)

This equation should be understood as the vanishing of every terms of the
expansion in the time differences
$\varepsilon_{k}=\gamma^{|k|}t_{k}-t^{\prime}_{-k}$. It is not difficult to
check that this is indeed the case.191919For instance, consider the expansion
in $\alpha,\beta$ in the following expression
$\displaystyle\begin{split}&\left(\oint_{0}+\oint_{\infty}\right)\dfrac{dz}{2i\pi}z^{-n-1}e^{\alpha
z}e^{\beta
z^{-1}}=\sum_{k,l=0}^{\infty}\dfrac{\alpha^{k}\beta^{l}}{k!l!}\left(\oint_{\infty}\dfrac{dz}{2i\pi}z^{k-l-n-1}+\oint_{0}\dfrac{dz}{2i\pi}z^{k-l-n-1}\right)\\\
&=\sum_{k,l=0}^{\infty}\dfrac{\alpha^{k}\beta^{l}}{k!l!}\left(-\oint_{0}\dfrac{dz}{2i\pi}z^{-k+l+n-1}+\oint_{0}\dfrac{dz}{2i\pi}z^{k-l-n-1}\right)=\sum_{k,l=0}^{\infty}\dfrac{\alpha^{k}\beta^{l}}{k!l!}\left(-\delta_{n,k-l}+\delta_{n,k-l}\right)=0.\end{split}$
(4.7) The simplicity of this problem comes from the fact that
$\rho^{(2,0)}(G)$ is purely a $u(1)$ factor, i.e. it does not contain the
modes $\alpha_{k}$ of the deformed Virasoro algebra, and so it trivially
commutes with the screening charges $\Psi_{\pm}$. This is reflected in the
simplicity of the refined tau functions (4.3) which are exponentials of an
expression linear in each time $t_{k}$.

#### Difference-differential equations

It is also instructive to verify explicitly that the first equations of the
refined hierarchy are satisfied by the tau functions computed in (4.3). At
first order, the equations $\text{Eq}_{\pm 1}^{(\emptyset,\emptyset)}$ read

$\text{Eq}_{\mp 1}^{(\emptyset,\emptyset)}:\quad\partial_{\pm
1}\log\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})=p\partial_{\pm
1}\log\tau(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}}),$
(4.8)

while the equation $\text{Eq}_{0}^{(\emptyset,\emptyset)}$ in (3.30) is
trivial since there is no dependence in $u$. These equations are satisfied as
a consequence of the property

$\partial_{k}\log\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|{\boldsymbol{\xi}})=\gamma^{2|k|}\partial_{k}\log\tau(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|\boldsymbol{\gamma}{\boldsymbol{\xi}})$
(4.9)

obeyed by the tau functions.

At orders $O(\varepsilon_{\pm 1})$, we observe that the equations
$\text{Eq}_{0}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}$ and
$\text{Eq}_{0}^{(\emptyset,{\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}$ written in (3.33)
are automatically satisfied as a consequence of the zero order equations
$\text{Eq}_{\pm 1}^{(\emptyset,\emptyset)}$. To write down the equations
(3.34) and (3.35) obtained for $n=\pm 1$ in a concise form, we use the
shortcut notations
$\tau=\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})$
and
$\tau^{\prime}=\tau(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})$

$\displaystyle\begin{split}\text{Eq}_{-1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}:\quad&p\left(1-p\dfrac{r_{1}}{2}\right)\tau\partial_{1}^{2}\tau^{\prime}-(1-pr_{1})\partial_{1}\tau\partial_{1}\tau^{\prime}-\dfrac{r_{1}}{2}\partial_{1}^{2}\tau\tau^{\prime}=\dfrac{1}{2}r_{1}\gamma^{-1}\left(\partial_{2}\tau\tau^{\prime}-\gamma^{-4}\tau\partial_{2}\tau^{\prime}\right),\\\
\text{Eq}_{1}^{(\emptyset,{\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&p\left(1-p\dfrac{r_{1}}{2}\right)\tau\partial_{-1}^{2}\tau^{\prime}-(1-pr_{1})\partial_{-1}\tau\partial_{-1}\tau^{\prime}-\dfrac{r_{1}}{2}\partial_{-1}^{2}\tau\tau^{\prime}=-\dfrac{r_{2}}{2r_{1}}\left(\partial_{-2}\tau\tau^{\prime}-\gamma^{-4}\tau\partial_{-2}\tau^{\prime}\right),\\\
\text{Eq}_{-1}^{(\emptyset,{\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&\partial_{1}\tau\partial_{-1}\tau^{\prime}-p\tau\partial_{1}\partial_{-1}\tau^{\prime}+\tau\tau^{\prime}=0,\\\
\text{Eq}_{1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}:\quad&\partial_{-1}\tau\partial_{1}\tau^{\prime}-p\tau\partial_{1}\partial_{-1}\tau^{\prime}+\tau\tau^{\prime}=0.\end{split}$
(4.10)

