Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains
We prove the unitarity of the separation of variables transform for SL(2, ℂ) spin chains by a method based on the use of Gustafson integrals.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 086, 24 pages
Unitarity of the SoV Transform
for SL(2,C) Spin Chains
Alexander N. MANASHOV
Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, 80805 München, Germany
E-mail: alexander.manashov@desy.de
Received March 30, 2023, in final form October 20, 2023; Published online November 04, 2023
https://doi.org/10.3842/SIGMA.2023.086
Abstract. We prove the unitarity of the separation of variables transform for SL(2,C) spin
chains by a method based on the use of Gustafson integrals.
Key words: spin chains; separation of variables; Gustafson’s integrals
2020 Mathematics Subject Classification: 33C70; 81R12
1 Introduction
Theory of quantum integrable models is an important part of modern theoretical physics.
The solution of such models relies on the Quantum inverse scattering method (QISM) which
includes such techniques as the algebraic Bethe ansatz (ABA) [50] and separation of variables
(SoV) [48, 49]. The ABA allows one to effectively calculate energies and eigenstates of inte-
grable models and to address more complicated problems such as calculating norms [32], scalar
products [51] and correlation functions [28, 31]. Models with infinite-dimensional Hilbert spaces
without a pseudo-vacuum state, the Toda chain [27] being the most famous example, are, how-
ever, beyond ABA’s grasp. The solution of such models relies on the SoV method proposed
by Sklyanin [48, 49]. The method consists in constructing a map between the original Hilbert
space, Horg, in which the model is formulated, and an auxiliary Hilbert space, HSoV. This map is
constructed in such a way that a multidimensional spectral problem associated with the original
Hamiltonian is reduced to a one-dimensional problem on an auxiliary Hilbert space which usually
takes the form of the Baxter T -Q relation. Technically constructing the SoV representation is
equivalent to finding the eigenfunctions of an element of the monodromy matrix associated with
the model. For the Toda chain it was done by Kharchev and Lebedev [29, 30]. Later, a regular
method for obtaining eigenfunctions for models with an R-matrix of the rank one1 was developed
in [9], and at present the SoV representation is known for a number of models [2, 10, 11, 14, 47].
In order to be sure that the spectral problems in the original and auxiliary Hilbert spaces are
equivalent, it is necessary to show that the corresponding map, HSoV 7→ Horg, is unitary (or that
the eigenfunctions form a complete set in Horg). If dimHorg < ∞ the problem can be solved,
at least in principle, by counting the dimensions of the Hilbert spaces. For the models with
infinite-dimensional Hilbert space, such as the Toda chain, the noncompact SL(2,C) spin chain,
etc., the task becomes more difficult. For the Toda chain, unitarity was first established by using
harmonic analysis of Lie groups techniques [46, 54]. However, this method is quite sophisticated
and can hardly be generalized to more complicated cases. The rigorous proof of the unitarity of
the SoV transform for the Toda chain based on the use of natural objects for the QISM was given
by Kozlowski [33]. This technique was later applied to the modular XXZ magnet [12]. Later
it was realized [15] that there exists a close relation between SL(2,R) symmetric spin chains
1In recent years, significant progress has been made in constructing SoV representations for higher rank finite-
dimensional models, see [3, 17, 22, 23, 24, 38, 39, 40, 41, 42, 43, 44, 53].
mailto:alexander.manashov@desy.de
https://doi.org/10.3842/SIGMA.2023.086
2 A.N. Manashov
and the multidimensional Mellin–Barnes integrals studied by Gustafson [25, 26] that allowed
to greatly simplify the proof of the unitarity of the SoV transform for SL(2,R) symmetric spin
chains [13].
In the present paper, we apply this technique to the analysis of the noncompact spin chains
with the SL(2,C) symmetry group. Such models appear in the studies of the Regge limit of
scattering amplitudes in gauge theories, in QCD in particular [1, 19, 35, 36, 37], see also [4, 5, 6, 7]
for recent developments. The SoV representation for the SL(2,C) spin chains2 was constructed
in [9] while the generalization of Gustafson integrals relevant for the SL(2,C) spin chains was
obtained recently in [16, 45]. Based on these results, we present below a proof of unitarity of
the SoV transform for a generic SL(2,C) spin chain.
The paper is organized as follows. In Section 2, we recall elements of the QISM relevant for
further analysis. The eigenfunctions of the elements of the monodromy matrix are constructed
in Section 3. In Section 4, we calculate several scalar products of the eigenfunctions and discuss
their properties. Section 5 contains the proof of unitarity of the SoV transform. Section 6 is
reserved for a summary and several appendices contain a discussion of technical details.
2 SL(2,C) spin chains
Spin chains are quantum mechanical systems whose dynamical variables are spin generators.
We consider models with spin generators belonging to the unitary continuous principal series
representation, T(sk,s̄k), of the unimodular group of complex two by two matrices. Namely, each
site of the chain is equipped with two sets of generators, holomorphic (Sα) and anti-holomorphic
ones
(
S̄α
)
,
S−
k = −∂zk , S0
k = zk∂zk + sk, S+
k = z2k∂zk + 2skzk,
S̄−
k = −∂z̄k , S̄0
k = z̄k∂z̄k + s̄k, S̄+
k = z̄2k∂z̄k + 2s̄kz̄k.
The generators Sα
k
(
S̄α
k
)
satisfy the standard sl(2) commutation relations, while the generators
at different sites and holomorphic and anti-holomorphic generators commute,
[
Sα
k , S̄
α′
k
]
= 0.
The parameters sk, s̄k specifying the representation take the form [21]
sk =
1 + nk
2
+ iρk, s̄k =
1− nk
2
+ iρk,
where nk is an integer or half-integer number and ρk is real, so that
sk + s̄∗k = 1 and sk − s̄k = nk ∈ Z/2.
The later condition comes from the requirement for the finite group transformations to be well
defined while the former one guarantees the unitary character of transformations and anti-
hermiticity of the generators,
(
Sα
k
)†
= −S̄α
k .
The Hilbert space of the model is given by the direct product of the Hilbert spaces at each
node. For a chain of length N , HN =
⊗N
k=1Hk, where Hk = L2(C).
In the QISM [34, 49, 50, 52], the dynamics of the model is determined by a family of mutually
commuting operators. Namely, one defines the so-called L-operators,
Lk(u) = u+ i
(
S0
k S−
k
S+
k −S−
0
)
, L̄k(ū) = ū+ i
(
S̄0
k S̄−
k
S̄+
k −S̄−
0
)
,
2To the best of our knowledge, the completeness of this representation has not yet been addressed.
Unitarity of the SoV Transform for SL(2,C) Spin Chains 3
which are the basic building blocks in the QISM. The complex variables u, ū are called spectral
parameters. The next important object – a monodromy matrix – is given by the product of L
operators
TN (u) = L1(u+ ξ1)L2(u+ ξ2) · · ·LN (u+ ξN ),
T̄N (ū) = L̄1
(
ū+ ξ̄1
)
L̄2
(
ū+ ξ̄2
)
· · · L̄N
(
ū+ ξ̄N
)
, (2.1)
where ξk, ξ̄k are the so-called impurity parameters.3 The entries of the monodromy matrix,
TN (u) =
(
AN (u) BN (u)
CN (u) DN (u)
)
,
are polynomials in u with the operator valued coefficients, e.g.,
AN (u) = uN + uN−1
(
iS0 + Ξ
)
+
N∑
k=2
uN−kak,
BN (u) = uN−1iS− +
N∑
k=2
uN−kbk, (2.2)
where Ξ =
∑N
k=1 ξk and S0, S− are the total generators,
Sα = Sα
1 + · · ·+ Sα
N .
The entries of the monodromy matrix form commuting operator families [18, 50]
[AN (u), AN (v)] = [BN (u), BN (v)] = [CN (u), CN (v)] = [DN (u), DN (v)] = 0.
In particular, each entry commutes with the corresponding total generator, Sα,[
S0, AN (u)
]
=
[
S0, DN (u)
]
= 0 and [S−, BN (u)] =
[
S+, CN (u)
]
= 0.
The same equations hold for the anti-holomorphic operators ĀN , B̄N , C̄N , D̄N and, of course,
the holomorphic and anti-holomorphic operators commute. Moreover it can be checked that if
the impurity parameters satisfy the constraint ξ̄k = ξ∗k for all k, the following relations between
holomorphic and anti-holomorphic operators hold:
(AN (u))† = ĀN (u∗), (BN (u))† = B̄N (u∗),
etc. This ensures that the operators ak and āk in the expansion of AN (u), (2.2), and ĀN (u),
are adjoint to each other a†k = āk
(
b†k = b̄k etc.
)
.
The commutativity of the operators AN (u), BN (u), CN (u), DN (u) implies that the following
families of self-adjoint operators:
AN =
{
iS0, iS̄0, ak + āk, i(ak − āk), k = 2, . . . , N
}
,
BN =
{
iS−, iS̄−, bk + b̄k, i(bk − b̄k), k = 2, . . . , N
}
,
(and similarly for others) are commutative and can be diagonalized simultaneously.4 The cor-
responding eigenfunctions provide a convenient basis – Sklyanin’s representation of Separated
Variables (SoV) – for the analysis of spin chain models [49].
