Reflection operator and hypergeometry II:
$SL(2,\mathbb{C})$ spin chain

P. Antonenko†∗, N. Belousov†∘, S. Derkachov†, P. Valinevich†

†Steklov Mathematical Institute, Fontanka 27,
St. Petersburg, 191023, Russia
∗Leonhard Euler International Mathematical Institute,
Pesochnaya nab. 10, St. Petersburg, 197022, Russia
∘National Research University Higher School of Economics,
Myasnitskaya 20, Moscow, 101000, Russia

###### Abstract

We consider noncompact open $SL(2,\mathbb{C})$ spin chain and construct
eigenfunctions of $B$-element of monodromy matrix for the simplest case of the
chain with one site. The reflection operator appearing in this construction
can be used to express eigenfunction for $n$ sites in terms of the
eigenfunction for $n-1$ sites, this general result is briefly announced. We
prove orthogonality and completeness of constructed eigenfunctions in the case
of one site, express them in terms of the hypergeometric function of the
complex field and derive the equation for the reflection operator with the
general $SL(2,\mathbb{C})$-invariant $\mathbb{R}$-operator.

###### Contents

  1. 1 Introduction
  2. 2 Open $SL(2,\mathbb{C})$ spin chain
    1. 2.1 Eigenfunctions for one site
    2. 2.2 Eigenfunctions for $n$ sites
  3. 3 Derivation of reflection operator
  4. 4 Eigenfunctions
    1. 4.1 Diagrammatic representations
    2. 4.2 Relation to hypergeometric function
    3. 4.3 Mellin-Barnes representation
  5. 5 Orthogonality
  6. 6 Completeness
  7. 7 Reflection equations
  8. A Diagram technique

##  Introduction

The closed spin chain with the symmetry group $SL(2,\mathbb{C})$ is an
integrable model with various applications in quantum field theory [L1, L2,
FK, DKM, DKO]. It was studied in the framework of the quantum inverse
scattering method [F], and many important objects of this method
($Q$-operators, $R$-operators, separated variables representation) are known
[DKM, DM2]. Separation of variables (SoV) [S1, S2] is performed by
diagonalization of element $B(u)$ of monodromy matrix. Unitarity of the SoV
transform for the closed $SL(2,\mathbb{C})$ spin chain or, equivalently,
orthogonality and completeness of eigenfunctions of $B(u)$ is proven in [M].

In this paper we consider open $SL(2,\mathbb{C})$ spin chain. To define this
model we follow Sklyanin’s approach [S3]. The construction of the monodromy
matrix is different from the previous case and involves $K$-matrix — a
solution to the reflection equation [C, S3, KS]. Eigenfunctions of the
$B$-element of monodromy matrix can be constructed recursively with a new
ingredient, the $\mathcal{K}$-operator that solves the reflection equation
with the Lax matrix of the model. Note that in the simplest situation, when
$K$-matrix and $\mathcal{K}$-operator are identities, eigenfunctions of the
$B$-element of monodromy matrix are constructed in [DMV].

In the present paper we focus on the simplest case of spin chain with one site
and prove orthogonality and completeness of eigenfunctions. Our main
computational tool is the diagram technique [DKM], which significantly
simplifies proofs and makes them visual.

##  Open $SL(2,\mathbb{C})$ spin chain

To describe the model we recall the main formulas for the principal series
representation of the group $SL(2,\mathbb{C})$ [GGV, Ch. III]. It is defined
on the space $\mathrm{L}^{2}(\mathbb{C})$ of square integrable functions on
$\mathbb{C}$ with the scalar product

$\langle\Phi|\Psi\rangle=\int\mathrm{d}^{2}z\;\overline{\Phi(z,\bar{z})}\,\Psi(z,\bar{z}).$
(2.1)

The integration measure is the Lebesque measure in $\mathbb{C}$:
$\mathrm{d}^{2}z=\mathrm{d}\operatorname{Re}z\;\mathrm{d}\operatorname{Im}z$,
and in what follows by $\int\mathrm{d}^{2}z$ we denote integration over the
whole complex plane.

To avoid misunderstanding, let us remark that below we denote complex
conjugation of a number $a\in\mathbb{C}$ as $a^{*}$, so that in generic
situation $\bar{a}$ does not mean complex conjugation of $a$. There are two
traditional exceptions: for the variable $z$ we use $\bar{z}\equiv z^{*}$, and
complex conjugated functions are denoted as $\overline{\Psi(z,\bar{z})}$.
Besides, for simplicity we display only the holomorphic argument of functions
$\Psi(z)\equiv\Psi(z,\bar{z})$.

The representation is parametrized by the pair of spins

$\bm{s}=(s,\bar{s})\,,\qquad 2(s-\bar{s})\in\mathbb{Z}.$ (2.2)

The group element

$g=\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\qquad ad-bc=1,$ (2.3)

is represented by the operator $T(g)$ acting on functions $\Psi(z)$ by the
formula

$[T(g)\,\Psi](z)=[d-bz]^{-2s}\,\Psi\biggl{(}\frac{-c+az}{d-bz}\biggr{)}.$
(2.4)

Here and in what follows for a pair $(a,\bar{a})\in\mathbb{C}^{2}$ such that
$a-\bar{a}\in\mathbb{Z}$ we denote the double power

$\displaystyle[z]^{a}\equiv
z^{a}\bar{z}^{\bar{a}}=|z|^{a+\bar{a}}\,{e}^{\mathrm{i}(a-\bar{a})\arg z}.$
(2.5)

The function (2.5) depends on both “holomorphic” parameter $a$ and
“antiholomorphic” one $\bar{a}$, but for brevity we display only the
“holomorphic” exponent. The numbers $a,\bar{a}$ are not complex conjugate in
general. We impose the condition $a-\bar{a}\in\mathbb{Z}$, so that the
function $[z]^{a}$ is single-valued in the whole complex plane. In addition,
for $\rho\in\mathbb{R}$ we denote

$[z]^{\rho+a}\equiv z^{\rho+a}\bar{z}^{\rho+\bar{a}}.$ (2.6)

The principal series representation is unitary if $s$ and $\bar{s}$ satisfy
the condition

$s^{\ast}+\bar{s}=1,$ (2.7)

where $s^{\ast}$ denotes the complex conjugation of $s$. Together with the
condition $2(s-\bar{s})\in\mathbb{Z}$ this gives the parametrization of spins

$s=\frac{n_{s}+1}{2}+\mathrm{i}\nu_{s},\qquad\bar{s}=\frac{-n_{s}+1}{2}+\mathrm{i}\nu_{s}\,,\qquad
n_{s}\in\mathbb{Z}+\sigma,\quad\nu_{s}\in\mathbb{R},$ (2.8)

where for the rest of the paper we fix the parameter $\sigma$

$\sigma\in\bigl{\\{}0,\tfrac{1}{2}\bigr{\\}}.$ (2.9)

Generators of the representation are defined in the standard way as the
coefficients of decomposition of the map $T(g)$ in the neighbourhood of unity.
They have the form

$\displaystyle\begin{aligned} &S=z\partial_{z}+s,\qquad
S_{-}=-\partial_{z},\qquad S_{+}=z^{2}\partial_{z}+2sz,\\\\[2.0pt]
&\bar{S}=\bar{z}\partial_{\bar{z}}+\bar{s},\qquad\bar{S}_{-}=-\partial_{\bar{z}},\qquad\bar{S}_{+}=\bar{z}^{2}\partial_{\bar{z}}+2\bar{s}\bar{z}.\end{aligned}$
(2.10)

We call $S,S_{\pm}$ holomorphic generators and $\bar{S},\bar{S}_{\pm}$ —
antiholomorphic ones. They obey the standard commutation relations of the Lie
algebra $sl(2,\mathbb{C})$

$\displaystyle[S_{+},S_{-}]=2S,\qquad[S,S_{\pm}]=\pm S_{\pm},$ (2.11)

and similarly for the antiholomorphic generators. Note that the holomorphic
generators commute with the antiholomorphic ones. Two types of generators are
conjugate to each other with respect to the scalar product (2.1).

$S^{\dagger}=-\bar{S},\qquad S_{-}^{\dagger}=-\bar{S}_{-},\qquad
S_{+}^{\dagger}=-\bar{S}_{+}$ (2.12)

due to the assumption (2.7).

In the theory of integrable systems an important role is played by Lax
matrices. The spin chain under consideration is defined by holomorphic and
antiholomorphic Lax matrices of the form

$\displaystyle L(u)=\begin{pmatrix}u+S&S_{-}\\\
S_{+}&u-S\end{pmatrix},\qquad\bar{L}(\bar{u})=\begin{pmatrix}\bar{u}+\bar{S}&\bar{S}_{-}\\\
\bar{S}_{+}&\bar{u}-\bar{S}\end{pmatrix}.$ (2.13)

Another useful notation for the Lax matrix reads

$\displaystyle L(u)\equiv L(u+s-1,u-s),$ (2.14)

where the parameters $u_{1}=u+s-1$, $u_{2}=u-s$ appear naturally in the
factorized expression

$\displaystyle L(u_{1},u_{2})$ $\displaystyle=\begin{pmatrix}1&0\\\
z&1\end{pmatrix}\begin{pmatrix}u_{1}&-\partial_{z}\\\
0&u_{2}\end{pmatrix}\begin{pmatrix}1&0\\\ -z&1\end{pmatrix}$
$\displaystyle=\begin{pmatrix}u_{1}+1+z\partial_{z}&-\partial_{z}\\\\[3.0pt]
z^{2}\partial_{z}+(u_{1}-u_{2}+1)z&u_{2}-z\partial_{z}\end{pmatrix}.$ (2.15)

For the construction of the open spin chain we also need $K$-matrices

$\displaystyle K(u)=\begin{pmatrix}\mathrm{i}\alpha&u-\frac{1}{2}\\\\[6.0pt]
\gamma^{2}\bigl{(}u-\frac{1}{2}\bigr{)}&\mathrm{i}\alpha\end{pmatrix},\qquad\bar{K}(\bar{u})=\begin{pmatrix}\mathrm{i}\bar{\alpha}&\bar{u}-\frac{1}{2}\\\\[6.0pt]
\bar{\gamma}^{2}\bigl{(}\bar{u}-\frac{1}{2}\bigr{)}&\mathrm{i}\bar{\alpha}\end{pmatrix},$
(2.16)

which solve the reflection equation [C, S3, KS]. The $K$-matrix considered in
the $SL(2,\mathbb{R})$ case [ABDK, eq. (1.12)] can be obtained by the change
$\gamma=i\beta$.

In the present work the attention is payed to the open $SL(2,\mathbb{C})$ spin
chain. The Hilbert space of the chain consisting of $n$ sites is given by the
direct product of $n$ copies of $\mathrm{L}^{2}(\mathbb{C})$, i.e. the space
of square integrable functions of $n$ variables $z_{1},\ldots,z_{n}$. To
$k$-th site one associates holomorphic Lax matrix

$\displaystyle
L_{k}(u)=\begin{pmatrix}u+z_{k}\partial_{z_{k}}+s&-\partial_{z_{k}}\\\\[4.0pt]
z_{k}^{2}\partial_{z_{k}}+2sz_{k}&u-z_{k}\partial_{z_{k}}-s\end{pmatrix}$
(2.17)

and the analogous antiholomorphic one $\bar{L}_{k}(\bar{u})$. Quantum
integrals of motion of the model are constructed with the help of the
monodromy matrix

$T(u)=L_{n}(u)\cdots L_{1}(u)K(u)L_{1}(u)\cdots
L_{n}(u)=\begin{pmatrix}A(u)&B(u)\\\ C(u)&D(u)\end{pmatrix},$ (2.18)

and its antiholomorphic counterpart $\bar{T}(\bar{u})$, which is defined by
means of $\bar{L}_{k}(\bar{u})$ and $\bar{K}(\bar{u})$ in the same way. The
monodromy matrix satisfies the reflection equation

$R(u-v)\,\bigl{(}T(u)\otimes\bm{1}\bigr{)}\,R(u+v-1)\,\bigl{(}\bm{1}\otimes
T(v)\bigr{)}\\\\[6.0pt] =\bigl{(}\bm{1}\otimes
T(v)\bigr{)}\,R(u+v-1)\,\bigl{(}T(u)\otimes\bm{1}\bigr{)}\,R(u-v).$ (2.19)

Here $R(u)$ is the Yang’s $R$-matrix that acts in the tensor product
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$

$R(u)=u+P,\qquad P\,a\otimes b=b\otimes a.$ (2.20)

From (2.19) and its antiholomorphic analogue we obtain the commutativity

$[B(u),B(v)]=0,\qquad[\bar{B}(\bar{u}),\bar{B}(\bar{v})]=0.$ (2.21)

Note that the holomorphic and antiholomorphic operators also commute with each
other

$[B(u),\bar{B}(\bar{v})]=0.$ (2.22)

Our main objective is the simultaneous diagonalization of operators $B(u)$ and
$\bar{B}(\bar{u})$. In this paper we study in detail the case $n=1$ and derive
the reflection operator, which is the key element in the construction of
eigenfunctions for arbitrary $n$. Most of the results and proofs are analogous
to the case of $SL(2,\mathbb{R})$ spin chain studied in the first paper of
this series [ABDK].