The first two equations are trivial in the self-dual limit, while the last two
correspond to the refined Toda equations (3.35). We observe that the r.h.s. of
the first two equations is vanishing as a consequence of the property (4.9).
This property can also be used to simplify the last two equations, and we
find202020We also use
$\displaystyle\begin{split}\dfrac{\partial^{2}\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})}{\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})}=p^{2}\dfrac{\partial^{2}\tau(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})}{\tau(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})}.\end{split}$
(4.11)

$\displaystyle\begin{split}\text{Eq}_{-1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)},\text{Eq}_{1}^{(\emptyset,{\leavevmode\hbox
to4.45pt{\vbox to4.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&(1-pr_{1})\partial_{\pm
1}^{2}\log\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}})=0,\\\
\text{Eq}_{1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)},\text{Eq}_{-1}^{(\emptyset,{\leavevmode\hbox
to4.45pt{\vbox to4.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&p\partial_{1}\partial_{-1}\log\tau(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}}|\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}})=1.\end{split}$
(4.12)

These remaining equations can be checked directly using the expression of the
tau function found in (4.3).

#### Remark

A slightly less trivial example is obtained from the following choice of
algebra elements

$G_{{\boldsymbol{\xi}},w}=x^{+}(w)e^{\sum_{k>0}\xi_{k}a_{-k}}.$ (4.13)

We compute the coproduct of this elements using (LABEL:coproduct)

$\Delta(G_{{\boldsymbol{\xi}},w})=G_{\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}},w}\otimes
G_{\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}}}+G_{\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}}-\gamma^{1/2}{\boldsymbol{w}}}\otimes
G_{\boldsymbol{\gamma}^{1/2}{\boldsymbol{\xi}},\gamma w},$ (4.14)

with the understanding
$\boldsymbol{\gamma}^{-1/2}{\boldsymbol{\xi}}-\boldsymbol{\gamma}^{1/2}\boldsymbol{w}=\\{\gamma^{-k/2}\xi_{k}-\gamma^{k/2}w^{k}\\}$,
the bilinear identities are actually linear in this case, with coefficients
involving the refined tau functions
$\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|{\boldsymbol{\xi}})$ studied
previously.

###  4.2 Grading operators

The study of the case where $G$ is defined using the grading operators
introduced in (LABEL:def_gradings) is enlightening. In fact, this analysis
will also be useful for our application to R-matrix tau functions discussed in
the next subsection. Note, however, that the grading operators are not genuine
elements of the quantum toroidal $\mathfrak{gl}(1)$ algebra, and so we cannot
apply our formalism directly. Instead, we need to check explicitly if their
representation of level $(2,0)$ obey the refined basic bilinear condition
(2.56) with the screening charges $\Psi_{\pm}$. It turns out that the grading
${\bar{d}}$ brings no difficulties since
$\rho^{(2,0)}({\bar{d}})={\hat{d}}^{(1)}+{\hat{d}}^{(2)}$ is a purely $u(1)$
factor which commutes with $\hat{q}=({\hat{d}}^{(1)}-{\hat{d}}^{(2)})\log q$,
and so with $\Psi_{\pm}$ as well.212121Alternatively, we can use the simpler
argument saying that only the ratio of weights $u_{1}/u_{2}$ enters in the
identification between the representation $\rho^{(2,0)}_{u_{1},u_{2}}$ and the
free field representation of q-Virasoro, and this ratio is invariant under
$\rho^{(2,0)}_{u_{1},u_{2}}({\bar{d}})=u_{1}\partial_{u_{1}}+u_{2}\partial_{u_{2}}$.

On the other hand, the case of the grading operator $d$ is more subtle, and
more interesting. Indeed, we observe that
$\rho^{(2,0)}_{u_{1},u_{2}}(\alpha^{d})=\alpha^{L_{0}}\otimes\alpha^{L_{0}}$
for some $\alpha\in\mathbb{C}$ does not commute with $\Psi_{\pm}$ but instead
it obeys the relations

$\displaystyle\begin{split}&\rho_{u^{\prime}_{1},u^{\prime}_{2}}^{(2,0)}(\alpha^{d})S_{+}(z)=S_{+}(\alpha
z)\alpha^{-2\beta\alpha_{0}}\rho_{u_{1},u_{2}}^{(2,0)}(\alpha^{d})\quad\Rightarrow\quad\rho_{u^{\prime}_{1},u^{\prime}_{2}}^{(2,0)}(\alpha^{d})\Psi_{+}(z)=\alpha^{n}\Psi_{+}\rho_{u_{1},u_{2}}^{(2,0)}(\alpha^{d})\quad\text{on}\
{\mathcal{F}}_{-1,n}\\\
&\rho_{u^{\prime}_{1},u^{\prime}_{2}}^{(2,0)}(\alpha^{d})S_{-}(z)=S_{-}(\alpha
z)\alpha^{2\alpha_{0}}\rho_{u_{1},u_{2}}^{(2,0)}(\alpha^{d})\quad\Rightarrow\quad\rho_{u^{\prime}_{1},u^{\prime}_{2}}^{(2,0)}(\alpha^{d})\Psi_{-}(z)=\alpha^{m}\Psi_{-}\rho_{u_{1},u_{2}}^{(2,0)}(\alpha^{d})\quad\text{on}\
{\mathcal{F}}_{m,-1}\end{split}$ (4.15)

