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-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.

Models of this type appear in gauge field theories and have been under intense investigations in the last two decades, see for a review [12,13]. Mathematical description of such systems is welldeveloped and known as Quantum Inverse Scattering Method (QISM), [14][15][16][17][18][19]. Noncompact spin chains have an infinite-dimensional local Hilbert space and most conveniently can be analysed within the Separation of Variable (SoV) framework [18]. It was shown by Sklyanin that eigenfunctions of the entries of a 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. † Fortunately, for the noncompact magnets of interest the corresponding eigenfunctions are known explicitly [23][24][25].
In the SL(2, R) spin chin framework, Gustafson's integrals follow from the identities for matrix elements of the shift operator [11]. Extension of this analysis to the SL(2, C) spin magnets leads to new integral identities [26,27] which, up to the expected modifications, are the exact replicas 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. refs. [28][29][30]. Completeness for the closed SL(2, R) spin chain follows from that for the Toda chain [28] 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 make it feasible one needs an independent derivation of the corresponding integrals. One of the purposes of this work is to provide such a derivation.
The paper is organized as follows: in sect. 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 [2]. We analyze analytic properties of these integrals in sect. 3 and derive two new integrals which are the SL(2, C) versions of Gustafson integrals [1] generalizing the second Barnes Lemma. In sect. 4 it will be shown that N = 1, 2 integrals take, after some rewriting, the form of the star-triangle relations derived in [31][32][33] as special limits of the elliptic star-triangle identity [34][35][36][37]. Sect. 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 the special case of the duality relation [38] for Dotsenko-Fateev (DF) integrals [39]. In sect. 6 we present an elementary proof of the above duality relation and give some evidence in favor that similar duality relations hold for the Γ-integrals. Sect. 7 is reserved for a summary.

Definitions and basic properties
Let u,ū be a pair of complex numbers of the form 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 = −ν 2 + n 2 /4 so that for imaginary ν, ||u|| 2 ≥ 0. The Γ function of the complex field C [40] is defined as 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 The Γ function appears in the generalization of Gustafson's integrals to the complex case. The corresponding integrals take the following form [26,27] and similarly for the barred variables. However, n r , m i , w m are allowed take integer or half-integer values, simultaneously. Accordingly, the sums in ( The integration contours in (2.4) separate the series of " ± " poles due to the Γ functions in the numerators. The poles are located at the points, where r ∈ [1, N ], j ∈ [1, N + 1] for the first integral and where r ∈ [1, N ], j ∈ [1, 2N + 2], for the second one. Let us discuss now the convergence properties of the integrals (2.4). Since the integrands are meromorphic functions and contours of integration avoid the poles it is sufficient to analyse the region of large u r = n r /2 + iν r , ||u r || 2 = ν 2 r + n 2 r /4 → ∞, only. With the help of Eq. (2.3) we simplify the denominators in the integrals (2.4) as follows where κ N = 1 for the integer case and κ N = (−1) N (N +1)/2 for the half-integer case. Finally, taking into account that for large u we conclude that the integrals (2.4) converge absolutely provided respectively. From now on we assume that these conditions are fulfilled.

Determinant representation
In this subsection we present the integrals (2.4) as determinants of one-fold integrals. Such a representation will be quite useful in what follows. ‡ For the derivation let us denote by Q(u|z, w) the function and by Q ik (z, w) its Mellin moments: (2.13) ‡ The determinant representation for elliptic hypergeometric integrals was constructed in ref. [41].
Here we introduced a short-hand notation for the integration measure (2.14) Let Q N (z, w) be a N × N matrix constructed from the Mellin moments, Rewriting the product on the r.h.s. of Eq. (2.8) as the product of two Vandermonde determinants and taking into account the symmetry of the integrand in (2.4a) with respect to the permutations u i ↔ u j we bring the first integral into the determinant form Proceeding in a similar way one gets the determinant representation for the second integral as follows . . . . . . . . .
, (2.18) where κ N is a phase factor, see Eq. (2.9), Let us note here that the conditions (2.11) are equivalent to the requirement of absolute convergence of the Mellin moments Q N N (z, w) and Q N N (z), respectively.