The operators BN and CN , (AN and DN ) are related to each other by the inversion trans-
formation, see [14] for detail, so it is sufficient to construct eigenfunctions for the operators BN
and AN . The eigenfunctions of BN for the homogeneous chain were constructed in [9] and
later on for the operator AN [14]. Extending this approach to the inhomogeneous case is rather
straightforward.
3As it can already be noticed any formula in the holomorphic sector has its exact copy in the anti-holomorphic
one. Therefore, from now on we write explicitly only holomorphic formulae tacitly implying its anti-holomorphic
counterparts.
4The impurity parameters must also satisfy the condition i
(
ξk − ξ̄k
)
= rk, where rk are half-integers.
4 A.N. Manashov
3 Eigenfunctions
In this section, we present explicit expressions for the eigenfunctions of the operators BN and AN
for a generic inhomogeneous spin chain with impurities. We start with the operator BN where
the construction follows the lines of [9] with minimal modifications.
3.1 BN operator
Let Λn be an integral (layer) operator which maps functions of n − 1 variables into functions
of n variables and depends on the spectral parameters x, x̄ and the complex vectors γ, γ̄ of
dimension 2n− 2
[Λn(x|γ)f ](z1, . . . , zn)
=
∫
· · ·
∫
Λn(x|γ)(z1, . . . , zn|w1, . . . , wn−1)f(w1, . . . , wn−1)
n−1∏
k=1
d2wk. (3.1)
The kernel is given by the following expression:
Λn(x|γ)(z1, . . . , zn|w1, . . . , wn−1) =
n−1∏
k=1
Dγ2k−1−ix(zk − wk)Dγ2k+ix(zk+1 − wk),
where the function Dα(z) (propagator) is defined as follows:
Dα(z) ≡ Dα,ᾱ(z, z̄) = z−αz̄−ᾱ. (3.2)
We will assume that the indices α, ᾱ satisfy the condition [α] ≡ α−ᾱ ∈ Z so that the propagator
is a single-valued function on the complex plane. It implies that the parameters γk and x have
the form
γk =
1
2
+
rk
2
+ iσk, γ̄k =
1
2
− rk
2
+ iσk, x =
im
2
+ ν, x̄ = − im
2
+ ν. (3.3)
The numbers {m, r1, . . . , r2N−2} are either integer or half-integer and depending on this we
call the corresponding variables integer (half-integer). The continuous parameters σk and ν are
subject to the constraints
Im(σ2k+1 − ν) > −1/2 and Im(σ2k + ν) > −1/2,
which guarantee the convergence of the integral (3.1) for a smooth function f with finite support.
In the case we are most interested in, γk + γ̄k = 1, the parameters σk ∈ R, and the variable ν
lies in the strip −1/2 < Im ν < 1/2.
The operators Λn possess two important properties:
(i) Let ρ be a map which takes M -dimensional vectors
γ = (γ1, . . . , γM ), γ̄ = (γ̄1, . . . , γ̄M )
to vectors of dimension M − 2 as follows:
ργ = (γ′2, γ
′
3, . . . , γ
′
M−1), ργ̄ = (γ̄′2, γ̄
′
3, . . . , γ̄
′
M−1),
where a′ ≡ 1 − a. It can be shown that the operators Λn and Λn−1 obey the following
exchange relation:
Λn(u|γ)Λn−1(v|ργ) = ωn(γ, u, v)Λn(v|γ)Λn−1(u|ργ). (3.4)
Unitarity of the SoV Transform for SL(2,C) Spin Chains 5
Here γ(γ̄) is (2n − 2)-dimensional vector and the factor ωn is given by the following
expression:
ωn(γ, u, v) =
n−1∏
m=1
Γ
[
γ2m−1 − iv, γ̄2m + iv̄
γ2m−1 − iu, γ̄2m + iū
]
=
n−1∏
m=1
Γ
[
γ̄2m−1 − iv̄, γ2m + iv
γ̄2m−1 − iū, γ2m + iu
]
, (3.5)
where
Γ
[
a1, a2, . . . , an
b1, b2, . . . , bm
]
≡
∏n
k=1 Γ[ak]∏m
k=1Γ[bk]
and Γ is the Gamma function of the complex field C [20]
Γ[u] ≡ Γ[u, ū] = Γ(u)/Γ(1− ū).
The relation (3.4) is a direct consequence of the exchange relation for the propagators,
see (A.1). Its proof is exactly the same as for the homogeneous spin chain. For more
details, see [9, 14].
(ii) Let us choose the vector γ as follows
γ = (s1 − iξ1, s2 + iξ2, s2 − iξ2, . . . , sN−1 + iξN−1, sN−1 − iξN−1, sN + iξN ),
γ̄ = (s̄1 − iξ̄1, s̄2 + iξ̄2, s̄2 − iξ̄2, . . . , s̄N−1 + iξ̄N−1, s̄N−1 − iξ̄N−1, s̄N + iξ̄N ), (3.6)
where sk and ξk are the spins and impurity parameters of the spin chain, respectively. For
such a choice of the vector γ, the operator BN (x) annihilates ΛN (x|γ) [8, 9]
BN (x)ΛN (x|γ) = 0. (3.7)
Let us define a function
Ψ(N)
p,x (z) ≡ Ψ(N)
p,x1,...,xN−1
(z1, . . . , zN )
= π−N2/2|p|N−1
∫
d2zUx1,...,xN−1(z1, . . . , zN |z)ei(pz+p̄z̄),
where the kernel Ux1,...,xN−1 is given by the product of the layer operators,
Ux1,...,xN−1 = ϖ(x|γ)ΛN (x1|γ)ΛN−1(x2|ργ)ΛN−2
(
x3|ρ2γ
)
· · ·Λ2
(
xN−1|ρN−2γ
)
,
and γ is given by (3.6). Equation (3.4) guarantees that Ux1,...,xN−1 ∼ Uxi1
,...,xiN−1
for any
permutation of x1, . . . , xN−1. The kernel Ux becomes totally symmetric for the following choice
of the prefactor ϖ(x|γ):
ϖ(x|γ) = ϖ(x1, . . . , xN−1|γ) =
N−1∏
m=1
m∏
k=1
ϖ1
(
xk|ρm−1γ
)
, (3.8)
where
ϖ1(x|γ) = ϖ1(x|γ1, . . . , γ2n) =
n∏
m=1
Γ[γ2m−1 − ix, γ̄2m + ix̄].
Thus the function Ψ
(N)
p,x1,...,xN−1 is a symmetric function of the variables x1, . . . , xN−1. Together
with (3.7) it implies that
BN (xk)Ψ
(N)
p,x1,...,xN−1
= 0 for k = 1, . . . , N − 1.
6 A.N. Manashov
Invariance of the kernel Ux1m...,xN−1(z1, . . . , zN |z) under shifts
Ux1...,xN−1(z1 + w, . . . , zN + w|z + w) = Ux1...,xN−1(z1, . . . , zN |z)
results in
iS−Ψ(N)
p,x1,...,xN−1
= pΨ(N)
p,x1,...,xN−1
, iS̄−Ψ(N)
p,x1,...,xN−1
= p̄Ψ(N)
p,x1,...,xN−1
. (3.9)
It follows then from equations (2.2), (3.7) and (3.9) that5
BN (u)Ψ(N)
p,x (z) = p
N−1∏
k=1
(u− xk)Ψ
(N)
p,x (z), B̄N (ū)Ψ(N)
p,x (z) = p̄
N−1∏
k=1
(ū− x̄k)Ψ
(N)
p,x (z).
For N = 1, the functions Ψ
(1)
p (z, z̄) = π−1/2ei(pz+p̄z̄) form the complete orthonormal system
in H1 = L2(C). The aim of this paper is to extend this statement to N > 1. Namely, we
will show in Section 5 that if the spins and impurities parameters of the spin chain obey the
“unitarity” condition,
γk + γ̄∗k = 1, (3.10)
for all k (γk has the form (3.3) with σk ∈ R ) then the set of functions
{
Ψ
(N)
p,x , xk = x̄∗k(νk ∈ R),
k = 1, . . . , N − 1
}
is complete in HN = (
⊗
L2(C))N .