As a result, the bilinear identities (LABEL:Hirota_2dToda) have to be modified
for the choice $G=\alpha^{d}$ by the introduction of an extra power of
$\alpha$ in front of the first integral

$\displaystyle\begin{split}&\alpha^{n}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\tau_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G)\tau_{uq^{-n}t^{-1}}({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G)\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{\prime})p^{k}z^{-k}}\tau_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G)\tau_{uq^{-n}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t}|G)=0,\\\
&\alpha^{m}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-m-1}e^{\sum_{k>0}(\gamma^{k}t_{k}-t^{\prime}_{k})z^{k}}\tau_{u}({\boldsymbol{t}}-[\gamma
z^{-1}]_{t,q},\bar{\boldsymbol{t}}|G)\tau_{uqt^{m}}({\boldsymbol{t}}^{\prime}+[z^{-1}]_{t,q},\bar{\boldsymbol{t}}^{\prime}|G)\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{-m-1}e^{-\sum_{k>0}(\gamma^{k}t_{-k}-t^{\prime}_{-k})\frac{1-q^{k}}{1-t^{k}}z^{-k}}\tau_{uq}({\boldsymbol{t}},\bar{\boldsymbol{t}}+[\gamma
z]|G)\tau_{ut^{m}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}-[z]|G)=0,\end{split}$
(4.16)

where we also used the simple coproduct of $G$ to remove the summations.

When $G=\alpha^{d}$, the refined tau function (3.1) can be computed using the
property (2.31) of $L_{0}$, it gives

$\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=e^{\sum_{k>0}k\alpha^{k}t_{k}t_{-k}},$
(4.17)

which is again independent of the weight $u$. We can use this expression to
check the bilinear identities. Focusing on the first one, we compute the
factors

$\displaystyle\begin{split}&\tau({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}}|G)\tau({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime}|G)=e^{\sum_{k>0}Q^{k}(\gamma^{k}t_{-k}-t^{\prime}_{-k})p^{k}z^{-k}},\\\
&\tau({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t}|G)\tau({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t}|G)=e^{-\sum_{k>0}Q^{k}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}},\end{split}$
(4.18)

which can then be inserted in the integrand of the bilinear identity

$\displaystyle\begin{split}&\alpha^{n}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}e^{\sum_{k>0}\alpha^{k}(\gamma^{k}t_{-k}-t^{\prime}_{-k})p^{k}z^{-k}}\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}\alpha^{k}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{\prime})p^{k}z^{-k}}=0.\end{split}$
(4.19)

Upon the rescaling of the integration variable $z\to\alpha z$ in the first
integral, we recover the situation of the previous subsection with the same
integrand in both integrals. For the same reason, the expansion in
$\varepsilon_{k}$ of these integrals is vanishing, in agreement with our
general result.

##  5 R-matrix and refined tau functions

In this section we study the generating function
$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$ of all refined tau
functions (3.1) which is built using the universal R-matrix ${\mathcal{R}}$ of
the quantum toroidal $\mathfrak{gl}(1)$ algebra. The bilinear identities for
this generator take the form of a two-term quadratic relation involving non-
commutative objects.

#### The R-matrix of the quantum toroidal $\mathfrak{gl}(1)$ algebra

The universal R-matrix takes values in a completion of
${\mathcal{E}}^{\prime}\otimes{\mathcal{E}}^{\prime}$, where
${\mathcal{E}}^{\prime}$ denotes the quantum toroidal $\mathfrak{gl}(1)$
algebra ${\mathcal{E}}$ complemented with the two grading operators $d$ and
${\bar{d}}$. Consider a PBW basis $\mathcal{B}^{\prime}$ of
${\mathcal{E}}^{\prime}$ with respect to a pairing $(a|b^{*})=\delta_{a,b}$
for $a,a^{*},b,b^{*}\in{\mathcal{E}}^{\prime}$, where ∗ denotes the dual
element (see [55, 56]). The universal R-matrix has the form

$\displaystyle\mathcal{R}=\sum_{b\in\mathcal{B}^{\prime}}b^{*}\otimes b.$
(5.1)

The degree elements $d,\bar{d}$ have a non-zero pairing only with the central
elements $c,\bar{c}$ respectively, thus we have

$\displaystyle\mathcal{R}=\gamma^{d\otimes c+c\otimes
d}\gamma^{{\bar{c}}\otimes{\bar{d}}+{\bar{d}}\otimes{\bar{c}}}\sum_{b\in\mathcal{B}}b^{*}\otimes
b.$ (5.2)

where $\mathcal{B}$ is the set of PBW basis elements excluding the degree
elements. Recall the coproduct $\Delta$, defined in (LABEL:coproduct), and the
opposite coproduct $\Delta^{op}$. The universal R-matrix ${\mathcal{R}}$ of
the quantum toroidal algebra satisfies222222We use the standard notation from
the literature on quantum integrable systems in which the subscripts $ij$ in
${\mathcal{R}}_{ij}$ indicate that the ${\mathcal{R}}$-matrix acts on the
$i$-th and $j$-th component of the tensor product.

