On Complex Gamma-Function Integrals
It was observed recently that relations between matrix elements of certain operators in the SL(2, ℝ) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2, ℂ) symmetry group and L₂(ℂ) as a local Hilbert space give rise to a new type of Γ-...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2020 |
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Інститут математики НАН України
2020
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | On Complex Gamma-Function Integrals. Sergey É. Derkachov and Alexander N. Manashov. SIGMA 16 (2020), 003, 20 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859910519095296000 |
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| author | Derkachov, Sergey É. Manashov, Alexander N. |
| author_facet | Derkachov, Sergey É. Manashov, Alexander N. |
| citation_txt | On Complex Gamma-Function Integrals. Sergey É. Derkachov and Alexander N. Manashov. SIGMA 16 (2020), 003, 20 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | It was observed recently that relations between matrix elements of certain operators in the SL(2, ℝ) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2, ℂ) symmetry group and L₂(ℂ) as a local Hilbert space give rise to a new type of Γ-function integrals. In this work, we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit, these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals.
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| first_indexed | 2025-12-17T12:04:19Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 003, 20 pages
On Complex Gamma-Function Integrals
Sergey É. DERKACHOV † and Alexander N. MANASHOV ‡†
† St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences,
Fontanka 27, 191023 St. Petersburg, Russia
E-mail: derkach@pdmi.ras.ru
‡ Institut für Theoretische Physik, Universität Hamburg, D-22761 Hamburg, Germany
E-mail: alexander.manashov@desy.de
Received October 15, 2019, in final form January 14, 2020; Published online January 18, 2020
https://doi.org/10.3842/SIGMA.2020.003
Abstract. It was observed recently that relations between matrix elements of certain oper-
ators in the SL(2,R) spin chain models take the form of multidimensional integrals derived
by R.A. Gustafson. The spin magnets with SL(2,C) symmetry group and L2(C) as a local
Hilbert space give rise to a new type of Γ-function integrals. In this work we present a direct
calculation of two such integrals. We also analyse properties of these integrals and show that
they comprise the star-triangle relations recently discussed in the literature. It is also shown
that in the quasi-classical limit these integral identities are reduced to the duality relations
for Dotsenko–Fateev integrals.
Key words: Mellin–Barnes integrals; star-triangle relation
2010 Mathematics Subject Classification: 33C70; 81R12
1 Introduction
The multidimensional integrals derived by R.A. Gustafson [27, 28] together with their q- and
elliptic analogues [27, 28, 29, 57, 58, 61] play an important role in different areas of physics
and mathematics such as the theory of multi-variable orthogonal polynomials [62], Selberg type
integrals and constant term identities [1, 22], and supersymmetric dualities in quantum field
theory [60]. Recently, a new field – noncompact spin magnets – was added to this list [15].
Models of this type appear in gauge field theories and have been under intense investigations in
the last two decades, see references [7, 8]. The mathematical description of such systems is well-
developed and known as the quantum inverse scattering method(QISM) [20, 37, 38, 52, 53, 54].
Noncompact spin chains have an infinite-dimensional local Hilbert space and most conveniently
can be analysed within the separation of variable (SoV) framework [53]. It was shown by
Sklyanin that eigenfunctions of the entries of the monodromy matrix provide a suitable basis for
solving the spectral problem for the spin chain Hamiltonian. Building such a basis for a generic
spin chain is in itself a nontrivial problem and until recently the solution was available only for
the spin chains with a rank-1 symmetry group. For the noncompact magnets of interest the
corresponding eigenfunctions are known explicitly [9, 12, 14]. It should be mentioned here that
recently there was a significant progress in constructing the SoV representation for compact
chains of higher ranks, see [10, 24, 25, 34, 35, 40, 49].
In the SL(2,R) spin chin framework, Gustafson’s integrals follow from identities for matrix
elements of the shift operator [15]. Extension of this analysis to the SL(2,C) spin magnets leads
to new integral identities [16, 17] which, up to the expected modification, are the exact replicas
This paper is a contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quan-
tum Field Theory. The full collection is available at https://www.emis.de/journals/SIGMA/elliptic-integrable-
systems.html
mailto:derkach@pdmi.ras.ru
mailto:alexander.manashov@desy.de
https://doi.org/10.3842/SIGMA.2020.003
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html
2 S.É. Derkachov and A.N. Manashov
of the SL(2,R) integrals. The analysis, however, essentially depends on the completeness of the
SoV representation. Proof of completeness is a rather complicated problem, see, e.g., [13, 36, 51].
Completeness for the closed SL(2,R) spin chain follows from that of the Toda chain [36] while
for the SL(2,C) magnets of the length N > 2 it is still an open question. A fruitful strategy
seems to be to make use of Gustafson’s integrals to prove completeness. To realize this one needs
an independent derivation of the corresponding integrals. One of the purposes of this work is
to provide such a derivation. We note also that complex gamma integrals were studied recently
by V.F. Molchanov and Yu.A. Neretin [43] and Yu.A. Neretin [44, 45].
The paper is organized as follows: in Section 2, after setting the notations, we prove two
integral identities which are direct analogs of Gustafson integrals associated with the classical
su(N) and sp(N) Lie algebras [28]. We analyze the analytic properties of these integrals in
Section 3 and derive two new integrals which are the SL(2,C) versions of Gustafson integrals [27]
generalizing the second Barnes lemma. In Section 4 it will be shown that the N = 1, 2 integrals
take, after some rewriting, the form of the star-triangle relations derived in [5, 32, 33] as special
limits of the elliptic star-triangle identity [6, 18, 56, 59]. Section 5 is devoted to the study of the
quasi-classical limit of the Γ-integrals. We show that in this limit the integrals are equivalent to
a special case of the duality relation [2] for Dotsenko–Fateev (DF) integrals [19]. In Section 6 we
present an elementary proof of the above duality relation and give some evidence which suggest
that similar duality relations hold for the Γ-integrals. Section 7 is reserved for a summary.
2 Gamma integrals
2.1 Definitions and basic properties
Let u, ū be a pair of complex numbers of the form
u =
n
2
+ ν, ū = −n
2
+ ν, (2.1)
where n is an integer and ν is complex number. We will use the notations [u] = u − ū = n for
the discrete part and 〈u〉 = ν for the continuous part and put ‖u‖2 = −uū = −ν2 + n2/4 so
that for imaginary ν, ‖u‖2 ≥ 0. The Γ function of the complex field C [23] is defined as
Γ(u, ū) =
Γ(u)
Γ(1− ū)
=
Γ(n/2 + ν)
Γ(1 + n/2− ν)
= (−1)n
Γ(−n/2 + ν)
Γ(1− n/2− ν)
= (−1)[u]Γ(ū, u).
