Completeness of the Hypergeometric Solutions of the qKZ Equation at Level Zero

V.Tarasov

July 6, 1998

Abstract. We discuss the hypergeometric solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation at level zero and show that they give all solutions of the qKZ equation. We completely describe linear relations between the hypergeometric solutions and give an interpretation of the relations in terms of the quantum group U_{q}(𝔰𝔩_{2}) at q = e^{πi/2}.

1. Introduction

In this paper we study the hypergeometric solutions of the quantized Knizhnik-Zamolodchikov (qKZ) equation at level zero, cf. (2.1) – (2.3). We are going to address two problems. The first one is to show that the hypergeometric solutions give all solutions of the qKZ equation. The second one is to describe completely linear relations between the hypergeometric solutions, in other words, to say in what cases the construction of hypergeometric solutions produces zero solutions of the qKZ equation.

The qKZ equation (2.1) at level zero for μ = 0 was introduced in [S1] as equations for form factors in the SU(2)-invariant Thirring model and a construction of he hypergeometric solutions was given therein. A simple counting of dimensions shows that the obtained hypergeometric solutions are not linearly independent. Linear relations between the hypergeometric solutions were found later in [S2] as a consequence of the deformed Riemann bilinear identity, and it was conjectured that there was no other linear relations between the hypergeometric solutions. This conjecture, in particular, implied that all solutions of the qKZ equation can be obtained as hypergeometric solutions.

For the rational 𝔰𝔩_{2}-type qKZ equation at generic level the hypergeometric solutions were contsructed in [TV], [MV] and it was shown that they exhausted all solutions of the qKZ equation. The general construction was adapted to the case of level zero for μ = 0 in [NPT] and Smirnov’s solution were obtained from it, but completeness of the hypergeometric solutions still remained an open question.

In this paper we consider the qKZ equation (2.1) at level zero both for μ ≠ 0 and μ = 0 . The completeness of the hypergeometric solutions in the case μ ≠ 0 is claimed in Theorem 4.2, the main part of its proof being given by a formula for a determinant of a matrix formed by the hypergeometric integrals, see Theorem 5.3, which is analogous to Theorem 5.14 in [TV]. Theorem 4.2 also implies that in the case μ ≠ 0 the hypergeometric solutions are linearly independent.

E-mail: tarasov@mpim-bonn.mpg.de , vt@pdmi.ras.ru

To prove the completeness of the hypergeometric solutions in the case μ = 0 , claimed in Theorem 4.3, we employ a new differential equation obeyed by the hypergeometric solutions of the qKZ equation, see Theorem 6.1. Linear relations between the hypergeometric solutions in the case μ = 0 are described in Theorem 4.4, its proof being given in Section 7. In Section 8 we give an interpretation of the relations between the hypergeometric solutions in terms of the quantum group U_{q}(𝔰𝔩_{2}) at q = e^{πi/2}.

The author thanks A.Nakayashiki and F.Smirnov for helpful discussions.

2. The qKZ equation at level zero

Let ℏ be a nonzero complex number. Let V = ℂv_{+} ⊕ ℂv_{−} and let R(z) ∈ End(V ⊗ V) be the following R-matrix:

R(z) = \frac{z + ℏP}{z + ℏ} ,

where P ∈ End(V ⊗ V) is the permutation operator. Let

σ^{+} = \binom{0 1}{0 0} ,  σ^{−} = \binom{0 0}{1 0} ,  σ^{3} = \binom{1 0}{0 −1}

be the Pauli matrices and

H = \frac{1 − σ^{3}}{2} = \binom{0 0}{0 1} .

As usual, for any X ∈ End(V) we set X_{i} = id ⊗ … ⊗ X_{i-th} ⊗ … ⊗ id , and similarly, if

R = ∑ r ⊗ r′ ∈ End(V ⊗ V) , then we set

R_{ij} = ∑ id ⊗ … ⊗ r_{i-th} ⊗ … ⊗ r′_{j-th} ⊗ … ⊗ id .

Fix a nonzero complex number p called a step and a number μ, such that 0 ⩽ Im μ < 2π. We consider the qKZ equation for a V^{⊗n}-valued function Ψ(z_{1}, …, z_{n}) :

Ψ(z_{1}, …, z_{m} + p, …, z_{n}) = K_{m}(z_{1}, …, z_{n})Ψ(z_{1}, …, z_{n}) , m = 1, …, n , (2.1)

(2.2) K_{m}(z_{1}, …, z_{n}) = R_{m,m−1}(z_{m} − z_{m−1} + p) … R_{m,1}(z_{m} − z_{1} + p) ×
× exp(μH_{m})R_{m,n}(z_{m} − z_{n}) … R_{m,m+1}(z_{m} − z_{m+1}) .

The number −2 + p/ℏ is called a level of the qKZ equation. The operators K_{1}, …, K_{n} are called the qKZ operators. Notice, that they depend on the parameter μ only via e^{μ}, so the assumption 0 ⩽ Im μ < 2π is not, in fact, a restriction.

Let Σ^{a} = ∑_{m=1}^{n} σ_{m}^{a}, a = ±, 3 . The operators Σ^{±}, Σ^{3} define an 𝔰𝔩_{2}-action in V^{⊗n}.

Denote by (V^{⊗n})_{ℓ} the weight subspace:

(V^{⊗n})_{ℓ} = {v ∈ V^{⊗n} | Σ^{3}v = (n − 2ℓ)v} ,

which is nontrivial only for ℓ = 0, , …, n, and by (V^{⊗n})_{ℓ}^{sing} the subspace of singular vectors:

(V^{⊗n})_{ℓ}^{sing} = {v ∈ (V^{⊗n})_{ℓ} | Σ^{+}v = 0} ,

which is nontrivial only for 2ℓ ⩽ n. For μ = 0 the qKZ operators K_{1}, …, K_{n} commute with the 𝔰𝔩_{2}-action in V^{⊗n} while for μ ≠ 0 they commute only with the action of Σ^{3} . Therefore, the space of solutions of the qKZ equation inherits a weight decomposition of the 𝔰𝔩_{2}-module V^{⊗n} in general, and all the structure of the 𝔰𝔩_{2}-module if μ = 0 .