$\displaystyle{\mathcal{R}}\Delta(G)=\Delta^{op}(G){\mathcal{R}},\qquad\forall
G\in{\mathcal{E}},$ (5.3) $\displaystyle(\Delta\otimes
1){\mathcal{R}}={\mathcal{R}}_{13}{\mathcal{R}}_{23},$ (5.4)
$\displaystyle(1\otimes\Delta){\mathcal{R}}={\mathcal{R}}_{13}{\mathcal{R}}_{12}.$
(5.5)

As a consequence of these equations ${\mathcal{R}}$ satisfies the Yang–Baxter
equation

$\displaystyle{\mathcal{R}}_{23}{\mathcal{R}}_{13}{\mathcal{R}}_{12}={\mathcal{R}}_{12}{\mathcal{R}}_{13}{\mathcal{R}}_{23}.$
(5.6)

We can further isolate the Cartan part in the universal R-matrix and write it
as

${\mathcal{R}}=\gamma^{d\otimes c+c\otimes
d}\gamma^{{\bar{c}}\otimes{\bar{d}}+{\bar{d}}\otimes{\bar{c}}}{\mathcal{K}}\bar{\mathcal{R}},\quad{\mathcal{K}}=e^{-\sum_{k>0}\frac{\gamma^{-k(c\otimes
1-1\otimes c)/2}}{c_{k}}a_{k}\otimes a_{-k}},$ (5.7)

where $a_{\pm k}$ are the modes of the currents $\psi^{\pm}(z)$ (2.17) and the
coefficients $c_{k}\in\mathbb{C}(q_{1},q_{2})$ are defined in (2.18). The
factor $\bar{\mathcal{R}}$ takes values in
${\mathcal{E}}^{-}\otimes{\mathcal{E}}^{+}$ where ${\mathcal{E}}^{\pm}$ are
the algebras generated multiplicatively by the modes $x_{k}^{\pm}$. When it
comes to the horizontal representation $\rho^{(1,0)}_{u}$, the presence of the
operators $\gamma^{{\bar{c}}\otimes{\bar{d}}+{\bar{d}}\otimes{\bar{c}}}$ in
(5.7) forces us to consider Fock spaces ${\mathcal{F}}_{u}$ in which the
spectral parameter $u$ varies as discussed in Section 2.3. To avoid this
issue, the ${\mathcal{L}}$-operator is usually defined by removing this factor
before evaluating the second tensor component in the horizontal representation

$\displaystyle\mathcal{L}(u):=\left(1\otimes\rho^{(1,0)}_{u}\right)\left(\gamma^{-{\bar{c}}\otimes{\bar{d}}-{\bar{d}}\otimes{\bar{c}}}{\mathcal{R}}\right)=\gamma^{-\bar{c}\otimes\hat{d}}\left(1\otimes\rho^{(1,0)}_{u}\right){\mathcal{R}}.$
(5.8)

This operator
$\mathcal{L}(u)\in{\mathcal{E}}^{\prime}\otimes\text{End}({\mathcal{F}}_{u})$
was considered in [32] where it played a key role in deriving the Bethe Ansatz
of the associated integrable model defined on the physical space of horizontal
representations. The transfer matrix of this integrable model is computed by
taking a twisted trace of a product of the horizontal-horizontal R-matrices

$\displaystyle
R(u_{2}/u_{1}):=\left(\rho_{u_{1}}^{(1,0)}\otimes\rho_{u_{2}}^{(1,0)}\right){\mathcal{R}}\in\text{End}({\mathcal{F}}_{u_{1}}\otimes{\mathcal{F}}_{u_{2}}).$
(5.9)

The matrix $R(u_{2}/u_{1})$ was studied in [57, 58, 59, 60], and it is known
in the rational limit as the Maulik–Okounkov R-matrix (see [61, 62, 63, 43]).
Applying
$1\otimes\left(\rho_{u_{1}}^{(1,0)}\otimes\rho_{u_{2}}^{(1,0)}\right)$ to the
Yang–Baxter equation (5.6) produces the RLL equation

$\displaystyle
R_{23}(u_{2}/u_{1}){\mathcal{L}}_{13}(u_{2}){\mathcal{L}}_{12}(u_{1})={\mathcal{L}}_{12}(u_{1}){\mathcal{L}}_{13}(u_{2})R_{23}(u_{2}/u_{1}),$
(5.10)

where we used the equations (5.8), (5.9) and cancelled the common factor
$\gamma^{\bar{c}(\hat{d}^{(1)}+\hat{d}^{(2)})}$ that commutes with
$R(u_{2}/u_{1})$.

