Homological Representations of Braid Groups and the Space of Conformal Blocks

Abstract

We compare homological representations of the braid groups and the monodromy representations of the KZ connection by means of hypergeometric integrals. Then we discuss a relationship to the space of conformal blocks.

1 Introduction

The purpose of this paper is to give a review on recent developments on the homological representations of the braid groups and the monodromy representations of the Knizhnik-Zamolodchikov (KZ) connection. The homological representations of the braid groups are defined as the action of the braid groups on the homology of abelian coverings of certain configuration spaces. They were extensively investigated by Bigelow [2] and Krammer [12]. It was shown independently by Bigelow and Krammer that they provide faithful representations of braid groups.

On the other hand, it was shown by Schechtman-Varchenko [14] and others that the solutions of the KZ equation are expressed by hypergeometric integrals. We consider the KZ equation with values in the space of null vectors in the tensor product of Verma modules of sl ₂(C) and show that a specialization of the homological representation is equivalent to the monodromy representation of such KZ equation for generic parameters. This result was obtained in [9, 10] and [11].

Then we describe a relation between homological representations and the space of conformal blocks in the case of complex simple Lie algebras. We study a period map from the homology of local systems over the configuration spaces to the space of conformal blocks. This construction is based on [4, 5, 15] and [11].

2 Local Systems on the Complement of Hyperplane Arrangement

Let be an arrangement of affine hyperplanes in the complex vector space C ⁿ. We consider the complement

First, we recall some basic definition for local systems. Let M be a smooth manifold and V a complex vector space. Given a linear representation of the fundamental group

$$\displaystyle{r:\pi _{1}(M,x_{0})\longrightarrow GL(V )}$$

there is an associated flat vector bundle E over M. The local system associated to the representation r is the sheaf of horizontal sections of the flat bundle E. Let $\pi: \widetilde{M} \rightarrow M$ be the universal covering. We denote by Z π ₁ the group ring of the fundamental group π ₁(M, x ₀). We consider the chain complex

$$\displaystyle{C_{{\ast}}(\widetilde{M} ) \otimes _{\mathbf{Z}\pi _{1}}V }$$

with the boundary map defined by ∂(c ⊗ v) = ∂c ⊗ v. Here Z π ₁ acts on $C_{{\ast}}(\widetilde{M} )$ via the deck transformations and on V via the representation r. The homology of this chain complex is called the homology of M with coefficients in the local system and is denoted by .

Let be a complex rank one local system over associated with a representation of the fundamental group

For an arrangement we denote by f _j a linear form defining the hyperplane H _j , $1\leqslant j\leqslant \ell$. We associate a complex number a _j = a(H _j ) called an exponent to each hyperplane and consider the multivalued function

$$\displaystyle{\varPhi = f_{1}^{a_{1} }\cdots f_{\ell}^{a_{\ell}}.}$$

The homology is isomorphic to Z ^⊕ℓ, where each generator corresponds to a hyperplane. By associating to the generator of corresponding to the hyperplane H _j the complex number $e^{2\pi \sqrt{-1}a_{j}}$ we obtain a homomorphism . Combining with the abelianization map we obtain a homomorphism

The associated local system is denoted by .

We shall investigate the homology of with coefficients in the local system . For our purpose the homology of locally finite chains also plays an important role.

We briefly summarize basic properties of the above homology groups. Let be an essential hyperplane arrangement. Namely, we suppose that maximal codimension of a non-empty intersection of some subfamily of is equal to n. We choose a smooth compactification Namely, is written as X∖D, where X is a smooth projective variety and D is a divisor with normal crossings.

We shall say that the local system is generic if and only if there is an isomorphism

where i _∗ is the direct image and i _! is the extension by 0. This means that the monodromy of along any divisor at infinity is not equal to 1. The following theorem was shown in [7].

Theorem 1

If the local system is generic in the above sense, then there is an isomorphism

Moreover, we have for any k ≠ n.

Let us suppose that each hyperplane in is defined over R. We set and denote by Δ _ν , $1\leqslant \nu \leqslant s$, the bounded chambers in . Let

be the inclusion map. We denote by the restriction of the local system on . In this situation we have the following theorem.

Theorem 2 ([11])

In addition to the condition that the local system is generic we suppose that there is an isomorphism

Then the homology with locally finite chains is spanned by the homology class of bounded chambers Δ _ν , $1\leqslant \nu \leqslant s$ .

3 Homological Representations of Braid Groups

We denote by B _n the braid group with n strands. We fix a positive integer n and a set of distinct n points in R ² as

$$\displaystyle{Q =\{ (0,0),\cdots \,,(n - 1,0)\},}$$

where we set p _ℓ = (ℓ − 1, 0), ℓ = 1, ⋯ , n. We take a 2-dimensional disk D in R ² containing Q in the interior. We fix a positive integer m and consider the configuration space of ordered distinct m points in Σ = D∖Q defined by

which is also denoted by . The symmetric group $\mathfrak{S}_{m}$ acts freely on by the permutations of distinct m points. The quotient space of by this action is by definition the configuration space of unordered distinct m points in Σ and is denoted by . We also denote this configuration space by .

In the original papers by Bigelow [2] and by Krammer [12] the case m = 2 was extensively studied, but for our purpose it is convenient to consider the case when m is an arbitrary positive integer such that $m\geqslant 2$.