In what follows we discuss solutions of the qKZ equation (2.1) taking values in a weight subspace (V^{⊗n})_{ℓ}. For μ = 0 we write down only solutions taking values in the subspace of singular vectors (V^{⊗n})_{ℓ}^{sing}. All other solutions can be obtained from them by the action of Σ^{−}.

In this paper we consider the case of the qKZ equation at level zero, that is, all over the paper we assume that

(2.3)  p = 2ℏ ,

unless otherwise stated. We also assume that Im p < 0 . This assumption is used in the definition of the hypergeometric integral, cf. (3.3).

3. Hypergeometric solutions of the qKZ equation at level zero

The hypergeometric solutions of the qKZ equation at level zero in the case μ = 0 are discussed in detail in [NPT]. The case μ ≠ 0 can be considered similarly. For this reason we give here only the most essential points of the construction. In general, the hypergeometric solutions of the rational qKZ equation are obtained in [TV]. Here we adapt the general construction from [TV] to the case p = 2ℏ.

For any function f(t_{1}, …, t_{ℓ}) we set

Asym f(t_{1}, …, t_{ℓ}) = ∑_{σ∈S_{ℓ}}sgn(σ)f(t_{σ_{1}}, …, t_{σ_{ℓ}}) .

Let M be a subset of {1, …, n} such that #M = ℓ. We write M = {m_{1}, …, m_{ℓ}} assuming that m_{1} < … < m_{ℓ}. Introduce a function w_{M} as follows:

w_{M} = Asym g_{M} , (3.1)

g_{M}(t_{1}, …, t_{ℓ}) = ∏_{a=1}^{ℓ} ( \frac{1}{t_{a} − z_{m_{a}}} ∏_{1⩽j<m_{a}}\frac{t_{a} − z_{j} − ℏ}{t_{a} − z_{j}} ) ∏_{1⩽a<b⩽ℓ}(t_{a} − t_{b} − ℏ) ,

and define a vector v_{M} ∈ (V^{⊗n})_{ℓ} by the rule:

(3.2)  v_{M} = v_{ε_{1}} ⊗ … ⊗ v_{ε_{n}} ,  ε_{m} = \{ \begin{matrix}+ if m ∉ M \\ − if m ∈ M \end{matrix}

Let \overline{F} F_{q}^{∧ℓ} be a space of functions F (t_{1}, …, t_{ℓ}; z_{1}, …, z_{n}) p-periodic in z_{1}, …, z_{n}:

F (t_{1}, …, t_{ℓ}; z_{1}, …, z_{m} + p, …, z_{n}) = F (t_{1}, …, t_{ℓ}; z_{1}, …, z_{n}) ,

and such that the product

F (t_{1}, …, t_{ℓ}; z_{1}, …, z_{n}) ∏_{m=1}^{n} ∏_{a=1}^{ℓ}(1 − exp(2πi(t_{a} − z_{m})/p))

is an antisymmetric polynomial in the exponentials e^{2πit_{1}/p}, …, e^{2πit_{ℓ}/p} of degree less than n in each of the variables. Notice that any F ∈ \overline{F} F_{q}^{∧ℓ} is a p-periodic function of t_{1}, …, t_{ℓ}:

F (t_{1}, …, t_{a} + p, …, t_{ℓ}; z_{1}, …, z_{n}) = F (t_{1}, …, t_{ℓ}; z_{1}, …, z_{n}) .

Denote by F_{q}^{∧ℓ} ⊂ \overline{F} F_{q}^{∧ℓ} a subspace of functions F such that the product

F (t_{1}, …, t_{ℓ}; z_{1}, …, z_{n}) exp(−2πi ∑_{a=1}^{ℓ}t_{a}/p) ∏_{m=1}^{n} ∏_{a=1}^{ℓ}(1 − exp(2πi(t_{a} − z_{m})/p))

is a polynomial in the exponentials e^{2πit_{1}/p}, …, e^{2πit_{ℓ}/p}.

Remark. Here we defined a minimal space \overline{F} F_{q}^{∧ℓ} of periodic functions required to obtain the hypergeometric solutions of the qKZ equation. In [NPT] a larger space of periodic functions is considered. But it is easy to see that it produces the same space of the hypergeometric solutions of the qKZ equation (2.1) for μ = 0 .

Let ϕ(t) be the phase function:

ϕ(t) = exp(μt/p) ∏_{m=1}^{n} \frac{Γ((t − z_{m}− ℏ)/p)}{Γ((t − z_{m})/p)} .

Lemma 3.1. For any ε > 0 the phase function ϕ(t) has the following asymptotics

ϕ(t) = (t/p)^{−n/2} exp(μt/p)(1 + o(1))

as t → ∞, ε < | arg (t/p)| < π − ε.

The statement follows from the Stirling formula.

Let C be a simple curve separating the sets ⋃_{m=1}^{n}(z_{m}+ℏ+pℤ_{⩽0}) and ⋃_{m=1}^{n}(z_{m}+pℤ_{⩾0}) ,

and going from −∞ to +∞. More precisely, the contour C admits a parametrization ρ : ℝ → ℂ such that ρ(u) → ±∞ and Im ρ(u) has finite limits as u → ±∞. Recall that we assume Im p < 0 and p = 2ℏ.