#### Universal refined tau function

The universal refined tau function is defined as the following evaluation of
the ${\mathcal{L}}$-operator

$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})=\left(1\otimes\bra{u}e^{\sum_{k>0}t_{k}J_{k}}\right){\mathcal{L}}(u)\left(1\otimes
e^{\sum_{k>0}t_{-k}J_{-k}}\ket{u}\right).$ (5.11)

This tau function is represented graphically in figure 2. The tau function
$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$ encodes the monodromy
matrix associated to the horizontal representation. A particular monodromy
matrix element labelled by two partitions $\lambda,\mu$ can be accessed by
taking the coefficient of $t_{\lambda},t_{-\mu}$ in
$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$. Therefore this tau
function is intimately related to the integrable model which was considered in
[32, 33].

In fact, this universal tau function can be seen as the generating function of
all refined tau functions of the form (3.1). Indeed, by inserting the
expansion (5.2) of the universal R-matrix into (5.8), the equation (5.11) can
be rewritten as a sum of refined tau functions over the elements of the PBW
basis (excluding grading elements)

$\displaystyle\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}):=\gamma^{d}\sum_{b\in\mathcal{B}}b^{*}\tau_{u}(\boldsymbol{\gamma}^{c}{\boldsymbol{t}},\bar{\boldsymbol{t}}|b)\in{\mathcal{E}}[{\boldsymbol{t}},\bar{\boldsymbol{t}}].$
(5.12)

The rescaling of the time parameters
$\boldsymbol{\gamma}^{c}{\boldsymbol{t}}=(\gamma^{c}t_{1},\gamma^{2c}t_{2},\cdots)$
is a consequence of the presence of the grading, and it is not essential in
our argumentation. Any element $G\in{\mathcal{E}}$ can be expanded in the PBW
basis as $G=\sum_{b\in\mathcal{B}}c_{b}b$ with some coefficients $c_{b}$.
Then, the pairing of the algebra can be used to recover the corresponding tau
function

$\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|G)=\sum_{b\in{\mathcal{B}}}c_{b}\tau_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|b)=(G|\gamma^{-d}\hat{\tau}_{u}(\boldsymbol{\gamma}^{-c}{\boldsymbol{t}},\bar{\boldsymbol{t}})).$
(5.13)

Alternatively, the universal tau functions can be obtained as the limit
$\kappa\to 0$ of a transfer matrix dressed with exponential of the times
$({\boldsymbol{t}},\bar{\boldsymbol{t}})$

$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})=\lim_{\kappa\to
0}\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|\kappa),\quad\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|\kappa)=1\otimes{\rm
tr~{}}_{{\mathcal{F}}_{u}}\left[\kappa^{L_{0}}e^{\sum_{k>0}t_{k}J_{k}}{\mathcal{L}}(u)e^{\sum_{k>0}t_{-k}J_{-k}}\right].$
(5.14)

These objects display some similarity with the master T-operators studied in
[64, 65, 66] which are tau functions of the KP hierarchy. However, the
definitions are slightly different since the sum over representations defining
the master T-operators is replaced here with a single horizontal Fock
representation. As a result, contrary to master T-operators, our operators
$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}|\kappa)$ do not commute
for different times. It would be interesting to study further these objects,
but we will leave this question for a future work and focus instead on the
universal tau functions
$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$.

$e^{\sum_{k>0}t_{-k}J_{-k}}\ket{u}$$\bra{u}e^{\sum_{k>0}t_{k}J_{k}}$ Figure 2:
Definition of the universal refined tau function from the
${\mathcal{L}}$-operator. The ${\mathcal{L}}$-operator is represented by the
cross.
$e^{\sum_{k>0}t^{\prime}_{-k}J_{-k}}\ket{u_{2}}$$\bra{u_{2}}e^{\sum_{k>0}t^{\prime}_{k}J_{k}}$$e^{\sum_{k>0}t_{-k}J_{-k}}\ket{u_{1}}$$\bra{u_{1}}e^{\sum_{k>0}t_{k}J_{k}}$
Figure 3: Element entering the basic bilinear condition.

#### Bilinear identities

The universal refined tau function
$\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})$ also satisfies a set
of bilinear identities of the form (3.16), but the property (5.4) of the
R-matrix allows us to replace the sum over elements $G_{\alpha}\otimes
G_{\alpha}^{\prime}$ coming from the coproduct by a product of universal tau
functions

