# New solvable two-matrix model and BKP tau function

E. N. Antonov
A. Yu. Orlov Petersburg Nuclear Physics Institute named by B.P.Konstantinov of
NRC “Kurchatov Institute”, 1 mkr. Orlova Roshcha, Gatchina, Leningradskaya
Obl, 188300, Russia, email<EMAIL_ADDRESS>Institute of Transmition
Information Problems RAS, Moscow 127051 Russia, Bolshoy Karetny per. 19,
build. 1; Institute of Oceanology, Nahimovskii Prospekt 36, Moscow 117997,
Russia, email<EMAIL_ADDRESS>
###### Abstract

We present exactly solvable modifications of the two-matrix Zinn-Justin-Zuber
model and write it as a tau function. The grand partition function of these
matrix integrals is written as the fermion expectation value. The perturbation
theory series is written out explicitly in terms of series in strict
partitions. The related string equations are presented.

##  1 Introduction

This note was initiated by the work on the generalized Kontsevich model [35]
and further discussions with the authors. The perturbation theory series for
the partition function of this model was written in a very compact form as a
sum over strict partitions of a pair of projective Schur functions. It was
followed by the work [1] where the similar series appeared for the different
model (BGW model). The sources of interest to series in projective Schur
functions can be found in [22],[52],[31],[1],[37],[3],[3],[34],[32],[36],
[33]. There are some earlier works on the series, see
[51],[40],[44],[47],[27],[12]. On the projective Schur functions and the
representation theory of the supersymmetric group $q(N)$ and the symmetric
group $S_{n}$, see [46],[16],[9],[48]. The appearance of these functions in
the integrable models was presented in [54] and [38].

If a matrix integral is a tau function as a function of its coupling
constants, we call it solvable. As far as we know, the first solvable (in this
sense) matrix model was presented in the preprint of [10]; see other examples
in [19], [20] and [21]. Then we should point out the work [25].

Here, we present and compare two families of solvable matrix integrals. The
second family is completely new and is related to the KP hierarchy on the root
system B (the BKP hierarchy, which was introduced in [8]).

##  2 Models of two unitary matrices

###  2.1 Standard and modified models of two unitary matrices

In [55] the following integral over two unitary matrices was studied

$I_{1}(\mathbf{t},\mathbf{t}^{*})=C\int_{\mathbb{U}_{N}\times\mathbb{U}_{N}}e^{c\,\mathrm{tr}\left(U_{1}^{-1}U_{2}^{-1}\right)+\sum_{n>0}\frac{1}{n}\left(t_{n}\mathrm{tr}U_{1}^{n}+t^{*}_{n}\mathrm{tr}U_{2}^{n}\right)}d_{*}U_{1}dU_{2}^{*}$
(1)

where $d_{*}U$ is the Haar measure of the unitary group $\mathbb{U}_{N}$. The
number $c$ and the sets
$\mathbf{t}=(t_{1},t_{2},\dots),\,\mathbf{t}^{*}=(t_{1}^{*},t_{2}^{*},\dots)$
play the role of coupling constants in the model, and $C$ is a normalization
constant: $CI(0,0)=1$. It is shown that $I_{1}$ can be written explicitly as a
series of the Schur polynomials as functions in the coupling constants over
partitions as follows:

$I_{N}(\mathbf{t},\mathbf{t}^{*})=\sum_{\lambda}s_{\lambda}(\mathbf{t})s_{\lambda}(\mathbf{t}^{*})\prod_{(i,j)\in\lambda}\frac{c}{(N-i+j)!}$
(2)

where $(i,j)$ are the coordinates of the node of the Young diagram $\lambda$.
The sum ranges over all Young diagrams of height do not accede $N$, that is,
$j=1,2,\dots$ and $i\leq N$. It can be shown that this series is an example of
the KP tau functions, also known as hypergeometric tau functions [21], [41],
which admit relatively simple determinant forms.

#### Modified family

. In [39] (see also Appendix A in [11]), the following generalization of the
integral (1) was introduced, namely, one can make a replacement for the term
responsible for the interaction between matrices $U_{1}$ and $U_{2}$:

$e^{c\,\mathrm{tr}U^{-1}_{1}U^{-1}_{2}}\,\to\,\tau\left(N;c\,U^{-1}_{1}U^{-1}_{2};f\right)$
(3)

where $\tau\left(c\,U^{-1}_{1}U^{-1}_{2};f\right)$ is defined by the choice of
the function $f$ as follows:

$\tau\left(N;c\,U^{-1}_{1}U^{-1}_{2};f\right):=\sum_{\lambda}\prod_{i<j\leq
N}(\lambda_{i}-\lambda_{j}-i+j)s_{\lambda}\left(c\,U^{-1}_{1}U^{-1}_{2}\right)\prod_{i\leq
N}f(\lambda_{i}-i+N)$ (4)

Examples:

${\rm if}\,\,f(x)=\frac{1}{\Gamma(x+1)}\,\,{\rm
then}\,\,\tau\left(N;c\,U^{-1}_{1}U^{-1}_{2};f\right)=e^{c\,\mathrm{tr}\left(U_{1}^{-1}U_{2}^{-1}\right)}$
(5) ${\rm if}\,\,f(x)=\frac{\Gamma(x+a)}{\Gamma(a)\Gamma(x+1)}\,\,{\rm
then}\,\,\tau\left(N;c\,U^{-1}_{1}U^{-1}_{2};f\right)=\mathrm{det}\left(1-c\,U_{1}^{-1}U_{2}^{-1}\right)^{-a}$
(6)

The wonderful property of such tau function is the following determinantal
representation:

$\int_{\mathbb{U}_{N}}\tau\left(c\,UU^{-1}_{1}U^{-1}U^{-1}_{2};f\right)d_{*}U=\frac{\mathrm{det}[\tau(1;c\,u^{-1}_{i}v^{-1}_{j};f)]_{i,j\leq
N}}{\prod_{i<j\leq N}(u^{-1}_{i}-u^{-1}_{j})(v^{-1}_{i}-v^{-1}_{j})}$ (7)

where

$\tau(1;c\,u^{-1}v^{-1};f)]=1+f(1)u^{-1}v^{-1}+f(2)u^{-2}v^{-2}+\cdots$ (8)

which allows us writing down the perturbation series for the generalized model
in the form of another matrix integral:

$I_{N}(\mathbf{t},\mathbf{t}^{*};f)=C\int_{\mathbb{U}_{N}\times\mathbb{U}_{N}}\tau\left(N;c\,U^{-1}_{1}U^{-1}_{2};f\right)e^{\sum_{n>0}\frac{1}{n}\left(t_{n}\mathrm{tr}U_{1}^{n}+t^{*}_{n}\mathrm{tr}U_{2}^{n}\right)}d_{*}U_{1}dU_{2}^{*}$
(9) $=\sum_{\lambda\atop\ell(\lambda)\leq
N}s_{\lambda}(\mathbf{t})s_{\lambda}(\mathbf{t}^{*})\prod_{i}f(\lambda_{i}-i+N)c^{\lambda_{i}}$
(10)

which is the Toda lattice [30] tau function [53],[49],[50] as presented in
[21] and carefully studied in [42]. In [11], the integral (9) was written as
the fermionic vacuum expectation value. See Appendix D, where the linear
equations (sometimes known as string ones) for (1) and (9) are written down.

