Correlation functions and quantum Knizhnik-Zamolodchikov equation on level -4

Hermann Boos^{1} Max-Planck Institut für Mathematik Vivatsgasse 7, 53111 Bonn, Germany

Vladimir Korepin C.N. Yang Institute for Theoretical Physics State University of New York at Stony Brook Stony Brook, NY 11794–3840, USA

Feodor Smirnov^{2} LPTHE, Tour 16, 1-er étage, 4, pl. Jussieu 75252, Paris Cedex 05, France

Abstract

This paper is continuation of our previous papers [12] and [13]. We discuss in more detail a new form of solution to the quantum Knizhnik-Zamolodchikov equation [qKZ] on level −4 obtianed in the paper [13] for the Heisenberg XXX spin chain. The main advantage of this form is it’s explicit reducibility to one-dimensional integrals. We argue that the deep mathematical reason for this is some special cohomologies of deformed Jacobi varieties. We apply this new form of solution to the correlation functions using the Jimbo-Miwa conjecture [3]. The formula (47) for the correlation functions obtained in this way is in a good agreement with the ansatz for the emptiness formation probability from the paper [12]. Our previous conjecture on a structure of correlation functions of the XXX model in the homogeneous limit through the Riemann zeta functions at odd arguments is a corollary of the formula (47).

1on leave of absence from the Institute for High Energy Physics, Protvino, 142284, Russia 2Membre du CNRS

1 Introduction.

In the paper [13] we have suggested a new form of solution to the quantum Knizhnik-Zamolodchikov equation [qKZ] on level -4 [2, 3]. The main idea was to use a duality between the solutions to the qKZ on level -4 and level 0. The latter solutions first appeared in a context of the form-factors of integrable models of quantum field theory [1]. They have much more simple structure in comparison with the qKZ solutions on level -4 because they all can be reduced to single integrals. The main point is to get solutions to the qKZ on level -4 from the known qKZ solutions on level 0 inverting some special matrix F of dimension \binom{2n}{n}−\binom{2n}{n−1} built up from different level 0 solutions [1, 7]. The matrix may be represented as a product of two matrices P and H which carry respectively transcendental and rational dependence on the rapidities β_{1}, …, β_{2n}. It is possible to invert the transcendental part using some facts about deformed hyper-elliptic integrals, in particular, the deformed Riemann bilinear relation [1, 4, 5]. But the problem to invert the rational part appeared to be rather complicated. In order to do this we need a theorem about a special form of solutions to the qKZ on level -4, namely, an ansatz given by the formula (20) from [13]. We have proved that the functions h̃ that appear in the above formula are polynomials of all arguments. We know how these polynomials look like for few first cases. Unfortunately, an explicit form of these polynomials in general case is still to be found. In this paper we are trying to make some steps in this direction, namely, we have found a representation for the polynomials h̃ in terms of contour integrals of a special form. The formula obtained has a cohomological meaning. In some sense it corresponds to a deformation of the cohomologies described by the theorem from the sixth section of [13] conjectured in [15] and proved by Nakayashiki in [16].

The next important issue is to apply the solution to the qKZ on level -4 to the correlation functions of the Heisenberg XXX spin chain. This is possible due to the nice conjecture suggested by Jimbo and Miwa in 1996 [3]. This conjecture says that there is a direct connection of the solution to the qKZ on level -4 with any correlation function for the Heisenberg XXZ model in the massless regime and, in particular, for the XXX model. The integral representation that follows from this conjecture was later confirmed by the algebraic Bethe ansatz approach [8], [9]. Our main statement is that using the Jimbo-Miwa conjecture together with our formulae for the solution to the qKZ on level -4 we come to the result that any correlation function of the XXX model can be expressed through the function G defined by the formulae (3.6) and (3.13) of the paper [12] with coefficients being rational functions like in the ansatz for the emptiness formation probability given by the formula (3.20) of the above mentioned paper. Our previous conjecture [10, 11, 14] that any correlation function in the homogeneous limit can be expressed in terms of the Riemann ζ-function at odd arguments with rational coefficients follows from the above statement which is valid for more general case when the spectral parameters are different.

The paper is organized as follows. In Section 2 we discuss in more detail the new form of solutions to the qKZ on level -4 which is similar to the Smirnov’s solution to the qKZ on level 0, namely, it also has a property of reducibility to one-dimensional integrals. In Section 3 we apply this soution to the correlation functions of the XXX model using the Jimbo-Miwa conjecture. Some particular cases are discussed in detail in the Appendix.

2 Solutions to qKZ on level -4

Let us remind the reader the main statement of our previous work [13], namely, the formula from the theorem of the section 5 for the solutions to the qKZ on level -4 counted by integers {k_{1}, ⋯, k_{n−1}}, with |k_{j}| ≤ n − 1, ∀j

g^{\{k_{1},··· ,k_{n−1}\}}(β_{1}, ⋯, β_{2n}) = \frac{e\frac{1}{2} ∑ β_{j}}{∑ e^{β_{j}}} ∏_{i<j} \frac{1}{ζ(β_{i} − β_{j})} ∫_{−∞}^{∞} dα_{1} ⋯ ∫_{−∞}^{∞} dα_{n−1} ∏_{i,j} φ(α_{i} − β_{j})

× det|e^{k_{i}α_{j}}|_{1≤i,j≤n−1} h̃(α_{1}, ⋯ α_{n−1}|β_{1}, ⋯, β_{2n})  (1)

where

φ(α) = Γ (\frac{1}{4} + \frac{α}{2πi}) Γ (\frac{1}{4} − \frac{α}{2πi})  (2)

ζ(β) = exp (− ∫_{0}^{∞} \frac{sin^{2} \frac{1}{2}(β + πi)k e^{− \frac{πk}{2}}}{k sinh(πk) cosh (\frac{πk}{2})} )  (3)

and h̃(α_{1}, ⋯ α_{n−1}|β_{1}, ⋯, β_{2n}) is a polynomial of all its arguments taking values in (ℂ^{2})^{⊗2n} which is skew-symmetric with respect to α_{1}, ⋯ α_{n−1}.

As was explained in [13] one can present h̃ in the following form:

h̃(α_{1}, ⋯, α_{n−1}|β_{1}, ⋯, β_{2n}) = ∑_{{1,··· ,2n}={i_{1},··· ,i_{n}}∪{j_{1},··· ,j_{n}}} v(α_{1}, ⋯ α_{n−1}|β_{i_{1}}, ⋯, β_{i_{n}}|β_{j_{1}}, ⋯, β_{j_{n}})

× ∏_{p,q=1}^{n} \frac{β_{i_{p}} − β_{j_{q}} + πi}{β_{i_{p}} − β_{j_{q}}} w_{ϵ_{1},··· ,ϵ_{2n}}^{†}(β_{1}, ⋯ β_{2n})

where ϵ_{i_{p}} = − and ϵ_{j_{q}} = +. Of course, the function w^{†} also depends on 2n external indices. Each of them takes two possible values ±1.

In order to find the function v we should solve two sets of equations. The first one follows from requirement that h̃ must belong to singlet subspace:

∑_{p=1}^{n+1} v(α_{1}, ⋯, α_{n−1}|β_{i_{1}}, ⋯, β_{i_{n−1}}, β_{j_{p}}|β_{j_{1}}, ⋯, β̂_{j_{p}}, ⋯, β_{j_{n+1}}) ∏_{q≠p} \frac{β_{j_{p}} − β_{j_{q}} − πi}{β_{j_{p}} − β_{j_{q}}} = 0 (4)

The second equation is equivalent to the fact that h̃ is obtained by inverting the matrix H:

∑_{{1,··· ,2n}={i_{1},··· ,i_{n}}∪{j_{1},··· ,j_{n}}} v(α_{1}, ⋯, α_{n−1}|β_{j_{1}}, ⋯, β_{j_{n}}|β_{i_{1}}, ⋯, β_{i_{n}})

× u(α′_{1}, ⋯, α′_{n−1}|β_{i_{1}}, ⋯, β_{i_{n}}|β_{j_{1}}, ⋯, β_{j_{n}}) ∏_{p,q=1}^{n} \frac{1}{β_{i_{p}} − β_{j_{q}}} = c(α_{1}, ⋯, α_{n−1}|α′_{1}, ⋯, α′_{n−1}) (5)

where

u(α_{1}, ⋯ α_{n−1}|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n}) = det(A_{i}(α_{j}|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n}))|_{i,j=1,··· ,n−1} (6)

and the polynomials A_{i}(α) depend on β_{j} as parameters and may be defined through the generating function:

∑_{i=1}^{n−1} γ^{n−i−1}A_{i}(α|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n}) = \frac{∏_{j=1}^{2n}(α − β_{j} + \frac{πi}{2})}{α − γ + πi} − \frac{∏_{j=1}^{2n}(α − β_{j} − \frac{πi}{2})}{α − γ − πi}+

+ \frac{πi ∏_{j=1}^{n}(α − β_{j} − \frac{πi}{2})(γ − β_{n+j} + \frac{πi}{2})}{(α − γ)(α − γ − πi)} + \frac{πi ∏_{j=1}^{n}(γ − β_{j} − \frac{πi}{2})(α − β_{n+j} + \frac{πi}{2})}{(α − γ)(α − γ + πi)}  (7)

The function c(α_{1}, ⋯ α_{n−1}|α′_{1}, ⋯ α′_{n−1}) is the ”intersection form”. Essential part of this ”intersection form” is det |c(α_{i}, α′_{j})| with

c(α_{1}, α_{2}) = \frac{∏_{j=1}^{2n}(α_{1} − β_{j} + \frac{πi}{2})}{α_{1} − α_{2} + πi} − \frac{∏_{j=1}^{2n}(α_{1} − β_{j} − \frac{πi}{2})}{α_{1} − α_{2} − πi} − \frac{∏_{j=1}^{2n}(α_{2} − β_{j} + \frac{πi}{2})}{α_{2} − α_{1} + πi} + \frac{∏_{j=1}^{2n}(α_{2} − β_{j} − \frac{πi}{2})}{α_{2} − α_{1} − πi}(8)

The rigorous definition of the ”intersection form” was given in the end of section 5 of [13]. Now we do not need it.

Let us consider v(α_{1}, ⋯ α_{n−1}|β_{i_{1}}, ⋯, β_{i_{n}}|β_{j_{1}}, ⋯, β_{j_{n}}) for different partitions as \binom{2n}{n} independent unknowns.

Actually, we have \binom{2n}{n−1} equations from (4). In order to analyse the number of equations which follow from the second requirement (5) we need the formula (17) from [13]

A_{k}(α|β_{i_{1}}, ⋯, β_{i_{n}}|β_{j_{1}}, ⋯, β_{j_{n}}) = s_{k}(α) + ∑_{l=1}^{n−1} c_{kl}(β_{i_{1}}, ⋯, β_{i_{n}}|β_{j_{1}}, ⋯, β_{j_{n}})s_{−l}(α)  (9)

where

s_{k}(α) = A_{k}(α|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n}), s_{−k}(α) = α^{n−k−1}, k = 1, … n − 1 (10)

and the matrix c_{kl} is symmetric. The formula (9) follows from the fact that

c(α_{1}, α_{2}) = ∑_{k=1}^{n−1} (s_{k}(α_{1})s_{−k}(α_{2}) − s_{k}(α_{2})s_{−k}(α_{1}))  (11)

and due to (8) it is symmetric function of all 2n variables β_{1}, …, β_{2n}. Using the equation (9) one can express the function u(α′_{1}, …, α′_{n−1}|β_{i_{1}}, …, β_{i_{n}}|β_{j_{1}}, …, β_{j_{n}}) for an arbitrary partition as a linear combination of the determinants det|s_{j_{k}}(α_{l})|_{1≤k,l≤n−1} with j_{k} = −(n − 1), …, −1, 1, …, (n − 1). One can show that due to the relation (11) only \binom{2n}{n} − \binom{2n}{n−1} of such determinants are linearly independent.

Thus two requirements (4) and (5) together provide us with sufficient number of linear equations, namely, \binom{2n}{n} which, in priciple, may be solved and their solution is unique.