Note that the functions Ψ
(N)
p,x are well defined for the complex parameters νk in the vicinity
of the real line. For further analysis, it will be useful to consider regularized functions, Ψ
(N),ϵ
p,x ,
by relaxing the last of the conditions (3.10) to γ2N−2 + γ̄∗2N−2 = 1 + 2ϵ. This can be achieved
by shifting the impurity parameter ξN → ξN − iϵ,6 i.e.,
Ψ(N),ϵ
p,x (z)
def
= Ψ(N)
p,x (z)
∣∣∣
ξN→ξN−iϵ
. (3.11)
3.2 AN operator
Construction of the eigenfunctions of the operator AN follows the scheme described in the
previous subsection. We define a layer operator Λ′
n which maps functions of n− 1 variables into
functions of n variables
[Λ′
n(x|γ)f ](z1, . . . , zn)
=
∫
· · ·
∫
Λ′
n(x|γ)(z1, . . . , zn|w1, . . . , wn−1)f(w1, . . . , wn−1)
n−1∏
k=1
d2wk,
where the kernel is given by the following expression:
Λ′
n(x|γ)(z1, . . . , zn|w1, . . . , wn−1)
= Dγ2n−1−ix(zn)
n−1∏
k=1
Dγ2k−1−ix(zk − wk)Dγ2k+ix(zk+1 − wk).
The layer operator Λ′
n depends on the spectral parameters x(x̄) and the vector γ(γ̄) of dimen-
sion 2n− 1 which have the form (3.3). These operators satisfy the exchange relation
Λ′
n(u|γ)Λ′
n−1(v|ργ) = ωn(γ, u, v)Λ
′
n(v|γ)Λ′
n−1(u|ργ),
and the factor ωn is defined in (3.5).
5We recall that the variables xk, x̄k, k = 1, . . . , N −1 take the form xk = ink/2+νk, x̄k = −ink/2+νk, where,
depending on the spin and impurities parameters, all nk are either integer or half-integer numbers.
6Of course, one also can regularize the function by shifting the parameter γ1 instead of γ2N−2, γ1+ γ̄∗
1 = 1+2ϵ.
Unitarity of the SoV Transform for SL(2,C) Spin Chains 7
.
z1 z2 z3 z1 z2 z3
γ 1
−
ix
1
γ
′
2
−
ix
2
γ ′3 +
ix
2
γ
2 +
ix
1
γ
4 +
ix
1
γ 3
−
ix
1
γ
′
2
−
ix
2
γ 1
−
ix
1
γ ′3 +
ix
2
γ
2 +
ix
1
γ
4 +
ix
1
γ 3
−
ix
1
γ 5
−
ix
1
γ
′
4
− i
x 2
γ3 − ix3
p
Figure 1. The diagrammatic representation for the function Ψ (left) and Φ (right) for N = 3. The
arrow from z to w with an index α stands for the propagator Dα(z − w), equation (3.2).
Let Φ
(N)
x (z) be the following function:
Φ(N)
x (z) ≡ Φ(N)
x1,...,xN
(z1, . . . , zN )
= π−N2/2ϖ(x|γ)
[
Λ′
N (x1|γ)Λ′
N−1(x2|ργ) . . .Λ′
1
(
xN |ρN−1γ
)]
(z1, . . . , zN ),
where γ is (2N −1)-dimensional vector and the prefactor ϖ is given by equation (3.8). For such
a choice of ϖ the function Φ
(N)
x is a symmetric function of the variables x1, . . . , xN .
It can be shown that the operator AN (x) annihilates the layer operator Λ′
N (x|γ),
AN (x)Λ′
N (x|γ) = 0,
for the following choice of the vector γ:
γ = (s1 − iξ1, s2 + iξ2, s2 − iξ2, . . . , sN + iξN , sN − iξN ),
γ̄ = (s̄1 − iξ̄1, s̄2 + iξ̄2, s̄2 − iξ̄2, . . . , s̄N + iξ̄N , s̄N − iξ̄N ).
Taking into account polynomiality of AN (u), see equation (2.2), one obtains
AN (u)Φ(N)
x (z) =
N∏
k=1
(u− xk)Φ
(N)
x (z), ĀN (ū)Φ(N)
x (z) =
N∏
k=1
(ū− x̄k)Φ
(N)
x (z).
Again, the variables xk, x̄k are integers (half-integers) for all k. We will show that these functions,
{Φ(N)
x (z), xk = x̄∗k, k = 1, . . . , N}, form a complete set in the Hilbert space HN .
4 Scalar products, momentum representation, etc.
The functions constructed in the previous section are given by multidimensional integrals. In
this section, we show that these integrals converge for the parameters νk in the vicinity of real
axis. To this end, it will be quite helpful, as was advocated in [9], to visualize the integrals as
Feynman diagrams. The examples for N = 3 are shown in Figure 1. It will be convenient to
convert diagrams (functions) to momentum space
Ψ(z1, . . . , zN ) = π−N
∫
· · ·
∫
Ψ̃(p1, . . . , pN )ei
∑N
k=1(pkzk+p̄k z̄k)d2p1 · · · d2pN .
8 A.N. Manashov
In momentum space the function Ψ
(N),ϵ
p,x , equation (3.11), takes the form
Ψ̃(N),ϵ
p,x (p1, . . . , pN ) = δ(2)
(
p−
N∑
k=1
pk
)
Ψ(N),ϵ
x (p1, . . . , pN ).
Let us remark here that the “ϵ” regularization is reduced to a multiplication by the factor
(pN p̄N )ϵ
Ψ(N),ϵ
x (p1, . . . , pN ) = (pN p̄N )ϵΨ(N)
x (p1, . . . , pN ). (4.1)
The function Ψ
(N),ϵ
x can be read from the Feynman diagram in Figure 1 as follows:
Ψ(N),ϵ
x (p1, . . . , pN ) =
∫
· · ·
∫
J ϵ
x({pk}, {ℓij})
∏
1≤j≤i≤N−2
d2ℓij , (4.2)
with the integrand J ϵ
x({pk}, {ℓij}) given by the product of the propagators, Dα(k). Up to
a momentum independent factor
J ϵ
x({pk}, {ℓij}) ≃
N−1∏
k=1
k∏
j=1
Dαkj
(ℓk,j − ℓk−1,j−1)Dβkj
(ℓk−1,j − ℓk,j),
where ℓk0 ≡ 0, ℓk−1,k ≡ p and ℓN−1,j = (p1 + · · ·+ pj). The indices αkj , βkj take the following
values:
αkj = γ
(N−k)
2j−1 + ixN−k, βkj = γ
(N−k)
2j − ixN−k,
where we introduced the notations:
a(1) = a′ = 1− a and a(k+1) = 1− a(k).
In many cases, Feynman diagrams can be evaluated diagrammatically. In particular, the compu-
tation of diagrams for the scalar product of Ψ (Φ) functions is based on the successive application
of the exchange relation (A.1) to the diagram.
Let us consider the scalar product of two functions Ψ
(N),ϵ
p,x and Ψ
(N),ϵ′
q,y(
Ψ(N),ϵ′
q,y ,Ψ(N),ϵ
p,x
)
= πδ2(p− q)(pp̄)ϵ+ϵ′Iϵ,ϵ
′
(x, y), (4.3)
where
Iϵ,ϵ
′
(x, y) =
1
π
(pp̄)−ϵ−ϵ′
∫
· · ·
∫
δ(2)
(
p−
∑
k
pk
)
Ψ(N),ϵ
x (p⃗)
(
Ψ(N),ϵ′
y (p⃗)
)† N∏
j=1
d2pj . (4.4)
The function Iϵ,ϵ
′
p (x, y) is given by the Feynman diagram shown in Figure 2 in Appendix A (left
panel), which is a multidimensional integral
Iϵ,ϵ
′
p (x, y) =
∫
· · ·
∫
Iϵϵ′
x,y(p, {ℓpr}|γ)
N−1∏
p,r=1
d2ℓpr (4.5)
with the integrand given by the product of the propagators. The diagram can be evaluated
in a closed form by successively applying the exchange relation (A.1), that is equivalent to
calculating the loop integrals in a certain order. The answer takes the form
Iϵ,ϵ
′
(x, y) = CN (γ)Γ
[
ϵ+ ϵ′ + iX − iȲ ∗
ϵ+ ϵ′
] ∏N−1
k,j=1 Γ[i(y
∗
k − x̄j)]∏N−1
k=1 ϕ̄N (x̄k)(ϕN (yk))∗
Unitarity of the SoV Transform for SL(2,C) Spin Chains 9
= CN (γ)Γ
[
ϵ+ ϵ′ + iX̄ − iY ∗
ϵ+ ϵ′
] ∏N−1
k,j=1 Γ[i(ȳ
∗
k − xj)]∏N−1
k=1 ϕN (xk)(ϕ̄N (ȳk))∗
, (4.6)
where X =
∑N−1
k=1 xk, Y =
∑N−1
k=1 Yk and
ϕN (x) = Γ
[
γ2N−3 − ix, γ
(1)
2N−4 − ix, γ2N−5 − ix, . . . , γ
(N−3)
N − ix
]
,
ϕ̄N (x̄) = Γ
[
γ̄2N−3 − ix̄, γ̄
(1)
2N−4 − ix̄, γ̄2N−5 − ix̄, . . . , γ̄
(N−3)
N − ix̄
]
.
For the sign factor CN (γ), we get
CN (γ1, γ2, . . . , γ2N−2) =
{
1, odd N,
(−1)
∑N−3
k=1
[
γ
(k−1)
2N−2−k−γ
(N−3)
N
]
, even N.