Denote by I^{⊗ℓ}(w, W) the following integral:

(3.3)  I^{⊗ℓ}(w, W) = \frac{1}{ℓ!} ∫_{C^{ℓ}} w(t_{1}, …, t_{ℓ}) W (t_{1}, …, t_{ℓ}) ∏_{a=1}^{ℓ} ϕ(t_{a}) dt_{a} .

We are interested in the integrals I^{⊗ℓ}(w_{M}, W) where a function w_{M} , #M = ℓ, is defined by (3.1) and W ∈ \overline{F} F_{q}^{∧ℓ}, and call them the hypergeometric integrals. If 0 < Im μ < 2π, then these integrals are clearly convergent. If Im μ = 0 , then the integral I^{⊗ℓ}(w_{M}, W) is convergent provided that either 2ℓ ⩽ n or W ∈ F_{q}^{∧ℓ}. In fact, the integral I^{⊗ℓ}(w_{M}, W) for arbitrary ℓ, w_{M} and W ∈ \overline{F} F_{q}^{∧ℓ} considered as a function of μ has a finite limit if μ tends from inside the strip 0 < Im μ < 2π to any point on the boundary lines Im μ = 0 and Im μ = 2π except 0 and 2πi. Further if necessary, the integrals I^{⊗ℓ}(w_{M}, W) should be understood in the described regularized sense.

For any W ∈ \overline{F} F_{q}^{∧ℓ} define a function Ψ_{W} (z_{1}, …, z_{n}) with values in (V^{⊗n})_{ℓ} as follows:

(3.4)  Ψ_{W} (z_{1}, …, z_{n}) = ∑_{#M=ℓ}I^{⊗ℓ}(w_{M}, W) v_{M} ,

where the vectors v_{M} are given by (3.2).

Theorem 3.2. Let 0 ⩽ Im μ < 2π and μ ≠ 0 . Then for any W ∈ \overline{F} F_{q}^{∧ℓ} the function Ψ_{W} is a solution of the qKZ equation (2.1) taking values in (V^{⊗n})_{ℓ}.

Theorem 3.3. Let μ = 0 and either 2ℓ ⩽ n or W ∈ F_{q}^{∧ℓ}. Then the function Ψ_{W} is a solution of the qKZ equation (2.1) taking values in (V^{⊗n})_{ℓ}^{sing}.

Corollary 3.4. Let μ = 0 , 2ℓ > n and W ∈ F_{q}^{∧ℓ}. Then Ψ_{W} = 0 .

Proof. The subspace (V^{⊗n})_{ℓ}^{sing} is trivial if 2ℓ > n, which implies the claim. □

Theorems 3.2 and 3.3 follow from the results on formal integral representations for solutions of the qKZ equation [R], [V]. Details of the proof in the case μ = 0 are available in [NPT]. The case μ ≠ 0 is similar.

Solutions Ψ_{W} of the qKZ equation are called the hypergeometric solutions.

4. Completeness of the hypergeometric solutions

Let us formulate accurately the meaning of completeness of the hypergeometric solutions of the qKZ equation. We will use the following compact notation: z = (z_{1}, …, z_{n}) .

Consider a vector space ℂ^{n} with coordinates z_{1}, …, z_{n} and a union of hyperplanes 𝔻 ⊂ ℂ^{n}:

𝔻 = {z ∈ ℂ^{n} | z_{k} − z_{m}+ ℏ ∈ pℤ , k, m = 1, …, n , k ≠ m} ,

called a discriminant. Say that z_{1}, …, z_{n} are generic if (z_{1}, …, z_{n}) ∉ 𝔻. The complement of the discriminant is a natural definition region for solutions of the qKZ equation (2.1).

Indeed, the qKZ operators K_{1}(z), …, K_{n}(z) are invertible for generic z since the R-matrix R(x) is invertible for x ≠ ±ℏ. Therefore, for generic z values of a solution Ψ of the qKZ equation (2.1) at points of a lattice z + pℤ, are uniquely determined by its value Ψ(z) at the initial point; the initial value can be an arbitrary vector in V^{⊗n} because for any j, m = 1, …, n

K_{j}(z_{1}, …, z_{m} + p, …, z_{n})K_{m}(z_{1}, …, z_{n}) = K_{m}(z_{1}, …, z_{j} + p, …, z_{n})K_{j}(z_{1}, …, z_{n}) .

The contour C used in the definition of the hypergeometric integral (3.3) exists for generic z. Therefore, if a periodic function W ∈ \overline{F} F_{q}^{∧ℓ} is regular in a vicinity of a generic point z, so is the corresponding hypergeometric solution Ψ_{W} of the qKZ equation. Hence, we can formulate completeness of the hypergeometric solutions of the qKZ equation as follows.

Theorem 4.1. Let z_{1}, …, z_{n} be generic. If 0 ⩽ Im μ < 2π and μ ≠ 0 , then

\{Ψ_{W} (z_{1}, …, z_{n}) | W ∈ \overline{F} F_{q}^{∧ℓ}\} = (V^{⊗n})_{ℓ} .

If μ = 0 and 2ℓ ⩽ n, then

\{Ψ_{W} (z_{1}, …, z_{n}) | W ∈ \overline{F} F_{q}^{∧ℓ}\} = (V^{⊗n})_{ℓ}^{sing}.