##  3 New models of two unitary matrices

Let us consider a modification of the models mentioned above. Let us consider
the following model:

$J_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=C_{N}\int_{\mathbb{U}_{N}\times\mathbb{U}_{N}}e^{c\,\mathrm{tr}\left(U_{1}^{-2}U_{2}^{-2}\right)+\sum_{n=1,3,5,\dots}\frac{2}{n}\left(p^{(1)}_{n}\mathrm{tr}U_{1}^{n}+p^{(2)}_{n}\mathrm{tr}U_{2}^{n}\right)}d\mu_{1}(U_{1})d\mu_{2}(U_{2})$
(11)

where

$d\mu_{i}(U)=\mathrm{det}\left(U^{-\frac{1}{2}(N^{2}-N)+\kappa_{i}}\right)d_{*}U_{i},\quad
i=1,2$ (12)

For the sake of simplicity, we choose $\kappa_{1}=\kappa_{2}=0$ (all
calculations below can be reproduced also in case $\kappa_{i}\neq 0$, but all
formulas take up noticeably more space in this case). Here, the constant
$C_{N}$ is chosen to ensure the normalization $J_{N}(0,0)=1$. In this model,
the parameters
$\mathbf{p}^{(1)}=(p^{(1)}_{1},p^{(1)}_{3},\dots),\,\mathbf{p}^{(2)}=(p^{(2)}_{1},p^{(2)}_{3},\dots)$
play the role of coupling constants.

To study (11) we apply the Harish-Chandra-Itzykson-Zuber relation:

$\int_{\mathbb{U}_{N}}e^{\mathrm{tr}\left(UAU^{-1}B\right)}d_{*}U=\frac{\mathrm{det}\left[e^{a_{k}b_{l}}\right]_{1\leq
k,l\leq N}}{\prod_{k<l}(a_{k}-a_{l})(b_{k}-b_{l})}$ (13)

where $a_{i}$ and $b_{i}$, $i=1,\dots,N$ are the eigenvalues of matrices $A$
and $B$. Then we rewrite $J_{N}$ as the integral in eigenvalues, also using
the symmetry property of the integrand:

$J_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=\tilde{C}_{N}\oint\dots\oint\prod_{i<j}\frac{(u_{i}-u_{j})(v_{i}-v_{j})}{(u_{i}+u_{j})(v_{i}+v_{j})}\prod_{i=1}^{N}e^{c\,\mathrm{tr}\left(u_{i}^{-2}v_{i}^{-2}\right)+\sum_{n=1,3,5,\dots}\frac{2}{n}\left(p^{(1)}_{n}u_{i}^{n}+p^{(2)}_{n}v_{i}^{n}\right)}\frac{du_{i}}{u_{i}}\frac{dv_{i}}{v_{i}}$
(14)

To obtain this form, we use (13) as well as the well-known explicit form of
the expression for the Haar measure $\mathbb{U}_{N}$, which contains the
factors
$d_{*}U_{1}\sim\prod_{i<j}|u_{i}-u_{j}|^{2}\prod_{k}\frac{du_{k}}{u_{k}}$ and
$d_{*}U_{2}\sim\prod_{i<j}|v_{i}-v_{j}|^{2}\prod_{k}\frac{dv_{k}}{v_{k}}$.

For the next step, we use the following analogue of the Cauchy-Littlewood
relation [29]:

$e^{\sum_{n=1,3,5,\dots}\frac{2}{n}p_{n}\mathrm{tr}\left(X^{n}\right)}=\sum_{\alpha\in
DP\atop\ell(\alpha)\leq
N}Q_{\alpha}(\mathbf{p})Q_{\alpha}\left(\mathbf{p}(X)\right)\prod_{i=1}^{\ell(\alpha)}2^{-\alpha_{i}}$
(15)

where $X$ is a matrix and where $Q_{\alpha}$ is the so-called projective Schur
function, see Appendix. This is a polynomial in the variables
$p_{1},p_{3},\dots$ and

$\alpha=(\alpha_{1},\alpha_{2},\dots,\alpha_{k}),\,\alpha_{1}>\cdots>\alpha_{k}\geq
0,\quad k=1,2,\dots$

is the multi index of this multivariable polynomial. Such sets are called
strict partitions, and the set of all strict partitions (or the same: the set
of Young diagrams with distinct lengths of rows
$\alpha_{1}>\alpha_{2}>\cdots$) we denote $DP$ as in [29]. The notation
$Q_{\alpha}\left(\mathbf{p}(X)\right)$ means that here the arguments
$p_{1},p_{3},\dots$ are not free parameters but chosen to be equal Newton sums
of the eigenvalues of the matrix $X$:

$p_{n}=p_{n}(X):=\mathrm{tr}\left(X^{n}\right),\quad n\,{\rm is\,odd}$ (16)

Thus, the polynomial $Q_{\alpha}\left(\mathbf{p}(X)\right)$ is a symmetric
function in the eigenvalues of $X$. For the sake of simplicity, we shall write
$Q_{\alpha}(X)$ instead of $Q_{\alpha}\left(\mathbf{p}(X)\right)$ having in
mind that a capital letter serves for a matrix.

At last, one can prove (see Appendix C) that

$\frac{1}{\left(2\pi i\right)^{2N}}\oint\dots\oint
Q_{\alpha}(U_{1})Q_{\beta}(U_{2})\prod_{i<j}\frac{(u_{i}-u_{j})(v_{i}-v_{j})}{(u_{i}+u_{j})(v_{i}+v_{j})}\prod_{i=1}^{N}2^{cu_{i}^{-2}v_{i}^{-2}}\frac{du_{i}}{u_{i}}\frac{dv_{i}}{v_{i}}$
(17)
$=\begin{cases}\delta_{\alpha,\beta}2^{2\ell(\alpha)}\prod_{i=1}^{\ell(\alpha)}\frac{c^{\alpha_{i}}}{(\frac{1}{2}\alpha_{i})!},\,{\rm
if\,each}\,\alpha_{i}\,{\rm is\,even}\\\ 0\quad{\rm otherwise}\end{cases}$

see Appendix. From (14),(15) and (18) we obtain

$J_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=\sum_{\alpha\in
DP\atop\ell(\alpha)\leq
N}Q_{2\alpha}(\mathbf{p}^{(1)})Q_{2\alpha}(\mathbf{p}^{(2)})\prod_{i=1}^{\ell(\alpha)}c^{2\alpha_{i}}f(\alpha_{i}),$
(18)

where

$f(\alpha_{i})=\frac{1}{\alpha_{i}!}$ (19)

compare to (1). The series in the right-hand side can be obtained by a
limiting procedure from the hypergeometric BKP tau function studied in [40].