(4.7)
Here [a] ≡ a− ā. Details of the calculation can be found in Appendix B.
Let us show now that integrations in (4.5) can be done in an arbitrary order. The integrand
in (4.5), Iϵϵ′
x,y(p, {ℓpr}|γ), is given by the product of the propagators Dα(k), with each index being
of the form α = 1
2 +
n
2 + iσ, momentum k being a linear combination of loop momenta, ℓij , and
the external momentum p. Since∣∣Dα(k)
∣∣ = ∣∣k−αk̄−ᾱ
∣∣ = |k|−1+2 Imσ = D1/2−Imσ(k)
then for the parameters γ satisfying the unitarity condition (3.10), and xk, yk having the form
xk = ink/2 + νk, yk = imk/2 + µk, (4.8)
one obtains for the modulus of the integrand∣∣Iϵϵ′
x,y(p, {ℓpr}|γ)
∣∣ = Iϵϵ′
x,y(p, {ℓpr}|γ) > 0,
where the underlined variables are: γ = (1/2, . . . , 1/2),
(x)k = Im(νk) = ϵk, (y)k = Im(µk) = ϵ′k.
Thus the integral of |Iϵϵ′
x,y(p, {ℓpr}|γ)| is a particular case of the integral (4.5) which was calcu-
lated by performing loop integrations in a certain order. Since all integrals converge under the
conditions
ϵkj ≡ ϵk + ϵ′j > 0 for k, j = 1, . . . , N − 1 and ϵ+ ϵ′ >
N−1∑
k=1
(ϵk + ϵ′k),
by Fubini theorem, the integral (4.5) exists and the integrations can be done in an arbitrary
order.
The following statements can immediately be deduced from this result:
� For any bounded function φ(p, x) with a finite support the function
Ψϵ
φ =
∫
· · ·
∫
φ(p, x)Ψ
(N),ϵ
p,xϵ d2pDx1 · · · DxN−1, (4.9)
where xϵ = (x1 + iϵ1, . . . , xN−1 + iϵN−1), xk = ink/2 + νk, ϵk > 0, ϵ >
∑N−1
k=1 ϵk and∫
Dxk ≡
∞∑
nk=−∞
∫ ∞
−∞
dνk,
belongs to the Hilbert space HN , ∥Ψϵ
φ∥2 < ∞, for sufficiently small ϵ.
10 A.N. Manashov
� It follows from the finiteness of the integral Iϵ,ϵ
′
p (x, y), equation (4.4), that the function
Ψ
(N),ϵ
x (p⃗), equation (4.2), exists almost for all p⃗ for the separated variables xk close to the
real axis:
Im νk =
1
2
Im(xk + x̄k) ∼ 0 for all k
and Ψ
(N),ϵ
x (p⃗) is a continuous function of νk in this region. Indeed, let us fix m < N
and put um = Reνm and vm = Im νm, |vm| < δ. One gets the following estimate for the
integrand (4.5):
|J ϵ
x({pk}, {ℓij})| < |J ϵ
x+
({pk}, {ℓij})|+ |J ϵ
x−({pk}, {ℓij})|, (4.10)
where x± are defined as follows: for k ̸= m (x±)k = xk and for k = m (x±)m = um±iδ. The
integrals of the functions on the right-hand side of (4.10) are finite for sufficiently small δ.
It follows then from the Lebesgue theorem that the function Ψ
(N),ϵ
x (p⃗) is continuous in the
variable νm.7
The scalar product of the functions Ψ
(N)
p,y and Φ
(N)
x constructed in Section 3.2 can be calculated
in a similar way. Note that there is no need to introduce “ϵ” regulator here. The corresponding
integral is absolutely convergent when Im(νk + µj) > 0 for all k, j (xk, yj given by (4.8)). The
scalar product takes the form(
Ψ(N)
p,y |Φ(N)
x
)
= CAB
N (γ)|p|N−1(−ip)−GN−iX(ip̄)−ḠN−iX̄
×
∏N
k=1
∏N−1
j=1 Γ[i(ȳ∗j − xk)](∏N
j=1 ϑN (xj)
)(∏N−1
j=1 ϑ̄N (ȳj)
)† , (4.11)
where
ϑN (x) =
N∏
k=1
Γ
[
γ
(k−1)
2N−k − ixj
]
, ϑ̄N (x̄) =
N∏
k=1
Γ
[
γ̄
(k−1)
2N−k − ix̄j
]
,
GN =
∑2N−1
k=N γ
(k)
k , X =
∑N
k=1 xk and
CAB
N (γ1, . . . , γ2N−1) =
{
1, odd N,
(−1)
∑N
k=1
[
γ
(k−1)
2N−k−γ
(N−1)
N
]
, even N.
Similar to the previous case one can argue that Φ
(N)
x is a continuous function of νk in the vicinity
of the real axis.
Finally, the scalar product of the functions Ψ
(N+1)
p,x (z1, . . . , zN+1) and Ψ
(N)
q1,y(z1, . . . , zN ) ⊗
Ψ
(1)
q2 (zN+1) which we need in the proof of Theorem 5.2, takes the form(
ΨN
q1,y ⊗Ψ(1)
q2 ,Ψ
(N+1)
p,x
)
= CNN+1(γ)πδ
(2)(p− q1 − q2)|p|N |q1|N−1
×(ip)−Ḡ∗
N+1(−ip̄)−G∗
N+1(iq2)
−γ′
2N (−iq̄2)
−γ̄′
2N (−iq1)
−GN (iq̄1)
−ḠN
×
(
1 +
q1
q2
)iȲ ∗ (
1 +
q̄1
q̄2
)iY ∗ (
−q2
q1
)iX (
− q̄2
q̄1
)iX̄
7Since the integrand is analytic function of νk Ψ
(N),ϵ
x (p⃗) is an analytic function of νk in the vicinity of the real
axis.
Unitarity of the SoV Transform for SL(2,C) Spin Chains 11
×
∏N−1
k=1
∏N
j=1 Γ [i(ȳ∗k − xj)](∏N
j=1
∏N−1
k=1 Γ
[
γ
(k−1)
2N−k − ixj
]) (∏N
k=1
∏N−1
j=1 Γ
[
γ̄
(k−1)
2N−k − iȳj
])† , (4.12)
where
GN =
2N−1∑
m=N
γ(m)
m , GN+1 = GN − γ
(N)
N =
2N−1∑
m=N+1
γ(m)
m (4.13)
and
CNN+1(γ1, . . . , γ2N ) =
{
1, for odd N,
(−1)
∑N−1
k=1
[
γ
(k−1)
2N−k−γ
(N−1)
N
]
, for even N.
The calculation is almost the same as in the previous cases so we omit the details.
5 SoV representation
In the previous section, we constructed the functions Ψ
(N)
p,x and Φ
(N)
x associated with the en-
tries BN and AN of the monodromy matrix (2.1). For a given vector Ψ ∈ HN , we define two
functions by projecting it on Ψ
(N)
p,x and Φ
(N)
x :
φ(p, x1, . . . , xN−1) =
(
Ψ(N)
p,x ,Ψ
)
, χ(x1, . . . , xN ) =
(
Φ(N)
x ,Ψ
)
.
These functions are symmetric functions of the variables x. It was shown by Sklyanin [49] that
the transformation Ψ 7→ φ(Ψ 7→ χ) reduces the original multidimensional spectral problem for
the transfer matrix to the set of one-dimensional spectral problems that greatly simplifies the
analysis. We want to show that the maps Ψ 7→ φ and Ψ 7→ χ can be extended to the isomorphism
between the Hilbert spaces, HN 7→ HSoV.
Let us define
(φ1, φ2)BN
=
∫
R×R
∫
Dσ
N−1
(φ1(p, x))
†φ2(p, x)µN−1 (x) d
2pdµB
N−1(x),
(χ1, χ2)AN
=
∫
Dσ
N
(χ1(x))
†χ2(x)dµ
A
N (x). (5.1)
The variables xk, x̄k take the form xk = ink/2 + νk, x̄k = −ink/2 + νk, where all nk are either
integers or half-integers,
nk ∈ Zσ ≡ Z+
σ
2
, σ = 0, 1,
and
Dσ
N ≡ (R× Zσ)N .
The measures are defined as follows:
dµ
B(A)
N (x) = µ
B(A)
N (x)
N∏
k=1
Dxk, µ
B(A)
N (x) = c
B(A)
N µN (x).
The symbol Dx stands for∫
Dx ≡
∑
n∈Zσ
∫ ∞
−∞
dν.
12 A.N. Manashov
The weight function µN (x) is given by the following expression:
µN (x1, . . . , xN ) =
∏
1≤k<j≤N
xkj x̄kj =
∏
1≤k<j≤N
(
ν2kj +
1
4
n2
kj
)
,
where xkj = xk − xj , νkj = νk − νj , nkj = nk − nj while the coefficients c
B(A)
N take the form(
cBN
)−1
=
1
2
(2π)N+1N !,
(
cAN
)−1
= (2π)NN !.