Given a point z ∈ ℂ^{n} consider a space \overline{F} F_{q}^{∧ℓ}(z) of functions in t_{1}, …, t_{ℓ} which have the form W (·; z) for some W ∈ \overline{F} F_{q}^{∧ℓ}. In other words, \overline{F} F_{q}^{∧ℓ}(z) is a space of functions F (t_{1}, …, t_{ℓ}) such that the product

F (t_{1}, …, t_{ℓ}) ∏_{m=1}^{n} ∏_{a=1}^{ℓ}(1 − exp(2πi(t_{a} − z_{m})/p))

is an antisymmetric polynomial in the exponentials e^{2πit_{1}/p}, …, e^{2πit_{ℓ}/p} of degree less than n in each of the variables. The space \overline{F} F_{q}^{∧ℓ}(z) is finite-dimensional and

dim \overline{F} F_{q}^{∧ℓ}(z) = dim(V^{⊗n})_{ℓ} = \binom{n}{ℓ}.

Define the hypergeometric map Ī_{z,ℓ} : \overline{F} F_{q}^{∧ℓ}(z) → (V^{⊗n})_{ℓ} by the rule:

(4.1)  Ī_{z,ℓ}W = ∑_{#M=ℓ}I^{⊗ℓ}(w_{M}, W) v_{M} ,

so that Ψ_{W} (z) = Ī_{z,ℓ}W (·; z) , cf. (3.4). Then we write down Theorem 4.1 in terms of the hypergeometric map.

Theorem 4.2. Let μ ≠ 0 . Then for generic z_{1}, …, z_{n} the hypergeometric map Ī_{z,ℓ} : \overline{F} F_{q}^{∧ℓ}(z) → (V^{⊗n})_{ℓ} is nondegenerate.

The proof is given in the next section.

Theorem 4.3. Let μ = 0 . Then for generic z_{1}, …, z_{n} we have im Ī_{z,ℓ} = (V^{⊗n})_{ℓ}^{sing}.

The proof is given in Section 6.

Since dim \overline{F} F_{q}^{∧ℓ}(z) = dim(V^{⊗n})_{ℓ}, the hypergeometric map I_{z} has a trivial kernel if μ ≠ 0 . So any nonzero periodic function W produces a nonzero solution Ψ_{W} , that is, the space of solutions of the qKZ equation (2.1) with values in (V^{⊗n})_{ℓ} is isomorphic to the space \overline{F} F_{q}^{∧ℓ} of periodic functions. On the contrary, for μ = 0 the hypergeometric map Ī_{z,ℓ} has a nontrivial kernel, which means that for certain periodic functions the corresponding hypergeometric solutions vanish, and we have a problem of describing those periodic functions which produces zero solutions of the qKZ equation.

Introduce the following functions:

(4.2)  Ξ^{(1)}(t) = Θ(t) − 1 ,  Θ(t) = ∏_{m=1}^{n} \frac{exp(2πi(t − z_{m})/p) + 1}{exp(2πi(t − z_{m})/p) − 1} ,

Ξ^{(2)}(t_{1}, t_{2}) = (Θ(t_{1})Θ(t_{2}) − 1) \frac{exp(2πi(t_{1} − t_{2})/p) − 1}{exp(2πi(t_{1} − t_{2})/p) + 1} + Θ(t_{1}) − Θ(t_{2}) .

It is easy to see that Ξ^{(1)} ∈ \overline{F} F_{q}^{∧1} and Ξ^{(2)} ∈ \overline{F} F_{q}^{∧2}. Define linear maps

X_{z,ℓ}^{(1)} : \overline{F} F_{q}^{∧(ℓ−1)}(z) → \overline{F} F_{q}^{∧ℓ}(z) ,

X_{z,ℓ}^{(2)} : \overline{F} F_{q}^{∧(ℓ−2)}(z) → \overline{F} F_{q}^{∧ℓ}(z) ,

X_{z,ℓ}^{(1)} F (t_{1}, …, t_{ℓ}) = Asym(Ξ^{(1)}(t_{1})F (t_{2}, …, t_{ℓ})) ,

X_{z,ℓ}^{(2)} F (t_{1}, …, t_{ℓ}) = Asym(Ξ^{(2)}(t_{1}, t_{2})F (t_{3}, …, t_{ℓ})) .

They obviously induce linear maps X_{ℓ}^{(1)} : \overline{F} F_{q}^{∧(ℓ−1)} → \overline{F} F_{q}^{∧ℓ} and X_{ℓ}^{(2)} : \overline{F} F_{q}^{∧(ℓ−2)} → \overline{F} F_{q}^{∧ℓ}.

Theorem 4.4. Let μ = 0 and 2ℓ ⩽ n. Then for generic z_{1}, …, z_{n} we have ker Ī_{z,ℓ} = im X_{z,ℓ}^{(1)} + im X_{z,ℓ}^{(2)} . For ℓ < a, we assume by convention that im X_{z,ℓ}^{(a)} = 0 .

The proof is given in Section 7.

The last theorem means that if μ = 0 and a periodic function W has the form W = X_{ℓ}^{(1)}W_{1} + X_{ℓ}^{(2)}W_{2} where W_{1}, W_{2} are some periodic functions in less numbers of variables, then Ψ_{W} = 0 , that is, the space of solutions of the qKZ equation (2.1) at μ = 0 with values in (V^{⊗n})_{ℓ}^{sing} is isomorphic to the quotient space \overline{F} F_{q}^{∧ℓ}/(im X_{ℓ}^{(1)}+ im X_{ℓ}^{(2)}). An interesting question is to see whether there is a distinguished subspace in \overline{F} F_{q}^{∧ℓ} complementary to im X_{ℓ}^{(1)}+ im X_{ℓ}^{(2)}.

Remark. The relation Ī_{z,ℓ} X_{z,ℓ}^{(2)} W = 0 was originally discovered by F.Smirnov in [S2] as a consequence of the deformed Riemann bilinear identity. In Section 8 we give another interpretation of this relation in terms of the representation theory of the quantum group U_{q}(𝔰𝔩_{2}) at q = e^{πi/2}.