If we generalize integral (11) with the help of replacement (3), namely

$J_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)};f)=C\int_{\mathbb{U}_{N}\times\mathbb{U}_{N}}\tau(N;cU_{1}^{-2}U^{-2}_{2};f)e^{\sum_{n=1,3,5,\dots}\frac{2}{n}\left(p_{n}^{(1)}\mathrm{tr}U_{1}^{n}+p^{(2)}_{n}\mathrm{tr}U_{2}^{n}\right)}d\mu_{1}(U_{1})d\mu_{2}(U_{2})$
(20)

we get (18).

#### Fermionic form for the grand partition function

The grand partition function for the sum (18) is

$J(\mathbf{p}^{(1)},\mathbf{p}^{(2)},\zeta)=\sum_{N}J_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)})\zeta^{N}=\sum_{N}\zeta^{N}\sum_{\alpha\in
DP\atop\ell(\alpha)\leq
N}Q_{2\alpha}(\mathbf{p}^{(1)})Q_{2\alpha}(\mathbf{p}^{(2)})\prod_{i=1}^{\ell(\alpha)}\frac{c^{2\alpha_{i}}}{\alpha_{i}!},$
(21)

where we insert (18),(19) for the model (11).

This series can be written as fermionic vacuum expectation value:

$J(\mathbf{p}^{(1)},\mathbf{p}^{(2)},\zeta)=\langle
0|\mathbb{T}\texttt{e}^{\zeta
F(\mathbf{p}^{(1)},\mathbf{p}^{(2)})}|0\rangle,\quad
F(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=-\frac{1}{4\pi^{2}}\oint\oint
e^{cu^{-2}v^{-2}}\phi^{(1)}(u,\mathbf{p}^{(1)})\phi^{(2)}(v,\mathbf{p}^{(2)})\frac{dudv}{uv}$
(22)

where

$\phi^{(i)}(\mathbf{p},z)=e^{\sum_{n=1,3,5,\dots}\frac{1}{n}p_{n}z^{n}}\phi^{(i)}(z),\quad
i=1,2,$ (23)

are neutral fermions:

$[\phi^{(i)}(z_{1}),\phi^{(i)}(z_{2})]_{+}=\sum_{n\,{\rm
odd}}(-1)^{n}\frac{z_{1}^{n}}{z_{2}^{n}},\quad[\phi^{(1)}(z_{1}),\phi^{(2)}(z_{2})]_{+}=0$
(24)

(Fermi fields $\phi^{(i)}(z,\mathbf{p})$ obey the same anticommutation
relations.) Here $\mathbb{T}$ is the chronological ordering of fermions in 2D
Eucledian space in the Taylor series of the exponential (which means the
ordering of the absolute values of arguments $|z_{a}|>|z_{b}|$ if
$\phi^{(i)}(z_{a})$ is placed to the left of $\phi^{(i)}(z_{a})$ and the sign
factor is taken into account). In further, we shall omit the symbol
$\mathbb{T}$ keeping it in mind as the well-known standard point. This
expression is rather similar to the vacuum expectation value written down in
[6] for the Fatteev-Frolov-Schwartz instanton sum in $\sigma$-model. Fermi
modes defined by $\phi^{(i)}(z)=\sum_{j\in\mathbb{Z}}\phi^{(i)}_{j}z^{j}$ obey

$[\phi^{(i_{1})}_{j_{a}},\phi^{(i_{2})}_{j_{b}}]_{+}=(-1)^{j_{a}}\delta_{i_{1},i_{2}}\delta_{j_{a}+j_{b}}$
(25)

We notice that
$F(0,0)=\sum_{i\in\mathbb{Z}}\frac{\phi^{(1)}_{2i}\phi^{(2)}_{2i}}{i!}$. Now,
attention, we use the approach of Kac-van de Leur [18] where the vacuum
vectors are chosen as follows:

$\phi^{(i)}_{0}|0\rangle=\frac{1}{\sqrt{2}}|0\rangle,\quad\langle
0|\phi^{(i)}_{0}=\frac{1}{\sqrt{2}}\langle 0|,\quad i=1,2$ (26)
$\phi_{n}|0\rangle=0,\quad\langle 0|\phi_{-n}=0,\quad n<0$ (27)

This means that

$\langle
0|\phi(z,\mathbf{p})|0\rangle=\frac{1}{\sqrt{2}}e^{\sum_{n=1,3,\dots}\frac{2}{n}p_{n}z^{n}}$
(28)

and for $N=1,2,3,\dots$ we get

$\langle
0|\phi^{(i)}(z_{1},\mathbf{p})\cdots\phi^{(i)}(z_{N},\mathbf{p})|0\rangle=2^{\frac{N}{2}}\prod_{i<j\leq
N}\frac{z_{i}-z_{j}}{z_{i}+z_{j}}e^{\sum_{m=1,3,\dots}\sum_{i=1}^{N}\frac{2}{m}p_{m}z_{i}^{m}}$
(29) $=2^{\frac{N}{2}}\prod_{i<j\leq
N}\frac{z_{i}-z_{j}}{z_{i}+z_{j}}\sum_{\alpha\in DP\atop\ell(\alpha)\leq
N}2^{-\ell(\alpha)}Q_{\alpha}(\mathbf{p})Q_{\alpha}(Z)$ (30)

where $z_{1},\dots,z_{N}$ are eigenvalues of an $N\times N$ matrix $Z$. The
first factor in (30) we got by the Wick theorem, and the second factor in (30)
was obtained from (15). However, it goes back to the work [54], which, in
turn, was based on [8]. The vacuum expectation value (22) is an example of the
two-component BKP tau function introduced in [8],[18]. The by-product of this
statement is the following bilinear equations (the same: Hirota equations) for
the integral (11). The Hirota equations for the two-component BKP tau function
can be found in [18]. The grand partition function (22) solves these bilinear
equations as well as any other BKP tau function. The special tau function is
selected by constraints written in the form of linear differential equations,
see below the paragraph on string equations.

Neutral Fermi fields $\phi^{i}(z)$ can be realized as anticommuting operators
$V(z,{\hat{\mathbf{p}}}^{(i)})$ acting in the bosonic Fock space $\cal{F}$
formed by polynomials in the variables $\mathbf{p}^{(1)},\mathbf{p}^{(2)}$
multiplied by $\frac{1}{2}(1+\eta_{1})(1+\eta_{2})$ where $\eta_{i},\,i=1,2$
are odd Grassmannian numbers: $\eta_{i}^{2}=0$. The (right) vacuum vector is
$\frac{1}{2}(1+\eta_{1})(1+\eta_{2})$, creation operators are $p^{(1)}_{n}$
and $p^{(2)}_{n}$, $n$ odd, and annihilation operators are equal to the
derivatives $n\partial_{p^{(1)}_{n}}$ and $n\partial_{p^{(2)}_{n}}$.