Proof of identities (2.4)
Calculating the integral (2.4a) we will assume that the parameters z k , w k satisfy the conditions These conditions imply the condition (2.11), but not follow from it and will be removed in the end of the calculation.
By virtue of (2.11) the integrals (2.13) are absolutely convergent for i, k = 1, . . . N . We evaluate the ν integral by residues closing the contour in the left half-plane. Recalling that u = n/2 + ν and u = −n/2 + ν one gets The first observation is that since the summand vanishes forp j < 0, only the poles at ν I,− rj (p), Eq. (2.6), contribute to the integral and, second, that under the assumptions (2.20) the sum over p converges uniformly on n. Therefore, evaluating n M ik (n) we swap the summation over n and p, change the summation variable from n top j and finally suppress the index j inp j : .

The final answer can be written in the form
After substituting this expression in (2.18), it can be rewritten as follows The infinite sums over {p}, {p} can be evaluated with the help of the following Milne's U (n) generalization of the Gauss summation formula [42],see also [2], Eq.
. (2.24) Using (2.24) we derive the following representation for the integral I . (2.25) After some simplifications it can be written as .
Taking into account that w k −w k is an integer one finds that the last product in (2.27) yields that cancels the second factor in (2.27). On the last step we use the Lemma 5.10 in Ref. [2] which claims (2.28)

It results in
so that we get the required result for I . (2.29) Finally by analytic continuation in ν k the assumptions (2.20) can be relaxed to the condition (2.11). One sees that the crucial point in the proof of (2.4a) is the factorization of the double sum arising after taking the ν integral by residues, Eq. (2.21), into the product of two infinite sums, see Eq. (2.22). After this property is established the further analysis follows the lines of Ref. [2] with minimal modifications. We note here that the factorization of sums was noticed by Ismagilov [43,44].
The proof of the second identity (2.4b) goes along the same lines so we will not go into details and discuss only the differences. First, we assume that the parameters z k satisfy the conditions After taking ν-integrals by residues and some rewriting one cast I (2) N into the following form Collecting all factors we obtain where sin π(z i ± z π(j) ) . (2.34) One can show that T N = 1 by taking the limit q → 1 of the identities (7.11) and (7.12) in ref. [2]. Finally, the conditions (2.30) can be relaxed to (2.11) by analytic continuation in ν k .

Limiting cases
In this section we derive two more integrals which are replicas of the integrals, (3.2) and (5.4) in ref. [1]. We show that in the complex case these integrals are intrinsically related to the integrals (2.4). This property is not seen in the SL(2, R) setup. We start our analysis with the integral (2.4a) and introduce the variables ζ and integer η by The r.h.s. of (2.4a) is a meromorphic function of ζ with poles located at the points ζ p = |1 + η/2| + p, p ∈ N + . For η = 0 and ζ ∼ 1 the r.h.s. of (2.4a) takes the form where z k , w k obey the constraint N +1 k=1 (z k + w k ) = N +1 k=1 (z k +w k ) = 1. At the same time, only the element Q N N of the matrix (2.15) becomes singular at this point. The corresponding integral diverges at large ν and n as ζ → 1. Indeed, taking into account Eq. (2.10) one gets Thus for ζ → 1 and therefore 5) where Q N −1 is the main N − 1 minor of the matrix Q N . Taking into account Eq. (3.4) and comparing with (3.2) one obtains the following identity which holds provided N +1 k=1 (z k + w k ) = N +1 k=1 (z k +w k ) = 1. The l.h.s. of (3.6) can be written in an integral form as This integral is an analog of the integral (3.2) in ref. [1]. Indeed, replacing the variable z N +1 by m=1 w m and using the relations (2.3) one can bring Eq. (3.7) into the form which is a replica of Eq. (3.2) in ref. [1] 1 The analysis of the second integral, Eq. (2.4b), goes by exactly 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 z k . The l.h.s. and r.h.s. of Eq. (2.4b) have a pole at ζ = 1 (for η = 0). Comparing the corresponding residues we get where 2N +2 k=1 z k = 2N +2 k=1z k = 1 and the subscript ± at the integral sign indicates that the sum goes over either integer n (plus) or half-integer n (minus). Again, introducing the variable γ = 2N +1 k=1 z k , one can rewrite this integral in a form identical to the integral (5.4) in ref. [1] . (3.10) Thus one sees that in the SL(2, C) setup the integrals (3.7), (3.8) and (3.9), (3.10) are intrinsically related to the integrals (2.4). For N = 2 the relation (3.10) was derived by Sarkissian and Spiridonov [46].