From the equations (5) we can deduce recurrent relations for v(α_{1}, ⋯ α_{n−1}|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n}) which look as follows:

v(α_{1}, ⋯ α_{n−2}, β|β_{1}, ⋯, β_{n−2}, β − \frac{πi}{2}, β + \frac{πi}{2}|β_{n−1}, ⋯, β_{2n−2}) = exact form

v(α_{1}, ⋯ α_{n−2}, β|β_{1}, ⋯, β_{n}, |β_{n+1}, ⋯, β_{2n−2}, β − \frac{πi}{2}, β + \frac{πi}{2}) = exact form

v(α_{1}, ⋯ α_{n−2}, β|β_{1}, ⋯, β_{n−1}, β − \frac{πi}{2}|β_{n}, ⋯, β_{2n−2}, β + \frac{πi}{2}) = exact form

v(α_{1}, ⋯ α_{n−2}, β|β_{1}, ⋯, β_{n−1}, β + \frac{πi}{2}|β_{n}, ⋯, β_{2n−2}, β − \frac{πi}{2}) =

= ∏_{j=1}^{n−2}(α_{j} − β) ∏_{k=1}^{n−1} (β − β_{k} − \frac{πi}{2}) (β − β_{n+k−1} + \frac{πi}{2})

× v(α_{1}, ⋯ α_{n−2}|β_{1}, ⋯, β_{n−1}|β_{n}, ⋯, β_{2n−2}) + exact form  (12)

where ”exact form” stands for an expression of the kind:

∏_{j=1}^{n−2} (α_{j} − β) ∑_{j=1}^{n−2}(−1)^{j}E(α_{j})m(α_{1}, ⋯, α̂_{j}, ⋯, α_{n−2}),

E(α) = ∏_{j=1}^{2n−2} (α − β_{k} + \frac{πi}{2}) − ∏_{j=1}^{2n−2} (α − β_{k} − \frac{πi}{2})

for some skew-symmetric polynomial of n − 2 variables m. The relations (12) look undetermined because of presence of unknown ”exact forms” in the RHS. However, knowing a priori degrees of polynomials one can show that this system defines v completely being in fact even overdetermined. Still we have no key for solving this system. At this point the idea arises of trying to find the polynomials v using original Jimbo-Miwa formula. Let us explain this point.

The polynomials v can be decomposed with respect to s_{a}:

v(α_{1}, ⋯, α_{n−1}) = ∑_{−(n−1)≤j_{1}<···<j_{n−1}≤n−1} K_{j_{1},··· ,j_{n−1}}det |s_{j_{p}}(α_{q})|_{p,q=1,··· ,n−1}  (13)

where irrelevant arguments are omitted. Let us look for K_{j_{1},··· ,j_{n−1}} in the following form:

K_{j_{1},··· ,j_{n−1}} = ∏_{p=1}^{n−1} sgn(j_{p})L(s_{−j_{1}}, ⋯, s_{−j_{n−1}})  (14)

where L is a skew-symmetric multi-linear functional. This functional has to satisfy certain requirements. Recall that it depends upon β_{j} as parameters, we shall take into account this dependence writing

K_{j_{1},··· ,j_{n−1}}(β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n})

Our main concern is the recurrence relations (12). Now, according to (12), we want to consider the case β_{i} = β + \frac{πi}{2}, β_{j} = β − \frac{πi}{2} for different choices of i and j. Notice that the formula for c(α_{1}, α_{2}) is symmetric with respect to β_{j}. So, whatever is the choice of β_{i} and β_{j} we have:

c^{(n)}(α_{1}, α_{2})|_{β_{i}=β+ \frac{πi}{2} ,β_{j}=β− \frac{πi}{2}} = (α_{1} − β)(α_{2} − β)c^{(n−1)}(α_{1}, α_{2}) + (α_{1} − β)E(α_{1}) − (α_{2} − β)E(α_{2})

where we wrote down c^{(n)} for the bilinear form depending on 2n variables β_{k} while c^{(n−1)} is similar form depending on 2n-2 β’s with β_{i} and β_{j} omitted. So, for any choice of β_{i}, β_{j} there is a change of basis

ŝ_{a}(α) = C_{ab} s_{b}(α)

such that C ∈ Sp(2n − 2) and

ŝ_{a}(α)|_{β_{i}=β+ \frac{πi}{2} ,β_{j}=β− \frac{πi}{2}} = (α − β)s_{a}^{(n−1)}(α), a = −(n − 2), ⋯, n − 2
ŝ_{−(n−1)}(α)|_{β_{i}=β+ \frac{πi}{2} ,β_{j}=β− \frac{πi}{2}} = 1,
ŝ_{n−1}(α)|_{β_{i}=β+ \frac{πi}{2} ,β_{j}=β− \frac{πi}{2}} = (α − β)E(α)

In the formulae (13, 14) we can change simultaneously all s_{a} to ŝ_{a}. The functional L depends on β_{j} as on parameters. The recurrence relations (12) are equivalent to the following ones:

L(ŝ_{a_{1}}, ⋯, ŝ_{a_{n−2}}, ŝ_{n−1})(β_{1}, ⋯, β_{n−2}, β − \frac{πi}{2}, β + \frac{πi}{2}|β_{n−1}, ⋯, β_{2n−2}) = 0

L(ŝ_{a_{1}}, ⋯, ŝ_{a_{n−2}}, ŝ_{n−1})(β_{1}, ⋯, β_{n}, |β_{n+1}, ⋯, β_{2n−2}, β − \frac{πi}{2}, β + \frac{πi}{2}) = 0

L(ŝ_{a_{1}}, ⋯, ŝ_{a_{n−2}}, ŝ_{n−1})(β_{1}, ⋯, β_{n−1}, β − \frac{πi}{2}|β_{n}, ⋯, β_{2n−2}, β + \frac{πi}{2}) = 0

L(ŝ_{a_{1}}, ⋯, ŝ_{a_{n−2}}, ŝ_{n−1})(β_{1}, ⋯, β_{n−1}, β + \frac{πi}{2}|β_{n}, ⋯, β_{2n−2}, β − \frac{πi}{2}) =

= ∏_{k=1}^{n−1} (β − β_{k} − \frac{πi}{2}) (β − β_{n+k−1} + \frac{πi}{2}) L(s_{a_{1}}^{(n−1)}, ⋯, s_{a_{n−2}}^{(n−1)})(β_{1}, ⋯, β_{n−1}|β_{n}, ⋯, β_{2n−2}) (15)

where a_{p} > −(n − 1) ∀p, the polynomials ŝ_{a} are constructed according to above procedure.

In fact the original Jimbo-Miwa formula satisfies some kind of similar recurrence relations. We cannot go into much details at this point, but careful development of this idea leads to the following formula for v. Let us define the function Φ(σ)

Φ(σ) = ∏_{j=1}^{2n} ϕ(σ − β_{j})  (16)

and the function ϕ(σ) satisfies properties which look similar to those of the function φ(σ) given by (2), namely, it also satisfies the property

ϕ(σ − β + 2πi) = ϕ(σ − β)\frac{σ − β + \frac{πi}{2}}{σ − β + \frac{3πi}{2}}  (17)

and the following expansion

ϕ(σ − β) = σ^{−1/2}(1 + c_{1}(β)σ^{−1} + c_{2}(β)σ^{−2} + ⋯)  (18)

is valid for |σ| ≫ 1 and β fixed. Combining these two conditions (17) and (18) one can define reccursively the coefficients c_{j}(β) to any given order j. One can also show that the function ϕ satisfies one more condition which will be very important for us, namely,

ϕ(σ)ϕ(σ + πi) = \frac{1}{σ + \frac{πi}{2}}

For the function Φ one has

Φ(σ + 2πi) = Φ(α)\frac{P (σ + \frac{πi}{2})}{P (σ + \frac{3πi}{2})}

Φ(σ)Φ(σ + πi) = \frac{1}{P (σ + \frac{πi}{2})}  (19)

where

P (σ) = ∏_{j=1}^{2n}(σ − β_{j})  (20)

Consider polynomials p_{1}, ⋯, p_{n−1} which are taken as a subset of s_{a}. Using the properties of the function Φ one can check that for such a polynomials

res_{∞}(p_{i}(σ)Φ(σ)) = 0  (21)

Then for every p_{i} let us define:

X_{i}(σ) = Δ^{−1}(p_{i}(σ)Φ(σ))  (22)

where by the defenition

(Δf)(σ) = f(σ + πi) − f(σ − πi)  (23)

The r.h.s. of eq. (22) is well defined as asymptotic series since p_{i}Φ does not have residue (21). Our first idea for definition of L(p_{1}, ⋯, p_{n−1}) was as follows:

L^{0}(p_{1}, ⋯, p_{n−1}) = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ⋯ ∫_{Γ_{n−1}} \frac{dσ_{n−1}}{2πi} h_{0}(σ_{1}, ⋯, σ_{n−1}) ∏_{j=1}^{n−1} X_{j}(σ_{j})Φ(σ_{j})

where

h_{0}(σ_{1}, ⋯, σ_{n−1}) = ∑_{π∈S_{n−1}}(−1)^{sgnπ}D(σ_{π{1}}, ⋯ σ_{π{n−1}}|β_{1}, ⋯, β_{2n})_{−···−+···+}  (24)

and the function D is given by the formula (5) of the paper [13]

D(α_{1}, ⋯ α_{n−1}|β_{1}, ⋯, β_{2n})_{−···−+···+} =

= ∏_{r>s} \frac{1}{α_{r} − α_{s} − πi} ∏_{k} ∏_{j>k} (α_{k} − β_{j} + \frac{πi}{2}) ∏_{j<k} (α_{k} − β_{j} − \frac{πi}{2})

× ∑_{l=1}^{n} (2 ∑ α_{k} + β_{l} + πi(2l − 1)) ∏_{j≥l} \frac{α_{j} − β_{j} − \frac{πi}{2}}{α_{j} − β_{j+1} + \frac{πi}{2}}  (25)

Due to the properties of D which ensure correct form of residues in Jimbo-Miwa [3] formulae one can expect that the formula (24) satisfies correct recurrence relations. It is almost the case, but some corrections are needed. The trouble is that h_{0} has poles at σ_{i} = σ_{j} ± πi. So, first of all we have to explain how the contours Γ_{j} are drawn. Let us use the following prescription:

Namely, we require that for any j |σ_{j}| ≪ |σ_{j+1}| where σ_{j} ∈ Γ_{j} and σ_{j+1} ∈ Γ_{j+1}. This definition, however brakes the skew-symmetry with respect to polynomials p_{i} which is crucial for us. Necessary improvement is done as follows. Let us introduce some notations. Let U be a set U = {1, …, n − 1} which is decomposed into the union of three subsets

U = S ⋃ \overline{S} ⋃ T

where S and \overline{S} are the subsets of U of the length p with 0 ≤ p ≤ [\frac{n−1}{2}] and T is their complementary subset of the length n − 1 − 2p

S = {i_{1} < i_{2} < … < i_{p}}
\overline{S} = \{ī_{1}, ī_{2}, …, ī_{p}\} i_{1} < ī_{1}, i_{2} < ī_{2}, …, i_{p} < ī_{p}
T = {j_{1} < j_{2} < … < j_{n−1−2p}}  (26)

Let us note that there is the ordering in the subset S while for the subset \overline{S} the ordering is not required. Then let us define an object

I_{S,\overline{S},T} ≡ ∫_{Γ_{i_{1}}} \frac{dσ_{i_{1}}}{2πi} … ∫_{Γ_{ip}} \frac{dσ_{i_{p}}}{2πi} ∫_{Γ_{j_{1}}} \frac{dσ_{j_{1}}}{2πi} … ∫_{Γ_{j_{n−1−2p}}} \frac{dσ_{j_{n−1−2p}}}{2πi}

∏_{q=1}^{p} X_{i_{q}}(σ_{i_{q}})pī_{q}(σ_{i_{q}})Φ(σ_{i_{q}}) ∏_{q=1}^{n−1−2p} X_{j_{q}}(σ_{j_{q}})Φ(σ_{j_{q}})h_{p}(σ_{i_{1}}, …, σ_{i_{p}}|σ_{j_{1}}, …, σ_{j_{n−1−2p}}) (27)

where the functions h_{p} are obtained from h_{0} by a procedure which will be described later.