Let

$V(z,\hat{\mathbf{p}}^{(i)})=\frac{\eta_{i}+\frac{\partial}{\partial\eta_{i}}}{\sqrt{2}}e^{\sum_{m\in\mathbb{Z}^{+}_{odd}}\tfrac{2}{m}z^{m}p^{(i)}_{m}}e^{-\sum_{m\in\mathbb{Z}^{+}_{odd}}z^{-m}\tfrac{\partial}{\partial
p^{(i)}_{m}}},\quad|z|=1,$ (31)

be the vertex operator as it was introduced in [18]. The symbol
$\hat{\mathbf{p}}^{(i)}$ denotes the set of two collections:
$\eta_{i},p^{(i)}_{1},p^{(i)}_{3},p^{(i)}_{5},\dots$ and
$\frac{\partial}{\partial\eta_{i}},\frac{\partial}{\partial
p^{(i)}_{1}},\frac{\partial}{\partial p^{(i)}_{3}},\dots$.

One can verify that vertex operators $V^{(i)}(z)$ satisfy relations (24).

The bosonization formulae result it the following representation for
$J(\mathbf{p},\mathbf{p}^{*},\zeta)$:

$J(\mathbf{p}^{(1)},\mathbf{p}^{(2)},\zeta)=g^{\rm
Bos}({\hat{\mathbf{p}}}^{(1)},{\hat{\mathbf{p}}}^{(2)},\zeta)\cdot 1,$ (32)

and

$g^{\rm
Bos}({\hat{\mathbf{p}}}^{(1)},{\hat{\mathbf{p}}}^{(2)},\zeta)=e^{\zeta\oint\oint
e^{cu^{-2}v^{-2}}V(u,{\hat{\mathbf{p}}}^{(1)})V(v,{\hat{\mathbf{p}}}^{(2)})\frac{dudv}{uv}}$
(33)

#### String equations

. The general construction of the algebra $W^{B}_{1+\infty}$ is presented in
[23],[24].

Following the standard procedure, one can expand the product of the two vertex
operators in the generators of $W^{B}_{1+\infty}$ algebra:

$\frac{1}{2}V(ze^{\tfrac{y}{2}},\hat{\mathbf{p}})V(-ze^{-\tfrac{y}{2}},\hat{\mathbf{p}})-\frac{1}{2}\frac{1+e^{-y}}{1-e^{-y}}=\frac{1}{4}\frac{e^{y}+1}{e^{y}-1}\left(\vdots
e^{\theta(ze^{\tfrac{y}{2}})+\theta(-ze^{-\tfrac{y}{2}})}\vdots-1\right)=$
(34)
$=\frac{1}{4}\frac{e^{y}+1}{e^{y}-1}\sum_{k>0}\frac{1}{k!}\vdots\left(\sum_{m\in\mathbb{Z}_{odd}}\theta_{m}(\hat{\mathbf{p}})z^{m}\left(e^{\tfrac{my}{2}}-e^{-\tfrac{my}{2}}\right)\right)^{k}\vdots=:\sum_{m\in\mathbb{Z},n\geq
0}\frac{1}{n!}y^{n}z^{m}\Omega_{\textsf{B}}(m,n,\hat{\mathbf{p}})$ (35)

where $\vdots a\vdots$ means the bosonic normal ordering (which means that all
derivatives $\partial_{p_{i}}$ are moved to the right of $p_{i}$) and where

$2\theta(z,\hat{\mathbf{p}}):=\sum_{m\in\mathbb{Z}^{+}_{odd}}\tfrac{2}{m}z^{m}p_{m}-\sum_{m\in\mathbb{Z}^{+}_{odd}}z^{-m}\tfrac{\partial}{\partial
p_{m}}$ (36)

Equivalently, one can write

$\Omega_{\textsf{B}}(m,n,\hat{\mathbf{p}})=-\mathop{\mathrm{res}}\limits_{z}V(-z,\hat{\mathbf{p}})z^{-\tfrac{m}{2}}\left(D^{n}z^{-\tfrac{m}{2}}V(z,\hat{\mathbf{p}})\right)\frac{dz}{z},\quad
D=z\frac{\partial}{\partial z}$ (37)

As follows from the left-hand side of this formula, $\Omega_{mn}$ vanishes
when $n$ and $m$ have the same parity. In particular, we have

$\Omega_{\textsf{B}}(0,1,\hat{\mathbf{p}})=\sum_{n>0}np_{n}\partial_{n}$
$\Omega_{\textsf{B}}(0,3,\hat{\mathbf{p}})=\frac{1}{2}\sum_{n>0}n^{3}p_{n}\partial_{n}+\frac{1}{2}\sum_{n>0}np_{n}\partial_{n}+4\sum_{n_{1},n_{2},n_{3}\,{\rm
odd}}p_{n_{1}}p_{n_{2}}p_{n_{3}}(n_{1}+n_{2}+n_{3})\partial_{n_{1}+n_{2}+n_{3}}+$
(38) $+3\sum_{n_{1}+n_{2}=n_{3}+n_{4}\,{\rm
odd}}p_{n_{1}}p_{n_{2}}n_{3}n_{4}\partial_{n_{3}}\partial_{n_{4}}+\sum_{n_{1},n_{2},n_{3}\,{\rm
odd}}p_{n_{1}+n_{2}+n_{3}}\partial_{n_{1}}\partial_{n_{2}}\partial_{n_{3}}$
(39)

The fermionic counterpart of (35) is much simpler:

$\frac{1}{2}:\phi(ze^{\tfrac{y}{2}})\phi(-ze^{-\tfrac{y}{2}}):=\frac{1}{2}\sum_{m,j\in\mathbb{Z}}z^{m}e^{\tfrac{y}{2}(m+2j)}(-1)^{j}:\phi_{m+j}\phi_{-j}:=\sum_{m\in\mathbb{Z}\atop
n\geq 0}\frac{1}{n!}y^{n}z^{m}\Omega_{\textsc{f}}(m,n)$ (40)

where $:a:$ serves for the fermionic normal ordering (namely, in formula (40)
one can replace each $:a:$ by $a-\langle 0|a|0\rangle$). Again, as follows
from the left-hand side of this formula, $\Omega_{mn}=0$ when $n$ and $m$ have
the same parity. One gets

$\Omega_{\textsc{f}}(m,n)=\frac{1}{2}\sum_{j\in\mathbb{Z}}\left(\tfrac{m}{2}+j\right)^{n}(-1)^{j}:\phi_{m+j}\phi_{-j}:$
(41) $=\mathop{\mathrm{res}}\limits_{z}\left(z^{-\frac{m}{2}}\cdot D^{n}\cdot
z^{-\frac{m}{2}}\cdot\phi(z)\right)\phi(-z)\frac{dz}{z},\quad
D=z\frac{\partial}{\partial z}.$ (42)