5. Proof of Theorem 4.2

Recall that we assume p = 2ℏ, so that z_{1}, …, z_{n} are generic if and only if

∏_{1⩽k<m⩽n}(e^{2πiz_{k}/p} + e^{2πiz_{m}/p}) ≠ 0.

Let M be a subset of {1, …, n}, M = {m_{1} < … < m_{ℓ}}. Introduce a function W_{M}(t_{1}, …, t_{ℓ}; z_{1}, …, z_{n}) as follows:

W_{M} = Asym G_{M} , (5.1)

G_{M}(t_{1}, …, t_{ℓ}) = ∏_{a=1}^{ℓ} ( \frac{1}{exp(2πi(t_{a} − z_{m_{a}})/p) − 1} ∏_{1⩽j<m_{a}}\frac{exp(2πi(t_{a} − z_{j})/p) + 1}{exp(2πi(t_{a} − z_{j})/p) − 1} ) ,

cf. (3.1). It is clear that W_{M} ∈ \overline{F} F_{q}^{∧ℓ}.

Proposition 5.1. For generic z_{1}, …, z_{n} functions W_{M}(·, z) , #M = ℓ, form a basis in the space \overline{F} F_{q}^{∧ℓ}(z) .

Proof. First observe that it suffices to prove the statement for ℓ = 1 . For ℓ = 1 the statement is implied by Lemma 5.2 after a substitution: u = e^{2πit/p}, x_{m} = y_{m} = −e^{2πiz_{m}/p}, m = 1, …, n. □

Lemma 5.2. Define a matrix M as follows: ∑_{m=1}^{n} M_{jk} u^{k−1} = ∏_{1⩽j<k}(u−x_{j}) ∏_{k<j⩽n}(u+

y_{j}) . Then

det M = ∏_{1⩽j<k⩽n}(x_{j} + y_{k}) .

The proof is similar to the proof of the Vandermonde determinant.

Theorem 5.3. Let 0 ⩽ Im μ < 2π and z_{1}, …, z_{n} be generic. Then

det [I^{⊗ℓ}(w_{M}, W_{N})]{M,N⊂\{1,...,n\} \atop #M=#N=ℓ} =

= ((ip)^{n(n−1)/2} (Γ(−1/2))^{n} (1 − e^{−μ})^{n/2} exp(μ ∑_{m=1}^{n}z_{m}/p))^{\binom{n−1}{ℓ−1}}×

× ∏_{1⩽k<m⩽n}(z_{k} − z_{m}− ℏ)^{−^{\binom{n−2}{ℓ−1}}}.

Here − 2π < arg (1 − e^{−μ}) ⩽ 0 .

Proof (idea). The proof of the theorem is in common with the proof of Theorem 5.14 in [TV]. Let us indicate here only the main points.

Denote by D(z) the determinant in question and by E(z) the right hand side of the formula to be proved. The determinant obeys a system of difference equations following from the qKZ equation:

D(z_{1}, …, z_{m} + p, …, z_{n}) = det^{(ℓ)} K_{m}(z_{1}, …, z_{n})D(z_{1}, …, z_{n}) , m = 1, …, n ,

where

det^{(ℓ)} K_{m} = e^{μ\binom{n−1}{ℓ−1}} ( ∏_{1⩽j<m} \frac{z_{m}− z_{j} − ℏ + p}{z_{m}− z_{j} + ℏ + p} ∏_{m<j⩽n} \frac{z_{m}− z_{j} − ℏ}{z_{m}− z_{j} + ℏ} )^{\binom{n−2}{ℓ−1}}

is a determinant of the restriction of the operator K_{m} to the weight subspace (V^{⊗n})_{ℓ}. This shows that the ratio D(z)/E(z) is a p-periodic function of z_{1}, …, z_{n} (quasiconstant).

To find this quasiconstant we look at the behaviour of the hypergeometric integrals I^{⊗ℓ}(w_{M}, W_{N}) as Im(z_{m} − z_{m+1}) tends to −∞ for all m = 1, …, n − 1 . We observe, that the leading term of the asymptotics of the matrix [I^{⊗ℓ}(w_{M}, W_{N})] is a diagonal matrix. Therefore, the leading term of the asymptotics of the determinant det[I^{⊗ℓ}(w_{M}, W_{N})] is given by the product of asymptotics of the diagonal entries I^{⊗ℓ}(w_{M}, W_{M}) . They can be calculated using the formula

∫ ; C̃ e^{u (μ−πi)}Γ(u − 1/2)Γ(−u) du = 2πi Γ(−1/2) (1 − e^{−μ})^{1/2}

where the contour C̃ is going from −i∞ to +i∞ separating the sets ℤ_{⩾0} and 1/2 + ℤ_{⩽0} . The formula is equivalent to the formula for the Barnes integral [WW].

Finally, we obtain that the ratio D(z)/E(z) is a quasiconstant which tends to 1 as Im(z_{m}− z_{m+1}) → −∞ for all m = 1, …, n − 1 . Such a quasiconstant necesserily equals 1 . The theorem is proved. □

Remark. Below we describe a differential equation with respect to μ obeyed by the hypergeometric solutions of the qKZ equation, see (6.1). It implies a differential equation with respect to μ for the determinant det[I^{⊗ℓ}(w_{M}, W_{N})], which allows to give another proof of Theorem 5.3 using the differential equation and asymptotics of the hypergeometric integrals I^{⊗ℓ}(w_{M}, W_{N}) as Re μ → ±∞.

Theorem 4.2 follows from the definition (4.1) of hypergeometric map, Proposition 5.1 and Theorem 5.3.

6. Proof of Theorem 4.3

Our proof of Theorem 4.3 is based on an additional differential equation which takes place for hypergeometric solutions of the qKZ equation.