###  Eigenfunctions for one site

In this section we collect our main results concerning the spin chain with one
site. In the case $n=1$ the operator $B(u)$ has the form

$B(u)=\biggl{(}u-\frac{1}{2}\biggr{)}\bigl{(}u^{2}-H^{s}\bigr{)},$ (2.23)

where $H^{s}$ can be written in terms of holomorphic generators (2.10)

$\displaystyle H^{s}$
$\displaystyle=S^{2}-\gamma^{2}S_{-}^{2}-2\mathrm{i}\alpha S_{-}$ (2.24)
$\displaystyle=(z^{2}-\gamma^{2})\partial_{z}^{2}+(2s+1)z\partial_{z}+2\mathrm{i}\alpha\partial_{z}+s^{2}.$

One can see that the spectral problem for $B(u)$ is equivalent to the spectral
problem for $H^{s}$. Similarly for its antiholomorphic counterpart

$\bar{B}(\bar{u})=\biggl{(}\bar{u}-\frac{1}{2}\biggr{)}\bigl{(}\bar{u}^{2}-\bar{H}^{\bar{s}}\bigr{)}$
(2.25)

and the operator

$\displaystyle\bar{H}^{\bar{s}}$
$\displaystyle=\bar{S}^{2}-\bar{\gamma}^{2}\bar{S}_{-}^{2}-2\mathrm{i}\bar{\alpha}\bar{S}_{-}$
(2.26)
$\displaystyle=(\bar{z}^{2}-\bar{\gamma}^{2})\partial_{\bar{z}}^{2}+(2\bar{s}+1)\bar{z}\partial_{\bar{z}}+2\mathrm{i}\bar{\alpha}\partial_{\bar{z}}+\bar{s}^{2}.$

Let us impose the condition on these operators

$(H^{s})^{\dagger}=\bar{H}^{\bar{s}},$ (2.27)

where the hermitian conjugation is taken with respect to the scalar product
(2.1). Operators $H^{s}$ and $\bar{H}^{\bar{s}}$ commute, so that under the
condition (2.27) we can construct commuting self-adjoint combinations
$H^{s}+\bar{H}^{\bar{s}}$, $\mathrm{i}(H^{s}-\bar{H}^{\bar{s}})$.

The holomorphic and antiholomorphic generators are conjugate to each other
(2.12): $S^{\dagger}=-\bar{S}$, $S_{-}^{\dagger}=-\bar{S}_{-}$. Then the
condition (2.27) is satisfied if

$\bar{\alpha}=\alpha^{\ast},\qquad\bar{\gamma}^{2}=(\gamma^{\ast})^{2}.$
(2.28)

Since the operator $\bar{H}^{\bar{s}}$ (2.26) depends on $\bar{\gamma}^{2}$,
without loss of generality we may assume

$\bar{\alpha}=\alpha^{\ast},\qquad\bar{\gamma}=\gamma^{*}.$ (2.29)

Under the above assumption we look for the joint eigenfunctions of operators
$H^{s}$ and $\bar{H}^{\bar{s}}$

$H^{s}\,\Psi_{\bm{x}}(z)=x^{2}\,\Psi_{\bm{x}}(z),\qquad\bar{H}^{\bar{s}}\,\Psi_{\bm{x}}(z)=\bar{x}^{2}\,\Psi_{\bm{x}}(z),$
(2.30)

where we denoted the spectral parameters $\bm{x}=(x,\bar{x})$.

The joint eigenfunctions are constructed with the help of the reflection
operator $\mathcal{K}(\bm{s},\bm{x})$ that acts on functions of $z,\bar{z}$
and satisfies two reflection equations with holomorphic Lax and $K$-matrices
(2.15), (2.16)

$\displaystyle\begin{aligned}
&\mathcal{K}(\bm{s},\bm{x})\,L(u+x-1,u-s)\,K(u)\,L(u+s-1,u-x)\\\\[4.0pt]
&=L(u+s-1,u-x)\,K(u)\,L(u+x-1,u-s)\,\mathcal{K}(\bm{s},\bm{x})\end{aligned}$
(2.31)

and their antiholomorphic counterparts

$\displaystyle\begin{aligned}
&\mathcal{K}(\bm{s},\bm{x})\,\bar{L}(\bar{u}+\bar{x}-1,\bar{u}-\bar{s})\,\bar{K}(\bar{u})\,\bar{L}(\bar{u}+\bar{s}-1,\bar{u}-\bar{x})\\\\[4.0pt]
&=\bar{L}(\bar{u}+\bar{s}-1,\bar{u}-\bar{x})\,\bar{K}(\bar{u})\,L(\bar{u}+\bar{x}-1,\bar{u}-\bar{s})\,\mathcal{K}(\bm{s},\bm{x}).\end{aligned}$
(2.32)

The first relation (2.31) is equivalent to three independent operator
equations on reflection operator, see (3.2). One of them includes the operator
$H^{s}$ (2.24)

$\displaystyle
H^{s}\,\mathcal{K}(\bm{s},\bm{x})=\mathcal{K}(\bm{s},\bm{x})\,H^{x}.$ (2.33)

Similarly, from (2.32) we have

$\bar{H}^{\bar{s}}\,\mathcal{K}(\bm{s},\bm{x})=\mathcal{K}(\bm{s},\bm{x})\,\bar{H}^{\bar{x}}.$
(2.34)

Acting on $1$ in the last two equations we obtain

$\displaystyle H^{s}\,\mathcal{K}(\bm{s},\bm{x})\cdot
1=\mathcal{K}(\bm{s},\bm{x})\,H^{x}\cdot
1=x^{2}\,\mathcal{K}(\bm{s},\bm{x})\cdot 1,$ (2.35)
$\displaystyle\bar{H}^{\bar{s}}\,\mathcal{K}(\bm{s},\bm{x})\cdot
1=\mathcal{K}(\bm{s},\bm{x})\,\bar{H}^{\bar{x}}\cdot
1=\bar{x}^{2}\,\mathcal{K}(\bm{s},\bm{x})\cdot 1.$ (2.36)

Thus, $\mathcal{K}(\bm{s},\bm{x})\cdot 1$ is the joint eigenfunction of the
operators $H^{s}$ and $\bar{H}^{\bar{s}}$.

All of this can be regarded as an induction step from the eigenfunction of
$B_{n=0}(u)$ to the eigenfunction of $B_{n=1}(u)$. The operator $B_{n=0}$ is
an element of the matrix $K(u)$, i.e. it is a number and its eigenfunction is
$1$. Acting by $\mathcal{K}(\bm{s},\bm{x})$ on $1$ we obtain an eigenfunction
of $B_{n=1}(u)$, which is the function of one complex variable $z$.

Now let us formulate our main results concerning the reflection operator and
eigenfunctions. It is convenient to denote

$\displaystyle
g=\frac{1}{2}+\frac{\mathrm{i}\alpha}{\gamma},\qquad\bar{g}=\frac{1}{2}+\frac{\mathrm{i}\bar{\alpha}}{\bar{\gamma}},$
(2.37)

and below we frequently use this pair of parameters instead of
$(\alpha,\bar{\alpha})$. Apart from the conditions on the parameters (2.8),
(2.29), in what follows we always assume

$\displaystyle x-\bar{x}\in\mathbb{Z}+\sigma,\qquad
g-\bar{g}\in\mathbb{Z}+\sigma,$ (2.38)

where $\sigma\in\\{0,1/2\\}$ is fixed by the spins (2.8). The second
assumption together with the conditions (2.29) gives the following
parametrization

$\;\;g=\frac{n_{g}+1}{2}+\mathrm{i}\nu_{g},\qquad\bar{g}=\frac{-n_{g}+1}{2}+\mathrm{i}\nu_{g}\,,\qquad
n_{g}\in\mathbb{Z}+\sigma,\;\;\;\nu_{g}\in\mathbb{R}.$ (2.39)

Consequently, the complex conjugation rule for the pair $(g,\bar{g})$

$\displaystyle g^{*}+\bar{g}=1$ (2.40)

is the same as for the spins $(s,\bar{s})$ (2.7).

In Section 3 we derive the explicit formula for the action of reflection
operator $\mathcal{K}(\bm{s},\bm{x})$ on a function $\Psi(z)$

$\displaystyle\begin{aligned}
\bigl{(}\mathcal{K}(\bm{s},\bm{x})\,\Psi\bigr{)}(z)&=\frac{1}{\pi\,\bm{\Gamma}(s-x)}\;[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\\\\[6.0pt]
&\times\int\mathrm{d}^{2}w\;\frac{[w+\gamma]^{x-g}\,[w-\gamma]^{x+g-1}}{[z-w]^{x-s+1}}\;\Psi(w).\end{aligned}$
(2.41)

Here $\bm{\Gamma}(a)$ is the gamma function of the complex field [GGR, Section
1.4], [N, Section 1.3]

$\bm{\Gamma}(a)=\frac{\Gamma(a)}{\Gamma(1-\bar{a})}.$ (2.42)

It depends on two parameters $(a,\bar{a})\in\mathbb{C}^{2}$ such that
$a-\bar{a}\in\mathbb{Z}$, but for brevity we display only the first one. In
addition, for $\rho\in\mathbb{R}$ we write

$\displaystyle\bm{\Gamma}(a+\rho)\equiv\frac{\Gamma(a+\rho)}{\Gamma(1-\bar{a}-\rho)}$
(2.43)

and for the products of gamma functions we use compact notation

$\displaystyle\bm{\Gamma}(a\pm
b,c,d)=\bm{\Gamma}(a+b)\,\bm{\Gamma}(a-b)\,\bm{\Gamma}(c)\,\bm{\Gamma}(d).$
(2.44)

The useful properties of this function can be found in Appendix A.

Define the integral operator $[\partial_{z}]^{a}$ acting on functions
$\Psi(z)$ by the formula

$\displaystyle([\partial_{z}]^{a}\,\Psi)(z)=\frac{1}{\pi\,\mathbf{\Gamma}(-a)}\int\mathrm{d}^{2}w\,\frac{1}{[z-w]^{a+1}}\,\Psi(w).$
(2.45)

Then the reflection operator (2.41) can be alternatively written in the form

$\displaystyle\mathcal{K}(\bm{s},\bm{x})=[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\,[\partial_{z}]^{x-s}\,[z+\gamma]^{x-g}\,[z-\gamma]^{x+g-1}.$
(2.46)

We remark that the operator $[\partial_{z}]^{2\ell-1}$ plays an important role
in the representation theory of $SL(2,\mathbb{C})$, namely, it intertwines
equivalent representations with spins $(\ell,\bar{\ell})$ and
$(1-\ell,1-\bar{\ell})$ [GGV, Ch. III, Section 3.2]. Moreover, it was used for
the construction of the general $SL(2,\mathbb{C})$-invariant solution of the
Yang-Baxter equation [DM1, Section 2.3].