Among operators (41), we have the Virasoro ones, whose fermionic form is

$L^{\textsc{f}}_{m}:=\Omega_{\textsc{f}}(-2m,1)=\frac{1}{2}\sum_{j\in\mathbb{Z}}\left(j-m\right)(-1)^{j}\phi_{j-2m}\phi_{-j}$
(43)

and operators

$\Omega_{\textsc{f}}(0,n)=\frac{1}{2}\sum_{j\in\mathbb{Z}}(-1)^{j}j^{n}:\phi_{j}\phi_{-j}:=\sum_{j=1,3,\dots}(-1)^{j}j^{n}\phi_{j}\phi_{-j},\quad
n\,{\rm odd}.$ (44)

It is known that

$\Omega_{\textsc{b}}(0,n,\hat{\mathbf{p}})Q_{\alpha}(\mathbf{p})=e_{\alpha}Q_{\alpha}(\mathbf{p}),\quad
e_{\alpha}=\sum_{i}\alpha_{i}^{n},\quad n=1,3,5,\dots$

and therefore we get the following string equation:

$\left(\Omega_{\textsc{b}}(0,n,\hat{\mathbf{p}}^{(1)})-\Omega_{\textsc{b}}(0,n,\hat{\mathbf{p}}^{(2)})\right)J_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=0,\quad
n=1,3,5,\dots$ (45)

Each of the equations (45) characterizes the series (18), where both of the
projective Schur functions in the products of pairs are labeled with the same
strict partition, but in no way does it characterize the prefactor
$\prod_{i}\frac{2^{-2\alpha_{i}}c^{2\alpha_{1}}}{\alpha_{i}!}$ in the right-
hand side of (18). To do it, we will write down constraints for the grand
partition function, which can be obtained by the usage of different
$BW_{1+\infty}$ elements. We describe this procedure in short.

First, for each given integer $m$, we introduce a special combination of
$\Omega_{\textsc{f}}(2m,n),\,n=1,3,\dots$ as follows:

${\cal{M}}^{(i)}_{\textsc{f}}(2m,G)=\frac{1}{2}\sum_{j\in\mathbb{Z}}y_{m,G}\left(j+m\right)(-1)^{j}:\phi^{(i)}_{2m+j}\phi^{(i)}_{-j}:$
(46) $=\mathop{\mathrm{res}}\limits_{z}:\left(z^{-m}\cdot y_{m,G}(D)\cdot
z^{-m}\cdot\phi^{(i)}(z)\right)\phi^{(i)}(-z):\frac{dz}{z}$ (47)

where we define $y_{m,G}$ as the following polynomial function of odd degree:

$y_{m,G}(x)=x\left(x^{2}-m^{2}\right)\left(x^{2}-(m+1)^{2}\right)\cdots\left(x^{2}-(2m-1)^{2}\right)G\left(x^{2}\right),\quad
x=m+j$ (48)

where $G(x^{2})$ is an arbitrary-chosen polynomial of $x^{2}$. This
combination of $BW_{1+\infty}$ elements is chosen to provide the property

${\cal{M}}^{(i)}_{\textsc{f}}(2m,G)|0\rangle=0,\quad i=1,2$ (49)

for any integer $m$ and for any polynomial $G(x^{2})$. Indeed, if $m$ is
negative, then each of $\phi_{j+2m}\phi_{-j}$ from the sum in the right-hand
side of (46) annihilates $|0\rangle$. If $m>0$, the situation is as follows:
we have

$\frac{1}{2}\sum_{j\in\mathbb{Z}}(-1)^{j}\phi^{(i)}_{2m+j}\phi^{(i)}_{-j}|0\rangle=\phi^{(i)}_{2m}\phi^{(i)}_{0}|0\rangle+\dots+(-1)^{m-1}\phi^{(i)}_{m+1}\phi^{(i)}_{m-1}|0\rangle\neq
0,$

however ${\cal M}^{(i)}_{\textsf{f}}(2m,G)$ annihilates the vacuum vector due
to the vanishing of $y_{m,G}(m+i)$ at points $i=0,i=1,\dots,i=m-1$, see (48).
Next, by (25) one can also verify that for $m\geq 0$ we obtain

$[{\cal{M}}^{(1)}_{\textsc{f}}(2m,G)-{\cal{M}}^{(2)}_{\textsc{f}}(-2m,\tilde{G}),F(0,0)]=0$
(50)

where $\tilde{G}(x^{2})$ is also a polynomial in $x^{2}$ and defined as
follows:

$\tilde{G}(x^{2})=G(x^{2})c_{\rm even}(x),\quad
c(x)=\frac{\Gamma(x+1+m)}{\Gamma(x+1)}=c_{\rm even}(x)+c_{\rm odd}(x)$ (51)

where polynomial $c_{\rm even}(x)=c_{\rm even}(-x)$ $c_{\rm odd}(x)=-c_{\rm
odd}(-x)$. Say, for $m=1$ we have $c(x)=x+1$, thus $c_{\rm even}=1$ and
$y_{1,G}(x)=y_{-1,\tilde{G}}(x)=x(x^{2}-1)G$.

Clearly, thanks to the symmetry between $\phi^{(1)}$ and $\phi^{(2)}$ in (21)
one can also write

$[{\cal{M}}^{(2)}_{\textsc{f}}(2m,G)-{\cal{M}}^{(1)}_{\textsc{f}}(-2m,\tilde{G}),F(0,0)]=0$

The properties (49)-(50) and the bosonization procedure result in (string)
equations on the grand partition function:

$\left({\cal{M}}_{\textsc{b}}(-2m,{\tilde{G}},\hat{\mathbf{p}}^{(2)})-{\cal{M}}_{\textsc{b}}(2m,G,\hat{\mathbf{p}}^{(1)})\right)J(\mathbf{p}^{(1)},\mathbf{p}^{(2)},\zeta)=0$
(52)

with

${\cal{M}}_{\textsc{b}}(2m,G,\hat{\mathbf{p}}^{(1,2)})=\mathop{\mathrm{res}}\limits_{z}\left(z^{-m}\cdot
y_{m,G}(D)\cdot z^{-m}\cdot
V(z,\hat{\mathbf{p}}^{(1,2)})\right)V(-z,\hat{\mathbf{p}}^{(1,2)})\frac{dz}{z}$
(53) ${\cal{M}}_{-\textsc{b}}(-2m,Gc_{\rm
even},\hat{\mathbf{p}}^{(2,1)})=\mathop{\mathrm{res}}\limits_{z}\left(z^{m}\cdot
y_{-m,Gc_{\rm even}}(D)\cdot z^{m}\cdot
V(z,\hat{\mathbf{p}}^{(2,1)})\right)V(-z,\hat{\mathbf{p}}^{(2,1)})\frac{dz}{z}$
(54)