L(p_{1}, ⋯, p_{n−1}) = ∑ ; U=S ⋃ \overline{S} ⋃ T ϵ(S, \overline{S}, T)I_{S,\overline{S},T}  (28)

where ϵ(S, \overline{S}, T) is the sign of the permutation π:

π : (1, …, n − 1) → (i_{1}, \overline{i}_{1}, i_{2}, \overline{i}_{2}, …, i_{p}, \overline{i}_{p}, j_{1}, …, j_{n−1−2p})

Let us return to h_{p}. Introduce the following notations: δ for the finite difference:

δ(f)(σ) = f(σ + πi) − f(σ),

R_{k} for normalized residue

R_{k} = \frac{1}{P (σ_{k} + \frac{πi}{2})}res_{σ_{k+1}=σ_{k}+πi}

and D_{l} is the “exact form” taken in the variable σ_{l}

D_{l}f(… |σ_{l}| …) =\frac{f(… |σ_{l}| …)}{P (σ_{l} + \frac{3πi}{2})} − \frac{f(… |σ_{l} − 2πi| …)}{P (σ_{l} − \frac{3πi}{2})}  (29)

Then

h_{p}(σ_{1}, σ_{3}, …, σ_{2p−1}|σ_{2p+1}, σ_{2p+2}, …, σ_{n−1}) = ∏_{k=1}^{p} δ_{σ_{2k−1}}^{−1}[∏_{k=1}^{p} R_{2k−1} h_{0}(σ_{1}, σ_{2}, …, σ_{n−1})

− ∑_{r=1}^{min(p,[ \frac{n−2}{2} ])}(−1)^{r} ∑_{1≤m_{1}<...<m_{r}≤p} ∏ ; \begin{matrix}1 ≤ k ≤ p \\ k ≠ \{m_{1}, …, m_{r}\} \end{matrix} R_{2k−1} ∑ ; \begin{matrix}1 ≤ l ≤ n − 1 \\ l ≠ \{2m_{1} − 1, 2m_{1}, …, 2m_{r} − 1, 2m_{r}\} \end{matrix}

(−1)^{l}D_{l}f_{r}(σ_{2m_{1}−1}, …, σ_{2m_{r}−1}|σ_{l}|σ_{1}, …, { \atop σ_{2m_{1}−1}}, σ̂_{2m_{1}}, …, σ̂_{l}, …, { \atop σ_{2m_{r}−1}}, σ̂_{2m_{r}}, …, σ_{n−1})] (30)

The functions f_{r} are chosen in such a way that

1. Taking ”primitive” ’s δ^{−1} is possible.

2. h_{p}(σ_{1}, σ_{3}, …, σ_{2p−1}|σ_{2p+1}, σ_{2p+2}, …, σ_{n−1}) is symmetric with respect to σ_{1}, σ_{3}, …, σ_{2p−1} and skew-symmetric with respect to σ_{2p+1}, σ_{2p+2}, …, σ_{n−1} or equivalently f_{r}(σ_{1}, σ_{3}, …, σ_{2r−1}|σ_{2r}|σ_{2r+1}, σ_{2r+2}, …, σ_{n−2r−2}) is symmetric w.r.t. σ_{1}, σ_{3}, …, σ_{2r−1} and skewsymmetric w.r.t. σ_{2r+1}, σ_{2r+2}, …, σ_{n−2r−2} 3. For the pair of variables σ_{2r}, σ_{2r+1} the only non-zero residues are in the points σ_{2r+1} = σ_{2r} + 3πi, σ_{2r+1} = σ_{2r} + πi and σ_{2r+1} = σ_{2r} − πi and

res_{σ_{2r+1}=σ_{2r}+3πi}f_{r}(σ_{1}, σ_{3}, …, σ_{2r−1}|σ_{2r}|σ_{2r+1}, σ_{2r+2}, …, σ_{n−2r−2}) =
res_{σ_{2r+1}=σ_{2r}}f_{r}(σ_{1}, σ_{3}, …, σ_{2r−1}|σ_{2r} + πi|σ_{2r+1}, σ_{2r+2}, …, σ_{n−2r−2})  (31)

Some additional requirements on these functions f_{r} must be satisfied. For the moment we were not able to construct these functions completely for arbitrary n, formulae for n = 2, 3, 4, 5, 6 are given in Appendix.

Notice two remarkable properties of the formulae (27 , 28). First, in order to achieve skewsymmetry we need to add some integral with coupled p_{i} and pī. But we do not need to consider more complicated integrals in which three or more polynomials are bounded together. Second, due to the fact that p_{i}∘p_{j} = 0 the ambiguity in definition of δ^{−1} in the formula for h_{p} is irrelevant. The definition of the pairing ∘ is defined by the formula (15) of the paper [13].

3 The correlation functions

Let us remind the reader the formula from [3] which relates a solution of the qKZ on level -4 to any correlation function of the XXX model:

⟨∏_{j=1}^{n} E_{ϵ_{j},ϵ′_{j}}⟩ = g_{n}(β_{1}, …, β_{n}, β_{n} + πi, …, β_{1} + πi)_{−ϵ′_{1},...,−ϵ′_{n},ϵ_{n},...,ϵ_{1}}  (32)

Let us also note that this solution should also satisfy some additional constraint, namely, that

g_{n}(…, β_{j−1}, β_{j}, β_{j} − πi, β_{j+2}, …)_{...,ϵ_{j−1},ϵ_{j},ϵ_{j+1},ϵ_{j+2},...} = δ_{ϵ_{j},−ϵ_{j+1}} g_{n−1}(…, β_{j−1}, β_{j+2}, …)_{...,ϵ_{j−1},ϵ_{j+2},...} (33)

One can check that the singlet solution to qKZ equations on level −4 that satisfy the property (33) can be written in the following:

g(β_{1}, ⋯, β_{2n})_{ϵ_{1},··· ,ϵ_{2n}} = \frac{e\frac{1}{2} ∑ β_{j}}{∑ e^{β_{j}}} ∏_{i<j} \frac{1}{ζ(β_{i} − β_{j})}

× ∑_{−(n−1)≤j_{1}<j_{2}<···<j_{n−1}≤n−1} P_{j_{1},··· ,j_{n−1}}^{ϵ_{1},··· ,ϵ_{2n}}(β_{1}, ⋯, β_{2n})det(I_{j_{l},m})_{l,m=1,··· ,n−1}  (34)

where polynomials

P_{j_{1},··· ,j_{n−1}}^{ϵ_{1},··· ,ϵ_{2n}}(β_{1}, ⋯, β_{2n}) = ∑_{{1,··· ,2n}={k_{1},··· ,k_{n}}∪{l_{1},··· ,l_{n}}} K_{j_{1},··· ,j_{n−1}}(β_{k_{1}}, ⋯, β_{k_{n}}|β_{l_{1}}, ⋯, β_{l_{n}})

× ∏_{p,q=1}^{n} \frac{β_{l_{q}} − β_{k_{p}} + πi}{β_{l_{q}} − β_{k_{p}}}w^{†}_{ϵ′_{1},··· ,ϵ′_{2n}}^{ϵ_{1},··· ,ϵ_{2n}}(β_{1}, ⋯, β_{2n})  (35)

and as above ϵ′_{k_{p}} = − and ϵ′_{l_{q}} = +. Here we have introduced the integrals:

I_{j,m} = ∫_{−∞}^{∞} dα ∏_{p=1}^{2n} φ(α − β_{p})s_{j}(α)e^{(n−1−2m)α}

where the functions φ and ζ are given by the formulae (2) and (3) respectively. We want to split β’a into pairs:

β_{k} = λ_{k} − \frac{πi}{2} + iδ_{k}, β_{2n−k+1} = λ_{k} + \frac{πi}{2} − iδ_{k}  (36)

and to take limit δ_{k} → 0. It is very convenient to keep δ_{k} different.

The function φ(α) satisfies the equation:

φ(α + \frac{πi}{2})φ(α − \frac{πi}{2}) = \frac{4π^{3}}{α sinh(α)}

In the limit δ_{k} → 0 the contour of integration is pinched by two poles which may give a singularity. Let us study the leading behaviour. For s_{−k} the situation is simple:

I_{−k,m} = π(4π)^{n} ∑_{p=1}^{n} \frac{1}{δ_{p}} ∏_{q≠p} \frac{1}{(λ_{p} − λ_{q}) sinh(λ_{p} − λ_{q})}e^{(n−1−2m)λ_{p}}λ_{p}^{n−1−k} + O(1)

so, the leading part of the integrals is expressed in terms of elementary functions.

For s_{k} with k > 0 the situation is more complicated. It is more convenient to use here the explicit formula for the polynomials

s_{k}(α) ≡ A_{k}(α|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n})

which follows form the formula (7) for the generating function

A_{k}(α|β_{1}, ⋯, β_{n}|β_{n+1}, ⋯, β_{2n}) =

∏_{j=1}^{n}(α − β_{j} − \frac{πi}{2})Q_{k}(α − \frac{πi}{2}|β_{n+1} − \frac{πi}{2}, ⋯, β_{2n} − \frac{πi}{2})+

+ ∏_{j=1}^{n}(α − β_{n+j} + \frac{πi}{2})Q_{k}(α + \frac{πi}{2}|β_{1} + \frac{πi}{2}, ⋯, β_{n} + \frac{πi}{2})  (37)

with the polynomials

Q_{k}(α|λ_{1}, ⋯, λ_{n}) = ∑_{l=1}^{k}((α + \frac{πi}{2})^{l} − (α − \frac{πi}{2})^{l})(−1)^{k−l}σ_{k−l}(λ_{1}, ⋯, λ_{n})  (38)

and the standard symmetric functions of n variables σ_{k}(λ_{1}, ⋯, λ_{n}) defined as follows

∏_{j=1}^{n}(λ + λ_{j}) = ∑_{l=0}^{n} λ^{l}σ_{k−l}(λ_{1}, …, λ_{n})

Actually, the multipliers ∏(α − β_{j} ± \frac{πi}{2}) in (37) cancel singularities of φ(α − β_{j}) closest to the real axis. So, the pinching does not take place and, hence, the result is finite. Denote:

I_{k,m} = I_{k,m}^{0} + ∑ iδ_{p}I_{k,m}^{p} + O(δ^{2})

For I_{k,m}^{0} we have:

I_{k,m}^{0} = ∫_{ℝ+i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α}Q_{k}(α − \frac{πi}{2}|λ_{1}, ⋯, λ_{n})+

+ ∫_{ℝ−i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α}Q_{k}(α + \frac{πi}{2}|λ_{1}, ⋯, λ_{n}) = 0

So, the leading contribution is of the first order in δ’s. Consider I_{k,m}^{p}:

I_{k,m}^{p} =

∫_{ℝ+i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α} (χ(α − λ_{p}) − \frac{1}{α − λ_{p}} − \frac{∂}{∂λ_{p}} ) Q_{k}(α − \frac{πi}{2}|λ_{1}, ⋯, λ_{n}) +

+ ∫_{ℝ−i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α} (χ(α − λ_{p}) + \frac{1}{α − λ_{p}} + \frac{∂}{∂λ_{p}} ) Q_{k}(α + \frac{πi}{2}|λ_{1}, ⋯, λ_{n})

where

χ(α) = \frac{d}{dα} (log \frac{φ(α − \frac{πi}{2})}{φ(α + \frac{πi}{2})}) = 2πi ∑_{k=1}^{∞}(−1)^{k} \frac{k}{α^{2} + (πk)^{2}}

Using the definition (38) of Q_{k} we present I_{k,m}^{p} as follows:

I_{k,m}^{p} = ∑_{l=1}^{k}(−1)^{k−l}σ_{k−l}(λ_{1}, ⋯, λ_{n})J_{l,m}^{p}

Then

J_{l,m}^{p} =

∫_{ℝ+i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α} (χ(α − λ_{p}) ((α − πi)^{l} − α^{l}) + πi\frac{(α − πi)^{l} − λ_{p}^{l}}{(α − λ_{p})(α − λ_{p} − πi)}) +

+ ∫_{ℝ−i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α} (χ(α − λ_{p}) (α^{l} − (α + πi)^{l}) − πi\frac{(α + πi)^{l} − λ_{p}^{l}}{(α − λ_{p})(α − λ_{p} + πi)})