In addition, for the derivative operator (2.45) we have the expected relation

$[\partial_{z}]^{a}\,[\partial_{z}]^{b}=[\partial_{z}]^{a+b},$ (2.47)

which is equivalent to the integral identity (A.3). Then from the formula
(2.46) the reflection operator clearly satisfies

$\displaystyle\mathcal{K}(\bm{s},\bm{x})\,\mathcal{K}(\bm{x},\bm{y})=\mathcal{K}(\bm{s},\bm{y}).$
(2.48)

Section 4 is devoted to the integral representations of the eigenfunctions.
According to the formulas (2.35), (2.36) the eigenfunctions can be written in
the form

$\Psi_{\bm{x}}(z)=\pi\,[2\gamma]^{s-x}\,\bm{\Gamma}(g-x)\;\mathcal{K}(\bm{s},\bm{x})\cdot
1.$ (2.49)

Due to the explicit expression for the reflection operator (2.41) and
reflection formula for the gamma function (A.9) it is equivalent to the
integral representation

$\displaystyle\begin{aligned}
&\Psi_{\bm{x}}(z)=[2\gamma]^{s-x}\,\mathbf{\Gamma}(g-x,1-s+x)\\\\[6.0pt]
&\quad\times\int\frac{\mathrm{d}^{2}w}{[z+\gamma]^{s-g}\,[z-\gamma]^{s+g-1}\,[w-z]^{x-s+1}\,[w+\gamma]^{g-x}\,[w-\gamma]^{1-x-g}}.\end{aligned}$
(2.50)

In Appendix A we explain how such integrals with power functions can be
depicted by diagrams. In this way various integral identities correspond to
simple transformations of diagrams.

Using diagram technique in Section 4.1 we prove that with the above
normalization the eigenfunctions have reflection symmetry

$\Psi_{\bm{x}}(z)=\Psi_{-\bm{x}}(z).$ (2.51)

Moreover, making certain diagrams transformations we derive the second
integral representation for them

$\displaystyle\begin{aligned}
\Psi_{\bm{x}}(z)&=[2\gamma]^{s-x}\,\mathbf{\Gamma}(g-x,1-s+x)\\\\[4.0pt]
&\times\int\frac{\mathrm{d}^{2}w}{[w-z]^{x+s}\,[w-\gamma]^{g-x}\,[w+\gamma]^{1-x-g}}.\end{aligned}$
(2.52)

With the help of this representation in Section 4.2 we show that the
eigenfunctions are related to the hypergeometric function of the complex field
[GGR, §2], [MN, Section 1.6] by the formula

$\displaystyle\begin{aligned}
\Psi_{\bm{x}}(z)&=\,\frac{\pi\,\mathbf{\Gamma}(g+x,g-x)}{\mathbf{\Gamma}(s+g)}\\\\[8.0pt]
&\times{}_{2}F_{1}^{\mathbb{C}}\left[\\!\left.\begin{array}[]{c}s+x|\bar{s}+\bar{x},\;s-x|\bar{s}-\bar{x}\\\\[3.0pt]
s+g|\bar{s}+\bar{g}\end{array}\right|\frac{1}{2}-\frac{z}{2\gamma}\right].\end{aligned}$
(2.53)

Last but not least in Section 4.3 we obtain one more integral representation
of Mellin-Barnes type

$\displaystyle\Psi_{\bm{x}}(z)=\frac{1}{2}\int\mathcal{D}\bm{y}\;\frac{[-1]^{y-s}\;\mathbf{\Gamma}(\varepsilon-y\pm
x,\,1-s-\varepsilon+y,\,g-\varepsilon+y)}{[2\gamma]^{\varepsilon-
y-s}\,[z+\gamma]^{s-g}\,[z-\gamma]^{g-\varepsilon+y}}$ (2.54)

where we assume $\varepsilon\in(0,1/2)$ and that both the spectral and
integration variables have the form

$\displaystyle\begin{aligned}
&\bm{x}=(x,\bar{x})=\biggl{(}\frac{k}{2}+\mathrm{i}\eta,-\frac{k}{2}+\mathrm{i}\eta\biggr{)},&&\quad
k\in\mathbb{Z}+\sigma,\quad\;\eta\in\mathbb{R},\\\\[6.0pt]
&\bm{y}=(y,\bar{y})=\biggl{(}\frac{m}{2}+\mathrm{i}\tau,-\frac{m}{2}+\mathrm{i}\tau\biggr{)},&&\quad
m\in\mathbb{Z}+\sigma,\quad\tau\in\mathbb{R}.\end{aligned}$ (2.55)

By integration over $\bm{y}$ we mean

$\displaystyle\int\mathcal{D}\bm{y}=\sum_{m\in\mathbb{Z}+\sigma}\,\int_{\mathbb{R}}\mathrm{d}\tau.$
(2.56)

In Section 5 we prove that eigenfunctions $\Psi_{\bm{x}}(z),\Psi_{\bm{y}}(z)$
with spectral parameters of the form (2.55) are orthogonal with respect to the
scalar product (2.1)

$\displaystyle\langle\Psi_{\bm{y}}|\Psi_{\bm{x}}\rangle=\mu^{-1}(\bm{x})\,\frac{\delta(\bm{x}-\bm{y})+\delta(\bm{x}+\bm{y})}{2}.$
(2.57)

Here we denoted the delta-function with spectral parameters

$\displaystyle\delta(\bm{x}-\bm{y})=\delta_{km}\,\delta(\eta-\tau)$ (2.58)

and the normalization coefficient

$\displaystyle\mu(\bm{x})=\frac{1}{4\pi^{4}}\Bigl{|}\frac{x}{\gamma}\Bigr{|}^{2}.$
(2.59)

The proof of the orthogonality relies on the diagram technique described in
Appendix A.

With the help of the Mellin-Barnes representation (2.54) in Section 5 we prove
the completeness relation

$\int\mathcal{D}\bm{x}\;\mu(\bm{x})\,\Psi_{\bm{x}}(z)\,\overline{\Psi_{\bm{x}}(w)}=\delta^{2}(z-w),$
(2.60)

where $\delta^{2}(z)$ is the delta-function in the complex plane

$\displaystyle\delta^{2}(z)=\delta(\operatorname{Re}z)\,\delta(\operatorname{Im}z).$
(2.61)

Notice that the measure in the completeness relation $\mu(\bm{x})$ agrees with
the orthogonality (2.57).

Finally, in Section 7 we show that the operator $\mathcal{K}(\bm{s},\bm{x})$
(2.46) also satisfies the reflection equation

$\displaystyle\begin{aligned}
&\mathcal{K}_{1}(\bm{x})\;\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})\;\mathcal{K}_{2}(\bm{y})\;{\mathbb{R}}_{12}(\bm{x},\bm{y})\\\\[6.0pt]
&\quad\;\;={\mathbb{R}}_{12}(\bm{x},\bm{y})\;\mathcal{K}_{2}(\bm{y})\;\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})\;\mathcal{K}_{1}(\bm{x})\end{aligned}$
(2.62)

with the $\mathbb{R}$-operators defined by the formulas

$\displaystyle{\mathbb{R}}_{12}(\bm{x},\bm{y})=\mathcal{P}_{12}\;[z_{1}-z_{2}]^{1-s-x}\,[\partial_{z_{1}}]^{y-x}\,[z_{1}-z_{2}]^{s+y-1},$
(2.63)
$\displaystyle\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})=\mathcal{P}_{12}\;[z_{1}-z_{2}]^{1-2s}\,[\partial_{z_{2}}]^{x-s}\,[\partial_{z_{1}}]^{y-s}\,[z_{1}-z_{2}]^{x+y-1},$
(2.64)

where $\mathcal{P}_{12}$ is the permutation operator

$\displaystyle\mathcal{P}_{12}\,\psi(z_{1},z_{2})=\psi(z_{2},z_{1}).$ (2.65)

Here the lower indices mean spaces of functions of $z_{1},z_{2}$ and in all
operators we suppressed the dependence on spin $\bm{s}$. Such
$\mathbb{R}$-operators represent special cases of the most general
$SL(2,\mathbb{C})$-invariant solution of the Yang-Baxter equation [DM1, §3].

The orthogonality and completeness of the eigenfunctions $\Psi_{\bm{x}}(z)$
are equivalent to unitarity of the transform

$\displaystyle[T\,\Phi](\bm{x})=\int\mathrm{d}^{2}z\;\overline{\Psi_{\bm{x}}(z)}\,\Phi(z)$
(2.66)

that maps the initial Hilbert space to the space of square integrable
functions with respect to the measure $\mu(\bm{x})\,\mathcal{D}\bm{x}$ and
invariant under reflection $\bm{x}\to-\bm{x}$. It will be interesting to
establish the relation between this transform and the complex analogue of the
index hypergeometric transform that also has the hypergeometric function in
the kernel, but acts on different Hilbert space [MN].

###  Eigenfunctions for $n$ sites

For the spin chain with $n$ sites operators $B(u),\bar{B}(\bar{u})$ act on
functions of $n$ complex variables $z_{1},\dots,z_{n}$. In this section we
state the formula for their joint eigenfunctions

$\displaystyle
B(u)\,\Psi_{\bm{x}_{1},\dots,\,\bm{x}_{n}}=\left(u-\frac{1}{2}\right)(u^{2}-x_{1}^{2})\ldots(u^{2}-x_{n}^{2})\,\Psi_{\bm{x}_{1},\dots,\,\bm{x}_{n}},$
(2.67)
$\displaystyle\bar{B}(\bar{u})\,\Psi_{\bm{x}_{1},\dots,\,\bm{x}_{n}}=\left(\bar{u}-\frac{1}{2}\right)(\bar{u}^{2}-\bar{x}_{1}^{2})\ldots(\bar{u}^{2}-\bar{x}_{n}^{2})\,\Psi_{\bm{x}_{1},\dots,\,\bm{x}_{n}}$
(2.68)

with pairs of spectral variables $\bm{x}_{k}=(x_{k},\bar{x}_{k})$ such that
$x_{k}-\bar{x}_{k}\in\mathbb{Z}+\sigma$.

Let us introduce the integral operator acting on functions $\Psi(z_{i},z_{j})$

$\displaystyle(\mathcal{R}_{ij}(\bm{x})\,\Psi)(z_{i},z_{j})=\int\mathrm{d}^{2}w\,\frac{[z_{i}-z_{j}]^{1-2s}}{[z_{i}-w]^{1-s+x}\,[w-z_{j}]^{1-s-x}}\,\Psi(w,z_{j}).$
(2.69)

We note that it interchanges the arguments in two holomorphic Lax matrices

$\displaystyle\begin{aligned}
&\mathcal{R}_{12}(\bm{x})\,L_{1}(u+s-1,u-s)\,L_{2}(u+s-1,u-x)\\\\[6.0pt]
&=L_{1}(u+s-1,u-x)\,L_{2}(u+s-1,u-s)\,\mathcal{R}_{12}(\bm{x}),\end{aligned}$
(2.70)

and the same relation holds for the antiholomorphic ones. This operator is
related to the general $SL(2,\mathbb{C})$-invariant solution of the Yang-
Baxter equation [DM1, §3].

With the help of the above $\mathcal{R}$-operator and reflection operator
(2.41) we define

$\displaystyle\begin{aligned}
\Lambda_{n}(\bm{x})&=\mathcal{R}_{n,n-1}(\bm{x})\,\mathcal{R}_{n-1,n-2}(\bm{x})\ldots\mathcal{R}_{21}(\bm{x})\\\\[4.0pt]
&\times\mathcal{K}_{1}(\bm{s},\bm{x})\,\mathcal{R}_{12}(\bm{x})\,\mathcal{R}_{23}(\bm{x})\ldots\mathcal{R}_{n-1,n}(\bm{x}).\end{aligned}$
(2.71)

Here $\mathcal{K}_{1}(\bm{s},\bm{x})$ is the reflection operator acting on
functions of $z_{1}$. Then the induction formula for the eigenfunctions reads

$\Psi_{\bm{x}_{1},\dots,\,\bm{x}_{n}}(z_{1},\dots,z_{n})=\Lambda_{n}(\bm{x}_{n})\,\Psi_{\bm{x}_{1},\dots,\,\bm{x}_{n-1}}(z_{1},\dots,z_{n-1}),$
(2.72)

or, equivalently,

$\Psi_{\bm{x}_{1},\ldots,\,\bm{x}_{n}}(z_{1},\dots,z_{n})=\Lambda_{n}(\bm{x}_{n})\,\Lambda_{n-1}(\bm{x}_{n-1})\cdots\Lambda_{1}(\bm{x}_{1})\cdot
1.$ (2.73)

We postpone the proof of this formula to the future work.