#### On the Pfaffian form

. Let us write $\mathbf{p}$ instead of $\mathbf{p}^{(1)}$. Let us choose
$p^{(2)}_{n}=p_{n}(y):=\sum_{i=1}^{2k}y_{i}^{n},\,n=1,3,\dots$. For this
choice one can apply the bosonization formula and the Wick theorem and get the
following pfaffian representation:

$J(\mathbf{p},\mathbf{p}(y),\zeta)=\prod_{i<j}^{2k}\left(\frac{y_{i}+y_{j}}{y_{i}-y_{j}}\right)\mathop{\mathrm{missing}}{Pf}\nolimits\left[J\left(\mathbf{p}^{(1)},\mathbf{p}(y_{i},y_{j}),\zeta\right)\right]_{i,j}$
(55)

where

$J\left(\mathbf{p},\mathbf{p}(y_{i},y_{j}),\zeta\right)=\sum_{N}\int_{U_{1}\times
U_{1}}e^{c\,\mathrm{tr}\left(U_{1}^{-2}U_{2}^{-2}\right)+\sum_{n=1,3,\dots}\frac{2}{n}\left(p^{(1)}_{n}\mathrm{tr}U_{1}^{n}\right)}\mathrm{det}\frac{(1-y_{i}U_{2})(1-y_{j}U_{2})}{(1+y_{i}U_{2})(1+y_{j}U_{2})}d_{*}U_{1}dU_{2}^{*}$
(56)

And by the same method, one can write down the answers as the Pfaffian of a
block $2k+2l\times 2k+2l$ matrix in case
$p^{(1)}_{n}=\sum_{i=1}^{2k}y_{i}^{n},\,p{(2)}_{n}=\sum_{i=1}^{2l}z_{i}^{n}\,n=1,3,\dots$.
We omit this specious expression and leave it as quite a tedious exercise for
the reader, see [13] where such calculations are made.

##  4 Cauchy-type interaction

Next, we consider the integral

$K_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=C\int_{\mathbb{U}_{N}\times\mathbb{U}_{N}}\mathrm{det}\left(1-cU_{1}^{-2}U_{2}^{-2}\right)^{-a}e^{\sum_{n=1,3,5,\dots}\frac{2}{n}\left(p^{(1)}_{n}\mathrm{tr}U_{1}^{n}+p^{(2)}_{n}\mathrm{tr}U_{2}^{n}\right)}d_{*}U_{1}d_{*}U_{2}$
(57)

This is a different version of the Cauchy matrix model [39],[11] and [7].

Instead of (14), we obtain

$K_{N}(\mathbf{p},\mathbf{p}^{*})=\tilde{C}\oint\dots\oint\prod_{i<j}\frac{(u_{i}-u_{j})(v_{i}-v_{j})}{(u_{i}+u_{j})(v_{i}+v_{j})}\prod_{i=1}^{N}\left(1-cu_{i}^{-2}v_{i}^{-2}\right)^{N-1-a}e^{\sum_{n=1,3,5,\dots}\frac{2}{n}\left(p_{n}u_{i}^{n}+p^{*}_{n}v_{i}^{n}\right)}\frac{du_{i}}{u_{i}}\frac{dv_{i}}{v_{i}}$
(58)

where $C$ and $\tilde{C}$ provide the condition $K_{N}(0,0)=1$.

Similarly to the formula (18) in the previous case, we get

$K_{N}(\mathbf{p},\mathbf{p}^{*})=\sum_{\alpha\in DP\atop\ell(\alpha)\leq
N}Q_{2\alpha}(\mathbf{p})Q_{2\alpha}(\mathbf{p}^{*})\prod_{i=1}^{\ell(\alpha)}\frac{2^{-2\alpha_{i}}c^{2\alpha_{i}}(1-c\alpha_{i})^{-a}}{\alpha_{i}!},$
(59)

In this case, the formulas (22) and (32) for the grand partition function are
the same; however, in Cauchy’s case, we have

$F(\mathbf{p},\mathbf{p}^{*})=-\frac{1}{4\pi^{2}}\oint\oint(1-cuv)^{-a}\phi^{(1)}(u,\mathbf{p})\phi^{(2)}(v,\mathbf{p}^{*})\frac{dudv}{uv}$
(60)

and

$F^{\rm
Bos}(\hat{\mathbf{p}},\hat{\mathbf{p}}^{*})=-\frac{1}{4\pi^{2}}\oint\oint(1-cuv)^{-a}V^{(1)}(u,\hat{\mathbf{p}})V^{(2)}(v,\hat{\mathbf{p}}^{*})\frac{dudv}{uv}$
(61)

Formula (55) where

$J\left(\mathbf{p},\mathbf{p}(y_{i},y_{j}),\zeta\right)=\sum_{N}\int_{U_{1}\times
U_{1}}e^{\sum_{n>0}\frac{1}{n}\left(p_{n}\mathrm{tr}U_{1}^{n}\right)}\mathrm{det}\left(1-cU_{1}^{-2}U_{2}^{-2}\right)^{-a}\mathrm{det}\frac{(1-y_{i}U_{2})(1-y_{j}U_{2})}{(1+y_{i}U_{2})(1+y_{j}U_{2})}d_{*}U_{1}dU_{2}^{*},$
(62)

is still correct.

String equations are the same, namely, (45) and (52),(53) and (54), which are
still true, however, now in (51) we have

$c(x)=\frac{\Gamma(x+m+1)\Gamma(x+a)}{\Gamma(x+1)\Gamma(x+m+a)}\frac{\Gamma(a+m)}{\Gamma(a)}=c_{\rm
even}(x)+c_{\rm odd}(x)$ (63)

For $m=1$, $G\equiv 1$, and $a=1$, we get $c_{\rm even}(x)=1$. Thus,
$y_{1,1}=x^{3}-x=y_{-1,1}$.

Acknowledgements. The authors are grateful A. Alexandrov, A.Morozov, A.
Mironov for attracting attention to [3],[35], and special thanks to Andrey
Mironov for fruitful discussions. The work was supported by the Russian
Science Foundation (Grant No.23-41-00049).

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##  Appendix A The Haar measure on $\mathbb{U}_{N}$

The Haar measure on $\mathbb{U}_{N}$ in its explicit form is written as

$d_{*}U=\frac{1}{(2\pi)^{N}}\prod_{1\leq i<k\leq
N}|e^{\theta_{i}}-e^{\theta_{k}}|^{2}\prod_{i=1}^{N}d\theta_{i},\quad-\pi<\theta_{1}<\cdots<\theta_{N}\leq\pi$
(64)

where $e^{\theta_{1}},\dots,e^{\theta_{N}}$ are eigenvalues of
$U\in\mathbb{U}_{N}$.