Here we have used the following identity

\frac{∂}{∂λ_{p}} σ_{k}(λ_{1}, ⋯, λ_{n}) = ∑_{l=0}^{k−1}(−λ_{p})^{k−l−1}σ_{l}(λ_{1}, …, λ_{n})

Let us denote

f(α) = α^{l} − λ_{p}^{l}

then using the identities:

χ(α + πi) + χ(α) = \frac{1}{α + πi} − \frac{1}{α}  (39)

χ(α) + χ(α − πi) = \frac{1}{α} − \frac{1}{α − πi}

one finds:

J_{l,m}^{p} = ∫_{ℝ+i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α} (f(α − πi)χ(α − λ_{p} − πi) + f(α)χ(α − λ_{p})) −

− ∫_{ℝ−i0} dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α} (f(α + πi)χ(α − λ_{p} + πi) + f(α)χ(α − λ_{p})) =

= 2 ( ∫_{ℝ+i0} − ∫_{ℝ−i0} ) dα ∏_{j=1}^{n} \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)α}χ(α − λ_{p}) (α^{l} − λ_{p}^{l}) =

= 4πi(4π^{3})^{n} ∑_{q≠p} ∏_{j≠q} ∏ \frac{4π^{3}}{sinh(α − λ_{j})}e^{(n−1−2m)λ_{q}}χ(λ_{p} − λ_{q})(λ_{p}^{l} − λ_{q}^{l})

Returning to I_{k,m}^{p}:

I_{k,m}^{p} = 4πi(4π^{3})^{n}(−1)^{k} ∑_{q≠p} ∏_{j≠q} \frac{1}{sinh(λ_{q} − λ_{j})}e^{(n−1−2m)λ_{q}}χ(λ_{p} − λ_{q})
× (σ_{k}(λ_{1}, ⋯, λ̂_{p}, ⋯, λ_{n}) − σ_{k}(λ_{1}, ⋯, λ̂_{q}, ⋯, λ_{n}))

Let Ĩ_{j,m} (|j| = 1, ⋯, n − 1) correspond to a leading term of I_{j,m} in δ’s. It can be rewritten in the following form:

Ĩ_{j,m} = ∑_{p=1}^{n} A_{j,p}B_{p,m}

where A and B are, respectively, 2(n − 1) × n and n × (n − 1) matrices with the following matrix elements:

A_{−k,p} = π(4π^{3})^{n} ∏_{q≠p} \frac{1}{λ_{p} − λ_{q}} λ_{p}^{n−1−k},

A_{k,p} = (−1)^{k−1}4π(4π^{3})^{n} ∑_{q≠p} δ_{p}δ_{q}χ(λ_{p} − λ_{q})(σ_{k}(λ_{1}, ⋯, λ̂_{p}, ⋯, λ_{n}) − σ_{k}(λ_{1}, ⋯, λ̂_{q}, ⋯, λ_{n}))

B_{p,m} = \frac{1}{δ_{p}} ∏_{j≠p} \frac{1}{sinh(λ_{p} − λ_{j})}e^{(n−1−2m)λ_{p}}

Notice the following simple property:

∑_{p=1}^{n} A_{j,p} = 0  (40)

Take a partition {j_{1}, ⋯, j_{n−1}} and introduce the following object

A_{j_{1},··· ,j_{l}} ≡ det(A_{j_{l},r})|_{r≠p}

Then the identity (40) implies that this definition does not depend on the choice of p. There is one more identity which replaces Riemann bilinear relation:

∑_{k=1}^{n−1} (A_{−k,p}A_{k,q} − A_{−k,q}A_{k,p}) = 0  (41)

Calculating minors of the matrix B one finds:

det(Ĩ_{j_{l},m})|_{l,m=1,···n−1} = 2^{\frac{(n−1)(n−2)}{2}} A_{j_{1},··· ,j_{l}} ∏_{p=1}^{n} \frac{1}{δ_{p}} \frac{∑(δ_{p}e^{λ_{p}}) exp(− ∑ λ_{p})}{∏_{i<j} sinh(λ_{i} − λ_{j})}  (42)

Let us return to the formula (34). It can be shown that

ζ(λ)ζ(−λ)ζ(λ − πi)ζ(−λ − πi) = C^{2} \frac{λ}{sinh(λ)}

where the constant

C = \frac{ζ(\frac{πi}{2})ζ(−\frac{πi}{2})}{Γ(\frac{1}{4})^{2}(\frac{π}{2})^{2}}

So, the formula (34) can be rewritten as follows:

g(λ_{1} − \frac{πi}{2} + iδ_{1}, ⋯, λ_{1} + \frac{πi}{2} − iδ_{1})_{ϵ_{1},··· ,ϵ_{2n}} = [∏_{p=1}^{n} \frac{1}{δ_{p}} ]\frac{2^{n(n−3)/2}}{C^{n(n−1)}} ∏_{i<j} \frac{1}{λ_{i} − λ_{j}}

× ∑_{−(n−1)≤j_{1}<j_{2}<···<j_{n−1}≤n−1} P_{j_{1},··· ,j_{n−1}}^{ϵ_{1},··· ,ϵ_{2n}}(λ_{1} − \frac{πi}{2} + iδ_{1}, ⋯, λ_{1} − \frac{πi}{2} + iδ_{1})A_{j_{1},··· ,j_{l}} (1 + O(δ)) (43)

A priori we know that the result of specification β_{k} = λ_{k} − \frac{πi}{2}, β_{2n−k−1} = λ_{k} + \frac{πi}{2} is finite. The only source of possible singularities in δ’s is the multiplier ∏ \frac{1}{δ_{p}} in RHS of (43). The sum is Tailor series in δ’s. Hence, in the sum

1. All the coefficients in front of those terms which are not divisible by ∏ δ_{p} must vanish.

2. In the final result only one term is important: that containing ∏ δ_{p}. It means, in particular that the terms divisible by δ_{p}^{2} for some p do not contribute into result.

Recall that A_{j_{1},··· ,j_{n−1}} is constructed out of A_{j,p}. Let us denote the number of positive j’s in the multiindex by l({j}). Obviously

A_{{j}} = O(δ^{2l({j})})  (44)

The second property shows that A_{{j}} for which

l(\{j\}) > [\frac{n}{2}]

do not contribute to the final result.

Now we need the following Assumption. If l(\{j\}) ≤ [\frac{n}{2}] then

P_{\{ϵ\}}^{\{j\}}(λ_{1} − \frac{πi}{2} + iδ_{1}, ⋯, λ_{1} − \frac{πi}{2} + iδ_{1}) = O(δ^{n−2l(\{j\})})  (45)

Unfortunately, for the moment we are not able to prove it. Let us mention at least some arguments why we believe in this Assumption? First, suppose the degree in δ is smaller then in (45). Then some huge cancellation must take place because all contributions from leading term of asymptotic described by A_{{j}} are singular in the limit. The estimation (45) is valid for n = 2, 3, these cases we were able to calculate by Maple. Certainly, the best way to prove the Assumption is to calculate the polynomials P_{{j}}^{{ϵ}}. These polynomials are related to the polynomials K_{j_{1},··· ,j_{n−1}} through the formula (35). The latter polynomials are given by the formulae (14), (27) and (28). The main problem here is to take the limit δ_{k} → 0 for the integrals (27) when the spectral parameters β_{k} are taken in the form (36). We leave this question for further publication.

Let us suppose that the Assumption is true. Then still important cancellations must occur because only one kind of terms of the order δ^{n} must remain in the Tailor series: that with ∏ δ_{p}. The result will look like:

g(λ_{1} − \frac{πi}{2}, ⋯, λ_{1} + \frac{πi}{2})_{ϵ_{1},··· ,ϵ_{2n}} =

= \begin{matrix}[ & \frac{n}{2} & ] \\ ∑ \\ m=0 \end{matrix} ∑_{k_{1},··· ,k_{2m}} Q_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(λ_{1}, ⋯, λ_{n})χ(λ_{k_{1}} − λ_{k_{2}}) ⋯ χ(λ_{k_{2m−1}} − λ_{k_{2m}})  (46)

where Q_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(λ_{1}, ⋯ λ_{n}) are rational functions with denominators containing only products of λ_{i} − λ_{j}. The requirement that terms in Tailor series with δ_{p}^{2} must disappear implies that k_{i} ≠ k_{j} ∀i, j. We have checked for n = 2 and n = 3 that the functions Q_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}} have additional multipliers, namely,

Q_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(λ_{1}, ⋯, λ_{n}) = ∏_{j=1}^{m}((λ_{k_{2j−1}} − λ_{k_{2j}})^{2} + π^{2})Ã_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(λ_{1}, ⋯, λ_{n})

where Ã_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}} are some rational functions. Doing a substitution like λ_{k} → π(z_{k} − \frac{i}{2}) and using the formula (32) we arrive at the formula

⟨∏_{j=1}^{n} E_{ϵ_{j},ϵ′_{j}}⟩ = \begin{matrix}[ & \frac{n}{2} & ] \\ ∑ \\ m=0 \end{matrix} ∑_{k_{1}≠···≠k_{2m}} A_{−ϵ′_{1},··· ,−ϵ′_{n},ϵ_{n},··· ,ϵ_{1}}^{k_{1},··· ,k_{2m}}(z_{1}, ⋯, z_{n})G(z_{k_{1}} − z_{k_{2}}) ⋯ G(z_{k_{2m−1}} − z_{k_{2m}}) (47)

where A_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(z_{1}, ⋯, z_{n}) are rational functions related in an obvious way to the functions Ã_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(λ_{1}, ⋯, λ_{n}) and the function G is defined by the formula (3.6) of the work [12]. We see wonderful agreement with the ansatz (3.20) from the above paper for the emptiness formation probability P_{n}(z_{1}, ⋯, z_{n}) which was based on quite different arguments.

4 Conclusion

In this paper we have applied a new form of solution to the qKZ on level -4 for the correlation functions of the XXX model. They are given by the formula (47) in a full agreement with the result of paper [12]. The mathematical reasons for a reducibility of solutions of the qKZ on level -4 and the correlation functions to one-dimensional integrals is the special kind of the cohomologies of the deformed Jacobi varieties. Our previous conjecture that in the homogeneous limit z_{j} → 0 both the emptiness formation probability and other correlation function may be expressed in terms of the Riemann zeta function at odd arguments and rational coefficients [10, 11] follows from the formula (47) and the expansion (3.13) of the paper [12].

Unfortuantely, there are still some technical problems that were not solved or solved only partially in this paper. First we are not completely satisfied with the formulae (14), (27) and (28) which are necessary for defining the polynomials K_{j_{1},··· ,j_{n−1}}. In particular, we have not succeeded in finding a general formula for the functions f_{r} from the formula (30). We have done it completely for n ≤ 6 (see Appendix). For a generic case we have only formulated some general conditions for these functions. We may hope that, probably, some more explicit formulae should exist for those polynomials.

Also the challenging problem is to define the rational functions A_{ϵ_{1},··· ,ϵ_{2n}}^{k_{1},··· ,k_{2m}}(z_{1}, ⋯, z_{n}) explicitly using the results of the Section 2.

We hope to do it in our further publication. We also hope to generalize all the above results to the case of the XXZ spin chain.

5 Acknowledgements

The authors would like to thank R. Flume, F. Göhmann and A. Klümper for useful discussions. This research has been supported by the following grants: NSF grant PHY-9988566, the Russian Foundation of Basic Research under grant # 01–01–00201, by INTAS under grants #00-00055 and # 00-00561. HEB would like to thank the administration of the Max-Planck Institute for Mathematics for hospitality and perfect conditions for the work.

6 Appendix

Here we show how the scheme generally described in the previous Section works for the cases n = 2, 3, …, 6.

1. The case n = 2

This case is rather trivial because the function D defined in (25) does not have any poles and

h_{0}(σ) = D(σ|β_{1}, β_{2}, β_{3}, β_{4})_{−−++}

The formula (28) contains only one term for p = 0 which is the one-fold integral:

L(p_{1}) = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi}X_{1}(σ_{1})Φ(σ_{1})h_{0}(σ_{1})  (48)

and the function X_{1} is defined by the formula (22) through a polynomial p_{1} which in this case is one of the polynomials s_{1} or s_{−1}.

2. The case n = 3

This is the first non-trivial case because one faces two-fold integrals for which the integration order is essential.