##  Derivation of reflection operator

The operator $\mathcal{K}(\bm{s},\bm{x})$ acts on functions
$\Psi(z)\equiv\Psi(z,\bar{z})$ and satisfies the holomorphic reflection
equation (2.31)

$\displaystyle\begin{aligned}
&\mathcal{K}(\bm{s},\bm{x})\,L(u+x-1,u-s)\,K(u)\,L(u+s-1,u-x)\\\\[4.0pt]
&=L(u+s-1,u-x)\,K(u)\,L(u+x-1,u-s)\,\mathcal{K}(\bm{s},\bm{x})\end{aligned}$
(3.1)

and similar antiholomorphic one (2.32). The equation (3.1) is a matrix
equation, the entries of matrices are operators polynomial in $u$. To obtain
the system of operator equations on $\mathcal{K}(\bm{s},\bm{x})$ one needs to
look at the coefficients of polynomials from both sides of relation (3.1).

Since the reflection equation (3.1) is the same as in $SL(2,\mathbb{R})$ case
[ABDK, Section 2.1], we obtain the same operator equations together with their
antiholomorphic counterparts

$\displaystyle\begin{aligned}
&\mathcal{K}\,N=N\,\mathcal{K},\qquad\mathcal{K}\,H^{x}=H^{s}\,\mathcal{K},\qquad\mathcal{K}\,I^{x,s}=I^{s,x}\,\mathcal{K},\\\
&\mathcal{K}\,\bar{N}=\bar{N}\,\mathcal{K},\qquad\mathcal{K}\,\bar{H}^{\bar{x}}=\bar{H}^{\bar{s}}\,\mathcal{K},\qquad\mathcal{K}\,\bar{I}^{\bar{x},\bar{s}}=\bar{I}^{\bar{s},\bar{x}}\,\mathcal{K},\end{aligned}$
(3.2)

where the holomorphic operators are given by the formulas

$\displaystyle N=(z^{2}-\gamma^{2})\partial_{z}+(s+x)z,$ (3.3) $\displaystyle
H^{s}=(z^{2}-\gamma^{2})\partial_{z}^{2}+(2s+1)z\partial_{z}+2\mathrm{i}\alpha\partial_{z}+s^{2},$
(3.4) $\displaystyle\begin{aligned}
I^{s,x}&=z(z^{2}-\gamma^{2})\partial_{z}^{2}+(2s+1)z^{2}\partial_{z}+x(z^{2}-\gamma^{2})\partial_{z}\\\\[4.0pt]
&+2\mathrm{i}\alpha
z\partial_{z}+s(s+x)z+N+\mathrm{i}\alpha(s+x),\end{aligned}$ (3.5)

and analogously for the antiholomorphic ones
$\bar{N},\bar{H}^{\bar{s}},\bar{I}^{\bar{s},\bar{x}}$. To solve the above
system we use the following ansatz for the reflection operator

$\mathcal{K}(\bm{s},\bm{x})=[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\;W(\bm{s},\bm{x})\;[z+\gamma]^{x-g}\,[z-\gamma]^{x+g-1},$
(3.6)

where $W(\bm{s},\bm{x})$ is an unknown operator. Here and in what follows we
denote

$g=\frac{1}{2}+\frac{\mathrm{i}\alpha}{\gamma}\,,\qquad\bar{g}=\frac{1}{2}+\frac{\mathrm{i}\bar{\alpha}}{\bar{\gamma}}\,,$
(3.7)

and notation for the double power $[z]^{a}$ is defined in (2.5). The right
hand side of (3.6) is the composition of $W(\bm{s},\bm{x})$ with operators of
multiplication by power functions. To have single-valued power functions we
assume

$\displaystyle x-\bar{x}\in\mathbb{Z}+\sigma,\qquad
g-\bar{g}\in\mathbb{Z}+\sigma,$ (3.8)

where $\sigma\in\\{0,1/2\\}$ is fixed by the spins $(s,\bar{s})$
parametrization (2.8). The motivation for such ansatz is explained at the end
of this section.

Substituting the ansatz (3.6) into the equations (3.2) we obtain the
equivalent system

$\displaystyle\begin{aligned}
&W\,\mathcal{N}^{x,s}=\mathcal{N}^{s,x}\,W,\qquad
W\,\mathcal{H}^{x,s}=\mathcal{H}^{s,x}\,W,\qquad
W\,\mathcal{I}^{x,s}=\mathcal{I}^{s,x}\,W,\\\\[6.0pt]
&W\,\bar{\mathcal{N}}^{\bar{x},\bar{s}}=\bar{\mathcal{N}}^{\bar{s},\bar{x}}\,W,\qquad
W\,\bar{\mathcal{H}}^{\bar{x},\bar{s}}=\bar{\mathcal{H}}^{\bar{s},\bar{x}}\,W,\qquad
W\,\bar{\mathcal{I}}^{\bar{x},\bar{s}}=\bar{\mathcal{I}}^{\bar{s},\bar{x}}\,W,\end{aligned}$
(3.9)

where each holomorphic operator $A^{s,x}=N,H^{s},I^{s,x}$ has been similarity
transformed

$\mathcal{A}^{s,x}=[z+\gamma]^{s-g}[z-\gamma]^{s+g-1}A^{s,x}\,[z+\gamma]^{g-s}[z-\gamma]^{1-s-g},$
(3.10)

and analogously for the antiholomorphic ones. Now our aim is to solve the
system (3.9) for $W(\bm{s},\bm{x})$. First, denote the generators of
$SL(2,\mathbb{C})$

$\displaystyle S^{s,x}=z\partial_{z}+\ell,\qquad
S_{-}^{s,x}=-\partial_{z},\qquad S_{+}^{s,x}=z^{2}\partial_{z}+2\ell z,$
(3.11)
$\displaystyle\bar{S}^{s,x}=\bar{z}\partial_{\bar{z}}+\bar{\ell},\qquad\bar{S}^{s,x}_{-}=-\partial_{\bar{z}},\qquad\bar{S}^{s,x}_{+}=\bar{z}^{2}\partial_{\bar{z}}+2\bar{\ell}\bar{z},$

with the pair of spins

$\ell=\frac{1+x-s}{2},\qquad\bar{\ell}=\frac{1+\bar{x}-\bar{s}}{2}.$ (3.12)

Next calculate the similarity transformed derivatives

$\displaystyle\begin{aligned}
&[z+\gamma]^{s-g}\,[z-\gamma]^{s+g-1}\,\partial_{z}\,[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\\\\[6.0pt]
&\;\;=\partial_{z}+\frac{(1-2s)z-2\mathrm{i}\alpha}{z^{2}-\gamma^{2}},\end{aligned}$
(3.13) $\displaystyle\begin{aligned}
&[z+\gamma]^{s-g}\,[z-\gamma]^{s+g-1}\,\partial_{z}^{2}\,[z+\gamma]^{g-s}[z-\gamma]^{1-s-g}\\\\[6.0pt]
&\;\;=\partial_{z}^{2}+2\,\frac{(1-2s)z-2\mathrm{i}\alpha}{z^{2}-\gamma^{2}}\,\partial_{z}+\frac{(2s-1)(2sz^{2}+\gamma^{2})-4\alpha^{2}+8s\mathrm{i}\alpha
z}{(z^{2}-\gamma^{2})^{2}}.\end{aligned}$ (3.14)

Using the above formulas we express the operators in the system (3.9) in terms
of introduced generators

$\displaystyle\begin{aligned}
&\mathcal{N}^{s,x}=S_{+}^{s,x}+\gamma^{2}S_{-}^{s,x}-2\mathrm{i}\alpha,\\\\[8.0pt]
&\begin{aligned}
\mathcal{H}^{s,x}&=-S_{+}^{s,x}S_{-}^{s,x}+(2-s-x)S^{s,x}\\\\[4.0pt]
&+2\mathrm{i}\alpha
S_{-}^{s,x}-\gamma^{2}(S_{-}^{s,x})^{2}+\frac{s^{2}+x^{2}-s-x}{2},\end{aligned}\\\\[6.0pt]
&\begin{aligned}
\mathcal{I}^{s,x}&=S_{+}^{s,x}S^{s,x}+\gamma^{2}S^{s,x}S_{-}^{s,x}+\frac{3-s-x}{2}\,S_{+}^{s,x}\\\\[4.0pt]
&+\frac{1+s+x}{2}\,\gamma^{2}S_{-}^{s,x}-2\mathrm{i}\alpha
S^{s,x}-\mathrm{i}\alpha.\end{aligned}\end{aligned}$ (3.15)

The expressions for $\bar{\mathcal{N}}^{\bar{s},\bar{x}}$,
$\bar{\mathcal{H}}^{\bar{s},\bar{x}}$, $\bar{\mathcal{I}}^{\bar{s},\bar{x}}$
are of the same form, one just needs to replace holomorphic generators and
parameters with antiholomorphic ones. Therefore, the solution of the system
(3.9) is given by the operator that intertwines introduced generators

$\displaystyle W\,S^{x,s}=S^{s,x}\,W,\qquad
W\,S_{\pm}^{x,s}=S_{\pm}^{s,x}\,W,$ (3.16) $\displaystyle
W\,\bar{S}^{\bar{x},\bar{s}}=\bar{S}^{\bar{s},\bar{x}}\,W,\qquad
W\,\bar{S}_{\pm}^{\bar{x},\bar{s}}=\bar{S}_{\pm}^{\bar{s},\bar{x}}\,W.$ (3.17)

Note that the interchange $s\leftrightarrows x$ corresponds to the
transformation of the spin $\ell\to 1-\ell$ (3.12). It is known that
$SL(2,\mathbb{C})$ representations with spins $(\ell,\bar{\ell})$ and
$(1-\ell,1-\bar{\ell})$ are equivalent [GGV, Ch. III, Section 3.2]. The
operator establishing their equivalence can be written in the form [DM1,
Section 2.3]

$W=[\partial_{z}]^{2\ell-1}=[\partial_{z}]^{x-s}.$ (3.18)

Here $[\partial_{z}]^{a}$ is an integral operator depending on two parameters
$(a,\bar{a})\in\mathbb{C}^{2}$ such that $a-\bar{a}\in\mathbb{Z}$. It is given
by the following formula:

$([\partial_{z}]^{a}\,\Psi)(z)=\frac{1}{\pi\,\mathbf{\Gamma}(-a)}\int\mathrm{d}^{2}w\,\frac{1}{[z-w]^{a+1}}\,\Psi(w),$
(3.19)

where $\mathbf{\Gamma}(-a)$ is the gamma function of the complex field

$\bm{\Gamma}(-a)=\frac{\Gamma(-a)}{\Gamma(1+\bar{a})}.$ (3.20)

Thus, the reflection operator admits the expression

$\displaystyle\mathcal{K}(\bm{s},\bm{x})=[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\,[\partial_{z}]^{x-s}\,[z+\gamma]^{x-g}\,[z-\gamma]^{x+g-1}.$
(3.21)

According to the definition (3.19), the explicit action of this operator on
$\Psi(z)$ is given by the formula

$\displaystyle\begin{aligned}
\bigl{(}\mathcal{K}(\bm{s},\bm{x})\,\Psi\bigr{)}(z)&=\frac{1}{\pi\,\bm{\Gamma}(s-x)}\;[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\\\\[6.0pt]
&\times\int\mathrm{d}^{2}w\;\frac{[w+\gamma]^{x-g}\,[w-\gamma]^{x+g-1}}{[z-w]^{x-s+1}}\;\Psi(w).\end{aligned}$
(3.22)

Note that in Section 2.1 we imposed the conditions $\bar{\alpha}=\alpha^{*}$,
$\bar{\gamma}=\gamma^{*}$ (2.29), which together with the assumption
$g-\bar{g}\in\mathbb{Z}+\sigma$ (3.8) give us the following parametrization

$g=\frac{n_{g}+1}{2}+\mathrm{i}\nu_{g},\qquad\bar{g}=\frac{-n_{g}+1}{2}+\mathrm{i}\nu_{g}\,,\qquad
n_{g}\in\mathbb{Z}+\sigma,\;\;\nu_{g}\in\mathbb{R},$ (3.23)

similar to the parametrization of the spins $(s,\bar{s})$ (2.8).