Let HB,σ
N , HA,σ
N be the Hilbert spaces of symmetric functions corresponding to the scalar
products (5.1):
HB,σ
N = L2(R× R)⊗ L2
sym
(
Dσ
N−1, dµ
B
N−1(x)
)
,
HA,σ
N = L2
sym
(
Dσ
N ,dµA
N (x)
)
.
Given that φ(p, x) and χ(x) are smooth and compactly supported functions on R2 × Dσ
N−1
and Dσ
N , respectively, we introduce transforms TB
N : φ 7→ Ψφ and TA
N : χ 7→ Ψχ,
Ψφ(z) ≡
[
TB
Nφ
]
(z) =
∫
R2
∫
Dσ
N−1
φ(p, x)Ψ(N)
p,x (z)d2pdµB
N−1(x), (5.2a)
Φχ(z) ≡
[
TA
Nχ
]
(z) =
∫
Dσ
N
χ(x)Φ(N)
p,x (z)dµA
N (x). (5.2b)
Note that the function Ψφ depends on the vector γ, equation (3.6), which appears in the defi-
nition of the function Ψ
(N)
p,x . That is TB
N ≡ TB
N (γ) and the same applies to the operator TA
N . In
order to not overload the notation, we do not display this dependence explicitly.
5.1 B system
We begin the proof of the unitarity of the transform TB
N with the following lemma.
Lemma 5.1. For any smooth fast decreasing function φ on R2 × Dσ
N−1, the function TB
Nφ
belongs to the Hilbert space HN and it holds∥∥TB
Nφ
∥∥2
HN
= ∥φ∥2HB,±
N
=
∫
D±
N
|φ(p, x)|2d2pdµB
N−1(x).
Proof. Let Ψϵ
φ be a function defined by equation (5.2a) with Ψ
(N)
p,x replaced by Ψ
(N),ϵ
p,xϵ , see
equations (4.1) and (4.9). It can be shown that Ψϵ
φ(p⃗) 7→
ϵ→0
Ψφ(p⃗) almost everywhere. Next,
taking into account equation (4.3) one gets
(Ψϵ
φ,Ψ
ϵ′
φ′)HN
= π
∫
d2p
∫
dµB
N−1(x)
×
∫
dµB
N−1(x
′)(pp̄)ϵ+ϵ′φ(p, x)(φ′(p, x′))†Iϵ,ϵ
′
(x, x′), (5.3)
with Iϵ,ϵ
′
(x, x′) given by equation (4.6). Let us assume that the function φ(φ′) has the form
φ(p, x1, . . . , xN−1) = κ(p)ϕ(x1, . . . , xN−1), (5.4)
where ϕ(x1, . . . , xN−1) is a symmetric function
ϕ(x1, . . . , xN−1) =
∑
SN−1
ϕ1(xi1) · · ·ϕN−1(xiN−1) (5.5)
Unitarity of the SoV Transform for SL(2,C) Spin Chains 13
and the sum goes over all permutations. We also assume that the functions ϕk(xk) = ϕk(nk, νk)
are local in nk, ϕ(nk, νk) = δnk,mk
ϕk(νk) and ϕk(νk) is an analytic function of νk in some
strip | Im νk| < δk which vanishes sufficiently fast at νk → ±∞. Such functions form a dense
subspace in the Hilbert space HB,σ
N . Since the momentum integral in (5.3) factorizes one has to
consider the integrals over xk = (nk, νk), x
′
k = (n′
k, ν
′
k), which have the form∫
dµB
N−1(x)
∫
dµB
N−1(x
′) · · ·
≡
N−1∏
j=1
∑
nj∈Z+σ
2
∑
n′
j∈Z+
σ
2
∫ ∞
−∞
· · ·
∫ ∞
−∞
µB
N−1(n⃗, ν⃗)µ
B
N−1(n⃗
′, ν⃗ ′)
N−1∏
k=1
dνkdν
′
k · · · . (5.6)
According to our assumptions, only finite number of terms contribute to the sum in (5.6). Let
us study behaviour of a particular term in the sum in the limit ϵ, ϵ′ 7→ 0. The functions ϕ, ϕ′ are
smooth and fast decreasing functions of ν, ν ′. The function Iϵ,ϵ
′
(x, x′) contains the factor Γ[ϵ+
ϵ′ + iX̄ − i(X ′)∗]/Γ[ϵ+ ϵ′] and the product of the Γ-functions
Γ
[
i((x̄′k)
∗ − xj)
]
= Γ
[
n′
k
2
− nj
2
+ i(ν ′k − νj) + ϵjk
]
=
Γ
(n′
k
2 − nj
2 + i(ν ′k − νj) + ϵjk
)
Γ
(
1 +
n′
k
2 − nj
2 − i(ν ′k − νj)− ϵjk
) , (5.7)
where ϵjk ≡ ϵj + ϵ′k. In the ϵ′k, ϵj → 0 this function becomes singular at ν ′k = νj if n′
k = nj . Let
us shift the contours of integrations over ν ′k variables to the upper half-plane, Im ν ′k = δ > ϵjk,
and pick up the residues at the corresponding poles. After this, we can send ϵ′k, ϵj 7→ 0. Let us
consider a generic contribution arising after this rearrangement. It has the form∫
Cδ
· · ·
∫
Cδ
M∏
k=1
dν ′ikf
(
x1, . . . , xN−1, S(x
′
1), . . . , S(x
′
N−1)
)
,
where S(x′k) = x′k if k ∈ (i1, . . . , iM ) and S(x′k) = xpk if k does not belong to this set. The
integrand f is given by the product of the functions ϕk, ϕ′
k, Γ-functions (5.7) and the fac-
tor A = Γ[ϵ+ ϵ′ + iX̄ − i(X ′)∗]/Γ[ϵ+ ϵ′]. All these factors are regular on the contours of in-
tegration. Moreover, if M ≥ 1 the last factor, A, tends to zero at ϵ, ϵ′ 7→ 0. Thus the only
non-vanishing contribution comes from the term with M = 0, i.e., when all x′k 7→ xik for
k = 1, . . . , N − 1. It takes the form
(Ψϵ
φ,Ψ
ϵ′
φ′)HN
=
∫
d2p
∫
dµB
N−1(x)φ(p, x)(φ
′(p, x))† +O(ϵ+ ϵ′)
that results in the following estimate for the norm of the function Ψϵ
φ:
∥Ψϵ
φ∥2HN
= K +O(ϵ),
where
K = ∥φ∥2HB,σ
N
≡
∫
R2
∫
Dσ
N−1
|φ(p, x)|2d2pdµB
N−1(x).
Since Ψϵ
φ(p⃗) 7→ Ψφ(p⃗) at ϵ → 0, it follows from Fatou’s theorem that ∥Ψφ∥2HN
< K. At the same
time, the inequality
∥Ψφ −Ψϵ
φ∥2HN
≥ 0
implies ∥Ψφ∥2HN
≥ K that results in ∥Ψφ∥2HN
= K.
14 A.N. Manashov
Since the set of functions (5.4), (5.5) is dense in the Hilbert spaces HB±
N , the transforma-
tion TB
N can be extended to the entire Hilbert space HB,±
N and equation (5.8a) holds for any
function φ ∈ HB,±
N . ■
Taking this result into account we formulate the following theorem.
Theorem 5.2. The map TB
N defined in equation (5.2a) can be extended to the linear bijective
isometry of the Hilbert spaces, HB,σ
N 7→ HN , i.e.,∥∥TB
Nφ
∥∥2
HN
= ∥φ∥2HB,σ
N
(5.8a)
and
R
(
TB
N
)
= HN . (5.8b)
Proof. Equation (5.8a) is a direct consequence of Lemma 5.1. It implies that
∥∥TB
N
∥∥ = 1, hence
R
(
TB
N
)
is a closed subspace in HN and HN = R
(
TB
N
)
⊕R
(
TB
N
)⊥
. Since R
(
TB
N
)⊥
= ker
(
TB
N
)∗
in order to prove (5.8b) it is enough to show that ker
(
TB
N
)∗
= 0. ■
We prove this statement using induction on N . For N = 1, the map TB
N=1 is a two-
dimensional Fourier transform, hence equation (5.8b) is true. Let us now assume that R
(
TB
N
)
=
HN and prove that it implies R
(
TB
N+1
)
= HN+1. As was stated above, it is sufficient to prove
that ker
(
TB
N+1
)∗
= 0. To this end, let us consider the map
SN =
(
TB
N+1
)∗ (
TB
N ⊗ TB
1
)
, HB,σ
N ⊗ L2(R2)
TB
N⊗TB
17−→ HN+1
(TB
N+1)
∗
7−→ HB,σ
N+1.
Since by the assumption TB
N⊗TB
1 is a bijective isometry ker SN = 0 if and only if ker
(
TB
N+1
)∗
= 0.