Star-triangle relation
In this section we show that the star-triangle relations with the Boltzmann weights given by a product of Γ-functions [31][32][33] follow in a rather straightforward way from the integrals (2.4). The star-triangle relation underlies exact solvability of various two dimensional lattice models, see refs. [47,48] for a review. We recall here the star-triangle relation inherent in the noncompact SL(2, C) spin chain magnets [23]. Let s α (z) ≡ s α ,ᾱ (z,z) be a function of the complex variables z = x + iy,z ≡ z * = x − iy, This is a single valued function on the complex plane and in physics literature it is usually called the propagator §. It satisfies two relations: • the chain relation • the star-triangle relation which holds provided α 1 + α 2 + α 3 =ᾱ 1 +ᾱ 2 +ᾱ 3 = 2.
In fact these two relations are equivalent: Eq. In refs. [31][32][33] new solutions of the star-triangle relation have been derived. They arise as certain limits of the elliptic star-triangle relation [34][35][36][37]. Below we show that these star-triangle relations can be derived from the integrals (2.4).
As in the previous sections we first consider the relation associated with the integral (2.4a). To this end we define the propagator . (4.4) § Let 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 somewhat unfortunate notation will not lead to confusion.
The variables u and α have the form u = n/2 + iνū = −n/2 + iν, α = m + σᾱ = −m + σ , (4.5) where [u] = n and 1 2 [α] = m are either both integer or half-integer numbers, u = ν is real and α = σ is a complex number. Under these conditions the arguments of the Γ-functions in (4.4) has the form (2.1). Slightly abusing the terminology we will call u (α) integer or half-integer 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 "holomorphic" arguments of functions, f (α) ≡ f (α,ᾱ), we accept that, wheneverx is not defined from a context, x + α ≡ (x + α, x +ᾱ), e.g.
The chain relation for the propagator S follows from the first integral (2.4a) for N = 1. Namely, substituting where u = n/2 + ν and sum goes over integers. The parity of the α 1 and z (α 2 and w) are always the same. The integral is well defined provided Re α k < 1 , k = 1, 2 and Re α 1 + α 2 > 1: the poles of the Γ functions in the integral (4.8) are separated by the integration contour if Re α k < 1 and the integral converges at large u if Re α 1 + α 2 > 1. The star-triangle relation for S α can be obtained from the integral identity (3.7) for N = 2. Let us make the following substitution in this equation. The condition k (z k + w k ) = 1 gives rise to the following restriction on the indices: k α k = kᾱ k = 2 and after some algebra one derives where, again, α k and z k are integer or half-integer simultaneously. For the special choice of the parameters, α k =ᾱ k , this relation coincides with the star-triangle relation derived in [31]. Proceeding with the second integral (2.4b) we define the propagator as the product of four Γ functions (4.11) Here, again the requirement for the arguments of Γ functions, 1−α 2 ± z 1 ± z 2 , to be integers imposes obvious restrictions on the relative parity of all variables. The propagator D α (z 1 , z 2 ) is an even function of z 1 , z 2 and invariant under z 1 ↔ z 2 permutation. Therefore, for each variable z k = n k /2+ν k one can restrict n k to positive (negative) values. Note that unlike the previous case the propagator D α is not shift invariant. Also, up to a phase factor depending on the parity of arguments, The chain relation for the propagator D α follows from the identity (2.4b). Indeed, after the substitution z 1(2) = (1 − α 1 )/2 ± z, z 3(4) = (1 − α 2 )/2 ± w the integral (2.4b) takes the form 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 = ν 2 + n 2 /4. (4.13) where k α k = kᾱ k = 2. As was mentioned earlier the parity of all variables have to be coordinated so that (4.13) encompasses four different identities I All α k are integer / have positive parity (a) z k , u -integer / positive parity (b) z k , u -half-integer / negative parity II α 1 and α 2 , α 3 have positive and negative parity, respectively (a) u and z 1 have positive and z 2 , z 3 negative parity (b) u and z 1 have negative and z 2 , z 3 positive parity The variant I a corresponds to the star-triangle relation obtained by A.P. Kels in [32,33]. An extension of the star-triangle relation of [32] following from relation (3.10) with N=2 was also considered by G.A.Sarkissian and V.P.Spiridonov [46]. In the first two cases if 0 < α =ᾱ < 1 the functions D α (z 1 , z 2 ) are real: D α (z 1 , z 2 ) > 0 for integer z 1 , z 2 (the case I a) and −D α (z 1 , z 2 ) > 0 for half-integer z 1 , z 2 (the case I b). In both cases these functions can be interpreted as the Boltzmann weights of lattice integrable models, for more details see ref. [32,33].