Theorem 6.1. Let Ψ be a hypergeometric solution of the qKZ equation (2.1), see (3.4). Then

(6.1) p ∂_{μ}Ψ = ( ∑_{m=1}^{n}z_{m} H_{m} + \frac{ℏ}{e^{μ} − 1} ( ∑_{m=1}^{n}H_{m} + ∑_{1⩽k<m⩽n}(e^{μ} σ_{k}^{−}σ_{m}^{+} + σ_{k}^{+}σ_{m}^{−})))Ψ .

The proof of the theorem will appear separately. The theorem is a particular specialization of a more general result about the hypergeometric solutions of the qKZ equations and the Knizhnik-Zamolodchikov differential equations [FTV].

Remark. One can check directly that the equation (6.1) is compatible with the qKZ equation (2.1):

L(z_{1}, …, z_{m} + p, …, z_{n})K_{m}(z_{1}, …, z_{n}) = p ∂_{μ}K_{m}(z_{1}, …, z_{n})L(z_{1}, …, z_{n}) ,

where L is the operator in the right hand side of equation (6.1).

Let U be a finite-dimensional vector space and A(μ) be an End(U)-valued function holomorphic in a neighbourhood of μ = 0 . Consider a differential equation:

(6.2)  μ∂_{μ}Ψ = A(μ)Ψ .

Set A_{0} = A(0) . Assume that the operator A_{0} is diagonalizable and let ψ ∈ U be an eigenvector of A_{0} : A_{0} ψ = αψ.

Proposition 6.2. There is a solution Ψ of the equation (6.2) which have the form

Ψ(μ) = μ^{α}(ψ + ∑_{k=1}^{∞} ∑_{j=0}^{i_{k}} μ^{k}(log μ)^{j} ψ_{jk})

where 0 = i_{0} ⩽ i_{1} ⩽ i_{2} ⩽ … are integers such that i_{k} + 1 ⩾ i_{k+1} for any k and i_{k} = i_{k+1} for large k, and ψ_{jk} ∈ U . The series is convergent for μ in a punctured neighbourhood of μ = 0 .

The proof is an exercise on the analytic theory of differential equations and we leave it to a reader.

Corollary 6.3. Assume that all nonzero eigenvalues of A_{0} have positive real parts. Let Ψ(μ) ∈ End(U) be a fundamental solution of the equation (6.2), det Ψ(μ) ≠ 0 , in a punctured neighbourhood of μ = 0 . Then there is a limit Ψ_{0} = lim_{μ→0} Ψ(μ) and

im Ψ_{0} = ker A_{0} .

Proof. According to Proposition 6.2 we can start from an eigenbasis of A_{0} in U and lift it to a fundamental solution of the equation (6.2). For this fundamental solution the claim of the corollary clearly holds. Since two fundamental solutions of the same differential equation differ only by multiplication from the right by a nondegenerate constant matrix, the corollary is proved. □

For the equation (6.1) taking into account that p = 2ℏ and H_{m} = σ_{m}^{−}σ_{m}^{+} we have

A_{0} = \frac{1}{2} Σ^{−} Σ^{+} = \frac{1}{8} (Σ^{3}(Σ^{3}+ 2) + 4Σ^{−} Σ^{+}) − \frac{1}{8} Σ^{3}(Σ^{3}+ 2)

where Σ^{a} = ∑_{m=1}^{n} σ_{m}^{a}, a = ±, 3 . Since the first term of the last expression is the

Casimir operator for 𝔰𝔩_{2} , the operator A_{0} restricted to the weight subspace (V^{⊗n})_{ℓ} has the following eigenvalues:

\frac{(n − 2k)(n − 2k + 2) − (n − 2ℓ)(n − 2ℓ + 2)}{4} = \frac{(ℓ − k)(n − k − ℓ + 1)}{2} ,

k = 0, …, min (ℓ, n − ℓ) , which are nonnegative half-integers. Besides, ker A_{0} ∩ (V^{⊗n})_{ℓ} = (V^{⊗n})_{ℓ}^{sing}.

Consider the hypergeometric map Ī_{z,ℓ}(μ) , cf. (4.1), where we write its dependence of μ explicitly. Theorem 6.1 means that Ī_{z,ℓ}(μ) obeys the equation (6.1), and according to Theorem 4.2 it is a fundamental solution of the equation (6.1) with values in (V^{⊗n})_{ℓ}. Hence, by Corollary 6.3 there is a limit lim_{μ→0} Ī_{z,ℓ}(μ) and im ( lim_{μ→0} Ī_{z,ℓ}(μ)) = (V^{⊗n})_{ℓ}^{sing}.

If 2ℓ ⩽ n, then lim_{μ→0} Ī_{z,ℓ}(μ) = Ī_{z,ℓ}(0) , which completes the proof of Theorem 4.3.

Remark. The fact that im Ī_{z,ℓ}(0) ⊂ (V^{⊗n})_{ℓ}^{sing} is well known, see [NPT] for details of its proof.

7. Proof of Theorem 4.4

It is known that im X_{z,ℓ}^{(1)} ⊂ ker Ī_{z,ℓ}, see [NPT] for details. The proof of the second inclusion im X_{z,ℓ}^{(2)} ⊂ ker Ī_{z,ℓ} is similar, though slightly more complicated. Due to Theorem 4.3 what remains to show is

dim(im X_{z,ℓ}^{(1)} + im X_{z,ℓ}^{(2)} ) = dim(V^{⊗n})_{ℓ} − dim(V^{⊗n})_{ℓ}^{sing} = dim Σ^{−}(V^{⊗n})_{ℓ−1}.

The second equality is due to the common knowledge:

(V^{⊗n})_{ℓ} = (V^{⊗n})_{ℓ}^{sing} ⊕ Σ^{−}(V^{⊗n})_{ℓ−1}.