In what follows we will, for brevity, display only the first argument of the Γ function, i.e.,
Γ(u) ≡ Γ(u, ū). Hereafter the following functional relations will be useful
Γ(u)Γ(1− u) = (−1)[u], Γ(u+ 1) = −uūΓ(u). (2.2)
The Γ function appears in the generalization of Gustafson’s integrals to the complex case.
The corresponding integrals take the following form [16, 17]
1
N !
∑
n1,...,nN∈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)
dν1
2πi
· · · dνN
2πi
=
N+1∏
k,j=1
Γ(zk + wj)
Γ
(
N+1∑
k=1
(zk + wk)
) , (2.3a)
On Complex Gamma-Function Integrals 3
1
2NN !
∑
n1,...,nN∈Z+σ
2
∫ i∞
−i∞
N∏
k=1
2N+2∏
m=1
Γ(zm ± uk)
N∏
k=1
Γ(±2uk)
∏
k<j
Γ(±uk ± uj)
dν1
2πi
· · · dνN
2πi
=
∏
j<k
Γ(zj + zk)
Γ
(
2N+2∑
k=1
zk
) , (2.3b)
where we put Γ(a± b) ≡ Γ(a + b)Γ(a− b), Γ(±a± b) ≡ Γ(a + b)Γ(a− b)Γ(−a + b)Γ(−a− b).
We will refer to the integrals in the first and second lines as I
(1)
N and I
(2)
N , respectively.
The variables uk, wm, zm have the form (2.1)
ur =
nr
2
+ νr, zj =
mj
2
+ xj , wm =
`m
2
+ ym
and similarly for the barred variables. However, nr, mi, `m are allowed take integer or half-integer
values, simultaneously. Accordingly, the sums in (2.3) go over integers (σ = 0) or half-integers
(σ = 1/2). For the first integral (2.3a) there is no difference between the integer/half-integer
cases since they are related by the change of variables: [ur], [wm], [zj ] 7→ [ur] + 1/2, [wm] −
1/2, [zj ] + 1/2 so that we will assume that the variables [uk], [wm], [zm] in the integral (2.3a)
are integers.
The integration contours in (2.3) separate the series of “± ” poles due to the Γ functions in
the numerators. The poles are located at
νI,+
rj (p) =
1
2
|nr −mj |+ xj + p, νI,−
rj (p) = −1
2
|nr + `j | − yj − p, p ≥ 0, (2.4)
where r ∈ {1, N}, j ∈ {1, N + 1} for the first integral and
νII,+
rj (p) =
1
2
|nr −mj |+ xj + p, νII,−
rj (p) = −1
2
|nr +mj | − xj − p, p ≥ 0,
where r ∈ {1, N}, j ∈ {1, 2N + 2}, for the second one.
Let us discuss now the convergence properties of the integrals (2.3). Since the integrands
are meromorphic functions and contours of integration avoid poles it is sufficient to analyse the
region of large ur = nr/2+iνr, ‖ur‖2 = ν2
r +n2
r/4→∞, only. With the help of (2.2) we simplify
the denominators in the integrals (2.3) as follows∏
1≤i<k≤N
1
Γ(ui − uk)Γ(uk − ui)
= (−1)(N+1)
∑
k[uk]
∏
1≤i<k≤N
‖ui − uk‖2 (2.5)
and
N∏
k=1
1
Γ(±2uk)
∏
1≤i<m≤N
1
Γ(±ui ± um)
= κN4N
N∏
k=1
‖uk‖2
∏
1≤i<m≤N
‖ui − um‖2‖ui + um‖2, (2.6)
where κN = 1 for the integer case and κN = (−1)N(N+1)/2 for the half-integer case. Finally,
taking into account that for large u
Γ(z − u)Γ(u+ w) = (−1)[z−u] Γ(u+ w)
Γ(u− z)
4 S.É. Derkachov and A.N. Manashov
= (−1)[z−u]uz+w−1(−ū)z̄+w̄−1
(
1 +O (1/‖u‖)
)
(2.7)
we conclude that the integrals (2.3) converge absolutely provided
N+1∑
j=1
Re(xj + yj) < 1 and
2N+2∑
j=1
Re(xj) < 1, (2.8)
respectively. From now on we assume that these conditions are satisfied.
2.2 Determinant representation
In this subsection we present the integrals (2.3) as determinants of 1-dimensional integrals. Such
a representation will be useful in what follows.1 For its derivation let us denote by Q(u|z, w)
the function
Q(u|z, w) =
N+1∏
k=1
(−1)[u]Γ(zk − u)Γ(u+ wk)
and by Qik(z, w) its Mellin moments
Qik(z, w) =
∫
Duui−1(−ū)k−1Q(u|z, w), i, k = 1, . . . , N. (2.9)
Here we introduced a short-hand notation for the integration measure∫
Du ≡
∞∑
n=−∞
∫ i∞
−i∞
dν
2πi
.
Let QN (z, w) be the following N ×N matrix constructed from the Mellin moments
QN (z, w) =
Q11(z, w) · · · Q1N (z, w)
...
. . .
...
QN1(z, w) · · · QNN (z, w)
. (2.10)
Rewriting the product on the r.h.s. of equation (2.5) as the product of two Vandermonde deter-
minants∏
1≤i<k≤N
‖ui − uk‖2 = ∆(u)∆(−ū), ∆(u) = det
1≤i,j≤N
ui−1
j
and taking into account the symmetry of the integrand in (2.3a) with respect to the permutations
ui ↔ uj we bring the first integral into the determinant form
I
(1)
N =
∫
Du1 · · · DuN∆(−ū)
N∏
k=1
Q(uk|z, w)uk−1
k = detQN (z, w). (2.11)
Proceeding in a similar way one gets the determinant representation for the second integral as
follows
I
(2)
N = κN det Q̃N (z) = κN det
∣∣∣∣∣∣∣
Q̃11(z) · · · Q̃1N (z)
...
. . .
...
Q̃N1(z) · · · Q̃NN (z)
∣∣∣∣∣∣∣ ,
1The determinant representation for elliptic hypergeometric integrals was constructed in [47].
On Complex Gamma-Function Integrals 5
where κN is a phase factor, see equation (2.6),
Q̃ik(z) = 2
∫
Duu2i−1(−ū)2k−1Q̃(u, z) and Q̃(u, z) =
2N+2∏
j=1
Γ(zj ± u).
Let us note here that the conditions (2.8) are equivalent to the requirement of absolute conver-
gence of the Mellin moments QNN (z, w) and Q̃NN (z), respectively.