Let us represent the function D from (25) in the following form:

D(σ_{1}, σ_{2}|β_{1}, …, β_{6})_{−−−+++} = \frac{g(σ_{1}, σ_{2})}{σ_{2} − σ_{1} − πi}  (49)

where g is a polynomial for which the dependence on the rapidities β_{1}, …, β_{6} is implied. Then according to the formula (24)

h_{0}(σ_{1}, σ_{2}) = \frac{g(σ_{1}, σ_{2})}{σ_{2} − σ_{1} − πi} − \frac{g(σ_{2}, σ_{1})}{σ_{1} − σ_{2} − πi}  (50)

The formula (28) contains two terms for p = 0 and p = 1 respectively:

L(p_{1}, p_{2}) = I_{0} + I_{1}  (51)

with polynomials p_{1} and p_{2} which belong to the set of polynomials s_{−2}, s_{−1}, s_{1}, s_{2}. The term I_{0} corresponds to the case when partitions

S = \overline{S} = ∅ T = (1, 2)

I_{0} ≡ I_{∅,∅,(1,2)} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})h_{0}(σ_{1}, σ_{2})  (52)

If one tries to transpose indices 1 and 2 in this formula:

I_{0}_{|_{1↔2}} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{2}(σ_{1})Φ(σ_{1})X_{1}(σ_{2})Φ(σ_{2})h_{0}(σ_{1}, σ_{2})

= − ∫_{Γ_{2}} \frac{dσ_{1}}{2πi} ∫_{Γ_{1}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})h_{0}(σ_{1}, σ_{2})

(53)

If it were not necessary to transpose the contours Γ_{1} and Γ_{2} the last expression would be −I_{0}. Due to the transposition of the contours we get one additional term:

I_{0}_{|_{1↔2}} = −I_{0} + J_{0}  (54)

where

J_{0} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi}B_{12}(σ_{1})R_{1}(h_{0}(σ_{1}, σ_{2}))  (55)

and

B_{12}(σ_{1}) = X_{1}(σ_{1})X_{2}(σ_{1} + πi) + X_{2}(σ_{1})X_{1}(σ_{1} + πi)  (56)

R_{1} = \frac{1}{P (σ_{1} + \frac{πi}{2})}Res_{σ_{2}=σ_{1}+πi}

Here we have used the property

Φ(σ)Φ(σ + πi) = \frac{1}{P (σ + \frac{πi}{2})}  (57)

In order to compensate this additional term J_{0} we need the second term I_{1} in the formula (51). In this case there is only one partition corresponding to p = 1 which satisfies the conditions (26)

S = (1) \overline{S} = (2) T = ∅

I_{1} ≡ I_{(1),(2),∅} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi}X_{1}(σ_{1})p_{2}(σ_{1})Φ(σ_{1})h_{1}(σ_{1})  (58)

where

h_{1}(σ_{1}) = δ_{σ_{1}}^{−1}[R_{1}h_{0}(σ_{1}, σ_{2})]  (59)

Using the formula (50) one gets

[R_{1}h_{0}(σ_{1}, σ_{2})] = \frac{g(σ_{1}, σ_{1} + πi)}{P (σ_{1} + \frac{πi}{2})}

One can check that due to the formula (25) the polynomial g(σ_{1}, σ_{1} + πi) has a divisor P (σ_{1} + \frac{πi}{2}) i.e. [R_{1}h_{0}(σ_{1}, σ_{2})] is a polynomial. Therefore the operator δ_{σ_{1}}^{−1} is well-defined in this case and may be calculated explicitly.

In order to check that the term I_{1} defined in (58) is suitable for compensation of the additional term J_{0} given by (55) let us substitute to the r.h.s. of (55)

R_{1}h_{0}(σ_{1}, σ_{2}) = δ_{σ_{1}}h_{1}(σ_{1})

which is an obvious consequnce of the formula (59). Then we get

J_{0} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi}B_{12}(σ_{1})δ_{σ_{1}}h_{1}(σ_{1})

= − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi}(X_{1}(σ_{1})p_{2}(σ_{1}) + X_{2}(σ_{1})p_{1}(σ_{1}))Φ(σ_{1})h_{1}(σ_{1})  (60)

Actually the last expression is just

−I_{1} − I_{1}_{|_{1↔2}}

and we come to the result that the r.h.s. of the formula (51) is the skew-symmetric function. 3. The case n = 4 First let us represent the function D from (25) in analogous to (49) form:

D(σ_{1}, σ_{2}, σ_{3}|β_{1}, …, β_{8})_{−−−−++++} =
\frac{g(σ_{1}, σ_{2}, σ_{3})}{(σ_{2} − σ_{1} − πi)(σ_{3} − σ_{1} − πi)(σ_{3} − σ_{2} − πi)}  (61)

where g is a polynomial of both variables σ_{1}, σ_{2}, σ_{3} and of the rapidities β_{1}, …, β_{8}. An explicit form of this polynomial is defined by the formula (25). For us it is important that this polynomial has the property that three polynomials g(σ_{1}, σ_{1} + πi, σ_{3}), g(σ_{1}, σ_{3}, σ_{1} + πi), g(σ_{3}, σ_{1}, σ_{1} + πi) contain divisor P (σ_{1} + \frac{πi}{2}) while the polynomial g(σ_{1}, σ_{1} + πi, σ_{1} + 2πi) is divisible by P (σ_{1} + \frac{πi}{2})P (σ_{1} + \frac{3πi}{2}). Then according to the formula (24)

h_{0}(σ_{1}, σ_{2}, σ_{3}) =

\frac{g(σ_{1}, σ_{2}, σ_{3})}{(σ_{21} − πi)(σ_{31} − πi)(σ_{32} − πi)} − \frac{g(σ_{2}, σ_{1}, σ_{3})}{(σ_{12} − πi)(σ_{31} − πi)(σ_{32} − πi)}+

\frac{g(σ_{2}, σ_{3}, σ_{1})}{(σ_{32} − πi)(σ_{12} − πi)(σ_{13} − πi)} − \frac{g(σ_{3}, σ_{2}, σ_{1})}{(σ_{23} − πi)(σ_{12} − πi)(σ_{13} − πi)}+

\frac{g(σ_{3}, σ_{1}, σ_{2})}{(σ_{13} − πi)(σ_{23} − πi)(σ_{21} − πi)} − \frac{g(σ_{1}, σ_{3}, σ_{2})}{(σ_{31} − πi)(σ_{23} − πi)(σ_{21} − πi)}  (62)

As in the previous case the formula (28) contains two terms for p = 0 and p = 1 respectively:

L(p_{1}, p_{2}, p_{3}) = I_{0} + I_{1}  (63)

with polynomials p_{1}, p_{2}, p_{3} from the set s_{−3}, s_{−2}, s_{−1}, s_{1}, s_{2}, s_{3}.

The term I_{0} corresponds to the case when partitions

S = \overline{S} = ∅ T = (1, 2, 3)

I_{0} ≡ I_{∅,∅,(1,2,3)} =

∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})X_{3}(σ_{3})Φ(σ_{3})h_{0}(σ_{1}, σ_{2}, σ_{3})  (64)

For the next term corresponding to p = 1 there are three contributions because one can find three different partitions that satisfy the conditions (26)

S = (1) \overline{S} = (2) T = (3)
S = (1) \overline{S} = (3) T = (2)
S = (2) \overline{S} = (3) T = (1)

and according to the formulae (27,28)

I_{1} = I_{(1),(2),(3)} − I_{(1),(3),(2)} + I_{(2),(3),(1)}  (65)

where

I_{(1),(2),(3)} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})p_{2}(σ_{1})Φ(σ_{1})X_{3}(σ_{3})Φ(σ_{3})h_{1}(σ_{1}|σ_{3})

I_{(1),(3),(2)} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})p_{3}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})h_{1}(σ_{1}|σ_{2})

I_{(2),(3),(1)} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{2}(σ_{2})p_{3}(σ_{2})Φ(σ_{2})X_{1}(σ_{1})Φ(σ_{1})h_{1}(σ_{2}|σ_{1})  (66)

and due to the formula (30)

h_{1}(σ_{1}|σ_{3}) = δ_{σ_{1}}^{−1}(R_{1}h_{0}(σ_{1}, σ_{2}, σ_{3}) − D_{3}f_{1}(σ_{1}|σ_{3}))  (67)

with the “exact form” given by (29)

D_{3}f_{1}(σ_{1}|σ_{3}) =\frac{f_{1}(σ_{1}|σ_{3})}{P (σ_{3} + \frac{3πi}{2})} − \frac{f_{1}(σ_{1}|σ_{3} − 2πi)}{P (σ_{3} − \frac{3πi}{2})}  (68)

In order to define the function f_{1} we need some facts about the structure of non-zero residues. Let us introduce the notation

Res_{σ_{2}=σ_{1}+πi}h_{0}(σ_{1}, σ_{2}, σ_{3}) ≡ V_{1}(σ_{1}|σ_{3})

then it is not difficult to check that

Res_{σ_{2}=σ_{1}−πi}h_{0}(σ_{1}, σ_{2}, σ_{3}) = V_{1}(σ_{1} − πi|σ_{3})

and

Res_{σ_{2}=σ_{1}}h_{0}(σ_{1}, σ_{2}, σ_{3}) = 0

Explicitly,

V_{1}(s_{1}|σ_{3}) = \frac{g(σ_{1}, σ_{1} + πi, σ_{3})}{(σ_{31} − πi)(σ_{31} − 2πi)} + \frac{g(σ_{3}, σ_{1}, σ_{1} + πi)}{(σ_{31} − πi)σ_{13}} − \frac{g(σ_{1}, σ_{3}, σ_{1} + πi)}{(σ_{31} − πi)σ_{13}}  (69)

We shall also need

Res_{σ_{3}=σ_{1}+2πi}V_{1}(σ_{1}|σ_{3}) ≡ V_{2}(σ_{1})  (70)

which due to (69) is

V_{2}(σ_{1}) = \frac{g(σ_{1}, σ_{1} + πi, σ_{1} + 2πi)}{πi}  (71)

It is easy to check that

Res_{σ_{3}=σ_{1}−πi}V_{1}(σ_{1}|σ_{3}) = −V_{2}(σ_{1} − πi)  (72)

and

Res_{σ_{3}=σ_{1}+πi}V_{1}(σ_{1}|σ_{3}) = Res_{σ_{3}=σ_{1}}V_{1}(σ_{1}|σ_{3}) = 0

Our claim is that

f_{1}(σ_{1}|σ_{3}) = \frac{V_{2}(σ_{3})}{σ_{13}}  (73)

Let us briefly explain this result. First we should check that the r.h.s. of (67) is well-defined. Indeed, using (70,72) one can establish that the expression

\frac{V_{1}(σ_{1}|σ_{3})}{P (σ_{1} + \frac{πi}{2})} − \frac{V_{2}(σ_{3})}{(σ_{13} − πi)P (σ_{3} + \frac{3πi}{2})} + \frac{V_{2}(σ_{3} − 2πi)}{(σ_{13} + 2πi)P (σ_{3} − \frac{3πi}{2})}

is a polynomial of σ_{1} i.e. the operator δ_{σ_{1}}^{−1} is well-defined for it. Here we also used the above property of the polynomial g which provides that the ratio \frac{V_{1}(σ_{1}|σ_{3})}{P (σ_{1}+ \frac{πi}{2} )} has poles only for the variable σ_{13} because the numerator V_{1}(σ_{1}|σ_{3}) is divisible by the poly-

nomial P (σ_{1} + \frac{πi}{2}).