Let ξ_{1}, …, ξ_{n} be the Grassman variables: ξ_{k} ξ_{m} = −ξ_{m}ξ_{k} for any k, m = 1, …, n. Consider the group algebra ℂ[ξ_{1}, …, ξ_{n}] . Set deg ξ_{1} = … = deg ξ_{n} = 1 , and denote by ℂ[ξ_{1}, …, ξ_{n}]_{k} the subspace of elements of degree k. By convention, set ℂ[ξ_{1}, …, ξ_{n}]_{k} = 0 for k < 0 . Introduce elements φ^{(1)}, φ^{(2)} ∈ ℂ[ξ_{1}, …, ξ_{n}] :

(7.1)  φ^{(1)} = ∑_{m=1}^{n} ξ_{m} ,  φ^{(2)} = ∑_{1⩽k<m⩽n}ξ_{k} ξ_{m} .

Consider isomorphisms of vector spaces:

(7.2)  \overline{F} F_{q}^{∧k}(z) → ℂ[ξ_{1}, …, ξ_{n}]_{k} ,  W_{M}(·, z) ↦ ξ_{m_{1}} … ξ_{m_{k}} ,

where M = {m_{1} < … < m_{k}}. Under these isomorphisms the operator X_{z,ℓ}^{(a)} trans-

lates modulo proportionality into multiplication by φ^{(a)} acting on ℂ[ξ_{1}, …, ξ_{n}]_{(ℓ−a)} , because the functions Ξ^{(1)}, Ξ^{(2)}, cf. (4.2), can be written as follows:

Ξ^{(1)} = 2 ∑_{m=1}^{n}W_{{m}} , Ξ^{(2)} = 4 ∑_{1⩽k<m⩽n}W_{{k,m}} .

Here the functions W_{{m}} and W_{{k,m}} are given by (5.1) for M = {m} and M = {k, m}, respectively.

Therefore, the problem is to show that

(7.3) dim(φ^{(1)}ℂ[ξ_{1}, …, ξ_{n}]_{(ℓ−1)} + φ^{(2)}ℂ[ξ_{1}, …, ξ_{n}]_{(ℓ−2)}) = dim Σ^{−}(V^{⊗n})_{ℓ−1}.

The cases ℓ = 0, 1 are trivial. The case ℓ = n = 2 is very easy. From now on we assume that ℓ ⩾ 2 and n ⩾ 3 (recall that ℓ ⩽ n). It is easy to see that

ℂ[ξ_{1}, …, ξ_{n}]_{k} = ℂ[ξ_{1}, …, ξ_{n−1}]_{k} ⊕ φ^{(1)}ℂ[ξ_{1}, …, ξ_{n−1}]_{k}

and

φ^{(1)}ℂ[ξ_{1}, …, ξ_{n−1}]_{k} = φ^{(1)}ℂ[ξ_{1}, …, ξ_{n}]_{k} .

Set φ̃ = ∑_{1⩽k<m<n}ξ_{k} ξ_{m}. Since φ^{(2)} = φ̃ + φ^{(1)}ξ_{n}, it is a simple exercise to replace

relation (7.3) by

(7.4) dim φ^{(1)}ℂ[ξ_{1}, …, ξ_{n−1}]_{(ℓ−1)} + dim φ̃ ℂ[ξ_{1}, …, ξ_{n−1}]_{(ℓ−2)} = dim Σ^{−}(V^{⊗n})_{ℓ−1}.

Calculating dimensions explicitly, we further reduce (7.4) to the following relation:

dim φ̃ ℂ[ξ_{1}, …, ξ_{n−1}]_{(ℓ−2)} = \{ \begin{matrix}dim ℂ[ξ_{1}, …, ξ_{n−1}]_{(ℓ−2)} & if & 2ℓ ⩽ n \\ dim ℂ[ξ_{1}, …, ξ_{n−1}]_{ℓ} & if & 2ℓ > n \end{matrix} ,

which another look is

(7.5)  dim φ̃ ℂ[ξ_{1}, …, ξ_{n−1}]_{(ℓ−2)} =
= min (dim ℂ[ξ_{1}, …, ξ_{n−1}]_{(ℓ−2)} , dim ℂ[ξ_{1}, …, ξ_{n−1}]_{ℓ}) .

Now it is convenient to change the variables. Set

ζ_{k} = ξ_{k} + … + ξ_{n−k−1} ,  ζ_{n−k} = ξ_{k+1} + … + ξ_{n−k} ,  1 ⩽ k < n/2 ,

and ζ_{n/2} = ξ_{n/2} if n is even. Then φ̃ = ∑_{1⩽k<n/2} ζ_{k} ζ_{n−k}.

Denote by ζ̂_{m} and φ̂ the operators of multiplication by ζ_{m} and φ̃, respectively. Introduce the Grassman derivations ∂_{1}, …, ∂_{n−1} corresponding to ζ_{1}, …, ζ_{n−1} :

∂_{k}1 = 0 ,  ∂_{k} ζ̂_{m} + ζ̂_{m} ∂_{k} = δ_{km} ,  k, m = 1, …, n − 1 .

Set D = ∑_{1⩽k<n/2} ∂_{k} ∂_{n−k}. Then the operators D, φ̂ and [D, φ̂] constitute an 𝔰𝔩_{2}

action in ℂ[ξ_{1}, …, ξ_{n−1}] , and the relation (7.5) results from the representation theory of 𝔰𝔩_{2} .