Now let us explain the motivation for the ansatz (3.6). In the first paper of
this series we obtained the formula for the reflection operator of open
$SL(2,\mathbb{R})$ spin chain [ABDK, Section 2.2]

$\displaystyle\mathcal{K}(s,x)=(2\gamma)^{s-x}\,(z+\gamma)^{g-s}\,\frac{\Gamma\bigl{(}(z-\gamma)\partial_{z}+x+g\bigr{)}}{\Gamma\bigl{(}(z-\gamma)\partial_{z}+s+g\bigr{)}}\,(z+\gamma)^{x-g}.$
(3.24)

Heuristically, we can introduce the complex power of the derivative over the
variable $z$ by the following formula [CDKK, eq. (2.8)]

$\partial_{z}^{a}=\frac{1}{(z-\gamma)^{a}}\,\frac{\Gamma\bigl{(}(z-\gamma)\partial_{z}+1\bigr{)}}{\Gamma\bigl{(}(z-\gamma)\partial_{z}+1-a\bigr{)}}.$
(3.25)

This operator mimics the properties of the usual derivative, see [CDKK]. Then
the expression (3.24) can be rewritten in the suggestive form

$\displaystyle\begin{aligned} \mathcal{K}(s,x)&=(2\gamma)^{s-x}\\\\[6.0pt]
&\times(z+\gamma)^{g-s}\,(z-\gamma)^{1-s-g}\;\partial_{z}^{x-s}\;(z-\gamma)^{x+g-1}\,(z+\gamma)^{x-g},\end{aligned}$
(3.26)

which motivates the ansatz (3.6). However, in the $SL(2,\mathbb{R})$ case we
consider $\gamma=\mathrm{i}\beta\in\mathrm{i}\mathbb{R}_{>0}$ and the Hilbert
space consists of the functions analytic in the upper half-plane
$\operatorname{Im}z>0$. The power functions $(z-\gamma)^{a}$ are not single-
valued in the upper half-plane, which makes the individual factors in the
formula (3.26) ill-defined.

Notice that the defining equation for the reflection operator (3.1) is
invariant with respect to the reflection $\gamma\to-\gamma$. Under this
reflection the parameter

$g=\frac{1}{2}+\frac{\mathrm{i}\alpha}{\gamma}$ (3.27)

also changes $g\to 1-g$. It is interesting that despite the defining equation
(3.1) in $SL(2,\mathbb{R})$ and $SL(2,\mathbb{C})$ cases is the same, only the
$SL(2,\mathbb{C})$ solution (3.21) is symmetric with respect to the reflection
$\gamma\to-\gamma$, whereas in the $SL(2,\mathbb{R})$ case the symmetry is
broken, as it can be seen from the definition (3.25).

##  Eigenfunctions

###  Diagrammatic representations

In this section we derive two integral representations for the one-site
eigenfunctions. Both of them can be represented by certain diagrams using the
technique described in Appendix A.

Figure 1: Diagrammatic representation of $\Psi_{\bm{x}}(z)$

As it is explained in Section 2.1, the eigenfunctions can be constructed using
the reflection operator (2.49)

$\displaystyle\Psi_{\bm{x}}(z)=\pi\,[2\gamma]^{s-x}\,\bm{\Gamma}(g-x)\;\mathcal{K}(\bm{s},\bm{x})\cdot
1.$ (4.1)

Below we prove that with such normalization the eigenfunctions are invariant
with respect to the reflection $\bm{x}\to-\bm{x}$.

The explicit action of the reflection operator is given by the formula (3.22)

$\displaystyle\begin{aligned}
\bigl{(}\mathcal{K}(\bm{s},\bm{x})\,\Psi\bigr{)}(z)&=\frac{1}{\pi\,\bm{\Gamma}(s-x)}\;[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\\\\[6.0pt]
&\times\int\mathrm{d}^{2}w\;\frac{[w+\gamma]^{x-g}\,[w-\gamma]^{x+g-1}}{[z-w]^{x-s+1}}\;\Psi(w).\end{aligned}$
(4.2)

Therefore, due to the reflection formula for the gamma function (A.9)

$\displaystyle[-1]^{s-x}=\bm{\Gamma}(s-x,1-s+x)$ (4.3)

we obtain the first integral representation for the eigenfunctions

$\displaystyle\begin{aligned}
&\Psi_{\bm{x}}(z)=[2\gamma]^{s-x}\,\mathbf{\Gamma}\left(g-x,1-s+x\right)\\\\[6.0pt]
&\quad\times\int\frac{\mathrm{d}^{2}w}{[z+\gamma]^{s-g}\,[z-\gamma]^{s+g-1}\,[w-z]^{x-s+1}\,[w+\gamma]^{g-x}\,[w-\gamma]^{1-x-g}}.\end{aligned}$
(4.4)

Up to gamma functions behind the integral it is depicted in Figure 1.

Figure 2: Reflection symmetry $\bm{x}\to-\bm{x}$

Now let us prove the reflection symmetry

$\Psi_{\bm{x}}(z)=\Psi_{-\bm{x}}(z)$ (4.5)

using diagram technique. The left diagram in Figure 2 is equivalent to the one
in Figure 1: we just divided the vertical line corresponding to the factor
$[2\gamma]^{s-x}$ into two parts

$[2\gamma]^{s-x}=[2\gamma]^{s+x}\,[2\gamma]^{-2x}.$ (4.6)

Using the reduced cross relation shown in Figure 8 we arrive at the second
diagram in Figure 2. Modulo coefficient with gamma functions, this diagram
depicts the function $\Psi_{-\bm{x}}(z)$. Restoring the gamma functions from
(4.4) and taking into account the gamma function factor coming from the
reduced cross relation (A.12) one obtains (4.5).

At last, we derive the second integral representation for the eigenfunctions.
Starting from the first representation (4.4) and using the same reduced cross
relation (A.12) two times we arrive at

$\displaystyle\begin{aligned}
\Psi_{\bm{x}}(z)&=[2\gamma]^{s-x}\,\mathbf{\Gamma}(g-x,1-s+x)\\\\[4.0pt]
&\times\int\frac{\mathrm{d}^{2}w}{[w-z]^{x+s}\,[w-\gamma]^{g-x}\,[w+\gamma]^{1-x-g}}.\end{aligned}$
(4.7)

The corresponding diagram transformations are shown in Figure 3.

Figure 3: Derivation of the second integral representation

###  Relation to hypergeometric function

By analogy with the $SL(2,\mathbb{R})$ spin chain [ABDK, Section 3], the one-
site eigenfunction (4.7) can be expressed in terms of the hypergeometric
function. In the $SL(2,\mathbb{C})$ case it is the hypergeometric function of
the complex field [GGR, § 2], [MN, Section 1.6]

${}_{2}F_{1}^{\mathbb{C}}\left[\\!\left.\begin{array}[]{c}a|\bar{a},\,b|\bar{b}\\\
c|\bar{c}\end{array}\right|v\right]=\frac{\mathbf{\Gamma}(c)}{\pi\,\mathbf{\Gamma}(b,c-b)}\int\mathrm{d}^{2}t\;\frac{[t]^{b-1}[1-t]^{c-b-1}}{[1-tv]^{a}},$
(4.8)

where $a-\bar{a},\,b-\bar{b},\,c-\bar{c}\in\mathbb{Z}$. It satisfies the pair
of hypergeometric differential equations [MN, Proposition 3.7]

$\displaystyle\Bigl{(}v(1-v)\partial_{v}^{2}+\bigl{[}c-(a+b+1)v\bigr{]}\partial_{v}-ab\Bigr{)}F(v)=0,$
(4.9)
$\displaystyle\Bigl{(}\bar{v}(1-\bar{v})\partial_{\bar{v}}^{2}+\bigl{[}\bar{c}-(\bar{a}+\bar{b}+1)\bar{v}\bigr{]}\partial_{\bar{v}}-\bar{a}\bar{b}\Bigr{)}F(v)=0,$
(4.10)

where $\bar{v}\equiv v^{*}$ is the complex conjugate of $v$. Notice that the
spectral problem for the operator $H^{s}$ (2.24)

$\displaystyle\Bigl{(}(z^{2}-\gamma^{2})\partial_{z}^{2}+(2s+1)z\partial_{z}+2\mathrm{i}\alpha\partial_{z}+s^{2}\Bigr{)}\Psi_{\bm{x}}(z)=x^{2}\,\Psi_{\bm{x}}(z)$
(4.11)

transforms into the first hypergeometric equation (4.9), if we change the
variable

$v=\frac{1}{2}-\frac{z}{2\gamma}$ (4.12)

and identify the parameters

$\displaystyle a=s+x,\qquad b=s-x,\qquad c=s+g,$ (4.13)

where as before $g=1/2+\mathrm{i}\alpha/\gamma$. Since
$\gamma^{*}=\bar{\gamma}$ (2.29), the same is true for the operator
$\bar{H}^{\bar{s}}$ (2.26) and the second hypergeometric equation (4.10) under
identification

$\bar{a}=\bar{s}+\bar{x},\qquad\bar{b}=\bar{s}-\bar{x},\qquad\bar{c}=\bar{s}+\bar{g},$
(4.14)

where $\bar{g}=1/2+\mathrm{i}\bar{\alpha}/\bar{\gamma}$.

Correspondingly, if we change the integration variable in the second integral
representation of the eigenfunctions (4.7)

$t=-\frac{2\gamma}{w-\gamma},\qquad\mathrm{d}^{2}w=\frac{[2\gamma]}{[t]^{2}}\,\mathrm{d}^{2}t,$
(4.15)

we arrive at the relation with the hypergeometric function defined above (4.8)

$\displaystyle\Psi_{\bm{x}}(z)$
$\displaystyle=\,\frac{\pi\,\mathbf{\Gamma}(g+x,g-x)}{\mathbf{\Gamma}(s+g)}$
$\displaystyle\times{}_{2}F_{1}^{\mathbb{C}}\left[\\!\left.\begin{array}[]{c}s+x|\bar{s}+\bar{x},\;s-x|\bar{s}-\bar{x}\\\\[3.0pt]
s+g|\bar{s}+\bar{g}\end{array}\right|\frac{1}{2}-\frac{z}{2\gamma}\right].$
(4.18)

This formula is similar to the expression for the one-site eigenfunction of
the open $SL(2,\mathbb{R})$ spin chain [ABDK, (1.32)] — parameters and the
argument of the hypergeometric function have the same form. Note also that the
symmetry of the hypergeometric function [MN, Proposition 3.3]

$\displaystyle{}_{2}F_{1}^{\mathbb{C}}\biggl{[}\\!\begin{array}[]{c}a|\bar{a},\,b|\bar{b}\\\
c|\bar{c}\end{array}\bigg{|}\,v\biggr{]}={}_{2}F_{1}^{\mathbb{C}}\biggl{[}\\!\begin{array}[]{c}b|\bar{b},\,a|\bar{a}\\\
c|\bar{c}\end{array}\bigg{|}\,v\biggr{]}$ (4.23)

corresponds to the reflection symmetry $\bm{x}\to-\bm{x}$ of the
eigenfunctions.

###  Mellin-Barnes representation

Denote for brevity the products of gamma functions

$\displaystyle\bm{\Gamma}(a\pm
b,c,d)=\bm{\Gamma}(a+b)\,\bm{\Gamma}(a-b)\,\bm{\Gamma}(c)\,\bm{\Gamma}(d).$
(4.24)

In this section we obtain one more representation for the eigenfunctions

$\Psi_{\bm{x}}(z)=\frac{1}{2}\int\mathcal{D}\bm{y}\;\frac{[-1]^{y-s}\;\mathbf{\Gamma}(\varepsilon-y\pm
x,\,1-s-\varepsilon+y,\,g-\varepsilon+y)}{[2\gamma]^{\varepsilon-
y-s}\,[z+\gamma]^{s-g}\,[z-\gamma]^{g-\varepsilon+y}}$ (4.25)

with arbitrary $\varepsilon\in(0,1/2)$. Here we assume that both the spectral
and integration variable have the form

$\displaystyle\bm{x}=(x,\bar{x})=\biggl{(}\frac{k}{2}+\mathrm{i}\eta,-\frac{k}{2}+\mathrm{i}\eta\biggr{)},$
$\displaystyle k\in\mathbb{Z}+\sigma,\quad\;\eta\in\mathbb{R},$ (4.26)
$\displaystyle\bm{y}=(y,\bar{y})=\biggl{(}\frac{m}{2}+\mathrm{i}\tau,-\frac{m}{2}+\mathrm{i}\tau\biggr{)},$
$\displaystyle m\in\mathbb{Z}+\sigma,\quad\tau\in\mathbb{R}$ (4.27)

and by integration over $\bm{y}$ we mean

$\displaystyle\int\mathcal{D}\bm{y}=\sum_{m\in\mathbb{Z}+\sigma}\,\int_{\mathbb{R}}\mathrm{d}\tau.$
(4.28)

Recall that the parameter $\sigma\in\\{0,1/2\\}$ is fixed from the very
beginning by the spins $(s,\bar{s})$ (2.8). Notice that the reflection
symmetry $\bm{x}\to-\bm{x}$ is evident from the above representation.