Then if (73) is correct then the expression in brackets of the r.h.s. of (67) is

\frac{V_{1}(σ_{1}|σ_{3})}{P (σ_{1} + \frac{πi}{2})} − \frac{V_{2}(σ_{3})}{σ_{13}P (σ_{3} + \frac{3πi}{2})} + \frac{V_{2}(σ_{3} − 2πi)}{(σ_{13} + 2πi)P (σ_{3} − \frac{3πi}{2})}

which up to the above polynomial of σ_{1} is as follows:

(\frac{1}{σ_{13} − πi} − \frac{1}{σ_{13}} )\frac{V_{2}(σ_{3})}{P (σ_{3} + \frac{3πi}{2})} = −δ_{σ_{1}}(\frac{V_{2}(σ_{3})}{(σ_{13} − πi)P (σ_{3} + \frac{3πi}{2})})

Now let us check the skew-symmetry of the formula (63). Actually it is enough to do for two transpositions 1 ↔ 2 and 2 ↔ 3. Let us check it, for instance, for the transposition 2 ↔ 3. Proceeding in analogous way as we did for the previous case for derivation of the formula (54) we can get for the first term I_{0} defined by (64)

I_{0} + I_{0}_{|_{2↔3}} = J_{0}  (74)

where now

J_{0} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})Φ(σ_{1})B_{23}(σ_{2})R_{2}h_{0}(σ_{1}, σ_{2}, σ_{3})  (75)

and

B_{23}(σ_{2}) = X_{2}(σ_{2})X_{3}(σ_{2} + πi) + X_{3}(σ_{2})X_{2}(σ_{2} + πi)  (76)

R_{2} = \frac{1}{P (σ_{2} + \frac{πi}{2})}Res_{σ_{3}=σ_{2}+πi}

We can rewrite (67) in the following form

R_{2}h_{0}(σ_{1}, σ_{2}, σ_{3}) = δ_{σ_{2}}h_{1}(σ_{2}|σ_{1}) + D_{1}f_{1}(σ_{2}|σ_{1})  (77)

Substituting this expression into the formula (75) we come to

J_{0} = J′_{0} + F_{1}  (78)

where

J′_{0} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})Φ(σ_{1})B_{23}(σ_{2})δ_{σ_{2}}h_{1}(σ_{2}|σ_{1})

= − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})Φ(σ_{1})(X_{2}(σ_{2})p_{3}(σ_{2}) + X_{3}(σ_{2})p_{2}(σ_{2}))Φ(σ_{2})h_{1}(σ_{2}|σ_{1})

= −I_{(2),(3),(1)} − I_{(2),(3),(1)}_{|_{2↔3}}  (79)

F_{1} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}X_{1}(σ_{1})Φ(σ_{1})B_{23}(σ_{2})D_{1}f_{1}(σ_{2}|σ_{1})

= − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi}B_{23}(σ_{2})\frac{p_{1}(σ_{1})f_{1}(σ_{2}|σ_{1})}{P (σ_{1} + \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})} = 0  (80)

Here we have got zero because the ratio

\frac{p_{1}(σ_{1})f_{1}(σ_{2}|σ_{1})}{P (σ_{1} + \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})}

can have poles only for the variable σ_{12} and since σ_{2} belongs to the contour Γ_{2} which contains the contour Γ_{1} those poles can be only outside of the contour Γ_{1}. Thus we come to the following result

J_{0} = −I_{(2),(3),(1)} − I_{(2),(3),(1)}_{|_{2↔3}}  (81)

It is left to note that the combination of two residual terms in the r.h.s. of (65), namely, I_{(1),(2),(3)} − I_{(1),(3),(2)} is already skew-symmetric w.r.t. the transposition 2 ↔ 3.

Thus we have checked the skew-symmetry for the transposition 2 ↔ 3. For another transposition 1 ↔ 2 it is a bit more difficult but also possible to do. The strategy is more or less repeats that for the transposition 2 ↔ 3. We shall not do it here.

4. The case n = 5

It is straightforward to write down the function D from (25) in the form like (49) for n = 3 and (61) for n = 4 defining a polynomial g(σ_{1}, σ_{2}, σ_{3}, σ_{4}) which also depends on the rapidities β_{1}, …, β_{10}. Again the main property of this polynomial is that all polynomails of the form g(…, σ_{1}, …, σ_{1} + πi, …) contain as a divisor P (σ_{1}+\frac{πi}{2}), the polynomials g(…, σ_{1}, …, σ_{1}+πi, …, σ_{1}+2πi) are divisible by P (σ_{1}+\frac{πi}{2})P (σ_{1}+\frac{3πi}{2}) etc.

Now the formula (28) will contain three terms for p = 0, p = 1 and p = 2:

L(p_{1}, p_{2}, p_{3}, p_{4}) = I_{0} + I_{1} + I_{2}  (82)

with polynomials p_{1}, p_{2}, p_{3}, p_{4} from the set s_{−4}, s_{−3}, s_{−2}, s_{−1}, s_{1}, s_{2}, s_{3}, s_{4}.

The main term I_{0} corresponds to the partitions

S = \overline{S} = ∅ T = (1, 2, 3, 4)

I_{0} ≡ I_{∅,∅,(1,2,3,4)} =

∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi} ∫_{Γ_{4}} \frac{dσ_{4}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})

· X_{3}(σ_{3})Φ(σ_{3})X_{4}(σ_{4})Φ(σ_{4})h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4})  (83)

where as above the function h_{0}(σ_{1}, …, σ_{4}) is defined by the formula (24).

For the next term with p = 1 there are six contributions corresponding to six different partitions that satisfy the conditions (26)

S = (1) \overline{S} = (2) T = (34)

S = (1) \overline{S} = (3) T = (24)
S = (1) \overline{S} = (4) T = (23)
S = (2) \overline{S} = (3) T = (14)
S = (2) \overline{S} = (4) T = (13)
S = (3) \overline{S} = (4) T = (12)

and according to the formulae (27,28)

I_{1} = I_{(1),(2),(34)} − I_{(1),(3),(24)} + I_{(1),(4),(23)} + I_{(2),(3),(14)} − I_{(2),(4),(13)} + I_{(3),(4),(12)}  (84)

where

I_{(i_{1}),(i_{2}),(i_{3} i_{4})} = ∫_{Γ_{i_{1}}} \frac{dσ_{i_{1}}}{2πi} ∫_{Γ_{i_{3}}} \frac{dσ_{i_{3}}}{2πi} ∫_{Γ_{i_{4}}} \frac{dσ_{i_{4}}}{2πi} X_{i_{1}}(σ_{i_{1}})p_{i_{2}}(σ_{i_{1}})Φ(σ_{i_{1}})

· X_{i_{3}}(σ_{i_{3}})Φ(σ_{i_{3}})X_{i_{4}}(σ_{i_{4}})Φ(σ_{i_{4}})h_{1}(σ_{i_{1}}|σ_{i_{3}}, σ_{i_{4}})  (85)

and due to the formula (30)

h_{1}(σ_{1}|σ_{3}, σ_{4}) = δ_{σ_{1}}^{−1}(R_{1}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) − D_{3}f_{1}(σ_{1}|σ_{3}|σ_{4}) + D_{4}f_{1}(σ_{1}|σ_{4}|σ_{3}))  (86)

with the “exact forms” D_{3} and D_{4} defined by (29). Let us note that the function h_{1}(σ_{1}|σ_{3}, σ_{4}) is manifestly anti-symmetric w.r.t. the transposition σ_{3} ↔ σ_{4}.

For the last term with p = 2 there are three contributions which correspond to the three possible partitions that satisfy the conditions (26)

S = (12) \overline{S} = (34) T = ∅
S = (12) \overline{S} = (43) T = ∅
S = (13) \overline{S} = (24) T = ∅

and

I_{2} = −I_{(12),(34),∅} + I_{(12),(43),∅} + I_{(13),(24),∅}  (87)

where

I_{(i_{1} i_{2}),(i_{3} i_{4}),∅} = ∫_{Γ_{i_{1}}} \frac{dσ_{i_{1}}}{2πi} ∫_{Γ_{i_{2}}} \frac{dσ_{i_{2}}}{2πi} X_{i_{1}}(σ_{i_{1}})p_{i_{3}}(σ_{i_{1}})Φ(σ_{i_{1}})

· X_{i_{2}}(σ_{i_{2}})p_{i_{4}}(σ_{i_{2}})Φ(σ_{i_{2}})h_{2}(σ_{i_{1}}, σ_{i_{2}})  (88)

and due to the formula (30)

h_{2}(σ_{1}, σ_{3}) = δ_{σ_{1}}^{−1}δ_{σ_{3}}^{−1}(R_{1}R_{3}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4})
− R_{1}[D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) − D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1})]
− R_{3}[D_{3}f_{1}(σ_{1}|σ_{3}|σ_{4}) − D_{4}f_{1}(σ_{1}|σ_{4}|σ_{3})])  (89)

Actually this formula for h_{2}(σ_{1}, σ_{3}) is symmetric w.r.t. the transposition σ_{1} ↔ σ_{3}.

In order to define the function f_{1} we need notation which are similar to those for the case n = 4. Let us write only non-zero residues

Res_{σ_{2}=σ_{1}+πi}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) ≡ V_{1}(σ_{1}|σ_{3}, σ_{4})
Res_{σ_{2}=σ_{1}−πi}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = V_{1}(σ_{1} − πi|σ_{3}, σ_{4})
Res_{σ_{3}=σ_{1}+2πi}V_{1}(σ_{1}|σ_{3}, σ_{4}) ≡ V_{2}(σ_{1}|σ_{4})
Res_{σ_{3}=σ_{1}−πi}V_{1}(σ_{1}|σ_{3}, σ_{4}) = −V_{2}(σ_{1} − πi|σ_{4})  (90)

Res_{σ_{4}=σ_{1}+3πi}V_{2}(σ_{1}|σ_{4}) ≡ V_{3}(σ_{1})
Res_{σ_{4}=σ_{1}+πi}V_{2}(σ_{1}|σ_{4}) ≡ V_{2,1}(σ_{1})
Res_{σ_{4}=σ_{1}−πi}V_{2}(σ_{1}|σ_{4}) = V_{3}(σ_{1} − πi)  (91)

The result for the function f_{1} looks as follows:

f_{1}(σ_{1}|σ_{3}|σ_{4}) = \frac{V_{2}(σ_{3}|σ_{4})}{σ_{13}} + (\frac{1}{σ_{13}} − \frac{1}{σ_{13} − πi})\frac{V_{3}(σ_{3})}{σ_{34} + 3πi}  (92)

An important property of the function f_{1} is that non-zero residues w.r.t. to the pair of the variables σ_{3}, σ_{4} are only in three points σ_{4} = σ_{3} + 3πi, σ_{4} = σ_{3} + πi, σ_{4} = σ_{3} − πi. Let us denote

Res_{σ_{4}=σ_{3}+3πi}f_{1}(σ_{1}|σ_{3}|σ_{4}) ≡ χ_{3}(σ_{1}|σ_{3})
Res_{σ_{4}=σ_{3}+πi}f_{1}(σ_{1}|σ_{3}|σ_{4}) ≡ χ_{1}(σ_{1}|σ_{3})  (93)

One can also check that

Res_{σ_{4}=σ_{3}−πi}f_{1}(σ_{1}|σ_{3}|σ_{4}) = χ_{3}(σ_{1}|σ_{3} − πi)  (94)

First we should be convinced that the expressions (86) and (89) for the functions h_{1} and h_{2} respectively are well-defined. For the case of the function h_{1} one can proceed in the same line as it was done above for n = 4. In order to check that (89) is well-defined also we need the following relation which can be verified straightforwardly using the above properties (93,94) of the function f_{1}

R_{1}(D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) − D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1})) = δ_{σ_{1}}(\frac{χ_{1}(σ_{3}|σ_{1} − πi)}{P (σ_{1} − \frac{πi}{2})P (σ_{1} + \frac{πi}{2})}

+ \frac{χ_{3}(σ_{3}|σ_{1} − πi)}{P (σ_{1} − \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})} + \frac{χ_{3}(σ_{3}|σ_{1} − 2πi)}{P (σ_{1} − \frac{3πi}{2})P (σ_{1} + \frac{πi}{2})})  (95)

Then the bracket in the r.h.s. of (89) can be rewritten as follows

R_{3}(R_{1}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) − D_{3}f_{1}(σ_{1}|σ_{3}|σ_{4}) + D_{4}f_{1}(σ_{1}|σ_{4}|σ_{3}))
− R_{1}(D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) − D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1}))  (96)

For the first term the operator δ_{σ_{1}}^{−1} is well-defined just because it is δ_{σ_{1}}R_{3}h_{1}(σ_{1}|σ_{3}, σ_{4}) according to the formula (86). For the second term in the formula (96) δ_{σ_{1}}^{−1} does exist due to the identity (95). Thus we have proved that the operator δ_{σ_{1}}^{−1} is well-defined for the whole expression (96). On the other hand it is manifestly symmetric w.r.t. the transposition σ_{1} ↔ σ_{3}. It means that the operator δ_{σ_{3}}^{−1} is well-defined as well.