2.3 Proof of identities (2.3)
Calculating the integral (2.3a) we will assume that the parameters zk, wk satisfy the conditions
N+1∑
k=1
Re(zk + wk) < 1 and
N+1∑
k=1
Re(z̄k + w̄k) < 1. (2.12)
These conditions imply the condition (2.8), but do not follow from it and will be removed at
the end of the calculation.
By virtue of (2.8) the integrals (2.9) are all absolutely convergent. We evaluate the integral
over ν by closing the contour in the left half-plane and then computing the sum over the residues.
Recalling that u = n/2 + ν and ū = −n/2 + ν one gets
Mik(n) =
∫ i∞
−i∞
ui−1(−ū)k−1Q(u|z, w)
dν
2πi
= (−1)(N+1)n
N+1∑
j=1
∞∑
p=0
(−1)p
p!p̄j !
(−wj − p)i−1(w̄j + p̄j)
k−1
×
N+1∏
m=1
Γ(zm + wj + p)
Γ(1− z̄m − w̄j − p̄j)
N+1∏
m 6=j
Γ(wm − wj − p)
Γ(1− w̄m + w̄j + p̄j)
,
where p̄j = p+ n+ `j .
The first observation is that since the summand vanishes for p̄j < 0, only the poles at νI,−
rj (p)
given in (2.4), contribute to the integral and, second, that under the assumptions (2.12) the sum
over p converges uniformly on n. Therefore, to evaluate
∑
nMik(n) we swap the summation
over n and p, change the summation variable from n to p̄j and finally suppress the index j in p̄j :
p̄j 7→ p̄
∑
n
Mik(n) =
N+1∑
j=1
∞∑
p=0
∞∑
n=−∞
(. . .) 7→
N+1∑
j=1
∞∑
p=0
∞∑
p̄j=−∞
(. . .) =
N+1∑
j=1
∞∑
p=0
∞∑
p̄=0
(. . .).
The final answer can be written in the form
Qik(z, w) =
N+1∑
j=1
(−1)(N+1)[wj ]
(
N+1∏
m=1
Γ(zm + wj)
Γ(1− z̄m − w̄j)
)N+1∏
m=1
m 6=j
Γ(wm − wj)
Γ(1− w̄m + w̄j)
×
∞∑
p=0
(−wj − p)i−1
N+1∏
m=1
(zm + wj)p
(1− wm + wj)p
×
∞∑
p̄=0
(w̄j + p̄)k−1
N+1∏
m=1
(z̄m + w̄j)p̄
(1− w̄m + w̄j)p̄
, (2.13)
6 S.É. Derkachov and A.N. Manashov
where (a)p is the Pochhammer symbol. Thus Qik =
N+1∑
j=1
Qik(j) =
N+1∑
j=1
Qi(j)Q̄k(j). Hence the
determinant can be represented as follows
detQ =
N+1∑
j1=1
· · ·
N+1∑
jN=1
det ‖Qik(jk)‖ =
∑
σ∈SN+1
det ‖Qik(jσ(k)‖
=
∑
σ∈SN+1
(
N∏
k=1
Q̄k(jσ(k))
)
det ‖Qi(jσ(k))‖,
where we take into account that det ‖Qik(jk)‖ = 0 whenever jm = jn for n 6= m. Then making
use of (2.13) one can bring (2.11) into the following form
I
(1)
N =
1
N !
∑
σ∈SN+1
(−1)
(N+1)
N∑
s=1
[wσ(s)]
N∏
k=1
N+1∏
j=1
Γ(zj + wσ(k))
Γ(1− z̄j − w̄σ(k))
N+1∏
m=1
m 6=σ(k)
Γ(wm − wσ(k))
Γ(1− w̄m + w̄σ(k))
×
∞∑
p1,...,pN=0
(∏
k<m
(wσ(k) + pk − wσ(m) − pm)
)
N+1∏
m=1
N∏
k=1
(zm + wσ(k))pk
(1− wm + wσ(k))pk
×
∞∑
p̄1,...,p̄N=0
(∏
k<m
(w̄σ(m) + p̄m − w̄σ(k) − p̄k)
)
N+1∏
m=1
N∏
k=1
(z̄m + w̄σ(k))p̄k
(1− w̄m + w̄σ(k))p̄k
.
The infinite sums over {p}, {p̄} can be evaluated by Milne’s U(n) Gauss summation [42], see
also [28, equation (5.8)] and [50],
∞∑
p1,...,pN=0
∏
1≤k<m≤N
(ασ(k) + pk − ασ(m) − pm)
N+1∏
i=1
N∏
k=1
(βi + ασ(k))pk
(1− αi + ασ(k))pk
=
Γ
(
1−
N+1∑
k=1
(αk + βk)
)
N∏
i=1
Γ(1 + ασ(i) − ασ(N+1))
N+1∏
i=1
Γ(1− βi − ασ(N+1))
. (2.14)
Using (6.5) we obtain the following representation for the integral I
(1)
N
I
(1)
N =
1
N !
∑
σ∈SN+1
(−1)
(N+1)
N∑
s=1
[wσ(s)]
N∏
k=1
N+1∏
i=1
Γ(zi + wσ(k))
Γ(1− z̄i − w̄σ(k))
N+1∏
m=1
m 6=σ(k)
Γ(wm − wσ(k))
Γ(1− w̄m + w̄σ(k))
×
∏
1≤m<j≤N
(wσ(m) − wσ(j))Γ
(
1−
N+1∑
k=1
(zk + wk)
) N∏
k=1
Γ(1 + wσ(k) − wσ(N+1))
N+1∏
k=1
Γ(1− zk − wσ(N+1)))
×
∏
1≤m<j≤N
(w̄σ(j) − w̄σ(m))Γ
(
1−
N+1∑
k=1
(z̄k + w̄k)
) N∏
k=1
Γ(1 + w̄σ(k) − w̄σ(N+1))
N+1∏
k=1
Γ(1− z̄k − w̄σ(N+1))
.
On Complex Gamma-Function Integrals 7
After some simplifications this can be written as
I
(1)
N =
Γ
(
1−
N+1∑
k=1
(z̄k + w̄k)
)
Γ
(
N+1∑
k=1
(zk + wk)
) N+1∏
i,k=1
Γ(zi + wk)
Γ(1− z̄i − w̄k)
RN
N ! sinπ
(
N+1∑
k=1
(zk + wk)
) ,
where
RN =
∑
σ∈SN+1
N+1∏
k=1
sinπ(zσ(k) + wσ(N+1))
N∏
k=1
sinπ(wσ(N+1) − wσ(k))
× (−1)
(N+1)
N∑
s=1
[wσ(s)] ∏
1≤k<j≤N
sinπ(w̄σ(j) − w̄σ(k))
sinπ(wσ(j) − wσ(k))
. (2.15)
Taking into account that wk − w̄k is an integer one finds that the last product in (2.15) yields
(−1)
(N−1)
N∑
s=1
[wσ(s)]
which cancels the second factor in (2.15). In the last step we use the [28,
Lemma 5.10] which states that
∑
σ∈SN+1
N+1∏
k=1
sinπ(βk + ασ(N+1))
N∏
k=1
sinπ(ασ(N+1) − ασ(k))
= N ! sinπ
N+1∑
k=1
(αk + βk).