$\displaystyle\begin{split}&\gamma^{nc}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-n-1}e^{-\sum_{k>0}(\gamma^{k}t_{k}-t_{k}^{\prime})\frac{1-t^{k}}{1-q^{k}}z^{k}}\hat{\tau}_{uq^{-n}t^{-1}}({\boldsymbol{t}}^{\prime}-[\gamma^{-2}z^{-1}],\bar{\boldsymbol{t}}^{\prime})\hat{\tau}_{u}({\boldsymbol{t}}+[\gamma^{-1}z^{-1}],\bar{\boldsymbol{t}})\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{-n-1}e^{\sum_{k>0}(\gamma^{k}t_{-k}-t_{-k}^{\prime})p^{k}z^{-k}}\hat{\tau}_{uq^{-n}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}+[z]_{q,t})\hat{\tau}_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{q,t})=0,\\\
&\gamma^{mc}\oint_{\infty}\dfrac{dz}{2i\pi}z^{-m-1}e^{\sum_{k>0}(\gamma^{k}t_{k}-t^{\prime}_{k})z^{k}}\hat{\tau}_{uqt^{m}}({\boldsymbol{t}}^{\prime}+[z^{-1}]_{t,q},\bar{\boldsymbol{t}}^{\prime})\hat{\tau}_{u}({\boldsymbol{t}}-[\gamma
z^{-1}]_{t,q},\bar{\boldsymbol{t}})\\\
+&\oint_{0}\dfrac{dz}{2i\pi}z^{-m-1}e^{-\sum_{k>0}(\gamma^{k}t_{-k}-t^{\prime}_{-k})\frac{1-q^{k}}{1-t^{k}}z^{-k}}\hat{\tau}_{ut^{m}}({\boldsymbol{t}}^{\prime},\bar{\boldsymbol{t}}^{\prime}-[z])\hat{\tau}_{uq}({\boldsymbol{t}},\bar{\boldsymbol{t}}+[\gamma
z])=0.\end{split}$ (5.15)

The derivation of these identities follow the same lines as what was done in
Appendix B to obtain the relations (3.16). However, the presence of grading
operators in the R-matrix renders the derivation a bit more complicated, and
we will discuss it briefly below. The replacement of the coproduct with a
product of universal tau-functions also occurs in the difference-differential
equations, and for instance the $(q,t)$-deformed Toda equations (3.35) take
the form

$\displaystyle\begin{split}\text{Eq}_{-1}^{(\emptyset,{\leavevmode\hbox
to4.45pt{\vbox to4.45pt{\pgfpicture\makeatletter\hbox{\hskip
0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})}:\quad&\partial_{-1}\hat{\tau}_{up}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}})\partial_{1}\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})-p\partial_{1}\partial_{-1}\hat{\tau}_{up}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}})\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})+\gamma^{c}\hat{\tau}_{uq}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}})\hat{\tau}_{ut^{-1}}({\boldsymbol{t}},\bar{\boldsymbol{t}})=0,\\\
\text{Eq}_{1}^{({\leavevmode\hbox to4.45pt{\vbox
to4.45pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{4.04762pt}\pgfsys@lineto{4.04762pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{4.04762pt}{4.04762pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}},\emptyset)}:\quad&\partial_{1}\hat{\tau}_{up^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}})\partial_{-1}\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})-p\partial_{1}\partial_{-1}\hat{\tau}_{up^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}})\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})+\gamma^{c}\hat{\tau}_{uq^{-1}}(\boldsymbol{\gamma}{\boldsymbol{t}},\boldsymbol{\gamma}\bar{\boldsymbol{t}})\hat{\tau}_{ut}({\boldsymbol{t}},\bar{\boldsymbol{t}})=0.\end{split}$
(5.16)

While the expression for the universal R-matrix is known [56], it is not easy
to use it to show directly that the universal refined tau function (5.11)
satisfies the identities (5.15) (see Theorem 4.16 in [56]). On the other hand,
for certain choices of representations, more explicit expressions of the
R-matrix have been obtained, and can be used to study these equations. In the
next subsection, we will examine two important cases, namely the vertical-
horizontal and horizontal-horizontal Fock R-matrices.

The starting point for the derivation of the bilinear identities (5.15) is
again the $(q,t)$-deformed basic bilinear condition (2.56). This equation can
be summed over the elements of the algebra to produce

$\displaystyle\sum_{b\in\mathcal{B}}b^{*}\left(\Psi_{\pm}\rho_{u_{1},u_{2}}^{(2,0)}(b)-\rho_{u_{1}^{\prime},u_{2}^{\prime}}^{(2,0)}(b)\Psi_{\pm}\right)=0.$
(5.17)

We can rewrite this identity in terms of ${\mathcal{R}}$ by recalling (5.2)

$\displaystyle(1\otimes\Psi_{\pm})(1\otimes\rho_{u_{1},u_{2}}^{(2,0)})\left(\gamma^{-d\otimes
c-c\otimes
d-{\bar{c}}\otimes{\bar{d}}-{\bar{d}}\otimes{\bar{c}}}\right){\mathcal{R}}=(1\otimes\rho_{u_{1}^{\prime},u_{2}^{\prime}}^{(2,0)})\left(\gamma^{-d\otimes
c-c\otimes
d-{\bar{c}}\otimes{\bar{d}}-{\bar{d}}\otimes{\bar{c}}}{\mathcal{R}}\right)(1\otimes\Psi_{\pm}).$
(5.18)