Quasi-classical limit
Let us consider the identities (4.8), (4.10) when the external variables become large. We replace the variables z k (w k ) by Lz k = L(x k + iy k ) in (4.8), (4.10) and take the limit L → ∞. The variables x k , y k can be considered as continuous in this limit, so that z k ∈ C,z k = z * k . It is not hard to check that the leading contribution to the integrals (4.8), (4.10) comes from the region where u k ∼ L, so that we replace u k → Lu k as well. Moreover in this limit S α turns into s α : and Du can be replaced by the integral over complex plain as follows where u = u x + iu y and d 2 z = du x du y . Taking into account (5.1) and (5.2) it is easy to check that Eqs. (4.8), (4.10) turn in this limit into the chain-relation (4.2) and the star-triangle relation (4.3).
In what follows we study the quasi-classical limits the integrals (2.4a) and (3.7) for general N . First of all we rewrite these identities in term of the propagator S α . To this end we make the substitution in Eq. (2.4a) and rewrite it in the following form Similarly, the identity (3.7) can be represented as follows and where k α k = kᾱ k = N .
In the quasi-classical limit these identities are reduced to the following two-dimensional integrals with power functions It is easy to check that in the quasi-classical limit the propagator D α (z, w) turns into [z 2 − w 2 ] −α . Therefore Eqs. (2.4b) and Eq. (3.9) do not produce new identities in this limit but reduce to the integrals (5.6) and (5.7) after the appropriate change of variables.
One can take a different point of view on the integrals (2.4a) and (2.4b) and consider them as the "quantized" version of the integrals (5.6). For α k =ᾱ k the integral (5.6) is a special case of a duality relation [38] for the Dotsenko-Fateev (DF) integrals [39], see next section, and one can hope that there exists a "quantized" version of the duality relation in a general situation.

Dotsenko-Fateev integrals
In this section we give an elementary proof of the following integral relation which is essentially the duality relation [38,49] for the DF integrals [39] provided the parameters satisfy the constraint α k =ᾱ k , see also [50]. For m = 0 this identity coincides with (5.6).
To evaluate the l.h.s. of Eq. (6.1) we go over to the symmetric variables x k = x k (y 1 , . . . , y n ) defined as follows It is evident that every point in x-space has n! preimages in y-space and it is not hard to calculate the corresponding Jacobian ∂x k ∂yi = n i<j [y i − y k ]. Introducing the notation t j = −z j one gets for the l.h.s. of Eq. (6.1) Then using the momentum space representation for the propagators in Eq. (6.3) and carrying out x k integrals one gets In order to get the last δ-function one represents e i j (kj t n j +kjt n j ) as dλe i(λ+λ) δ (2) (λ − j k j t n j ) and rescales k j → λk j . The delta functions cut out a m dimensional surface in n + m + 1 dimensional The solutions can be written in the form where the vector λ j is the solution of the linear equations, The solution has the form and taking into account that one gets after some algebra Finally going over to the variables u k , m i=1 (u i + t) = x 1 + x 2 t + x 3 t 2 + · · ·+ x n t m−1 + t m and changing t j → −z j one gets the r.h.s. of Eq. (6.1). We have learned from the discussions with A. Litvinov that similar proof of the duality relation (6.1) is presented in his lectures on conformal field theory [51].
Finally, we make a conjecture that the duality relation (6.1) admits a generalization to the "quantized" cases. Namely, these relations take the form (6.14) where z ′ j = 1/2 − w j , w ′ j = 1/2 − z j and Γ(1/2 − z j ± u i ), (6.15) where κ k = 1, (−1) k(k+1)/2 for the integer and half-integer cases,respectively. For m = 0 these integrals are equivalent to the integrals (2.4) and in the quasi-classical limit both of them reproduce the duality relation (6.1). For n = m = 1 the relations (6.14) and (6.15) follow from the star-triangle relations (4.10) and (4.13). For few first n and m the integrals (6.14), (6.15) go through numerical tests. Closing this section we note that quite similar duality relations were observed recently in the so-called conformal Fishnet model [52].

Summary
In refs. [26,27] 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 [2]. 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 refs. [31][32][33] and in the quasi-classical limit are reduced to the special (m = 0) case of the duality relation for the DF integrals [38,39,49]. We also conjecture that these duality relations, after appropriate modifications, hold for the integrals with complex Γ functions.