For the proof we need the formula [N, eq. (3.2)]

$\displaystyle\int\mathcal{D}\bm{y}\;\bm{\Gamma}(a+y,b-y)\,[z]^{-y-a}=2\pi\,\bm{\Gamma}(a+b)\,\frac{1}{[1+z]^{a+b}},$
(4.29)

where it is assumed that $a-\bar{a},\,b-\bar{b}\in\mathbb{Z}+\sigma$ and

$\displaystyle\operatorname{Re}(a+\bar{a})>0,\qquad\operatorname{Re}(b+\bar{b})>0,$
(4.30)

so the contour $\tau\in\mathbb{R}$ separates series of poles from gamma
functions.

To obtain the Mellin-Barnes representation (4.25) we start with the formula
for the eigenfunction (4.4)

$\displaystyle\begin{aligned}
&\Psi_{\bm{x}}(z)=[2\gamma]^{s-x}\,\mathbf{\Gamma}(g-x,1-s+x)\\\\[6.0pt]
&\quad\times\int\frac{\mathrm{d}^{2}w}{[z+\gamma]^{s-g}\,[z-\gamma]^{s+g-1}\,[w-z]^{x-s+1}\,[w+\gamma]^{g-x}\,[w-\gamma]^{1-x-g}}.\end{aligned}$
(4.31)

First, change the variable $z\to-(z-\gamma)/(w-\gamma)$ in the identity (4.29)
and with its help rewrite the power function

$\displaystyle\begin{aligned}
\frac{1}{[w-z]^{x-s+1}}&=\frac{1}{2\pi\,\mathbf{\Gamma}(x-s+1)}\\\\[6.0pt]
&\times\int\mathcal{D}\bm{y}\;\frac{[-1]^{y-s}\,\mathbf{\Gamma}(1-s-\varepsilon+y,\,x+\varepsilon-y)}{[z-\gamma]^{1-s-\varepsilon+y}\,[w-\gamma]^{x+\varepsilon-y}}.\end{aligned}$
(4.32)

Here we assume $\varepsilon\in(0,1/2)$ in accordance with (4.30) and the
parametrizations of $\bm{s},\bm{x}$ (2.8), (4.26). Then from (4.31) we obtain
the multiple integral

$\displaystyle\begin{aligned}
&\Psi_{\bm{x}}(z)=\frac{[2\gamma]^{s-x}\,\bm{\Gamma}(g-x)}{2\pi}\\\\[6.0pt]
&\;\times\int\mathrm{d}^{2}w\;\mathcal{D}\bm{y}\;\frac{[-1]^{y-s}\,\mathbf{\Gamma}(1-s-\varepsilon+y,\,x+\varepsilon-y)}{[z+\gamma]^{s-g}\,[z-\gamma]^{g+y-\varepsilon}\,[w+\gamma]^{g-x}\,[w-\gamma]^{1+\varepsilon-
g-y}}.\end{aligned}$ (4.33)

Integrating over $w$ using the chain relation (A.3)

$\int\mathrm{d}^{2}w\;\frac{1}{[w+\gamma]^{g-x}\,[w-\gamma]^{1+\varepsilon-
g-y}}\\\\[6.0pt] =\frac{\pi\,[-1]^{g-x}}{\bm{\Gamma}(1+\varepsilon-
g-y,\,g-x,\,1-\varepsilon+x+y)}\;\frac{1}{[2\gamma]^{\varepsilon-x-y}}$ (4.34)

we arrive at the desired formula (4.25).

##  Orthogonality

Recall the scalar product (2.1)

$\displaystyle\langle\Phi|\Psi\rangle=\int\mathrm{d}^{2}z\,\overline{\Phi(z)}\,\Psi(z),$
(5.1)

where integration is performed over the whole complex plane. In this section
we prove that the eigenfunctions $\Psi_{\bm{x}}(z)$ are orthogonal with
respect to this scalar product

$\langle\Psi_{\bm{y}}|\Psi_{\bm{x}}\rangle=2\pi^{4}\,\Bigl{|}\frac{\gamma}{x}\Bigr{|}^{2}\bigl{(}\delta(\bm{x}-\bm{y})+\delta(\bm{x}+\bm{y})\bigr{)},$
(5.2)

where the spectral parameters are of the type

$\displaystyle\bm{x}=(x,\bar{x})=\biggl{(}\frac{k}{2}+\mathrm{i}\eta,-\frac{k}{2}+\mathrm{i}\eta\biggr{)},$
$\displaystyle\quad k\in\mathbb{Z}+\sigma,\quad\;\eta\in\mathbb{R},$ (5.3)
$\displaystyle\bm{y}=(y,\bar{y})=\biggl{(}\frac{m}{2}+\mathrm{i}\tau,-\frac{m}{2}+\mathrm{i}\tau\biggr{)},$
$\displaystyle\quad m\in\mathbb{Z}+\sigma,\quad\tau\in\mathbb{R}$

and for brevity we denote

$\delta(\bm{x}-\bm{y})=\delta_{km}\,\delta(\eta-\tau).$ (5.4)

Note that the eigenfunctions $\Psi_{\bm{x}}$ possess the symmetry
$\bm{x}\to-\bm{x}$ (see Section 4.1). Therefore, the scalar product also must
be invariant under reflection $\bm{x}\to-\bm{x}$. So, it is sufficient to
prove the formula (5.2) in the sector $\bm{x}\neq-\bm{y}$, where the second
delta-function $\delta(\bm{x}+\bm{y})$ disappears, and at the end restore full
expression by symmetry.

For the proof we use the integral representation (4.4)

$\displaystyle\begin{aligned}
&\Psi_{\bm{x}}(z)=[2\gamma]^{s-x}\,\mathbf{\Gamma}\left(g-x,1-s+x\right)\\\\[4.0pt]
&\quad\times\int\frac{\mathrm{d}^{2}w}{[z+\gamma]^{s-g}\,[z-\gamma]^{s+g-1}\,[w-z]^{x-s+1}\,[w+\gamma]^{g-x}\,[w-\gamma]^{1-x-g}}\end{aligned}$
(5.5)

and diagram technique explained in Appendix A. Using the rule for complex
conjugation of double powers

$([z]^{a})^{\ast}=[z]^{\bar{a}^{\ast}}=z^{\bar{a}^{\ast}}\bar{z}^{a^{\ast}}$
(5.6)

and conditions on the parameters (2.7), (2.40)

$\displaystyle s^{*}+\bar{s}=1,\qquad g^{*}+\bar{g}=1$ (5.7)

one obtains the diagrammatic representation for the scalar product
$\langle\Psi_{\bm{y}}|\Psi_{\bm{x}}\rangle$ in the top left corner in Figure 4
(modulo coefficient behind the integral).

Figure 4: Derivation of the orthogonality relation

The first transformation in Figure 4 is the cancellation of lines connecting
the central vertex with $\gamma$ and $-\gamma$ and the use of chain relation
(A.3) depicted in Figure 7. On the next step we use the star-triangle relation
(A.10) also shown in Figure 7. In the last transformation we again use two
chain relations and arrive at the lower left corner:

$\int\mathrm{d}^{2}w\,[w-\gamma]^{-1+\bar{y}^{\ast}+x}=\int\mathrm{d}^{2}w\,[w]^{-1+\bar{y}^{\ast}+x}.$
(5.8)

For $\bm{x}$ and $\bm{y}$ of the form (5.3) the integral (5.8) can be easily
calculated in polar coordinates $w=re^{i\varphi}$, which gives

$\int\mathrm{d}^{2}w\,[w]^{-1+\bar{y}^{\ast}+x}=2\pi^{2}\,\delta_{km}\,\delta(\eta-\tau)=2\pi^{2}\,\delta(\bm{x}-\bm{y}).$
(5.9)

Taking into account the factors with gamma functions appearing during diagram
transformations and using the reflection formula (A.9)

$\mathbf{\Gamma}(a)\,\mathbf{\Gamma}(1-a)=[-1]^{a}=(-1)^{a-\bar{a}}$ (5.10)

we obtain the expression for the scalar product

$\langle\Psi_{\bm{y}}|\Psi_{\bm{x}}\rangle=\frac{2\pi^{4}\,[-1]^{x+y}\,[2\gamma]}{\mathbf{\Gamma}(1+x+y)\,\mathbf{\Gamma}(1-x-y)}\,\delta(\bm{x}-\bm{y}).$
(5.11)

The complex gamma function satisfies the difference equation (A.7)

$\mathbf{\Gamma}(a+1)=-a\bar{a}\,\mathbf{\Gamma}(a).$ (5.12)

Using it together with the reflection formula (5.10) we rewrite (5.11) as

$\displaystyle\langle\Psi_{\bm{y}}|\Psi_{\bm{x}}\rangle$
$\displaystyle=-\frac{2\pi^{4}\,[2\gamma]\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}{(x+y)(\bar{x}+\bar{y})}\,\delta(\bm{x}-\bm{y})$
(5.13)
$\displaystyle=2\pi^{4}\,\Bigl{|}\frac{\gamma}{x}\Bigr{|}^{2}\,\delta(\bm{x}-\bm{y}).$
(5.14)

Derivation of (5.14) is based on the tacit assumption $\bm{x}\neq\bm{y}$, only
in this case all steps in Figure 4 have sense. This can be clearly seen from
the expression in front of delta-function in (5.13). This expression collects
all factors appearing in intermediate steps of calculation and it is evidently
singular at the point $\bm{x}=-\bm{y}$.

Removing the condition $\bm{x}\neq-\bm{y}$ and restoring the whole answer from
(5.14) by symmetry we obtain the orthogonality relation (5.2).

##  Completeness

In this section we prove the completeness relation

$\int\mathcal{D}\bm{x}\;\mu(\bm{x})\,\Psi_{\bm{x}}(z)\,\overline{\Psi_{\bm{x}}(w)}=\delta^{2}(z-w),$
(6.1)

where as before we denote

$\bm{x}=(x,\bar{x})=\biggl{(}\frac{k}{2}+\mathrm{i}\eta,-\frac{k}{2}+\mathrm{i}\eta\biggr{)},\qquad\int\mathcal{D}\bm{x}=\sum\limits_{k\in\mathbb{Z}+\sigma}\,\int_{\mathbb{R}}\mathrm{d}\eta$
(6.2)

and $\delta^{2}(z)$ is the two-dimensional delta function in the complex plane

$\delta^{2}(z)=\delta(\operatorname{Re}z)\,\delta(\operatorname{Im}z).$ (6.3)

The integration measure

$\mu(\bm{x})=\frac{1}{4\pi^{4}}\,\Bigl{|}\frac{x}{\gamma}\Bigr{|}^{2}$ (6.4)

follows from the orthogonality relation (5.2) and it is useful to rewrite it
in equivalent form

$\mu(\bm{x})=\frac{\bm{\Gamma}(s\pm x,\,1-s\pm
x)}{4\pi^{4}\,[2\gamma]\,\mathbf{\Gamma}(\pm 2x)}$ (6.5)

with the help of difference equation (A.7) and reflection formula (A.9) for
the gamma function.

To prove the completeness relation (6.1) we use Mellin-Barnes representation
for the eigenfunctions (4.25)

$\Psi_{\bm{x}}(z)=\frac{1}{2}\,\int\mathcal{D}\bm{y}\;\frac{[-1]^{y-s}\,\mathbf{\Gamma}(\varepsilon-y\pm
x,\,1-s-\varepsilon+y,\,g-\varepsilon+y)}{[2\gamma]^{\varepsilon-
y-s}\,[z+\gamma]^{s-g}\,[z-\gamma]^{g-\varepsilon+y}},$ (6.6)

where $\varepsilon\in(0,1/2)$ and the integration variable is of the type

$\displaystyle\bm{y}=(y,\bar{y})=\biggl{(}\frac{m}{2}+\mathrm{i}\tau,-\frac{m}{2}+\mathrm{i}\tau\biggr{)},\qquad
m\in\mathbb{Z}+\sigma,\quad\tau\in\mathbb{R}.$ (6.7)

In what follows we treat $\varepsilon$ as the regularization parameter and at
the end of the calculation take the limit $\varepsilon\to 0_{+}$.