##  Appendix B Partitions. The Schur and the projective Schur functions

We recall that a non-increasing set of non-negative integers
$\lambda_{1}\geq\cdots\geq\lambda_{k}\geq 0$, we call partition
$\lambda=(\lambda_{1},\dots,\lambda_{l})$, and $\lambda_{i}$ are called parts
of $\lambda$. The sum of parts is called the weight $|\lambda|$ of $\lambda$.
The number of nonzero parts of $\lambda$ is called the length of $\lambda$, it
will be denoted $\ell(\lambda)$. See [29] for details. Partitions will be
denoted by Greek letters: $\lambda,\mu,\dots$. The set of all partitions is
denoted by $\mathop{\mathrm{missing}}{P}\nolimits$. Partitions with distinct
parts are called strict partitions, we prefer the letters $\alpha,\beta$ to
denote them. The set of all strict partitions will be denoted by
$\mathop{\mathrm{missing}}{DP}\nolimits$.

To define the projective Schur function $Q_{\alpha}$ where
$\alpha\in\mathop{\mathrm{missing}}{DP}\nolimits$, at the first step, we
define the set of functions $\\{q_{i},i\geq 0\\}$ by

$e^{\sum_{m>0,{\rm odd}}\frac{2}{m}p_{m}x^{m}}=\sum_{m\geq
0}x^{m}q_{m}(\mathbf{p}_{\rm odd})$ (65)

where now $\mathbf{p}_{\rm odd}=(p_{1},p_{3},\dots)$. Next, we define the
following skew-symmetric matrix:

$Q_{ij}(\mathbf{p}_{\rm odd}):=\begin{cases}q_{i}(\mathbf{p}_{\rm
odd})q_{j}(\mathbf{p}_{\rm
odd})+2\sum_{k=1}^{j}(-1)^{k}q_{i+k}(\mathbf{p}_{\rm
odd})q_{j-k}(\mathbf{p}_{\rm odd})\quad\text{if }(i,j)\neq(0,0),\\\
0\quad\text{if }(i,j)=(0,0),\end{cases}$ (66)

In particular, $Q_{(j,0)}(\mathbf{p}_{\rm odd})=-Q_{(0,j)}(\mathbf{p}_{\rm
odd})=q_{j}(\mathbf{p}_{\rm odd})\ \text{ for }j\geq 1.$ For a strict
partition $\alpha=(\alpha_{1},\dots,\alpha_{2r})$ where $\alpha_{2r}\geq 0$
the projective Schur function is defined

$Q_{\alpha}(\mathbf{p}_{\rm
odd}):=\mathop{\mathrm{missing}}{Pf}\nolimits\left[Q_{\alpha_{i}\alpha_{j}}(\mathbf{p}_{\rm
odd})\right]_{1\leq i,j\leq 2r},\quad Q_{\emptyset}:=1$ (67)

Let $X$ be a matrix. If we put
$p_{m}=p_{m}(X)=\mathrm{tr}\left(X^{m}-(-X)^{m}\right)$ which we call odd
power sum variables, $m$ odd, we write $Q_{\alpha}(\mathbf{p}_{\rm
odd}(X))=Q_{\alpha}(X)$.

##  Appendix C Neutral fermions [17]. Scalar product of the projective Schur
functions [40]

For
$\alpha=(\alpha_{1},\dots,\alpha_{k})\in\mathop{\mathrm{missing}}{DP}\nolimits$
introduce

$\displaystyle\Phi_{\alpha}$ $\displaystyle:=$ $\displaystyle
2^{\frac{k}{2}}\phi_{\alpha_{1}}\cdots\phi_{\alpha_{k}}$ (68)
$\displaystyle\Phi_{-\alpha}$ $\displaystyle:=$
$\displaystyle(-1)^{\sum_{i=1}^{k}\alpha_{i}}2^{\frac{k}{2}}\phi_{-\alpha_{k}}\cdots\phi_{-\alpha_{1}}$
(69)

We have $\langle
0|\Phi_{-\alpha}\Phi_{\beta}|0\rangle=2^{\ell(\alpha)}\delta_{\alpha,\beta}$.

The fermionic formula for projective Schur functions was obtained in [54]:

$Q_{\alpha}(\mathbf{x})\Delta^{*}(\mathbf{x})=2^{-\frac{N}{2}}\langle
0|\phi(-x_{1}^{-1})\cdots\phi(-x_{N}^{-1})\Phi_{\alpha}|0\rangle=2^{-\frac{N}{2}}\langle
0|\Phi_{-\alpha}\phi(x_{1})\cdots\phi(x_{N})|0\rangle$ (70)

We get

$\langle
0|\phi(-v_{1}^{-1})\cdots\phi(-v_{N}^{-1})=\Delta^{*}(v)\sum_{\alpha}2^{-\frac{N}{2}-\ell(\alpha)}Q_{\alpha}(v)\langle
0|\Phi_{-\alpha}$ (71)
$\phi(u_{1})\cdots\phi(u_{N})|0\rangle=\Delta^{*}(u)\sum_{\alpha}2^{-\frac{N}{2}-\ell(\alpha)}Q_{\alpha}(u)\Phi_{\alpha}|0\rangle$
(72)

Following [40] consider $f(x)=\sum_{i\geq 0}f_{i}x^{i}$ and write

$2^{-N}\sum_{\alpha}\Phi_{\alpha}|0\rangle\langle
0|\Phi_{-\alpha}\prod_{i=1}^{\ell(\alpha)}f_{\alpha_{i}}=$ (73)
$\frac{1}{(2\pi
i)^{N}}\oint\cdots\oint\phi(u_{1})\cdots\phi(u_{N})|0\rangle\langle
0|\phi(-v_{N}^{-1})\cdots\phi(-v_{1}^{-1})\prod_{i=1}^{N}\frac{du_{i}dv_{i}}{u_{i}v_{i}}f(u_{i}^{-1}v_{i}^{-1})=$
(74) $=\frac{1}{(2\pi
i)^{N}}\oint\cdots\oint\Delta^{*}(u)\Delta^{*}(v)Q_{\alpha}(u)Q_{\beta}(v)\sum_{\alpha,\beta}\Phi_{\alpha}|0\rangle\langle
0|\Phi_{-\beta}2^{-N-\ell(\alpha)-\ell(\beta)}\prod_{i=1}^{N}\frac{du_{i}dv_{i}}{u_{i}v_{i}}f(u_{i}^{-1}v_{i}^{-1})$
(75)

Then we obtain (17):

$\frac{1}{(2\pi
i)^{N}}\oint\cdots\oint\Delta^{*}(u)\Delta^{*}(v)Q_{\alpha}(u)Q_{\beta}(v)2^{-\ell(\alpha)-\ell(\beta)}\prod_{i=1}^{N}\frac{du_{i}dv_{i}}{u_{i}v_{i}}f(u_{i}^{-1}v_{i}^{-1})=\prod_{i=1}^{N}f_{\alpha_{i}}$
(76)