It results in
RN = N ! sinπ
N+1∑
k=1
(zk + wk),
so that we get the required result for I
(1)
N
I
(1)
N =
N+1∏
k,j=1
Γ(zk + wj)
Γ
(
N+1∑
k=1
(zk + wk)
) .
Finally by analytic continuation in the νk the assumptions (2.12) can be relaxed to the condi-
tion (2.8).
One sees that the crucial point in the proof of (2.3a) is the factorization of the double
sum arising after evaluating the integral over ν by residue theorem into the product of two
infinite sums, see equation (2.13). We remark that this factorization was first noticed by Ismagi-
lov [30, 31], see also [45]. After this property is established the further analysis follows along
the lines of [28].
The proof of the identity (2.3b) proceeds along the same lines so we will not go into details
and only highlight the essential differences. First, we assume that the parameters zk satisfy the
conditions
2N+2∑
k=1
Re(zk) < 1 and
2N+2∑
k=1
Re(z̄k) < 1. (2.16)
8 S.É. Derkachov and A.N. Manashov
After evaluating the integrals over ν using residue calculus, and carrying out some minor sim-
plifications, we obtain
I
(2)
N =
2NκN
N !
(−1)N
∑
π
({ ∞∑
y1,...,yN=0
N∏
j=1
(zπ(j) + yj)
∏
1≤i<j≤N
(
zπ(i) + yi ± (zπ(j) + yj)
)
×
N∏
k=1
(−1)yk
yk!
Γ(2zπ(k) + yk)
2N+2∏
j=1
j 6=π(k)
Γ(zj ± (zπ(k) + yk))
}
×
{ ∞∑
ȳ1,...,ȳN=0
N∏
j=1
(z̄π(j) + ȳj)
∏
1≤i<j≤N
(
z̄π(i) + ȳi ± (z̄π(j) + ȳj)
)
×
N∏
k=1
1
ȳk!
1
Γ(1− 2z̄π(k) − ȳk)
2N+2∏
j=1
j 6=π(k)
1
Γ(1− z̄j ± (z̄π(k) + ȳk))
})
,
where π is an injective map from {1, . . . , N} to {1, . . . , 2N + 2}. The sums over yj , ȳj can be
evaluated with the help of the following hypergeometric series summation formula, see [26, 28]
∞∑
y1,...,yn=−∞
n∏
j=1
zj + yj
zj
∏
1≤i<j≤n
zi + yi ± (zj + yj)
zi ± zj
2n+2∏
i=1
n∏
j=1
Γ(wi ± (zj + yj))
Γ(wi ± zj)
=
Γ
(
1−
2n+2∑
k=1
wi
)
2n+2∏
i=1
n∏
j=1
Γ(1− wi ± zj)
N∏
j=1
Γ(1± 2zj)
∏
1≤i<j≤n
Γ(1± zi ± zj)
∏
1≤i<j≤2n+2
Γ(1− wi − wj)
.
Collecting all factors we obtain
I
(2)
N =
∏
1≤i<j≤2N+2
Γ(zi + zj)
Γ
(
1−
∑
k
zk
) × TN ,
where
TN =
(−1)N
2NN !
∏
1≤i<j≤2N+2
sinπ(zi + zj)
sinπ
(∑
k
zk
) ∑
π
∏
1≤i<j≤N
sin2 π(zπ(i) ± zπ(j))
N∏
j=1
sinπ(2zπ(j))
N∏
j=1
2N+2∏
i=1
i 6=π(j)
sinπ(zi ± zπ(j))
.
One can show that TN = 1 by taking the limit q → 1 of the identities in [28, equations (7.11)
and (7.12)]. Finally, the conditions (2.16) can be relaxed to (2.8) by analytic continuation in
the νk.
3 Limiting cases
In this section we derive two more integrals which are replicas of the integrals [27, equations (3.2)
and (5.4)]. We show that in the complex case these integrals are intrinsically related to the
integrals (2.3). This property is not seen in the SL(2,R) setup.
On Complex Gamma-Function Integrals 9
We start our analysis with the integral (2.3a) and introduce the variables, complex ζ and
integer η, as follows
ζ +
η
2
=
N+1∑
k=1
(zk + wk), ζ − η
2
=
N+1∑
k=1
(z̄k + w̄k). (3.1)
The r.h.s. of (2.3a) is a meromorphic function of ζ with poles located at the points ζp =
|1 + η/2|+ p, p ∈ N+. For η = 0 and ζ close to 1 the r.h.s. of (2.3a) takes the form
I
(1)
N =
1
1− ζ
N+1∏
k,j=1
Γ(zk + wj) + · · · , (3.2)
where zk, wk obey the constraint
N+1∑
k=1
(zk +wk) =
N+1∑
k=1
(z̄k + w̄k) = 1. At the same time, only the
element QNN of the matrix (2.10) becomes singular at this point. The corresponding integral
diverges at large ν and n as ζ → 1. Indeed, taking into account equation (2.7) one finds
Q(u|z, w) = (−1)
∑
k[zk]uζ−N−1(−ū)ζ−N−1
(
1 +O(1/‖u‖)
)
.
Thus for ζ → 1
QN,N (z, w) ' (−1)
∑
k[zk] 1
π
∫
r>Λ
1
r4−2ζ
dxdy + · · · = 1
1− ζ
(−1)
∑
k[zk] + · · · (3.3)
and therefore
I
(1)
N =
ζ→1
QNN (z, w)× det Q̂N−1(z, w) + finite terms.
Here Q̂N−1 is the main N − 1 minor of the matrix QN . Comparing the residues at ζ = 1 in
equation (3.3) and (3.2) we obtain the following identity
det Q̂N−1(z, w) = (−1)
∑
k[zk]
N+1∏
k,j=1
Γ(zk + wj), (3.4)
which holds provided
N+1∑
k=1
(zk + wk) =
N+1∑
k=1
(z̄k + w̄k) = 1. Rewriting the l.h.s. of (3.4) in an
integral form one obtains
1
(N − 1)!