The adjoint operator
(
TB
N+1
)∗
is a bounded operator which acts on a vector Ψ ∈ HN+1 by
projecting it on the eigenfunction Ψ
(N+1)
p,x ,(
TB
N+1
)∗
Ψ =
(
Ψ(N+1)
p,x ,Ψ
)
HN+1
=
(
Ψ(N+1)
p,x ,PN+1Ψ
)
HN+1
≡ φ(p, x), (5.9)
where PN+1 is the projector on R
(
TB
N+1
)
. It follows from (5.9) that
∥φ∥2HB,σ
N+1
=
∫
R2
∫
Dσ
N
|φ(p, x)|2d2pdµB
N (x) = ∥PN+1Ψ∥2HN+1
≤ ∥Ψ∥2HN+1
. (5.10)
For ϕ ∈ HB,σ
N ⊗ L2
(
R2
)
, the function Ψϕ =
(
TB
N ⊗ TB
1
)
ϕ reads
Ψϕ(z) =
∫
R2⊗R2
∫
Dσ
N−1
Ψ(N)
q1,x(z1, . . . , zN )Ψ(1)
q2 (zN+1)ϕ(q1, q2, x)d
2q1d
2q2dµ
B
N−1(x). (5.11)
Replacing Ψ
(N)
q1,x 7→ Ψ
(N),ϵ
q1,x in (5.11), we define a new function, Ψϵ
ϕ. According to Lemma 5.1,
Ψϵ
ϕ −→
ϵ→0+
Ψϕ in HN+1 for smooth rapidly decreasing functions, we obtain
φ(p, x) = [SNϕ](p, x) =
(
Ψ(N+1)
p,x ,Ψϕ
)
HN+1
= lim
ϵ→0+
(
Ψ(N+1)
p,x ,Ψϵ
ϕ
)
HN+1
≡ lim
ϵ→0+
φϵ(p, x), (5.12)
where
φϵ(p, x) =
∫
R2×R2
∫
Dσ
N−1
Sϵ
N (p, x|q1, q2, x′)ϕ(q1, q2, x′)d2q2d2q1dµB
N−1(x
′). (5.13)
Unitarity of the SoV Transform for SL(2,C) Spin Chains 15
The kernel Sϵ
N reads
Sϵ
N (p, x|q1, q2, x′) =
(
Ψ(N+1)
p,x ,Ψ
(N)
q1,x′
ϵ
⊗Ψ(1)
q2
)
, (5.14)
see equation (4.12), and x′ϵ =
(
x′1 + iϵ1, . . . , x
′
N−1 + iϵN−1
)
. We assume that function ϕ takes
the form
ϕ(q1, q2, x1, . . . , xN−1) = κ1(q1)κ2(q2)
∑
SN−1
ϕ1(xi1) · · ·ϕN−1(xiN−1), (5.15)
where the sum goes over all permutations and that the functions ϕk are local in “n” variable, that
is ϕk(xk) = ϕk(nk, νk) = δnkmk
ϕnk
(νk) and ϕnk
are compactly supported. The function φ(p, y)
does not decrease sufficiently fast for large yk in order to justify changing the order of integration
after substituting φϵ(p, y) in the form (5.12), (5.13) into (5.10). To overcome this difficulty, we
following the lines of [13], consider the integral
IZ(φ) =
∫
R2
∫
Dσ
N
|φ(p, y)|2ΩZ(y)d
2pdµB
N (y),
where
ΩZ(y) =
N∏
k=1
Γ [Z + iyk, Z − iyk]
Γ [Z,Z]
, Z = Z̄ =
1
2
+ iM.
For y∗k = ȳk the factor Ω is a pure phase, |ΩZ(y)| = 1 and ΩZ(y) 7→ 1 when M → ∞, y is fixed.
Since the integral (5.10) is convergent,
∥φ∥2HB,σ
N+1
= lim
M→∞
∫
R2
∫
Dσ
N
|φ(p, y)|2ΩZ(y)d
2pdµB
N (y).
It follows from equations (5.13), (5.14) and (4.12) that for compactly supported functions ϕk
the function f(ν) = |φϵ(p, y)|2 is an analytic function of νk in the vicinity of the real axis for
sufficiently large νk. Thus, we can write
IZ(φ) = lim
ω→0
IωZ(φ) = lim
ω→0
∫
R2
∫
Dσ,ω
N
|φ(p, y)|2ΩZ−ω(y)d
2pdµB
N (y), (5.16)
where the integration contours over νk are deformed in order to separate the poles due to the
Gamma functions, Γ [Z − ω ± iyk], in the factor Ω. The integral IωZ(φ) is an analytic function
of ω. Substituting φ(p, y) in (5.16) in the form (5.13), one can show that for Reω > 1 the
integrals over y decay fast enough to allow the change of the order of integration over x, x′
and y. Thus, we obtain
IωZ(φ) = lim
ϵ,ϵ′→0+
∫
R2×R2
∫
Dσ
N−1×Dσ
N−1
δ(2)(q1 + q2 − q′1 − q′2)ϕ(q1, q2, x)
(
ϕ(q′1, q
′
2, x
′)
)†
×
∣∣∣∣q′1q1
∣∣∣∣N−1 ∣∣∣∣q1 + q2
q1q′2
∣∣∣∣2(1 + q′1
q′2
)iX′ (
1 +
q̄′1
q̄′2
)iX̄′ (
1 +
q1
q2
)−iX (
1 +
q̄1
q̄2
)−iX̄
×
(
q1
q′1
)GN
(
q̄1
q̄′1
)ḠN
(
q′2
q2
)γ2N
(
q̄′2
q̄2
)γ̄2N
R(x, x′)J (ϵ)
ω (Z, ζ, x, x′)
× d2q1d
2q2d
2q′1d
2q′2dµ
B
N−1(x)dµ
B
N−1(x
′), (5.17)
16 A.N. Manashov
where ζ =
q1q
′
2
q2q′1
, GN is defined in equation (4.13),
R(x, x′) =
N∏
k=1
N−1∏
j=1
Γ
[
γ̄
(k−1)
2N−k − ix′j
]
/Γ
[
γ̄
(k−1)
2N−k − ixj
]
and
J (ϵ,ϵ′)
ω (Z, ζ, x, x′) = π2
∫
Dω,σ
N
ζ iY ζ̄ iȲ
N∏
j=1
Γ[Z − ω ± iyj ]
Γ2(Z)
×
N−1∏
k=1
Γ[i(x̄k − ȳj)]Γ[i(yj − x′k)]dµ
B
N (y). (5.18)
We recall that the variables νk, ν
′
k, (xk = ink/2 + νk, x
′
k = in′
k/2 + ν ′k) have small negative
(positive) imaginary parts, Im νk = −ϵk, Im ν ′k = ϵ′k, which must be send to zero at the end of
the calculation.
The integral (5.18) can be obtained in the closed form with the help of equation (C.2). Indeed,∏
1≤j ̸=k≤N
1
Γ[i(yk − yj)]
= µN (y)(−1)
∑
k<j [i(yk−yj)]
and
N∏
j=1
N−1∏
k=1
Γ[i(x̄k − ȳj)] =
N∏
j=1
N−1∏
k=1
Γ[i(xk − yj)](−1)
∑N
j=1
∑N−1
k=1 [i(yj−xk)],
where yk = imk/2 + νk, ȳk = −imk/2 + νk and we recall that [iyk] = i(yk − ȳk) = −mk. Taking
into account that
(−1)
∑
k<j [i(yk−yj)](−1)
∑N
j=1
∑N−1
k=1 [i(yj−xk)] = (−1)
∑
1≤k<j≤N−1[i(xk−xj)],
one finds that the integral (5.18) is nothing else as Gustafson’s integral (C.2) [uk → iyk for all k,
{z1, . . . , zN} 7→ {ix1, . . . , ixN−1, Z−ω} and {w1, . . . , wN} 7→ {−ix′1, . . . ,−ix′N−1, Z−ω}]. Thus,
we obtain for J
(ϵ,ϵ′)
ω ,
J (ϵ,ϵ′)
ω (Z, ζ, x, x′) = π(−1)
∑
k<j [i(xk−xj)] ζZ−ω+iX
(1 + ζ)2(Z−ω)+i(X−X′)
ζ̄Z̄−ω+iX̄
(1 + ζ̄)2(Z̄−ω)+i(X̄−X̄′)
× Γ[2Z − 2ω]
Γ2[Z]
N−1∏
k=1
Γ [Z − ω + ixk, Z − ω − ix′k]
Γ[Z,Z]
×
N−1∏
k,j=1
Γ[i(xk − x′j)]. (5.19)
Let us substitute this expression into (5.17) and calculate the corresponding limits. First of all,
since all factors containing ω are regular at ω, ϵk, ϵ
′
k → 0 one can interchange the limits and first
send ω → 0.
At M → ∞ the integral over q, q′ is dominated by the contribution from the stationary point
at ζ = 1,
Γ[1 + 2iM ]
Γ2
[
1
2 + iM
] ∫ d2ζ
(
ζζ̄
)iM+ 1
2(
(1 + ζ)
(
1 + ζ̄
))1+2iM
φ(ζ) =
M→∞
πφ(1)
(
1 +O
(
1
M1/2
))
.