To write down the complex conjugated eigenfunction
$\overline{\Psi_{\bm{x}}(w)}$ recall the conditions on the parameters (2.7),
(2.40)

$\displaystyle s^{*}+\bar{s}=1,\qquad g^{*}+\bar{g}=1$ (6.8)

and how the double powers (2.5) are conjugated

$\displaystyle([z]^{a})^{*}=[z]^{\bar{a}^{*}}.$ (6.9)

Note also that assuming $a,\bar{a}$ are of the form

$a=\frac{n}{2}+\mathrm{i}\nu,\qquad\bar{a}=-\frac{n}{2}+\mathrm{i}\nu,\qquad
n\in\mathbb{Z},\quad\nu\in\mathbb{R}$ (6.10)

from the definition of the gamma function (A.4) and the formula (A.8) we have
complex conjugation rule

$\mathbf{\Gamma}(\rho+a)^{\ast}=\bm{\Gamma}(\rho-\bar{a})=[-1]^{a}\,\mathbf{\Gamma}(\rho-a),\qquad\rho\in\mathbb{R}.$
(6.11)

Due to all of the above remarks we have

$\displaystyle\overline{\Psi_{\bm{x}}(w)}=\frac{1}{2}\,\int\mathcal{D}\bm{y}^{\prime}\,\frac{[-1]^{y^{\prime}-g}\,\mathbf{\Gamma}(\varepsilon+y^{\prime}\pm
x,\,s-\varepsilon-y^{\prime},\,1-g-\varepsilon-y^{\prime})}{[2\gamma]^{\varepsilon+y^{\prime}+s-1}\,[w+\gamma]^{g-s}\,[w-\gamma]^{1-g-\varepsilon-y^{\prime}}},$
(6.12)

where integration variable is again of the type

$\bm{y}^{\prime}=(y^{\prime},\bar{y}^{\prime})=\biggl{(}\frac{m^{\prime}}{2}+\mathrm{i}\tau^{\prime},-\frac{m^{\prime}}{2}+\mathrm{i}\tau^{\prime}\biggr{)},\qquad
m^{\prime}\in\mathbb{Z}+\sigma,\quad\tau^{\prime}\in\mathbb{R}.$ (6.13)

Substituting the explicit formulas (6.5), (6.6) and (6.12) into the left hand
side of the completeness relation (6.1) and taking limit $\varepsilon\to
0_{+}$ one obtains the multiple integral

$\displaystyle\begin{aligned}
&\int\mathcal{D}\bm{x}\;\mu(\bm{x})\,\Psi_{\bm{x}}(z)\,\overline{\Psi_{\bm{x}}(w)}\\\\[6.0pt]
&\quad=\frac{1}{16\pi^{4}}\;\lim\limits_{\varepsilon\to
0_{+}}\,\int\mathcal{D}\bm{y}\;\frac{[-1]^{y-s}\;\mathbf{\Gamma}(1-s-\varepsilon+y,\,g-\varepsilon+y)}{[z+\gamma]^{s-g}\,[z-\gamma]^{g-\varepsilon+y}}\\\\[10.0pt]
&\quad\times\int\mathcal{D}\bm{y}^{\prime}\;\frac{[-1]^{y^{\prime}-g}\;\mathbf{\Gamma}(s-\varepsilon-y^{\prime},\,1-g-\varepsilon-y^{\prime})}{[2\gamma]^{2\varepsilon-y+y^{\prime}}\,[w+\gamma]^{g-s}\,[w-\gamma]^{1-\varepsilon-
g-y^{\prime}}}\\\\[10.0pt]
&\quad\times\int\mathcal{D}\bm{x}\;\frac{\mathbf{\Gamma}(s\pm
x,\,1-s-4\varepsilon\pm x,\,\varepsilon-y\pm x,\,\varepsilon+y^{\prime}\pm
x)}{\mathbf{\Gamma}(\pm 2x)},\end{aligned}$ (6.14)

where in the last $\bm{x}$-integral we introduced additional regularization

$\bm{\Gamma}(1-s\pm x)=\lim\limits_{\varepsilon\to
0_{+}}\bm{\Gamma}(1-s-4\varepsilon\pm x).$ (6.15)

In order to integrate over $\bm{x}$ we need the complex analog of
$\mathrm{sp}(1)$ Gustafson integral [DM3, eq. (2.3b)], [SS, eq. (65)]

$\int\mathcal{D}\bm{x}\;\frac{\prod\limits_{j=1}^{4}\bm{\Gamma}(a_{j}\pm
x)}{\mathbf{\Gamma}(\pm 2x)}=\frac{4\pi\prod\limits_{1\leq j<k\leq
4}\mathbf{\Gamma}(a_{j}+a_{k})}{\mathbf{\Gamma}(a_{1}+a_{2}+a_{3}+a_{4})},$
(6.16)

where the integration variable is of the same type as before (6.2) and the
parameters $a_{j},\bar{a}_{j}$ are of the form

$\displaystyle
a_{j}=\frac{n_{j}}{2}+\nu_{j},\qquad\bar{a}_{j}=-\frac{n_{j}}{2}+\nu_{j},\qquad
n_{j}\in\mathbb{Z}+\sigma,\quad\nu_{j}\in\mathbb{C}.$ (6.17)

The identity (6.16) holds if the contour of integration $\eta\in\mathbb{R}$
separates the series of poles [DM3]

$\mathrm{i}\eta=\frac{|k-n_{j}|}{2}+\nu_{j}+p,\qquad\mathrm{i}\eta=-\frac{|k+n_{j}|}{2}-\nu_{j}-p,\qquad
p\in\mathbb{Z}_{\geq 0}$ (6.18)

and $\nu_{j}$ obey the following inequality

$\operatorname{Re}(\nu_{1}+\nu_{2}+\nu_{3}+\nu_{4})<1.$ (6.19)

In our case this condition is satisfied since in the $\bm{x}$-integral (6.14)
we have

$\displaystyle a_{1}=s,\quad a_{2}=1-s-4\varepsilon,\quad
a_{3}=\varepsilon-y,\quad a_{4}=\varepsilon+y^{\prime},$ (6.20)

so that according to (6.17)

$\displaystyle\nu_{1}=\frac{1}{2}+\mathrm{i}\nu_{s},\quad\nu_{2}=\frac{1}{2}-\mathrm{i}\nu_{s}-4\varepsilon,\quad\nu_{3}=\varepsilon-\mathrm{i}\tau,\quad\nu_{4}=\varepsilon+\mathrm{i}\tau^{\prime},$
(6.21)

and the needed inequality holds for small enough $\varepsilon>0$

$\operatorname{Re}(\nu_{1}+\nu_{2}+\nu_{3}+\nu_{4})=1-2\varepsilon<1.$ (6.22)

Hence, integrating over $\bm{x}$ with the help of the Gustafson integral
(6.16) and using reflection formula for the gamma functions (A.9) we obtain

$\displaystyle\begin{aligned}
&\int\mathcal{D}\bm{x}\;\frac{\mathbf{\Gamma}(s\pm x,\,1-s-4\varepsilon\pm
x,\,\varepsilon-y\pm x,\,\varepsilon+y^{\prime}\pm
x\bigr{)}}{\mathbf{\Gamma}(\pm 2x)}\\\\[4.0pt]
&\quad=4\pi\,[-1]^{y-y^{\prime}}\,\mathbf{\Gamma}\bigl{(}1-4\varepsilon,\,2\varepsilon\pm(y-y^{\prime})\bigr{)}\\\\[6.0pt]
&\quad\times\mathbf{\Gamma}(s+\varepsilon-y,\,s+\varepsilon+y^{\prime},\,1-s-3\varepsilon-y,\,1-s-3\varepsilon+y^{\prime}).\end{aligned}$
(6.23)

The next step is to rewrite three gamma functions from the second line

$\mathbf{\Gamma}\bigl{(}1-4\varepsilon,\,2\varepsilon\pm(y-y^{\prime})\bigr{)}\\\\[6.0pt]
=\frac{4\varepsilon}{(2\varepsilon+y-y^{\prime})(2\varepsilon-y+y^{\prime})}\,\frac{\Gamma(1-4\varepsilon)}{\Gamma(1+4\varepsilon)}\,\mathbf{\Gamma}\bigl{(}2\varepsilon+\Delta\pm(y-y^{\prime})\bigr{)}$
(6.24)

using the pair of variables $\Delta=1$, $\bar{\Delta}=0$. Let us prove that
the first fraction in (6.24) produces the delta-function

$\displaystyle\lim_{\varepsilon\to
0_{+}}\frac{4\varepsilon}{(2\varepsilon+y-y^{\prime})(2\varepsilon-y+y^{\prime})}=2\pi\,\delta_{m\,m^{\prime}}\,\delta(\tau-\tau^{\prime})=2\pi\,\delta(\bm{y}-\bm{y}^{\prime}).$
(6.25)

First, we decompose it into the sum

$\displaystyle\frac{4\varepsilon}{(2\varepsilon+y-y^{\prime})(2\varepsilon-y+y^{\prime})}=\frac{1}{2\varepsilon+y-y^{\prime}}+\frac{1}{2\varepsilon-y+y^{\prime}}$
$\displaystyle\quad=-\frac{\mathrm{i}}{-\mathrm{i}\frac{m-m^{\prime}}{2}+\tau-\tau^{\prime}-2\mathrm{i}\varepsilon}+\frac{\mathrm{i}}{-\mathrm{i}\frac{m-m^{\prime}}{2}+\tau-\tau^{\prime}+2\mathrm{i}\varepsilon}.$
(6.26)

If $m\neq m^{\prime}$ then in the limit $\varepsilon\to 0_{+}$ both summands
in (6.26) do not contain singularities in $\tau,\tau^{\prime}\in\mathbb{R}$
and cancel each other. In the remaining case $m=m^{\prime}$ we use
Sokhotski–Plemelj theorem

$\lim\limits_{\varepsilon\to
0_{+}}\left(\frac{1}{\tau-\mathrm{i}\varepsilon}-\frac{1}{\tau+\mathrm{i}\varepsilon}\right)=2\pi\mathrm{i}\,\delta(\tau).$
(6.27)

Hence, the result for both cases is given by the formula (6.25).

The rest part of the right hand side in (6.24) does not contain poles on the
contours of integration over $\tau$ and $\tau^{\prime}$ in the limit
$\varepsilon\to 0_{+}$, as well as the remaining gamma functions in the right
hand side of (6.23) and in (6.14).

Therefore, putting (6.23), (6.24), (6.25) together, integrating the delta-
function and using reflection formula for the gamma functions (A.9) we derive
from (6.14) the following expression

$\displaystyle\begin{aligned}
&\int\mathcal{D}\bm{x}\;\mu(\bm{x})\,\Psi_{\bm{x}}(z)\,\overline{\Psi_{\bm{x}}(w)}\\\
&\qquad=\frac{1}{2\pi^{2}}\frac{[z+\gamma]^{g-s}\,[w+\gamma]^{s-g}}{|w-\gamma|^{2}}\int\mathcal{D}\bm{y}\,\left[\frac{w-\gamma}{z-\gamma}\right]^{y+g}.\end{aligned}$
(6.28)

The last step is to use the relation

$\displaystyle\int\mathcal{D}\bm{y}\,\left[\frac{v}{u}\right]^{y+g}=2\pi^{2}\,|v|^{2}\,\delta^{2}(u-v),$
(6.29)

which can be easily proved in polar coordinates $v/u=r\mathrm{e}^{i\varphi}$.
After that we obtain the answer for the completeness relation (6.1).

##  Reflection equations

The boundary $K$-matrix

$K(u)=\begin{pmatrix}\mathrm{i}\alpha&u-\frac{1}{2}\\\\[6.0pt]
\gamma^{2}\bigl{(}u-\frac{1}{2}\bigr{)}&\mathrm{i}\alpha\end{pmatrix}$ (7.1)

satisfies the reflection equation

$\bigl{(}\bm{1}\otimes
K(v)\bigr{)}\,R(u+v-1)\,\bigl{(}K(u)\otimes\bm{1}\bigr{)}\,R(u-v)\\\\[6.0pt]
=R(u-v)\,\bigl{(}K(u)\otimes\bm{1}\bigr{)}\,R(u+v-1)\,\bigl{(}\bm{1}\otimes
K(v)\bigr{)}$ (7.2)

with the rational $R$-matrix (2.20) acting in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$

$\displaystyle R(u)=u+P,\qquad P\,a\otimes b=b\otimes a.$ (7.3)

This equation is usually depicted as in Figure 5 that we read from left to
right: each reflection of lines from the “floor” corresponds to the $K$-matrix
and each intersection of lines corresponds to the $R$-matrix. The numbers
$1,2$ correspond to different matrix spaces $\mathbb{C}^{2}$ from the tensor
product.