∫
Du1 · · · DuN−1
N−1∏
p=1
(−1)np
N+1∏
j=1
N−1∏
k=1
Γ(zj − uk)Γ(uk + wj)∏
k<j
Γ(uk − uj)Γ(uj − uk)
= (−1)
∑
m[zm]
N+1∏
k,j=1
Γ(zk + wj). (3.5)
This integral is an analog of the integral [27, equation (3.2)]. Indeed, replacing the variable zN+1
by zN+1 = 1 −
N∑
k=1
zk −
N+1∑
m=1
wm and using the relation (2.2) one can bring equation (3.5) into
the form which is a replica of [27, equation (3.2)]
1
(N − 1)!
∫
Du1 · · · DuN−1
N−1∏
k=1
N∏
m=1
∏N+1
j=1 Γ(zm − uk)Γ(uk + wj)
N−1∏
m=1
Γ(γ + um)
∏
k<j
Γ(uk − uj)Γ(uj − uk)
10 S.É. Derkachov and A.N. Manashov
=
N∏
k=1
N+1∏
j=1
Γ(zk + wj)
N+1∏
j=1
Γ(γ − wj)
, (3.6)
where γ =
N∑
k=1
zk +
N+1∑
m=1
wm.
The analysis of the second integral, equation (2.3b), proceeds along the same lines so we give
only a brief account. Similar to (3.1) we define variables, η and ζ, by ζ + η/2 =
2N+2∑
k=1
zk. The
l.h.s. and r.h.s. of equation (2.3b) have a pole at ζ = 1 (for η = 0). Comparing the corresponding
residues we get
1
2N−1(N − 1)!
∫
±
Du1 · · · DuN−1
N−1∏
k=1
2N+2∏
j=1
Γ(zj ± uk)
N−1∏
k=1
Γ(±2uk)
∏
j<k
Γ(±uk ± uj)
= ±
∏
1≤j<k≤2N+2
Γ(zj + zk), (3.7)
where
2N+2∑
k=1
zk =
2N+2∑
k=1
z̄k = 1 and the subscript ± at the integral sign indicates that the sum
goes over either integer n (plus) or half-integer n (minus). Introducing the variable γ =
2N+1∑
k=1
zk,
one can rewrite this integral in the form identical to the integral [27, equation (5.4)]
21−N
(N − 1)!
∫
±
Du1 · · · DuN−1
N−1∏
k=1
2N+1∏
j=1
Γ(zj ± uk)
N−1∏
k=1
Γ(γ ± uk)Γ(±2uk)
∏
j<k
Γ(±uk ± uj)
=
∏
1≤j<k≤2N+1
Γ(zj + zk)
2N+3∏
k=1
Γ(γ − zk)
. (3.8)
Thus in the SL(2,C) setup the integrals (3.5), (3.6) and (3.7), (3.8) are intrinsically related to
the integrals (2.3). For N = 2 the relation (3.8) was derived by Sarkissian and Spiridonov [55].
4 Star-triangle relation
In this section we show that the star-triangle relations with the Boltzmann weights given by
a product of Γ-functions [5, 32, 33] follow in a rather straightforward way from the integrals (2.3).
The star-triangle relation underlies exact solvability of various two-dimensional lattice mod-
els, see references [3, 4] for a review. Here we recall such a relation inherent to the noncom-
pact SL(2,C) spin chain magnets [12].
Let sα(z) ≡ sα,ᾱ(z, z̄) be a function of the complex variables z = x+ iy, z̄ ≡ z∗ = x− iy,
sα(z) = [z]−α ≡ z−αz̄−ᾱ, [α] = α− ᾱ ∈ Z.
On Complex Gamma-Function Integrals 11
The function sα(z) is a single valued function on the complex plane and in the physics literature
it is usually called a propagator.2 It satisfies two identities:
• the chain relation
1
π
∫
sα1(z1 − z)sα2(z − z2)d2z =
Γ(1− α1)Γ(1− α2)
Γ(2− α1 − α2)
sα1+α2−1(z1 − z2), (4.1)
• the star-triangle relation
1
π
∫ 3∏
k=1
sαk(zk − z)d2z
=
(
3∏
k=1
Γ(1− αk)
)
s1−α1(z2 − z3)s1−α2(z3 − z1)s1−α3(z1 − z2), (4.2)
which holds provided α1 + α2 + α3 = ᾱ1 + ᾱ2 + ᾱ3 = 2.
In fact these two relations are equivalent: equation (4.2) is reduced to equation (4.1) in the
limit z3 → ∞ and, vice versa, equation (4.2) can be derived from equation (4.1) by a SL(2,C)
transformation. The relation (4.2) underlies the integrability of the noncompact SL(2,C) spin
chain magnets.
In [5, 32, 33] new solutions of the star-triangle relation have been derived from the elliptic
star-triangle relation [6, 18, 56, 59]. Below we show that these star-triangle relations can be
derived from the integrals (2.3).
First we consider the relation associated with the integral (2.3a). To this end we define the
propagator
Sα(u) = (−1)[α/2+u]Γ
(
1− α
2
+ u
)
Γ
(
1− α
2
− u
)
= (−1)[α/2+u] Γ
(
1−α
2 + u
)
Γ
(
1+ᾱ
2 − ū
) Γ
(
1−α
2 − u
)
Γ
(
1+ᾱ
2 + ū
) . (4.3)
The variables u and α have the form
u = n/2 + iν, ū = −n/2 + iν, α = m+ σ, ᾱ = −m+ σ,
where [u] = n and 1
2 [α] = m are either both integer or half-integer numbers and 〈u〉 = ν,
〈α〉 = σ are real complex numbers, respectively. Under these conditions the arguments of the
Γ-functions in (4.3) have the form (2.1). Slightly abusing the terminology we call the u (α) even
or odd depending on the character of n (m) and refer to this property as parity. Also, in order
to avoid possible misunderstanding due to our agreement to indicate only the “holomorphic”
arguments of functions, f(α) ≡ f(α, ᾱ), we accept that, whenever x̄ is not explicitly defined,
x+ α ≡ (x+ α, x+ ᾱ), e.g., Γ(1/2 + z) ≡ Γ(1/2 + z, 1/2 + z̄).
The propagator Sα inherits many properties of sα:
• for even(odd) α the propagator is an even(odd) function of u
Sα(−u) = (−1)[α]Sα(u),
• for imaginary 〈α〉 = σ, Sα(u)(Sα(u))† = 1, i.e., the propagator reduces to a phase factor,
while for [α] = 0 and 〈α〉 real, Sα(u) is real and positive.
2Let us stress here that [z]α denotes the power function while [α] without any superscript stands for the
“integer” part of α, [α] = α− ᾱ. We hope that this slightly unfortunate notation does not lead to confusion.