Let us check the skew-symmetry of (82) for the transposition 3 ↔ 4. An analog of the formulae (54) for n = 3 and (74) for n = 4 for the first term I_{0} defined by (83) is

I_{0} + I_{0}_{|_{3↔4}} = J_{0}  (97)

with a new

J_{0} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})B_{34}(σ_{3})

· R_{3}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4})  (98)

and analogously to (56) and (76)

B_{34}(σ_{2}) = X_{3}(σ_{3})X_{4}(σ_{3} + πi) + X_{4}(σ_{3})X_{3}(σ_{3} + πi)  (99)

R_{3} = \frac{1}{P (σ_{3} + \frac{πi}{2})}Res_{σ_{4}=σ_{3}+πi}

As we did above we rewrite (86) in the following form

R_{3}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}) = δ_{σ_{3}}h_{1}(σ_{3}|σ_{1}, σ_{2}) + D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) − D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1})  (100)

Substituting this expression into the formula (98) we get

J_{0} = J′_{0} + F_{1} + F_{2}  (101)

where

J′_{0} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})B_{34}(σ_{3})δ_{σ_{3}}h_{1}(σ_{3}|σ_{1}, σ_{2})

= − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})

· (X_{3}(σ_{3})p_{4}(σ_{3}) + X_{4}(σ_{3})p_{3}(σ_{3}))Φ(σ_{3})h_{1}(σ_{3}|σ_{1}, σ_{2})

= −I_{(3),(4),(12)} − I_{(3),(4),(12)}_{|_{3↔4}}  (102)

F_{1} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})B_{34}(σ_{3})D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2})

= − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{2}(σ_{2})Φ(σ_{2})B_{34}(σ_{3})\frac{p_{1}(σ_{1})f_{1}(σ_{3}|σ_{1}|σ_{2})}{P (σ_{1} + \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})} = 0  (103)

The reason why we get zero here is similar to that of the previous case, namely, because the ratio

\frac{p_{1}(σ_{1})f_{1}(σ_{3}|σ_{1}|σ_{2})}{P (σ_{1} + \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})}

can have poles only for the variables σ_{12} and σ_{13}. In both cases these poles are outside of the contour Γ_{1}.

The situation with the third term in (101) F_{2} is different

F_{2} = − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})B_{34}(σ_{3})D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1})

= ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{2}} \frac{dσ_{2}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})Φ(σ_{1})B_{34}(σ_{3})\frac{p_{2}(σ_{2})f_{1}(σ_{3}|σ_{2}|σ_{1})}{P (σ_{2} + \frac{πi}{2})P (σ_{2} + \frac{3πi}{2})}  (104)

This is not zero because now there is a contribution from the poles for the variable σ_{12}. Using the properties of the function f_{1} (93,94) we can get

∫_{Γ_{2}} \frac{dσ_{2}}{2πi} \frac{p_{2}(σ_{2})f_{1}(σ_{3}|σ_{2}|σ_{1})}{P (σ_{2} + \frac{πi}{2})P (σ_{2} + \frac{3πi}{2})}

= D_{1}(p_{2}(σ_{1})\frac{χ_{3}(σ_{3}|σ_{1} − πi)}{P (σ_{1} − \frac{πi}{2})} − p_{2}(σ_{1} + 2πi)\frac{χ_{3}(σ_{3}|σ_{1})}{P (σ_{1} + \frac{5πi}{2})})

− p_{2}(σ_{1})(\frac{χ_{1}(σ_{3}|σ_{1} − πi)}{P (σ_{1} − \frac{πi}{2})P (σ_{1} + \frac{πi}{2})} + \frac{χ_{3}(σ_{3}|σ_{1} − πi)}{P (σ_{1} − \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})}

+ \frac{χ_{3}(σ_{3}|σ_{1} − 2πi)}{P (σ_{1} − \frac{3πi}{2})P (σ_{1} + \frac{πi}{2})})

= D_{1}(p_{2}(σ_{1})\frac{χ_{3}(σ_{3}|σ_{1} − πi)}{P (σ_{1} − \frac{πi}{2})} − p_{2}(σ_{1} + 2πi)\frac{χ_{3}(σ_{3}|σ_{1})}{P (σ_{1} + \frac{5πi}{2})})

− p_{2}(σ_{1})δ_{σ_{1}}^{−1}R_{1}(D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) − D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1}))  (105)

The last equality follows from the relation (95). Substituting this result for the integration over σ_{2} into (104) and using the fact that the term D_{1}(…) in (105) does not contribute one arrives at

F_{2} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})p_{2}(σ_{1})Φ(σ_{1})B_{34}(σ_{3})

· δ_{σ_{1}}^{−1}R_{1}(−D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) + D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1}))  (106)

Thus we come to the following result

J_{0} = −I_{(3),(4),(12)} − I_{(3),(4),(12)}_{|_{3↔4}} + F_{2}  (107)

with F_{2} given by (106).

Now let us note that the part of (84)

−I_{(1),(3),(24)} + I_{(1),(4),(23)} + I_{(2),(3),(14)} − I_{(2),(4),(13)}

is already skew-symmetric w.r.t. the transposition 3 ↔ 4. It is left to treat the very first term in (84)

I_{(1),(2),(34)} Proceeding in a similar way as we treated above the term I_{0} by the derivation of the formula (98) and using the formula (86) for the function h_{1} we can obtain

I_{(1),(2),(34)} + I_{(1),(2),(34)}_{|_{3↔4}} + F_{2} = J_{1}  (108)

where

J_{1} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})p_{2}(σ_{1})Φ(σ_{1})B_{34}(σ_{3})δ_{σ_{1}}^{−1}(R_{1}R_{3}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4})

+ R_{1}[−D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}) + D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1})]

+ R_{3}[−D_{3}f_{1}(σ_{1}|σ_{3}|σ_{4}) + D_{4}f_{1}(σ_{1}|σ_{4}|σ_{3})])  (109)

Actually, expression in the curled bracket coincides with that from the formula (89). Therefore

J_{1} = ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})p_{2}(σ_{1})Φ(σ_{1})B_{34}(σ_{3})δ_{σ_{3}}h_{2}(σ_{1}, σ_{3})

= − ∫_{Γ_{1}} \frac{dσ_{1}}{2πi} ∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{1}(σ_{1})p_{2}(σ_{1})Φ(σ_{1})

· (X_{3}(σ_{3})p_{4}(σ_{3}) + X_{4}(σ_{3})p_{3}(σ_{3}))Φ(σ_{3})h_{2}(σ_{1}, σ_{3})

= −I_{(13),(24),∅} − I_{(13),(24),∅}_{|_{3↔4}}  (110)

What is left is to note that the part of (87) which contains the residual two terms

−I_{(12),(34),∅} + I_{(12),(43),∅}

is already skew-symmetric w.r.t. the transposition 3 ↔ 4.

Thus we have checked the skew-symmetry of the whole expression (82) for the transposition 3 ↔ 4. For other transpositions 1 ↔ 2 and 2 ↔ 3 this can also be done.

5. The case n = 6

As in the previous case the formula (28) contains three terms for p = 0, p = 1 and p = 2:

L(p_{1}, p_{2}, p_{3}, p_{4}, p_{5}) = I_{0} + I_{1} + I_{2}  (111)

where p_{1}, p_{2}, p_{3}, p_{4}, p_{5} belong to the set of polynomials s_{−5}, …, s_{−2}, s_{−1}, s_{1}, s_{2}, …, s_{5}.

The main term I_{0} corresponds to the partitions

S = \overline{S} = ∅ T = (1, 2, 3, 4, 5)

I_{0} ≡ I_{∅,∅,(1,2,3,4,5)} =

∫_{Γ_{1}} \frac{dσ_{1}}{2πi} … ∫_{Γ_{5}} \frac{dσ_{5}}{2πi}X_{1}(σ_{1})Φ(σ_{1})X_{2}(σ_{2})Φ(σ_{2})X_{3}(σ_{3})Φ(σ_{3})

· X_{4}(σ_{4})Φ(σ_{4})X_{5}(σ_{5})Φ(σ_{5})h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5})  (112)

where as above the function h_{0}(σ_{1}, …, σ_{5}) is defined by the formula (24).

For the next term with p = 1 there are ten contributions corresponding to ten partitions

S = (1) \overline{S} = (2) T = (345)

S = (1) \overline{S} = (3) T = (245)

S = (1) \overline{S} = (4) T = (235)

S = (1) \overline{S} = (5) T = (234)

S = (2) \overline{S} = (3) T = (145)

S = (2) \overline{S} = (4) T = (135)

S = (2) \overline{S} = (5) T = (134)

S = (3) \overline{S} = (4) T = (125)

S = (3) \overline{S} = (5) T = (124)

S = (4) \overline{S} = (5) T = (123)

and according to the formulae (27,28)

I_{1} = I_{(1),(2),(345)} − I_{(1),(3),(245)} + I_{(1),(4),(235)} − I_{(1),(5),(234)} + I_{(2),(3),(145)}

− I_{(2),(4),(135)} + I_{(2),(5),(134)} + I_{(3),(4),(125)} − I_{(3),(5),(124)} + I_{(4),(5),(123)}  (113)

where

I_{(i_{1}),(i_{2}),(i_{3} i_{4} i_{5})} = ∫_{Γ_{i_{1}}} \frac{dσ_{i_{1}}}{2πi} ∫_{Γ_{i_{3}}} \frac{dσ_{i_{3}}}{2πi} ∫_{Γ_{i_{4}}} \frac{dσ_{i_{4}}}{2πi} ∫_{Γ_{i_{5}}} \frac{dσ_{i_{5}}}{2πi} X_{i_{1}}(σ_{i_{1}})p_{i_{2}}(σ_{i_{1}})Φ(σ_{i_{1}})

· X_{i_{3}}(σ_{i_{3}})Φ(σ_{i_{3}})X_{i_{4}}(σ_{i_{4}})Φ(σ_{i_{4}})X_{i_{5}}(σ_{i_{5}})Φ(σ_{i_{5}})h_{1}(σ_{i_{1}}|σ_{i_{3}}, σ_{i_{4}}, σ_{i_{5}})  (114)

and due to the formula (30)

h_{1}(σ_{1}|σ_{3}, σ_{4}, σ_{5}) = δ_{σ_{1}}^{−1}(R_{1}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5})

− D_{3}f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) + D_{4}f_{1}(σ_{1}|σ_{4}|σ_{3}, σ_{5}) − D_{5}f_{1}(σ_{1}|σ_{5}|σ_{3}, σ_{4}))  (115)

Obviously the function h_{1}(σ_{1}|σ_{3}, σ_{4}, σ_{5}) is fully anti-symmetric for the variables σ_{3}, σ_{4}, σ_{5}.