Unitarity of the SoV Transform for SL(2,C) Spin Chains 17
Taking this into account and expanding the first factor in the second line in (5.19), one gets for
equation (5.17)
Iω=0(Z) = lim
ϵ,ϵ′→0+
∫
Dσ
N−1×Dσ
N−1
ϕ(q1, q2, x)
(
ϕ(q1, q2, x
′)
)†
π(−1)
∑
k<j [i(xk−xj)]iN−N ′
×R(x, x′)
(
1 +
q1
q2
)i(X′−X)(
1 +
q̄1
q̄2
)i(X̄′−X̄)(M
2
)2i(V−V ′)
×
N−1∏
k,j=1
Γ[i(xk − x′j) + ϵkj ]d
2q1d
2q2dµ
B
N−1(x)dµ
B
N−1(x
′) + · · · , (5.20)
where ellipses stand for terms vanishing at M → ∞ and
xk =
ink
2
+ νk, x′k =
in′
k
2
+ ν ′k, ϵkj = ϵk + ϵ′j ,
X =
N−1∑
k=1
xk, V =
N−1∑
k=1
νk, N =
N−1∑
k=1
nk,
etc. The analysis of this integral is similar to the analysis of the integral (5.3).8 In the limit
ϵ, ϵ′ → 0 the poles of the Gamma functions, xk = x′j , approach the integration contour, while
all other factors remain regular. Let us shift the integration contour in xk to the upper complex
half-plane picking up the residues at the poles at xk = x′j . We recall that the Gamma functions
develop poles only when nk = n′
j , otherwise they are regular at νk = ν ′j . Afterwards, we can
send ϵ, ϵ′ → 0. The answer is given by the sum of terms∫
· · ·
∫
M
i
∑m
k=1(νik−ν′jk
) × fm(x, x′)dνi1 · · · dνimdν ′1 · · · dν ′N−1,
where fm(x, x′) is a smooth function. Note, the contours of integration over ν variables lay in
the upper half-plane, so that |M i
∑m
k=1(νik−ν′jk
)| < 1 in the integration region. Since the func-
tions fm(x, x′) are smooth functions all such terms with m > 0 vanish after integration in the
limit M → ∞. Thus the only contribution with m = 0, i.e., when xk = x′kj , survives in this
limit. Then one obtains after some algebra
∥φ∥2HB,σ
N+1
= ∥SNϕ∥2HB,σ
N+1
=
∫
R2×R2
∫
Dσ
N−1
|ϕ(q1, q2, x)|2d2q1d2q2dµB
N−1(x)
= ∥ϕ∥2HB,σ
N ⊗L2(R2)
.
Since the space of functions (5.15) dense in HB,σ
N ⊗L2
(
R2
)
this relation can be extended to the
whole Hilbert space. Thus one concludes that kerSN = 0, and, hence, ker
(
TB
N+1
)∗
= 0.
5.2 A system
Using the results of the previous section it becomes quite easy to prove the unitarity of TA
N
transform. First, we prove an analogue of the Lemma 5.1.
Lemma 5.3. For any smooth fast decreasing function χ on Dσ
N the function TA
Nχ, equa-
tion (5.2b), belongs to the Hilbert space HN and it holds∥∥TA
Nχ
∥∥2
HN
= ∥χ∥2HA,σ
N
=
∫
Dσ
N
|χ(x)|2dµA
N (x). (5.21)
8We do it assuming that the functions ϕk(xk) have the properties discussed around equation (5.4).
18 A.N. Manashov
Proof. The proof is similar to the proof of the Lemma 5.1. It suffices to prove (5.21) for
functions of the form
χ(x1, . . . , xN ) =
∑
SN
χ1(xi1) · · ·χN (xiN ), χk(xk) = χk(nk, νk) = δnkmk
χk(νk). (5.22)
We assume that the functions χk(ν) are analytic in some strip near the real axis. Let us calculate
the projection
φχ(p, y) =
(
Ψ(N)
p,y ,Φχ
)
= lim
ϵ→0
∫
Dσ
N
(
Ψ(N)
p,y ,Φ
(N)
x+iϵ
)
χ(x)dµA
N (x). (5.23)
Here we have given the variables xk → xk + iϵk, ϵk = ϵ̄k > 0 small imaginary parts which allows
us to change the order of integration. In order to show that ∥φχ∥HB,σ
N
= ∥χ∥
HA,σ
N
we write
∥φχ∥2HB,σ
N
=
∫
R2
∫
Dσ
N−1
|φ(p, y)|2d2pdµB
N−1(y)
= lim
σ→0
∫
R2
e−σ|p|2
(∫
Dσ
N−1
|φ(p, y)|2dµB
N−1(y)
)
d2p.
Using the representation (5.23) for φχ(p, y), we first evaluate the y-integral.
9 This integral coin-
cides with the so-called SL(2,C) Gustafson integral and can be evaluated in a closed form (C.1)
resulting in
∥φχ∥2HB,σ
N
=
1
π
lim
σ→0
lim
ϵ,ϵ′→0+
∫
R2
∫
Dσ
N×Dσ
N
e−σ|p|2 iN−N ′
pi(X
′−X)−1+E+E ′
p̄i(X̄
′−X̄)−1+E+E ′
× χ(x)(χ(x′))†
(−1)
∑
k<j [i(x
′
k−x′
j)]∏N
j=1 ϑN (xj)(ϑN (x′j))
†
∏N
k,j=1 Γ[i(x
′
k − xj) + ϵjk]
Γ[i(X ′ −X) + E + E ′]
× dµA
N (x)dµA
N (x′)d2p,
where X =
∑N
k=1 xk, N =
∑N
k=1 nk, E =
∑N
k=1 ϵk, ϵjk = ϵj + ϵ′k, etc. For the momentum
integral, one gets
πδNN ′σi(V−V ′)−E−E ′
Γ(i(V ′ − V) + E + E ′),
where Γ is Euler’s gamma function. Thus
∥φχ∥2HB,σ
N
= lim
σ→0
lim
ϵ,ϵ′→0+
∫
Dσ
N×Dσ
N
(−1)
∑
k<j [i(x
′
k−x′
j)]δNN ′σi(V−V ′)
N∏
k,j=1
Γ[i(x′k − xj) + ϵjk]
× Γ(1 + i(V − V ′))
χ(x)(∏N
j=1 ϑN (xj)
)( χ(x′)(∏N
j=1 ϑN (x′j)
))†
dµA
N (x)dµA
N (x′),
where we put ϵk, ϵ
′
k = 0 in all nonsingular factors. The analysis of this integral in the σ, ϵ, ϵ′ → 0
limit is exactly the same as in Theorem 5.2, see discussion around equation (5.20), and results in
∥Φχ∥2HN
= ∥φχ∥2HB,σ
N
=
∫
Dσ
N
|φ(x)|2dµA
N (x). (5.24)
Since the space of the functions (5.22) is dense in HA,σ
N , the relation (5.24) extends to the whole
Hilbert space. ■
9The x, x′, y integral can be interchanged since the integral of modulus is convergent.
Unitarity of the SoV Transform for SL(2,C) Spin Chains 19
Finally, we formulate the analog of Theorem 5.2 for the map TA
N .
Theorem 5.4. The map TA
N defined in equation (5.2b) can be extended to the linear bijective
isometry of the Hilbert spaces, HA,σ
N 7→ HN , i.e.,∥∥TA
Nχ
∥∥2
HN
= ∥φ∥2HA,σ
N
and
R
(
TA
N
)
= HN . (5.25)
Proof. As in the Theorem 5.2, we only need to prove equation (5.25). As was discussed, earlier
equation (5.25) is equivalent to the statement that ker
(
TA
N
)∗
= 0 or to the assertion kerSN = 0,
where SN =
(
TA
N
)∗
TB
N . In order to prove this, it suffices to show that ∥SNφ∥HA,σ
N
= ∥φ∥HB,σ
N
.
The proof of this statement repeats step by step the proof given in the Theorem 5.2, and on the
technical level is reduced to the evaluation of the integral (5.18). ■
6 Summary
In this work, we consider a generic inhomogeneous SL(2,C) spin chain with impurities and
construct the eigenfunctions of the B and A entries of the monodromy matrix. We prove the
unitarity of the SoV transform associated with these systems or, equivalently, the completeness
of the corresponding systems in the Hilbert space of the model. Namely, the following identities
hold in the sense of distributions:∫
R2
∫
Dσ
N−1
Ψ(N)
p,x (z)
(
Ψ(N)
p,x (z′)
)†
d2pdµB
N (x) =
N∏
k=1
δ2(zk − z′k),
∫
Dσ
N
Φ(N)
x (z)
(
Φ(N)
x (z′)
)†
dµA
N (x) =
N∏
k=1
δ2(zk − z′k),
and ∫
CN
Ψ(N)
p,x (z)
(
Ψ
(N)
p′,x′(z)
)† N∏
k=1
d2zk =
(
µB
N (x)
)−1
δ2(p− p′)δN−1(x, x′),
∫
CN
Φ(N)
x (z)
(
Φ
(N)
x′ (z)
)† N∏
k=1
d2zk =
(
µA
N (x)
)−1
δN (x, x′),
where
δN (x, x′) =
1
N !