12 S.É. Derkachov and A.N. Manashov
The chain relation for the propagator S follows from the integral (2.3a) for N = 1. Namely,
making the substitution
z1 = (1− α1)/2 + z, w1 = (1− α1)/2− z,
z2 = (1− α2)/2 + w, w2 = (1− α2)/2− w
in (2.3a) one obtains
∞∑
n=−∞
∫ i∞
−i∞
Sα1(z − u)Sα2(u− w)
dν
2πi
=
Γ(1− α1)Γ(1− α2)
Γ(2− α1 − α2)
Sα1+α2−1(z − w), (4.4)
where u = n/2 + ν and sum goes over integers. The parity of the α1 and z (α2 and w) is always
the same. The integral (4.4) is well defined provided Re〈αk〉 < 1, k = 1, 2 and Re〈α1 +α2〉 > 1:
the poles of the Γ functions in the integral (4.4) are separated by the integration contour if
Re〈αk〉 < 1 and the integral converges at large u if Re〈α1 + α2〉 > 1.
The star-triangle relation for Sα can be obtained from the integral identity (3.5) for N = 2.
Let us make the following substitution
zi 7→
1− αi
2
+ zi, z̄i 7→
1− ᾱi
2
+ z̄i, wi 7→
1− αi
2
− zi, w̄i 7→
1− ᾱi
2
− z̄i
in that equation. Taking into account that the condition
∑
k(zk + wk) = 1 gives rise to the
following restriction on the indices:
∑
k αk =
∑
k ᾱk = 2, one derives after some algebra
∞∑
n=−∞
∫ i∞
−i∞
3∏
k=1
Sαk(zk − u)
dν
2πi
=
[
3∏
k=1
Γ(1− αk)
]
S1−α1(z2 − z3)S1−α2(z3 − z1)S1−α3(z1 − z2). (4.5)
Here, again, the αk and zk are even or odd simultaneously. For the special choice of the param-
eters, αk = ᾱk, this relation coincides with the star-triangle relation [5, equation (22)].
Proceeding with the second integral (2.3b) we define the propagator as the product of four Γ
functions
Dα(z1, z2) = Γ
(
1− α
2
± z1 ± z2
)
=
Γ
(
1−α
2 ± z1 ± z2
)
Γ
(
1+ᾱ
2 ± z̄1 ± z̄2
) .
The requirement for the arguments of Γ functions, 1−α
2 ± z1± z2, to be integers imposes obvious
restrictions on the relative parity of all variables. Obviously, the propagator Dα(z1, z2) is an
even function of z1, z2 and invariant under the z1 ↔ z2 permutation. Therefore, for each variable
zk = nk/2 + νk one can restrict nk to positive (negative) values. Note that unlike the previous
case the propagator Dα is not shift invariant. Also, up to a phase factor depending on the parity
of arguments, Dα(z1, z2) ∼ Sα(z1 − z2)Sα(z1 + z2).
The chain relation for the propagator Dα follows from the identity (2.3b). Indeed, after the
substitution z1(2) = (1− α1)/2± z, z3(4) = (1− α2)/2± w the integral (2.3b) takes the form
2
∫
±
Du‖u‖2Dα1(z, u)Dα2(u,w) = ±Γ(1− α1)Γ(1− α2)
Γ(2− α1 − α2)
Dα1+α2−1(z, w).
Here the subscripts ± indicate that the sum goes over all integers (“+”) or half-integers (“−”)
and we also recall that for u = n/2 + iν, ‖u‖2 = ν2 + n2/4.
On Complex Gamma-Function Integrals 13
Next, substituting z2i−1 = (1−αi)/2+zi, z2i = (1−αi)/2−zi, for i = 1, 2, 3 in equation (3.8)
for N = 2 one gets
2
∫
±
Du‖u‖2
3∏
k=1
Dαk(zk, u)
=
(
3∏
k=1
Γ(1− αk)
)
D1−α1(z2, z3)D1−α2(z1, z3)D1−α3(z1, z2), (4.6)
where
∑
k αk =
∑
k ᾱk = 2. As was mentioned earlier the parity of all variables have to be
coordinated so that (4.6) encompasses four different identities:
(1) All αk are integer / have positive parity
(a) zk, u are even,
(b) zk, u are odd,
(2) α1 and α2, α3 are even and odd, respectively
(a) u and z1 are even and z2, z3 are odd,
(b) u and z1 are odd and z2, z3 are even.
Variant (1a) corresponds to the star-triangle relation obtained by Kels [32, equation (17)]. An
extension of the star-triangle relation of [32] which follows from the relation (3.8) with N = 2
was also considered by Sarkissian and Spiridonov [55].
In Cases (1a) and (1b) the functions Dα(z1, z2) are real if 0 < α = ᾱ < 1: Dα(z1, z2) > 0
for even z1, z2 (Case (1a)) and −Dα(z1, z2) > 0 for odd z1, z2 (Case (1b)). In both cases these
functions can be interpreted as the Boltzmann weights of lattice integrable models, for more
details see references [32, 33].
5 Quasi-classical limit
Let us consider the identities (4.4), (4.5) when the external variables become large. We replace
the variables zk(wk) in (4.4), (4.5) by Lzk = L(xk + iyk) and take the limit L → ∞. The
variables xk, yk can be considered in this limit as continuous ones, so that zk ∈ C, z̄k = z∗k. It
is easy to check that the leading contribution to the integrals (4.4), (4.5) comes from the region
where uk ∼ L, so that we replace uk 7→ Luk as well. Moreover in this limit Sα turns into sα:
L2〈α〉Sα(Lz) 7→
L→∞
z−αz̄−ᾱ = sα(z). (5.1)
The sum over n in
∫
Du can be replaced by the integral so that
1
L2
∞∑
n=−∞
∫ i∞
−i∞
dν
2πi
7→
L→∞
1
π
∫
d2u, (5.2)
where u = ux+iuy and d2z = duxduy. Taking into account equations (5.1) and (5.2) one checks
that in the limit L→∞ equations (4.4), (4.5) become the chain and the star-triangle relations,
equations (4.1) and (4.2), respectively.
In what follows we study the quasi-classical limit the integrals (2.3a) and (3.5) for general N .
First of all we rewrite these identities in term of the propagator Sα. After the change of variables
zk 7→
1− αk
2
+ zk, wk 7→
1− αk
2
− zk, k = 1, . . . , N + 1.
14 S.É. Derkachov and A.N. Manashov
Equation (2.3a) takes the following form
1
N !