For the last term with p = 2 there are fifteen contributions corresponding to all possible partitions that satisfy the conditions (26)

S = (12) \overline{S} = (34) T = (5)

S = (12) \overline{S} = (43) T = (5)

S = (12) \overline{S} = (35) T = (4)

S = (12) \overline{S} = (53) T = (4)

S = (12) \overline{S} = (45) T = (3)

S = (12) \overline{S} = (54) T = (3)

S = (13) \overline{S} = (24) T = (5)

S = (13) \overline{S} = (25) T = (4)

S = (13) \overline{S} = (45) T = (2)

S = (13) \overline{S} = (54) T = (2)

S = (14) \overline{S} = (25) T = (3)

S = (14) \overline{S} = (35) T = (2)

S = (23) \overline{S} = (45) T = (1)

S = (23) \overline{S} = (54) T = (1)

S = (24) \overline{S} = (35) T = (1)

and

I_{2} = −I_{(12),(34),(5)} + I_{(12),(43),(5)} + I_{(12),(35),(4)} − I_{(12),(53),(4)} − I_{(12),(45),(3)} + I_{(12),(54),(3)} + I_{(13),(24),(5)} − I_{(13),(25),(4)} + I_{(13),(45),(2)} − I_{(13),(54),(2)}

+ I_{(14),(25),(3)} − I_{(14),(35),(2)} − I_{(23),(45),(1)} + I_{(23),(54),(1)} + I_{(24),(35),(1)}  (116)

where

I_{(i_{1} i_{2}),(i_{3} i_{4}),(i_{5})} = ∫_{Γ_{i_{1}}} \frac{dσ_{i_{1}}}{2πi} ∫_{Γ_{i_{2}}} \frac{dσ_{i_{2}}}{2πi} ∫_{Γ_{i_{5}}} \frac{dσ_{i_{5}}}{2πi} X_{i_{1}}(σ_{i_{1}})p_{i_{3}}(σ_{i_{1}})Φ(σ_{i_{1}})

X_{i_{2}}(σ_{i_{2}})p_{i_{4}}(σ_{i_{2}})Φ(σ_{i_{2}})X_{i_{5}}(σ_{i_{5}})Φ(σ_{i_{5}})h_{2}(σ_{i_{1}}, σ_{i_{2}}|σ_{i_{5}})  (117)

and due to the formula (30)

h_{2}(σ_{1}, σ_{3}|σ_{5}) = δ_{σ_{1}}^{−1}δ_{σ_{3}}^{−1}(R_{1}R_{3}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5})

− R_{1}[D_{1}f_{1}(σ_{3}|σ_{1}|σ_{2}, σ_{5}) − D_{2}f_{1}(σ_{3}|σ_{2}|σ_{1}, σ_{5}) + D_{5}f_{1}(σ_{3}|σ_{5}|σ_{1}, σ_{2})] − R_{3}[D_{3}f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) − D_{4}f_{1}(σ_{1}|σ_{4}|σ_{3}, σ_{5}) + D_{5}f_{1}(σ_{1}|σ_{5}|σ_{3}, σ_{4})]

+ D_{5}f_{2}(σ_{1}, σ_{3}|σ_{5}))  (118)

Like in the case n = 5 the formula for h_{2}(σ_{1}, σ_{3}|σ_{5}) is symmetric w.r.t. the transposition σ_{1} ↔ σ_{3}.

In order to define the functions f_{1} and f_{2} we need some residues which, in principle, may be computed explicitly using the formula (24)

Res_{σ_{2}=σ_{1}+πi}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5}) = V_{1}(σ_{1}|σ_{3}, σ_{4}, σ_{5})

Res_{σ_{2}=σ_{1}−πi}h_{0}(σ_{1}, σ_{2}, σ_{3}, σ_{4}, σ_{5}) = V_{1}(σ_{1} − πi|σ_{3}, σ_{4}, σ_{5})

Res_{σ_{3}=σ_{1}+2πi}V_{1}(σ_{1}|σ_{3}, σ_{4}, σ_{5}) = V_{2}(σ_{1}|σ_{4}, σ_{5})

Res_{σ_{3}=σ_{1}−πi}V_{1}(σ_{1}|σ_{3}, σ_{4}, σ_{5}) = −V_{2}(σ_{1} − πi|σ_{4}, σ_{5})

Res_{σ_{4}=σ_{1}+3πi}V_{2}(σ_{1}|σ_{4}, σ_{5}) = V_{3}(σ_{1}|σ_{5}) Res_{σ_{4}=σ_{1}+πi}V_{2}(σ_{1}|σ_{4}, σ_{5}) = V_{2,1}(σ_{1}|σ_{5}) Res_{σ_{4}=σ_{1}−πi}V_{2}(σ_{1}|σ_{4}, σ_{5}) = V_{3}(σ_{1} − πi|σ_{5})

Res_{σ_{5}=σ_{4}+πi}V_{2}(σ_{1}|σ_{4}, σ_{5}) = Ṽ_{2}(σ_{1}|σ_{4}) Res_{σ_{5}=σ_{1}+4πi}V_{3}(σ_{1}|σ_{5}) = V_{4}(σ_{1}) Res_{σ_{5}=σ_{1}+2πi}V_{3}(σ_{1}|σ_{5}) = V_{3,2}(σ_{1}) Res_{σ_{5}=σ_{1}+πi}V_{3}(σ_{1}|σ_{5}) = V_{3,1}(σ_{1})

Res_{σ_{5}=σ_{1}−πi}V_{3}(σ_{1}|σ_{5}) = −V_{4}(σ_{1} − πi) Res_{σ_{4}=σ_{3}+3πi}Ṽ_{2}(σ_{3}|σ_{4}) = V_{4}(σ_{3}) Res_{σ_{4}=σ_{3}+2πi}Ṽ_{2}(σ_{3}|σ_{4}) = V_{3,2}(σ_{3}) Res_{σ_{4}=σ_{3}+πi}Ṽ_{2}(σ_{3}|σ_{4}) = Ṽ_{2,1}(σ_{3}) Res_{σ_{4}=σ_{3}} Ṽ_{2}(σ_{3}|σ_{4}) = Ṽ_{2,0}(σ_{3})

Res_{σ_{4}=σ_{3}−πi}Ṽ_{2}(σ_{3}|σ_{4}) = V_{3,1}(σ_{3} − πi) Res_{σ_{4}=σ_{3}−2πi}Ṽ_{2}(σ_{3}|σ_{4}) = −V_{4}(σ_{3} − 2πi)

The result for the function f_{1} looks as follows:

f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) = \frac{V_{2}(σ_{1}|σ_{4}, σ_{5})}{σ_{13}}

− (\frac{1}{σ_{13}} − \frac{1}{σ_{13} − πi})p(σ_{3}|σ_{4}, σ_{5}) − (\frac{1}{σ_{13} − πi} − \frac{1}{σ_{13} − 2πi})q(σ_{3}|σ_{4}, σ_{5})  (119)

where

p(σ_{3}|σ_{4}, σ_{5}) = \frac{V_{3}(σ_{3}|σ_{4})}{σ_{35} + 3πi} − \frac{V_{3}(σ_{3}|σ_{5})}{σ_{34} + 3πi}−

− \frac{V_{4}(σ_{3})}{(σ_{35} + 4πi)(σ_{54} − πi)} + \frac{V_{4}(σ_{3})}{(σ_{34} + 4πi)(σ_{45} − πi)}+

+ \frac{V_{3,2}(σ_{3})}{(σ_{35} + 2πi)(σ_{45} − πi)} − \frac{V_{3,2}(σ_{3})}{(σ_{34} + 2πi)(σ_{54} − πi)}  (120)

and

q(σ_{3}|σ_{4}, σ_{5}) = \frac{V_{4}(σ_{3})}{(σ_{35} + 3πi)(σ_{45} − πi)} − \frac{V_{4}(σ_{3})}{(σ_{34} + 3πi)(σ_{54} − πi)}−

− \frac{V_{4}(σ_{3})}{(σ_{35} + πi)(σ_{45} − πi)} \frac{P (σ_{3} + \frac{3πi}{2})}{P (σ_{3} + \frac{7πi}{2})} + \frac{V_{4}(σ_{3})}{(σ_{34} + πi)(σ_{54} − πi)} \frac{P (σ_{3} + \frac{3πi}{2})}{P (σ_{3} + \frac{7πi}{2})}  (121)

The answer for the function f_{2} is

f_{2}(σ_{1}, σ_{3}|σ_{5}) = \frac{V_{4}(σ_{5})}{(σ_{15} − πi)(σ_{35} − πi)P (σ_{5} + \frac{7πi}{2})} − \frac{V_{4}(σ_{5})}{σ_{15}σ_{35}P (σ_{5} − \frac{3πi}{2})}

+ \frac{V_{3,1}(σ_{5} − πi)}{σ_{15}σ_{35}P (σ_{3} − \frac{πi}{2})} + \frac{V_{3,2}(σ_{5})}{(σ_{15} − πi)(σ_{35} − πi)P (σ_{5} + \frac{5πi}{2})}

+ \frac{Ṽ_{2,0}(σ_{5})}{σ_{15}σ_{35}P (σ_{5} + \frac{πi}{2})} + \frac{Ṽ_{2,1}(σ_{5})}{σ_{15}σ_{35}P (σ_{5} + \frac{3πi}{2})}  (122)

Let us note that the function f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) is anti-symmetric w.r.t. the transposition of the last two variables σ_{4} ↔ σ_{5} while the function f_{2}(σ_{1}, σ_{3}|σ_{5}) is symmetric w.r.t. the transposition of the first two variables σ_{1} ↔ σ_{3}.

Let us also note that the function f_{1} defined by (119) has a similar to the case n = 5 property, namely, that the only non-zero residues for the pair of the variables σ_{3}, σ_{4} are as follows

Res_{σ_{4}=σ_{3}+3πi}f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) = χ_{3}(σ_{1}|σ_{3}|σ_{5})
Res_{σ_{4}=σ_{3}+πi}f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) = χ_{1}(σ_{1}|σ_{3}|σ_{5})
Res_{σ_{4}=σ_{3}−πi}f_{1}(σ_{1}|σ_{3}|σ_{4}, σ_{5}) = χ_{3}(σ_{1}|σ_{3} − πi|σ_{5})  (123)

where the functions χ_{1} and χ_{3} may be calculated explicitly. Of course, such a property is true for the pair of variables σ_{3}, σ_{5} also.

Similar to the case n = 5 arguments can be used in order to check that the expressions (115) and (118) are well-defined because due to the property (123) a similar to (95) relation is valid also

R_{1}(D_{1}f_{1}(σ_{4}|σ_{1}|σ_{2}, σ_{3}) − D_{2}f_{1}(σ_{4}|σ_{2}|σ_{1}, σ_{3})) = δ_{σ_{1}}(\frac{χ_{1}(σ_{4}|σ_{1} − πi|σ_{3})}{P (σ_{1} − \frac{πi}{2})P (σ_{1} + \frac{πi}{2})}

+ \frac{χ_{3}(σ_{4}|σ_{1} − πi|σ_{3})}{P (σ_{1} − \frac{πi}{2})P (σ_{1} + \frac{3πi}{2})} + \frac{χ_{3}(σ_{4}|σ_{1} − 2πi|σ_{3})}{P (σ_{1} − \frac{3πi}{2})P (σ_{1} + \frac{πi}{2})})  (124)

In order to check the skew-symmetry of the answer (111) one can follow the same way like for the previous case n = 5. All necessary relations may be straightforwardly generalized for the case n = 6 also. For instance, the integral formula (105) may be generalized as follows

∫_{Γ_{3}} \frac{dσ_{3}}{2πi}X_{3}(σ_{3})Φ(σ_{3})D_{3}f_{1}(σ_{4}|σ_{3}|σ_{1}, σ_{2})

= D_{1}(p_{3}(σ_{1})\frac{χ_{3}(σ_{4}|σ_{1} − πi|σ_{2})}{P (σ_{1} − \frac{πi}{2})} − p_{3}(σ_{1} + 2πi)\frac{χ_{3}(σ_{4}|σ_{1}|σ_{2})}{P (σ_{1} + \frac{5πi}{2})})

− D_{2}(p_{3}(σ_{2})\frac{χ_{3}(σ_{4}|σ_{2} − πi|σ_{1})}{P (σ_{2} − \frac{πi}{2})} − p_{3}(σ_{2} + 2πi)\frac{χ_{3}(σ_{4}|σ_{2}|σ_{1})}{P (σ_{2} + \frac{5πi}{2})})

− p_{3}(σ_{1})δ_{σ_{1}}^{−1}[\frac{Res_{σ3=σ1+πi}}{P (σ_{1} + \frac{πi}{2})} (D_{1}f_{1}(σ_{4}|σ_{1}|σ_{3}, σ_{2}) − D_{3}f_{1}(σ_{4}|σ_{3}|σ_{1}, σ_{2}))]

− p_{3}(σ_{2})δ_{σ_{2}}^{−1}[\frac{Res_{σ3=σ2+πi}}{P (σ_{2} + \frac{πi}{2})} (D_{2}f_{1}(σ_{4}|σ_{2}|σ_{1}, σ_{3}) − D_{3}f_{1}(σ_{4}|σ_{3}|σ_{1}, σ_{2}))]  (125)

Only one complication in comparison with n = 5 appears because of the second term with D_{2}(…) in the r.h.s. of (125) that makes a non-zero contribution after further integration. Because of this fact we had to include an additional terms with the ratio P (σ_{3} + \frac{3πi}{2})/P (σ_{3} + \frac{7πi}{2}) in (121) which appeared to be necessary for the correct definition of the function f_{1}.

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