∑
w∈SN
δN (x′ − wx), wx = (xw1 , . . . , xwN )
and
δN (x′ − x) =
N∏
k=1
δ2(x′k − xk), δ2(x′ − x) = δnn′δ(ν − ν ′).
The method relies heavily on the use of multidimensional Mellin–Barnes integrals which
generalize integrals calculated by R.A. Gustafson [26]. The attractive feature of our approach
is that it does not depends on the details of the spin chain such as spins and inhomogeneity
parameters. We believe that this technique can also be used to prove the unitarity of the SoV
transform for the open SL(2,C) spin chain.
20 A.N. Manashov
A The diagram technique
Throughout this paper, we used a diagrammatic representation for the functions under con-
sideration. The calculation of relevant scalar products is, most conveniently, performed dia-
grammatically with the help of a few simple identities. Below, we give some of these rules (see
also [9]).
(i) An arrow with the index α directed from w to z stands for a propagator Dα(z − w) =
[z − w]−α:
w z
α
= [z − w]−α
(ii) The Fourier transform reads∫
d2zei(pz+p̄z̄)Dα(z) = πiα−ᾱa(α)D1−α(p),
where the function a(α) ≡ 1/Γ[α] = Γ(1− ᾱ)/Γ(α).
(iii) Chain rule∫
d2w
[z1 − w]α[w − z2]β
= π
a(α, β)
a(γ)
1
[z1 − z2]γ
,
where γ = α+ β − 1. Its diagrammatic form is
= πa(α)a(β)a(γ)
α β γ
(iv) Star-triangle relation
α
β
= πa(α, β, γ)
γ 1− α
1− β 1− γ
(v) Exchange relation
=
α
1− α′
β
1− β′α
′ −
α
1− α
α′ β′
1− β
β
−
β
′
a(α, β̄)a(α′, β̄′) , (A.1)
where α+ β = α′ + β′.
B Scalar products
Here, we discuss the calculation of scalar products of Ψ
(N),ϵ
p,x and Φ
(N)
x functions. The diagrams
for the scalar products (4.3), (4.11) are shown in Figure 2. The leftmost vertex on both diagrams
has only two propagators attached to it. We call such a vertex – free vertex. On the first step
one integrates over the free vertex (on both diagrams) using the chain relation for propagators
and move the resulting line to the right with the help of the exchange relation. After that two
new free vertices appear and one repeat the same procedure again. In this way one can integrate
over all vertices on the left edge of both diagrams (they are shown by black blobs). Keeping
trace of all factors arising in the process, one represent the initial diagram D as
DN
(
{x1, x2, . . . }, {y1, y2, . . . }, {γ1, γ2, . . . }
)
Unitarity of the SoV Transform for SL(2,C) Spin Chains 21
z = 0
Figure 2. Examples of diagrams for scalar products, equations (4.3), (4.11) for N = 4.
= f(x1, y1, γ)D
′
N
(
{x2, . . . }, {y2, . . . }, {γ3, . . . }
)
. (B.1)
Taking into account that the function Ψ
(N)
p,x and Φ
(N)
x are symmetric functions of the separated
variables it follows from (B.1) that
DN
(
{x1, x2, . . . }, {y1, y2, . . . }, {γ1, γ2, . . . }
)
= CN (γ)
∏
k,j
f(xk, yj , γ). (B.2)
The factor CN (γ) does not depend on x, y variables. The easiest way to fix it is to evaluate both
sides of (B.2) for special values of x, y. For example, one can take xk → x and yk → x̄∗. Both
sides, in this limits, contain divergent factors, Γ[i(ȳ∗j − xk)] which cancel out. It is easy check
that the result of the integration over any free vertex in this limit (after removing this singular
factor) gives one. Therefore, the equation on CN (γ) for the scalar product (4.6) takes the form
1 = CN (γ)(χ(x)(χ̄(x̄∗))∗)N−1 = CN (γ)(−1)(N−1)
∑N−3
k=0
[
γ
(k)
2N−3−k−ix
]
.
Since
[
γ
(k)
m − ix
]
is an integer number, one gets that CN = 1 for odd N , while for even N
N−3∑
k=0
[
γ
(k)
2N−3−k − ix
]
=
N−3∑
k=0
([
γ
(k)
2N−3−k − γ
(N−3)
N
]
+
[
γ
(N−3)
N − ix
])
=
N−3∑
k=0
[
γ
(k)
2N−3−k − γ
(N−3)
N
]
+ (N − 2)
[
γ
(N−3)
N − ix
]
.
Taking into account that the last term in the above equation is an even number, one gets
that CN (γ) is given by the expression (4.7). For the second diagram, the analysis follows exactly
the same lines.
C Gustafson’s integral reduction
The extension of the first Gustafson integral [26, Theorem 5.1] to the complex case was obtained
in [16]. It takes the form
N∏
j=1
∑
nj∈Z+σ
2
∫ i∞
−i∞
∏N+1
m=1
∏N
k=1 Γ(zm − uk)Γ(uk + wm)∏
m<j Γ(um − uj)Γ(uj − um)
N∏
p=1
dνp
2πi
22 A.N. Manashov
=
N !
∏N+1
k,j=1 Γ(zk + wj)
Γ
(∑N+1
k=1 (zk + wk)
) , (C.1)
where Γ is the Gamma function of the complex field C [20]
Γ(u) ≡ Γ(u, ū) =
Γ(u)
Γ(1− ū)
=
1
a(u)
.
The variables uk, wm, zm have the form
uk =
nk
2
+ νk, zm =
nm
2
+ xm, wm =
ℓm
2
+ ym,
ūk = −nk
2
+ νk, z̄m = −nm
2
+ xm, w̄m = −ℓm
2
+ ym.
and the integration contours over νk separate the series of poles associated with the Γ-functions:
Γ(zm − uk) and Γ(uk + wm), see [16] for more detail. The integral converges for
N+1∑
m=1
Re(zm + wm) < 1.
Let us put
zN+1 = M
(
1
2
+ ix
)
, z̄N+1 = M
(
−1
2
+ ix
)
,
wN+1 = M ′
(
1
2
+ ix′
)
, w̄N+1 = M ′
(
−1
2
+ ix′
)
and send M,M ′ → ∞ keeping M/M ′ = ξ fixed, so that wN+1/zN+1 7→ ζ and w̄N+1/z̄N+1 7→ ζ̄.
Dividing both sides of (C.1) by (Γ(zN+1)Γ(wN+1))
N , we get in this limit
1
N !
N∏
j=1
∑
nj∈Z+σ
2
∫ i∞
−i∞
[ζ]U
∏N
m,k=1 Γ(zm − uk)Γ(uk + wm)∏
m<j Γ(um − uj)Γ(uj − um)
N∏
p=1
dνp
2πi
=
[ζ]Z
[1 + ζ]Z+W
N∏
k,j=1
Γ(zk + wj), (C.2)
where | arg ζ| < π, Z =
∑N
k=1 zk, W =
∑N
k=1wk and we recall that [ζ]U ≡ ζU ζ̄Ū .
Acknowledgements
The author is grateful to S.É. Derkachov for fruitful discussions and T.A. Sinkevich for critical
remarks.
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1 Introduction
2 SL(2,C) spin chains
3 Eigenfunctions
3.1 B_N operator
3.2 A_N operator
4 Scalar products, momentum representation, etc.
5 SoV representation
5.1 B system
5.2 A system
6 Summary
A The diagram technique
B Scalar products
C Gustafson's integral reduction
References
|
| id | nasplib_isofts_kiev_ua-123456789-212045 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T08:37:29Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Manashov, Alexander N. 2026-01-23T10:11:38Z 2023 Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains. Alexander N. Manashov. SIGMA 19 (2023), 086, 24 pages 1815-0659 2020 Mathematics Subject Classification: 33C70; 81R12 arXiv:2303.11461 https://nasplib.isofts.kiev.ua/handle/123456789/212045 https://doi.org/10.3842/SIGMA.2023.086 We prove the unitarity of the separation of variables transform for SL(2, ℂ) spin chains by a method based on the use of Gustafson integrals. The author is grateful to S. E. Derkachov for fruitful discussions and T.A. Sinkevich for critical remarks. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains Article published earlier |
| spellingShingle | Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains Manashov, Alexander N. |
| title | Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains |
| title_full | Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains |
| title_fullStr | Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains |
| title_full_unstemmed | Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains |
| title_short | Unitarity of the SoV Transform for SL(2, ℂ) Spin Chains |
| title_sort | unitarity of the sov transform for sl(2, ℂ) spin chains |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212045 |
| work_keys_str_mv | AT manashovalexandern unitarityofthesovtransformforsl2cspinchains |