∫
Du1 · · · DuN
∏
1≤i<j≤N
‖ui − uj‖2
N∏
k=1
N+1∏
m=1
Sαm(zm − uk)
= (−1)
N∑
m=1
m[αm+1]
N+1∏
k=1
Γ(1− αk)
Γ
(
N + 1−
N+1∑
k=1
αk
) ∏
1≤i<k≤N+1
Sαi+αk−1(zi − zk).
Similarly, the identity (3.5) can be represented as follows
1
(N − 1)!
∫
Du1 · · · DuN−1
∏
1≤i<j≤N−1
‖ui − uj‖2
N∏
k=1
N+1∏
m=1
Sαm(zm − uk)
= (−1)
N∑
m=1
m[αm+1]
(
N+1∏
k=1
Γ(1− αk)
) ∏
1≤i<k≤N+1
Sαi+αk−1(zi − zk),
and where
∑
k αk =
∑
k ᾱk = N .
In the quasi-classical limit these identities are reduced to the following two-dimensional inte-
grals with power functions
1
N !
∫ ∏
1≤i<j≤N
|ui − uj |2
N∏
k=1
N+1∏
m=1
[zm − uk]−αmd2u1 · · · d2uN
= πN (−1)
N∑
m=1
m[αm+1]
N+1∏
k=1
Γ(1− αk)
Γ
(
N + 1−
N+1∑
k=1
αk
) ∏
1≤i<k≤N+1
[zi − zk]1−αi−αk (5.3)
and
1
(N − 1)!
∫ ∏
1≤i<j≤N−1
|ui − uj |2
N∏
k=1
N+1∏
m=1
[zm − uk]−αmd2u1 · · · d2uN−1
= πN−1(−1)
N∑
m=1
m[αm+1]
(
N+1∏
k=1
Γ(1− αk)
) ∏
1≤i<k≤N+1
[zi − zk]1−α1−αk . (5.4)
It can be checked that in the quasi-classical limit the propagator Dα(z, w) turns into
[
z2−w2
]−α
.
Therefore equations (2.3b) and equation (3.7) do not produce new identities in this limit but
reduce to the integrals (5.3) and (5.4) after an appropriate change of variables.
One can take a different point of view on the integrals (2.3a) and (2.3b) and consider them as
the “quantized” version of the integrals (5.3). For αk = ᾱk the integral (5.3) is the special case
of the duality relation [2] for the Dotsenko–Fateev (DF) integrals [19] considered in the next
section. Therefore one can hope that there exists a “quantized” version of the duality relation
in a general situation.
On Complex Gamma-Function Integrals 15
6 Dotsenko–Fateev integrals
In this section we give an elementary proof of the following relation
1
πnn!
∫ n∏
i<k
[yi − yk]
n∏
i=1
n+m+1∏
j=1
[yi − zj ]−αjd2y1 · · · d2yn =
(−1)
n+m∑
k=1
k[αk+1] n+m+1∏
j=1
Γ(1− αj)
Γ
(
1 + n−
n+m+1∑
j=1
αj
)
×
n+m+1∏
i<j
[zj − zi]1−αi−αj
1
πmm!
∫ m∏
i<k
[ui − uk]
m∏
i=1
n+m+1∏
j=1
[ui − zj ]−1+αjd2u1 · · · d2um, (6.1)
which is, provided the parameters satisfy the constraint αk = ᾱk, the duality relation [2, 21] for
the DF integrals [19], see also [46]. For m = 0 this identity coincides with (5.3).
To evaluate the integral on the l.h.s. of (6.1) we go over to the variables xk = xk(y1, . . . , yn)
which are essentially the elementary symmetric functions
n∏
i=1
(yi + t) = x1 + x2t+ x3t
2 + · · ·+ xnt
n−1 + tn.
Every point in x-space has n! preimages in y-space and the Jacobian of the transformation is
J =
∣∣∂xk
∂yi
∣∣2 =
n∏
i<j
[yi − yk]. Introducing the notation tj = −zj one gets for the integral on the
l.h.s. of (6.1)
π−n
∫ n+m+1∏
j=1
[
x1 + x2tj + x3t
2
j + · · ·+ xnt
n−1
j + tnj
]−αjd2x1 · · · d2xn. (6.2)
Then using the momentum space representation for the propagators
[z]−α =
1
π
iα−ᾱΓ(1− α)
∫
ei(pz+z̄p̄)[p]α−1d2p
in (6.2) and carrying out all integrals over xk one gets
i
∑
j(αj−ᾱj)
πm+1
n+m+1∏
k=1
Γ(1− αk)
n+m+1∏
i=1
∫
d2pi[pi]
αk−1
n∏
k=1
δ(2)
∑
j
pjt
k−1
j
ei
∑
j(pjt
n
j +p̄j t̄
n
j )
=
n+m+1∏
k=1
Γ(1− αk)
πmΓ
(
n+ 1−
∑
j αj
) n+m+1∏
i=1
∫
d2pi
[pi]1−αi
n∏
k=1
δ(2)
∑
j
pjt
k−1
j
δ(2)
∑
j
pjt
n
j − 1
.
In order to get the last δ-function one represents ei
∑
j(pjt
n
j +p̄j t̄
n
j ) as
∫
ei(λ+λ̄)δ(2)(λ−
∑
j pjt
n
j )dλ
and rescales pj → λpj . The delta functions cut out a m-dimensional subspace in the (n+m+1)-
dimensional space defined by the linear equations
n+m+1∑
j=1
pjt
k−1
j = 0, k = 1, . . . , n and
n+m+1∑
j=1
pjt
n
j = 1. (6.3)
The solution depends on m free variables, u1, . . . , um, and takes the form
pj(u1, . . . , um) = λj(u1 + tj) · · · (um + tj), (6.4)
16 S.É. Derkachov and A.N. Manashov
where λj =
∏
k 6=j
(tj − tk)−1, j = 1, . . . , n + m + 1. That pj , (6.4), satisfy equations (6.3) follows
from an identity due to Milne [41, Lemma 1.33], see also [48, equation (2.3)],
N∑
k=1
(b1 − tk) · · · (bN − tk)
tk
∏
j 6=k
(tj − tk)
=
b1 · · · bN
t1 · · · tN
− 1. (6.5)
To see it is enough to put N = m + n + 1, bk = −uk for k ≤ m and bk = −u, for k > m in
the above identity and compare the coefficients at the powers uk, k = 0, . . . , n on both sides of
(6.5).