After a careful treatment of the grading operators, we end up with232323The
evaluation of $\gamma^{d\otimes c+c\otimes
d+{\bar{c}}\otimes{\bar{d}}+{\bar{d}}\otimes{\bar{c}}}$ under the
representation $\rho^{(2,0)}_{u_{1},u_{2}}$ brings the factors
$\gamma^{\bar{c}(\hat{d}^{(1)}+\hat{d}^{(2)})}$ and $\gamma^{d\otimes 2}$
which commute with $\Psi_{\pm}$, and $\gamma^{c\otimes{\mathcal{L}}_{0}}$
which follows from
$\rho^{(2,0)}_{u_{1},u_{2}}(d)={\mathcal{L}}_{0}-\sum_{k>0}\dfrac{\beta_{-k}\beta_{k}}{kc_{k}(q_{3}^{k}-q_{3}^{-k})},\quad{\mathcal{L}}_{0}=\sum_{k>0}\dfrac{1-t^{k}}{1-q^{k}}\alpha_{-k}\alpha_{k},$
(5.19) the $u(1)$-part trivially commuting with $\Psi_{\pm}$. This extra
factor $\gamma^{c{\mathcal{L}}_{0}}$ obeys non-trivial commutation relations
with the screening currents, namely
$\gamma^{c{\mathcal{L}}_{0}}S_{+}(z)\gamma^{-c{\mathcal{L}}_{0}}=S_{+}(\gamma^{c}z)\gamma^{-2\beta\alpha_{0}c}$
and
$\gamma^{c{\mathcal{L}}_{0}}S_{-}(z)\gamma^{-c{\mathcal{L}}_{0}}=S_{-}(\gamma^{c}z)\gamma^{2\alpha_{0}c}$.
After integrating out and evaluating over the Fock space
${\mathcal{F}}_{-1,n}$ and ${\mathcal{F}}_{m,-1}$, it brings the extra factors
$\gamma^{nc}$ and $\gamma^{mc}$ in the following equations.

$\displaystyle(\gamma^{cn}\otimes\Psi_{+})(1\otimes\rho_{u_{1},u_{2}}^{(2,0)}){\mathcal{R}}$
$\displaystyle=(1\otimes\rho_{u_{1}^{\prime},u_{2}^{\prime}}^{(2,0)}){\mathcal{R}}(1\otimes\Psi_{+}),$
(5.20)
$\displaystyle(\gamma^{cm}\otimes\Psi_{-})(1\otimes\rho_{u_{1},u_{2}}^{(2,0)}){\mathcal{R}}$
$\displaystyle=(1\otimes\rho_{u_{1}^{\prime},u_{2}^{\prime}}^{(2,0)}){\mathcal{R}}(1\otimes\Psi_{-}),$
(5.21)

where $n$ and $m$ refer to the labels of the Fock spaces in (2.55). By
recalling $\rho_{u_{1},u_{2}}^{(2,0)}$ from (2.38), the identity (5.5) for the
universal R-matrix and the definition (5.8) of the operator ${\mathcal{L}}(u)$
we can write

$\displaystyle(1\otimes\rho_{u_{1},u_{2}}^{(2,0)}){\mathcal{R}}=\gamma^{\bar{c}(\hat{d}^{(1)}+\hat{d}^{(2)})}{\mathcal{L}}_{13}(u_{2}){\mathcal{L}}_{12}(u_{1}).$
(5.22)

Inserting this on both sides of (5.20) and (5.21), we find after cancelling
the grading operators

$\displaystyle(\gamma^{cn}\otimes\Psi_{+}){\mathcal{L}}_{13}(u_{2}){\mathcal{L}}_{12}(u_{1})$
$\displaystyle={\mathcal{L}}_{13}(u_{2}^{\prime}){\mathcal{L}}_{12}(u_{1}^{\prime})(1\otimes\Psi_{+}),$
(5.23)
$\displaystyle(\gamma^{cm}\otimes\Psi_{-}){\mathcal{L}}_{13}(u_{2}){\mathcal{L}}_{12}(u_{1})$
$\displaystyle={\mathcal{L}}_{13}(u_{2}^{\prime}){\mathcal{L}}_{12}(u_{1}^{\prime})(1\otimes\Psi_{-}).$
(5.24)

In order to deal with the zero modes of $\Psi_{\pm}$ it is useful to introduce
two operators $\Phi_{n}^{+}$ and $\Phi_{m}^{-}$

$\displaystyle\Phi^{+}_{n}:=e^{-\beta\hat{q}}\Psi_{+}=t^{\hat{d}^{(2)}-\hat{d}^{(1)}}\Psi_{+},\qquad\Phi^{-}_{m}:=e^{\hat{q}}\Psi_{-}=q^{\hat{d}^{(1)}-\hat{d}^{(2)}}\Psi_{-}.$
(5.25)

These operators now have an index since they act in the Fock spaces labelled
as follows

$\Phi^{+}_{n}:{\mathcal{F}}_{-1,n}\to{\mathcal{F}}_{-1,n},\quad\Phi^{-}_{m}:{\mathcal{F}}_{m,-1}\to{\mathcal{F}}_{m,-1}.$
(5.26)