##  Appendix D Fermionic representation and string equations for the modified
$I_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)};f)$

$I_{N}(\mathbf{p}^{(1)},\mathbf{p}^{(2)};f)=\langle
N,-N|e^{F(\mathbf{p}^{(1)},\mathbf{p}^{(2)})}|0,0\rangle$ (77)

and

$F(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=-\frac{1}{4\pi^{2}}\oint\oint
f(u^{-1}v^{-1})\psi^{(1)}(u,\mathbf{p}^{(1)})\psi^{{\dagger}(2)}(v,\mathbf{p}^{(2)})\frac{dudv}{uv}=\sum_{j}f_{j}\psi^{(1)}_{j}\psi^{{\dagger}(2)}_{-j-1}$
(78)

where $f(z)=\sum_{j}f_{j}z^{j}$, $\psi(z)=\sum_{j}z^{j}\psi_{j}$ and
$\psi^{\dagger}(z)=\sum_{j}z^{j}\psi^{\dagger}_{-j-1}$, and for $j>N$ we have
$\langle N|\psi_{j}|=0$ and $\langle 0|\psi^{\dagger}=0$. We have

$\psi(ze^{\tfrac{1}{2}y})\psi^{\dagger}(ze^{-\tfrac{1}{2}y})=\sum_{m,n}\Omega_{\textsc{f}}(m,n)z^{-m}y^{n}$
(79)

where

$\Omega_{\textsc{f}}=\sum_{j}(m+\tfrac{1}{2})^{n}\psi_{j}\psi^{\dagger}_{j+m}$
(80)

By $\hat{\mathbf{p}}$ we denote the collections $(p_{1},p_{2},\dots)$ and
$(\partial_{p_{1}},\partial_{p_{2}},\dots)$ The bosonic version is

$\Omega_{\textsc{b}}(m,n,\hat{\mathbf{p}})=-\mathop{\mathrm{res}}\limits_{z}\vdots\left(V(z,\hat{\mathbf{p}})z^{-\frac{1}{2}}\left(z^{m}D^{n}z^{\frac{1}{2}}V(z,-\hat{\mathbf{p}}\right)\right)\vdots
dz$ (81)

Apart of string equations typical for any “diagonal” series (9), string
equations which select the prefactor $f$ in (9) are rather different from the
case of two-matrix model with Hermitian matrices. We have

$L=\sum_{i}x(i)\psi^{(1)}_{i}\psi^{{\dagger}(1)}_{i+m},\quad
M=\sum_{i}y(i)\psi^{(2)}_{i}\psi^{{\dagger}(2)}_{i-m},\quad m>0$ (82)

and ask

$M|0,0\rangle=0,\quad m>0$ (83)

We have

$\sum_{i}y(i)\psi^{(2)}_{i}\psi^{{\dagger}(2)}_{i-m}|0,0\rangle=y(0)\psi^{(2)}_{0}\psi^{{\dagger}(2)}_{-m}|0,0\rangle+\cdots+y_{(}m-1)\psi^{(2)}_{m-1}\psi^{{\dagger}(2)}{-1}|0,0\rangle=0$

thus, $y(i)=0$ for $i=0,\dots,m-1$. Therefore, one can take

$y(i)=i(i-1)\dots(i-m+1)$ (84)

Then, the condition

$[L+M,\sum_{i}f_{i}\psi^{(1)}_{i}\psi^{{\dagger}(2)}_{-i-1}]=0$ (85)

yields

$x(i+m)f_{i+m}=y(i)f_{i}$ (86)

Say, for $f_{i}=\frac{1}{\Gamma(i+1)}$ and $m=1$ one can take $x(i)=i(i-1)$
and $y(i)=i$ (which is the Virasoro generator denoted $L_{-1}$).

The bosonized versions are as follows

$L^{\textsc{b}}({\hat{\mathbf{t}}}^{(1)})I(\mathbf{t}^{(1)},\mathbf{t}^{(2)};f)=M^{\textsc{b}}({\hat{\mathbf{t}}}^{(2)})I(\mathbf{t}^{(1)},\mathbf{t}^{(2)};f)$
(87)

The explicit examples will be written down in a more detailed text.

##  Appendix E Hirota equations for two-component $BKP$

We have the following fermionic form for the two-component BKP tau function:

$\tau(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=\langle
0|e^{\sum_{i=1,2}\sum_{m>0,{\rm odd}}\frac{2}{m}p_{m}^{(i)}\sum_{j\in
Z}\phi^{(i)}_{j-m}\phi^{(i)}_{-j}}h|0\rangle$ (88)

where $h$ is of form

$h=e^{\sum_{i,j\in\mathbb{Z}}\left(a_{i,j}:\phi^{(1)}\phi^{(1)}:+b_{i,j}:\phi^{(2)}\phi^{(2)}:+f_{i,j}\phi^{(1)}\phi^{(2)}\right)}$
(89)

The fermionic form of the two-component hierarchy is as follows:

$\sum_{i=1,2}\sum_{j\in\mathbb{Z}}(-1)^{j}\phi^{(i)}_{j}h\otimes\phi^{(i)}_{-j}h-\frac{1}{2}h\phi^{(i)}_{0}\otimes
h\phi^{(i)}_{0}$ (90)

The bilinear Hirota equations are the bosonized version of (90). They can be
found in [18]. In short, one gets

$\mathop{\mathrm{res}}\limits_{z}\left(\tau\left(\mathbf{p}^{(1)}+\epsilon^{(1)}-[z],\mathbf{p}^{(2)}+\epsilon^{(2)}\right)\tau\left(\mathbf{p}^{(1)}+[z],\mathbf{p}^{(2)})\right)\right)\frac{dz}{z}+$
(91)
$\mathop{\mathrm{res}}\limits_{z}\left(\tau\left(\mathbf{p}^{(1)}+\epsilon^{(1)},\mathbf{p}^{(2)}+\epsilon^{(2)}-[z]\right)\tau\left(\mathbf{p}^{(1)},\mathbf{p}^{(2)})+[z]\right)\right)\frac{dz}{z}-\tau(\mathbf{p}^{(1)}+\epsilon^{(1)},\mathbf{p}^{(2)}+\epsilon^{(2)})\tau(\mathbf{p}^{(1)},\mathbf{p}^{(2)})=0$

where the sets
$\epsilon^{(i)}=\left(\epsilon^{(i)}_{1},\epsilon^{(i)}_{3},\dots\right)$ are
the sets of arbitrary parameters, and where $[z]$ denotes the set
$(2z^{-1},2z^{-3},2z^{-5},\dots)$. The condition of equality to zero for all
coefficients of the Taylor series in these parameters gives bilinear
differential equations for tau functions called us the Hirota differential
equations.