Making the change of variables
pj = λj(u1 + tj) · · · (um + tj) + sj , j = 1, . . . , n+ 1,
pj = λj(u1 + tj) · · · (um + tj), j = n+ 2, . . . , n+m+ 1,
and taking into account that
n∏
k=1
δ2
n+m+1∑
j=1
pjt
k−1
j
δ2
n+m+1∑
j=1
pjt
n
j − 1
=
n+1∏
k=1
δ2
n+1∑
j=1
sjt
k−1
j
=
n+1∏
j=1
δ2 (sj)∏
1≤i<j≤n+1
[ti − tj ]
and
n+m+1∏
j=1
d2pj =
∏
n+2≤i<j≤n+m+1
[ti − tj ]
n+m+1∏
k=n+2
[λk]
1
m!
m∏
j=1
d2uj
n+1∏
j=1
d2sj
one gets after some algebra
n+m+1∏
k=1
Γ(1− αk)
Γ
(
n+ 1−
∑
j αj
)
∏
j<k
(−1)αk−ᾱk [tj − tk]1−αk−αj
× 1
πmm!
n+m+1∏
j=1
∫ m∏
k=1
(uk + tj)
αj−1d2u1 · · · d2um.
It coincides with the r.h.s. of equation (6.1) after changing tj → −zj . We have learned from
the discussions with Litvinov that similar proof of the duality relation (6.1) is presented in his
lectures on conformal field theory [39].
Finally, we make a conjecture that the duality relation (6.1) admits a generalization to the
“quantized” cases. Namely, these relations take the form
1
n!
∫
Du1 · · · Dun
∏
i<j
‖ui − uj‖2
n∏
i=1
n+m+1∏
j=1
(−1)[ui]Γ(zj − ui)Γ(ui + wj)
=
n+m+1∏
i,j=1
Γ(zi + wj)
Γ
(∑
j(zj + wj)−m
)(−1)m
∑
j [zj−wj ] 1
m!
∫
Du1 · · · Dum
×
∏
i<j
‖ui − uj‖2
m∏
i=1
n+m+1∏
j=1
(−1)[ui]Γ(z′j − ui)Γ(w′j + ui), (6.6)
On Complex Gamma-Function Integrals 17
where z′j = 1/2− wj , w′j = 1/2− zj and
2n
n!
∫
±
Du1 · · · Dun‖uk‖2
∏
1≤i<j≤n
‖ui ± uj‖2
n∏
i=1
2(n+m+1)∏
j=1
Γ(zj ± ui)
= κn+m
2(n+m+1)∏
i<j
Γ(zi + zj)
Γ
(∑
j zj −m
) 2m
m!
∫
±
Du1 · · · Dum‖uk‖2
×
∏
1≤i<j≤m
‖ui ± uj‖2
m∏
i=1
2(n+m+1)∏
j=1
Γ(1/2− zj ± ui), (6.7)
where κk = 1, (−1)k(k+1)/2 for the integer and half-integer cases, respectively. For m = 0
these integrals are equivalent to the integrals (2.3) and in the quasi-classical limit both of them
reproduce the duality relation (6.1). For n = m = 1 the relations (6.6) and (6.7) follow from the
star-triangle relations (4.5) and (4.6). For few first n and m the integrals (6.6), (6.7) go through
numerical tests. Closing this section we note that quite similar duality relations were observed
recently in the so-called conformal fishnet model [11].
7 Summary
In [16, 17] a generalization of Gustafson integrals to the complex case have been obtained. The
derivation of these integrals rely on the completeness of the SoV representation for the SL(2,C)
magnets, that is not yet proven. In this work we presented a direct calculation of two Γ function
integrals. We expect that these integral identities will be helpful in proving the completeness of
the SoV representation for the SL(2,C) spin chains.
The complex integrals are, up to appropriate modification of the Γ function and integration
measure, exact copies of the integrals obtained by R.A. Gustafson [28]. However, the analytic
properties of these integrals are different. We have shown that several, apparently distinct in
the SL(2,R) context integrals are intrinsically related to each other in the SL(2,C) formulation.
We have also shown that the complex Γ integrals for the lowest N underlie the star-triangle
identities derived in references [5, 32, 33] and in the quasi-classical limit are reduced to the special
(m = 0) case of the duality relation for the DF integrals [2, 19, 21]. We also conjecture an appro-
priate modification of these duality relations to hold for the integrals with complex Γ functions.
Acknowledgements
We are grateful to A.V. Litvinov and V.P. Spiridonov for fruitful discussions. We would also
like to express our sincere gratitude to the referees for their numerous useful comments and
suggestions. This study was supported by the Russian Science Foundation project No 19-11-
00131 and Deutsche Forschungsgemeinschaft (A.M.), grant MO 1801/1-3.
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https://arxiv.org/abs/math.CA/0411044
https://doi.org/10.1090/S0002-9947-99-02551-9
https://arxiv.org/abs/q-alg/9707005
1 Introduction
2 Gamma integrals
2.1 Definitions and basic properties
2.2 Determinant representation
2.3 Proof of identities (2.3)
3 Limiting cases
4 Star-triangle relation
5 Quasi-classical limit
6 Dotsenko–Fateev integrals
7 Summary
References
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| id | nasplib_isofts_kiev_ua-123456789-210607 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:19Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Derkachov, Sergey É. Manashov, Alexander N. 2025-12-12T10:40:53Z 2020 On Complex Gamma-Function Integrals. Sergey É. Derkachov and Alexander N. Manashov. SIGMA 16 (2020), 003, 20 pages 1815-0659 2010 Mathematics Subject Classification: 33C70; 81R12 arXiv:1908.01530 https://nasplib.isofts.kiev.ua/handle/123456789/210607 https://doi.org/10.3842/SIGMA.2020.003 It was observed recently that relations between matrix elements of certain operators in the SL(2, ℝ) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2, ℂ) symmetry group and L₂(ℂ) as a local Hilbert space give rise to a new type of Γ-function integrals. In this work, we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit, these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals. We are grateful to A.V. Litvinov and V.P. Spiridonov for fruitful discussions. We would also like to express our sincere gratitude to the referees for their numerous useful comments and suggestions. This study was supported by the Russian Science Foundation project No 19-1100131 and Deutsche Forschungsgemeinschaft (A.M.), grant MO 1801/1-3. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Complex Gamma-Function Integrals Article published earlier |
| spellingShingle | On Complex Gamma-Function Integrals Derkachov, Sergey É. Manashov, Alexander N. |
| title | On Complex Gamma-Function Integrals |
| title_full | On Complex Gamma-Function Integrals |
| title_fullStr | On Complex Gamma-Function Integrals |
| title_full_unstemmed | On Complex Gamma-Function Integrals |
| title_short | On Complex Gamma-Function Integrals |
| title_sort | on complex gamma-function integrals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210607 |
| work_keys_str_mv | AT derkachovsergeye oncomplexgammafunctionintegrals AT manashovalexandern oncomplexgammafunctionintegrals |