We replace $\Psi_{\pm}$ with these operators in (5.23) and (5.24) and use
(LABEL:weights)

$\displaystyle(\gamma^{cn}\otimes\Phi^{+}_{n}){\mathcal{L}}_{13}(q^{-n}t^{-1}u){\mathcal{L}}_{12}(u)$
$\displaystyle={\mathcal{L}}_{13}(q^{-n}u){\mathcal{L}}_{12}(t^{-1}u)(1\otimes\Phi^{+}_{n}),$
(5.27)
$\displaystyle(\gamma^{cm}\otimes\Phi^{-}_{m}){\mathcal{L}}_{13}(t^{m}qu){\mathcal{L}}_{12}(u)$
$\displaystyle={\mathcal{L}}_{13}(t^{m}u){\mathcal{L}}_{12}(qu)(1\otimes\Phi_{m}^{-}).$
(5.28)

These two equations can be interpreted as RLL equations as mentioned in the
introduction. The two equations (5.27) and (5.28) give rise to two bilinear
relations for the universal refined tau function (5.11). We sandwich (5.27)
and (5.28) between the states (see figure 3)

$\displaystyle
1\otimes\bra{u_{1}}e^{\sum_{k>0}t_{k}J_{k}}\otimes\bra{u_{2}}e^{\sum_{k>0}t^{\prime}_{k}J_{k}},\qquad
1\otimes e^{\sum_{k>0}t_{-k}J_{-k}}\ket{u_{1}}\otimes
e^{\sum_{k>0}t^{\prime}_{-k}J_{-k}}\ket{u_{2}}.$

Finally, the action on these states with $\Phi^{\pm}$ can be calculated in the
same way as what was done in Appendix B.

#### Remark

We can use the intertwining property
${\mathcal{R}}\Delta=\Delta^{op}{\mathcal{R}}$ of the R-matrix, to derive some
interesting identities on the refined tau functions. For instance, using the
two Cartan currents, we find

$\displaystyle\begin{split}&\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})\psi^{+}(z)=e^{\sum_{k>0}t_{-k}z^{-k}\gamma^{ck/2}(1-q_{2}^{k})(1-q_{3}^{k})}\psi^{+}(\gamma^{-1}z)\hat{\tau}_{u}({\boldsymbol{t}}+[\gamma^{2-c/2}z^{-1}]_{2}-[\gamma^{-c/2}z^{-1}]_{2},\bar{\boldsymbol{t}}),\\\
&\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}+[\gamma^{c/2}z]_{1}-[\gamma^{2+c/2}z]_{1})\psi^{-}(z)=e^{\sum_{k>0}t_{k}z^{k}\gamma^{-ck/2}(1-q_{1}^{k})(1-q_{3}^{k})}\psi^{-}(\gamma
z)\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}),\end{split}$ (5.29)

with
${\boldsymbol{t}}\pm[z]_{\alpha}={\boldsymbol{t}}\pm[z]\mp[q_{\alpha}z]=\\{t_{k}\pm\frac{1}{k}z^{k}(1-q_{\alpha}^{k})\\}$,
while the currents $x^{\pm}(z)$ produce the two relations

$\displaystyle\begin{split}&\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})x^{+}(z)-e^{\sum_{k>0}t_{k}z^{k}(1-q_{1}^{k})(1-q_{3}^{k})}x^{+}(\gamma
z)\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})\\\
&=ue^{\sum_{k>0}t_{k}z^{k}(1-q_{1}^{k})}\hat{\tau}_{u}({\boldsymbol{t}}-[z^{-1}]_{2},\bar{\boldsymbol{t}})-ue^{-\sum_{k>0}t_{-k}z^{-k}\gamma^{-ck}(1-q_{2}^{k})}\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}+[\gamma^{ck}z]_{1},\bar{\boldsymbol{t}})\psi^{-}(\gamma^{c/2}z),\\\
&x^{-}(z)\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})-e^{-\sum_{k>0}t_{-k}z^{-k}\gamma^{-k}(1-q_{2}^{k})(1-q_{3}^{k})}\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}})x^{-}(\gamma
z)\\\
&=u^{-1}e^{\sum_{k>0}t_{-k}z^{-k}\gamma^{k}(1-q_{2}^{k})}\hat{\tau}_{u}({\boldsymbol{t}},\bar{\boldsymbol{t}}-[\gamma
z]_{1})-u^{-1}e^{-\sum_{k>0}t_{k}z^{k}\gamma^{(1+c)k}(1-q_{1}^{k})}\psi^{+}(\gamma^{c/2}z)\hat{\tau}_{u}({\boldsymbol{t}}+[\gamma^{1-c}z^{-1}]_{2},\bar{\boldsymbol{t}}).\end{split}$
(5.30)

###  5.1 Vertical refined tau function

$\ket{u}$$\bra{\gamma^{-1}u}$$\langle\\!\bra{\mu}$$\ket{\lambda}\\!\rangle$$\Phi[u,v,0]$$\Phi^{\ast}[\gamma^{-1}u,\gamma