8. Kernel of the hypergeometric map at μ = 0 and U_{q}(𝔰𝔩_{2}) at q = e^{πi/2}

Consider the quantum group U_{q}(𝔰𝔩_{2}) with generators e, f, k subject to relations

ke = q^{2}ek ,  kf = q^{−2}fk ,  [e,f] = \frac{k − k^{−1}}{q − q^{−1}} ,

and a coproduct Δ:

Δ(e) = 1 ⊗ e + e ⊗ k ,  Δ(f) = k^{−1} ⊗ f + f ⊗ 1 ,  Δ(k) = k ⊗ k ,

Make the space V into a U_{q}(𝔰𝔩_{2})-module, the generators e, f, k acting as σ^{+}, σ^{−}, q^{σ^{3}}, respectively. Then the generator f is represented in the U_{q}(𝔰𝔩_{2})-module V^{⊗n} by the operator F (q) ∈ End(V^{⊗n}) :

F (q) = ∑_{m=1}^{n} q^{−σ_{1}^{3}} … q^{−σ_{m−1}^{3}} σ_{m}^{−} .

Looking at the limit q → e^{πi/2} define operators F^{(1)}, F^{(2)} ∈ End(V^{⊗n}) :

F^{(1)} = − iF (e^{πi/2}) = ∑_{m=1}^{n}(−i)^{m} σ_{1}^{3} … σ_{m−1}^{3} σ_{m}^{−} ,

F^{(2)} = − lim_{q→i} \frac{F (q)^{2}}{1 + q^{2}} = ∑_{1⩽k<m⩽n}(−i)^{k+m} σ_{k}^{−}σ_{k+1}^{3} … σ_{m−1}^{3} σ_{m}^{−} ,

the overall normalization factors being fixed just for the aesthetic reason. There is a faithful representation of ℂ[ξ_{1}, …, ξ_{n}] in V^{⊗n} given by

(8.1)  ξ_{m} ↦ (−i)^{m} σ_{1}^{3} … σ_{m−1}^{3} σ_{m}^{−} ,  m = 1, …, n ,

the operators F^{(1)}, F^{(2)} representing the elements φ^{(1)}, φ^{(2)} ∈ ℂ[ξ_{1}, …, ξ_{n}] , cf. (7.1).

This representation is equivalent to the left regular representation of ℂ[ξ_{1}, …, ξ_{n}] , the intertwiner being given as follows:

(8.2)  ξ_{m_{1}} … ξ_{m_{k}} ↦ (−i)^{∑_{a=1}^{k} m_{a}}v_{M} ,  1 ⩽ m_{1} < … < m_{k} ⩽ n ,

where M = {m_{1}, …, m_{k}} and the vector v_{M} is defined by (3.2). Actually, the representation (8.1) is the Jordan-Wigner transformation.

Remark. For generic q it is known that dim F (q)(V^{⊗n})_{k} = dim Σ^{−}(V^{⊗n})_{k} for any k = 0, …, n. Consider a subspace lim_{q→i}(F (q)(V^{⊗n})_{ℓ−1}). It clearly contains both

F^{(1)}(V^{⊗n})_{ℓ−1} and F^{(2)}(V^{⊗n})_{ℓ−2} . Hence, the equality (7.3) implies that

(8.3)  F^{(1)}(V^{⊗n})_{ℓ−1} + F^{(2)}(V^{⊗n})_{ℓ−2} = lim_{q→i}(F (q)(V^{⊗n})_{ℓ−1}) .

Denote by C_{z} the composition of isomorphisms (7.2) and (8.2):

C_{z} : ⨁_{k=0}^{n} \overline{F} F_{q}^{∧k}(z) → V^{⊗n} , C_{z} : W_{M}(·, z) ↦ (−i)^{∑_{a=1}^{k} m_{a}}v_{M} ,

Under this map the operator X_{z,ℓ}^{(a)} translates modulo proportionality into the operator

F^{(a)} acting on (V^{⊗n})_{(ℓ−a)} . So, by Theorem 4.4 and relation (8.3) the kernel of the

hypergeometric map Ī_{z,ℓ} corresponds to the limit of the subspace F (q)(V^{⊗n})_{ℓ−1} as q tends to i.

Appearance of the maps C_{z,k} in the descriptiopn of solutions of the qKZ equation is not accidental. Utilizing the general picture developed in [TV], [MV] one can see that for generic step p the space of solutions of the qKZ equation (2.1) at μ ≠ 0 with values in V^{⊗n} is naturally isomorphic over the field of quasiconstants to the U_{q}(𝔰𝔩_{2})-module V^{⊗n} desribed at the beginning of this section, q being equal to e^{πiℏ/p}. The isomorphism is based on the construction of hypergeometric solutions of the qKZ equation and its realization in the case p = 2ℏ, studied here, is the map C_{z}.

At the same time, for generic step p, the space of solutions of the qKZ equation (2.1) at μ = 0 with values in (V^{⊗n})^{sing} is isomorphic over quasiconstants to either the subspace (V^{⊗n})^{sing,q} of singular vectors with respect to the quantum group U_{q}(𝔰𝔩_{2}) or the quotient space V^{⊗n}/F (q)V^{⊗n}. These models of the space of solutions are equivalent since V^{⊗n} = (V^{⊗n})^{sing,q} ⊕ F (q)V^{⊗n} for generic q.

The results of this paper means that for p = 2ℏ, so that q = e^{πi/2}, the second model survives with the minimal required modification; namely, the space of solutions of the qKZ equation (2.1) at μ = 0 is isomorphic over quasiconstants to the quotient space

V^{⊗n}/ lim_{q→i}(F (q)V^{⊗n}) = V^{⊗n}/(F^{(1)}V^{⊗n} + F^{(2)}V^{⊗n}) .

Unfortunately, the first model does not survive at p = 2ℏ, at least for even n, because in this case the subspaces lim_{q→i}(V^{⊗n})^{sing,q} and lim_{q→i}(F (q)V^{⊗n}) intersect nontrivially,

and therefore, their sum does not coincide with V^{⊗n}.

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