$1$$2$

$=$$1$$2$

Figure 5: Reflection equation

Defining equation

$\displaystyle\begin{aligned}
&\mathcal{K}(\bm{s},\bm{x})\,L(u+x-1,u-s)\,K(u)\,L(u+s-1,u-x)\\\\[6.0pt]
&=L(u+s-1,u-x)\,K(u)\,L(u+x-1,u-s)\,\mathcal{K}(\bm{s},\bm{x})\end{aligned}$
(7.4)

for the reflection operator

$\displaystyle\mathcal{K}(\bm{s},\bm{x})$
$\displaystyle=[z+\gamma]^{g-s}\,[z-\gamma]^{1-s-g}\,[\partial_{z}]^{x-s}\,[z+\gamma]^{x-g}\,[z-\gamma]^{x+g-1}$
(7.5)

also can be pictured by Figure 5 with the first line “carrying” infinite-
dimensional space of functions $\psi(z)$, on which reflection operator and
elements of Lax matrices act.

It is natural to ask, whether there is a variant of reflection equation with
the same reflection operator and both spaces being infinite-dimensional. The
answer is given by the following identity

$\displaystyle\begin{aligned}
&\mathcal{K}_{1}(\bm{x})\;\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})\;\mathcal{K}_{2}(\bm{y})\;{\mathbb{R}}_{12}(\bm{x},\bm{y})\\\\[6.0pt]
&\quad\;\;={\mathbb{R}}_{12}(\bm{x},\bm{y})\;\mathcal{K}_{2}(\bm{y})\;\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})\;\mathcal{K}_{1}(\bm{x}).\end{aligned}$
(7.6)

Here the lower indices mean spaces of functions of $z_{1},z_{2}$ and in all
operators we suppressed the dependence on spin $\bm{s}$. The
$\mathbb{R}$-operators are defined as

$\displaystyle{\mathbb{R}}_{12}(\bm{x},\bm{y})=\mathcal{P}_{12}\;[z_{12}]^{1-s-x}\,[\partial_{z_{1}}]^{y-x}\,[z_{12}]^{s+y-1},$
(7.7)
$\displaystyle\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})=\mathcal{P}_{12}\;[z_{12}]^{1-2s}\,[\partial_{z_{2}}]^{x-s}\,[\partial_{z_{1}}]^{y-s}\,[z_{12}]^{x+y-1}$
(7.8)

where $z_{12}\equiv z_{1}-z_{2}$ and $\mathcal{P}_{12}$ is the permutation
operator

$\displaystyle\mathcal{P}_{12}\,\psi(z_{1},z_{2})=\psi(z_{2},z_{1}).$ (7.9)

Notice that for the two previous equations (7.2) and (7.4) we have
antiholomorphic counterparts, whereas in the last one (7.6) both holomorphic
and antiholomorphic variables come into play.

The proof of the identity (7.6) is long, but straightforward: one just uses
many times chain and star-triangle relations (A.3), (A.10), which can be
written in terms of operators as

$\displaystyle[\partial_{z}]^{a}\,[\partial_{z}]^{b}=[\partial_{z}]^{a+b},$
(7.10)
$\displaystyle[z+w]^{a}\,[\partial_{z}]^{a+b}\,[z+w]^{b}=[\partial_{z}]^{b}\,[z+w]^{a+b}\,[\partial_{z}]^{a}.$
(7.11)

Now let us explain how the identity (7.6) can be guessed and where do such
$\mathbb{R}$-operators come from. The operators
${\mathbb{R}}_{12}(\bm{x},\bm{y})$ and
$\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})$ are special cases of the more
general $\mathbb{R}$-operator [DM1, Section 3.1]

$\displaystyle\widehat{\mathbb{R}}_{12}(\bm{u}_{1},\bm{u}_{2}|\bm{v}_{1},\bm{v}_{2})=\mathcal{P}_{12}\;[z_{12}]^{u_{2}-v_{1}}\,[\partial_{z_{2}}]^{u_{1}-v_{1}}\,[\partial_{z_{1}}]^{u_{2}-v_{2}}\,[z_{12}]^{u_{1}-v_{2}}$
(7.12)

that permutes two Lax matrices

$\displaystyle\widehat{\mathbb{R}}_{12}\,L_{1}(u_{1},u_{2})\,L_{2}(v_{1},v_{2})=L_{2}(v_{1},v_{2})\,L_{1}(u_{1},u_{2})\,\widehat{\mathbb{R}}_{12}$
(7.13)

and their antiholomorphic counterparts. Indeed, from the above definitions we
have

$\displaystyle{\mathbb{R}}_{12}(\bm{x},\bm{y})=\widehat{\mathbb{R}}_{12}(\bm{u}+\bm{s}-\bm{1},\,\bm{u}-\bm{x}\,|\,\bm{u}+\bm{s}-\bm{1},\,\bm{u}-\bm{y}),$
(7.14)
$\displaystyle\widetilde{\mathbb{R}}_{12}(\bm{x},\bm{y})=\widehat{\mathbb{R}}_{12}(\bm{u}+\bm{x}-\bm{1},\,\bm{u}-\bm{s}\,|\,\bm{u}+\bm{s}-\bm{1},\,\bm{u}-\bm{y})$
(7.15)

where we denoted $\bm{1}=(1,1)$. Using these operators and reflection operator
satisfying (7.4) one can rearrange the product of matrices

$\displaystyle\begin{aligned}
&L_{2}(u+y-1,u-s)\;L_{1}(u+x-1,u-s)\;K(u)\\\\[6.0pt] &\hskip 56.9055pt\times
L_{1}(u+s-1,u-x)\;L_{2}(u+s-1,u-y)\\\\[10.0pt] &\to\quad
L_{2}(u+s-1,u-y)\;L_{1}(u+s-1,u-x)\;K(u)\\\\[6.0pt] &\hskip 56.9055pt\times
L_{1}(u+x-1,u-s)\;L_{2}(u+y-1,u-s)\end{aligned}$ (7.16)

in two different ways, which correspond to the left and right hand sides of
the identity (7.6). Hence, we guess that operators from opposite sides of
(7.6) are equal.

## Acknowledgements

The work of P. Antonenko and S. Derkachov (Sections 3–6, Appendix A) was
supported by the Russian Science Foundation under grant No. 23-11-00311. The
work of N. Belousov (Sections 2, 7) was supported by the Russian Science
Foundation under grant No. 24-21-00466.

##  Appendix A Diagram technique

Figure 6: Diagrammatic representation of $[z-w]^{-a}$

In the paper functions and scalar products are represented by diagrams. They
consist of directed lines (Figure 6) that correspond to the function

$\frac{1}{[z-w]^{a}}\equiv\frac{1}{(z-w)^{a}(\bar{z}-\bar{w})^{\bar{a}}}=\frac{(\bar{z}-\bar{w})^{a-\bar{a}}}{|z-w|^{2a}},$
(A.1)

where we always assume $a-\bar{a}\in\mathbb{Z}$. The flip of the arrow gives
an additional sign factor

$\frac{1}{[z-w]^{a}}=\frac{(-1)^{a-\bar{a}}}{[w-z]^{a}}=\frac{[-1]^{a}}{[w-z]^{a}}.$
(A.2)

Figure 7: The chain and star-triangle relations, $a+b+c=2$

There are several useful integral identities, which correspond to
transformations of diagrams. The proofs can be found in [DKM, Appendix A]. The
first identity is the chain relation depicted in Figure 7

$\int\mathrm{d}^{2}w\,\frac{1}{[z_{1}-w]^{a}\,[w-z_{2}]^{b}}=\frac{\pi\,[-1]^{c}}{\mathbf{\Gamma}(a,b,c)}\frac{1}{[z_{1}-z_{2}]^{a+b-1}},$
(A.3)

where $c=2-a-b,\ \bar{c}=2-\bar{a}-\bar{b}$. The bold vertex corresponds to
the integration over $\mathbb{C}$. From the right we have gamma function of
the complex field [GGR, Section 1.4], [N, Section 1.3]

$\bm{\Gamma}(a)=\frac{\Gamma(a)}{\Gamma(1-\bar{a})}$ (A.4)

and we denote its products as

$\displaystyle\bm{\Gamma}(a,b)=\bm{\Gamma}(a)\bm{\Gamma}(b).$ (A.5)

Notice that $\bm{\Gamma}(a)$ depends on two parameters
$(a,\bar{a})\in\mathbb{C}^{2}$ such that $a-\bar{a}\in\mathbb{Z}$, but for
brevity we display only the first one. Moreover, for $\rho\in\mathbb{R}$ we
write

$\displaystyle\bm{\Gamma}(a+\rho)\equiv\frac{\Gamma(a+\rho)}{\Gamma(1-\bar{a}-\rho)}.$
(A.6)

From the well-known properties of the ordinary gamma function it is easy to
prove the following relations

$\displaystyle\bm{\Gamma}(a+1)=-a\bar{a}\,\bm{\Gamma}(a),$ (A.7)
$\displaystyle\bm{\Gamma}(a)=[-1]^{a}\,\bm{\Gamma}(\bar{a}),$ (A.8)
$\displaystyle\bm{\Gamma}(a)\,\bm{\Gamma}(1-a)=[-1]^{a}.$ (A.9)

The second diagram identity is the star-triangle relation shown in Figure 7

$\displaystyle\begin{aligned}
&\int\mathrm{d}^{2}w\,\frac{1}{[w-z_{1}]^{a}[w-z_{2}]^{b}[w-z_{3}]^{c}}\\\\[6.0pt]
&=\frac{\pi}{\mathbf{\Gamma}(a,b,c)}\frac{1}{[z_{1}-z_{2}]^{1-c}[z_{3}-z_{1}]^{1-b}[z_{2}-z_{3}]^{1-a}}\,,\end{aligned}$
(A.10)

where again $a+b+c=2$ and $\bar{a}+\bar{b}+\bar{c}=2$.

From the star-triangle relation one can deduce the so-called cross relation

$\displaystyle\begin{aligned}
&\frac{1}{[z_{2}-z_{1}]^{a^{\prime}-a}}\int\mathrm{d}^{2}w\,\frac{\mathbf{\Gamma}(a,\bar{b})}{[w-z_{1}]^{a}[w-z_{2}]^{1-a^{\prime}}\,[w-z_{3}]^{b}\,[w-z_{4}]^{1-b^{\prime}}}\\\\[6.0pt]
&=\frac{1}{[z_{4}-z_{3}]^{b-b^{\prime}}}\int\mathrm{d}^{2}w\,\frac{\mathbf{\Gamma}(a^{\prime},\bar{b}^{\prime})}{[w-z_{1}]^{a^{\prime}}\,[w-z_{2}]^{1-a}\,[w-z_{3}]^{b^{\prime}}\,[w-z_{4}]^{1-b}},\end{aligned}$
(A.11)

depicted in Figure 8, where we assume $a+b=a^{\prime}+b^{\prime}$,
$\bar{a}+\bar{b}=\bar{a}^{\prime}+\bar{b}^{\prime}$. Its limiting case gives
the reduced cross relation

$\displaystyle\begin{aligned}
\frac{1}{[z_{2}-z_{1}]^{a^{\prime}-a}}&\int\mathrm{d}^{2}w\,\frac{\mathbf{\Gamma}(a,b)}{[w-z_{1}]^{a}\,[w-z_{2}]^{1-a^{\prime}}\,[w-z_{3}]^{b}}\\\\[6.0pt]
=&\int\mathrm{d}^{2}w\,\frac{\mathbf{\Gamma}(a^{\prime},b^{\prime})}{[w-z_{1}]^{a^{\prime}}\,[w-z_{2}]^{1-a}\,[w-z_{3}]^{b^{\prime}}},\end{aligned}$
(A.12)

where again $a+b=a^{\prime}+b^{\prime}$,
$\bar{a}+\bar{b}=\bar{a}^{\prime}+\bar{b}^{\prime}$.

Figure 8: Cross and reduced cross relations, $a+b=a^{\prime}+b^{\prime}$

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