We identify R ² with the complex plane C. The quotient space $\mathbf{C}^{m}/\mathfrak{S}_{m}$ defined by the action of $\mathfrak{S}_{m}$ by the permutations of coordinates is analytically isomorphic to C ^m by means of the elementary symmetric polynomials. Now the image of the hyperplanes defined by t _i = p _ℓ , ℓ = 1, ⋯ , n, and the diagonal hyperplanes t _i = t _j , $1\leqslant i\leqslant j\leqslant m$, are complex codimension one irreducible subvarieties of the quotient space $D^{m}/\mathfrak{S}_{m}$. This allows us to give a description of the first homology group of as

(1)

where the first n components correspond to meridians of the images of hyperplanes t _i = p _ℓ , ℓ = 1, ⋯ , n, and the last component corresponds to the meridian of the image of the diagonal hyperplanes t _i = t _j , $1\leqslant i\leqslant j\leqslant m$, namely, the discriminant set. We consider the homomorphism

(2)

defined by α(x ₁, ⋯ , x _n , y) = (x ₁ + ⋯ + x _n , y). Composing with the abelianization map , we obtain the homomorphism

(3)

Let be the covering corresponding to $\mathop{\mathrm{Ker}}\nolimits \beta$. Now the group Z ⊕Z acts as the deck transformations of the covering π. We identify the group ring of Z ⊕Z with the ring of Laurent polynomials R = Z[q ^±1, t ^±1]. We consider the homology group

as an R-module by the action of the deck transformations.

As is explained in the case of m = 2 in [2] it can be shown that H _n, m is a free R-module of rank

$$\displaystyle{ d_{n,m} = \left (\begin{array}{c} m + n - 2\\ m \end{array} \right ). }$$

(4)

A basis of H _n, m as a free R-module is discussed in relation with the homology of local systems in the next sections. Let denote the mapping class group of the pair (D, Q), which consists of the isotopy classes of homeomorphisms of D which fix Q setwise and fix the boundary ∂D pointwise. The braid group B _n is naturally isomorphic to the mapping class group . Now a homeomorphism f representing a class in induces a homeomorphism , which is uniquely lifted to a homeomorphism of . This homeomorphism commutes with the deck transformations.

Therefore, for $m\geqslant 2$ we obtain a representation of the braid group

$$\displaystyle{ \rho _{n,m}: B_{n}\longrightarrow \mathop{ \mathrm{Aut}}\nolimits _{R}H_{n,m} }$$

(5)

which is called the homological representation of the braid group or the Lawrence-Krammer-Bigelow (LKB) representation. Let us remark that in the case m = 1 the above construction gives the reduced Burau representation over Z[q ^±1].

4 Homology of Local Systems on Configuration Spaces

Let us consider the configuration space of ordered distinct n points in the complex plane defined by

$$\displaystyle{ X_{n} =\{ (z_{1},\cdots \,,z_{n}) \in \mathbf{C}^{n}\;\ z_{ i}\neq z_{j}\ \mbox{ if}\ i\neq j\}. }$$

The configuration space X _n is also denoted by as in the previous section. The fundamental group of X _n is the pure braid group with n strands denoted by P _n . For a positive integer m we consider the projection map

$$\displaystyle{ \pi _{n,m}: X_{n+m}\longrightarrow X_{n} }$$

(6)

given by π _n, m (z ₁, ⋯ , z _n , t ₁, ⋯ , t _m ) = (z ₁, ⋯ , z _n ), which defines a fiber bundle over X _n . For p ∈ X _n the fiber π _n, m ⁻¹(p) is denoted by X _n, m , which is also written as . Let (z ₁, ⋯ , z _n ) be the coordinates for p. Then, X _n, m is the complement of hyperplanes defined by

$$\displaystyle{ t_{i} = z_{\ell},\quad 1\leqslant i\leqslant m,\ 1\leqslant \ell\leqslant n,\quad t_{i} = t_{j},\quad 1\leqslant i <j\leqslant m. }$$

(7)

We call these hyperplanes H _iℓ , $1\leqslant i\leqslant m,\ 1\leqslant \ell\leqslant n$, and D _ij , $1\leqslant i <j\leqslant m$. Such arrangement of hyperplanes is called a discriminantal arrangement. The symmetric group $\mathfrak{S}_{m}$ acts on X _n, m by the permutations of the coordinates functions t ₁, ⋯ , t _m . We put $Y _{n,m} = X_{n,m}/\mathfrak{S}_{m}$, which is also denoted by .

Identifying R ² with the complex plane C, we have the inclusion map

(8)

which is a homotopy equivalence. By taking the quotient by the action of the symmetric group $\mathfrak{S}_{m}$, we have the inclusion map

(9)

which is also a homotopy equivalence.

We fix p = (z ₁, z ₂⋯ , z _n ) as a base point. We consider a rank one local system associated with a representation r: π ₁(X _n, m , x ₀) ⟶ C ^∗. 

Let us consider the compactification

$$\displaystyle{ i_{0}: X_{n,m}\longrightarrow (\mathbf{C}P^{1})^{m} =\mathop{\underbrace{ \mathbf{C}P^{1} \times \cdots \times \mathbf{C}P^{1}}}\limits _{ m}. }$$

Then we take blowing-ups at multiple points $\pi: (\widehat{\mathbf{C}P^{1}})^{m}\longrightarrow (\mathbf{C}P^{1})^{m}$ and obtain a smooth compactification $i: X_{n,m} \rightarrow (\widehat{\mathbf{C}P^{1}})^{m}$ with normal crossing divisors. We are able to write down the condition explicitly by computing the monodromy of the local system along divisors at infinity.

We consider the local system associated with the multivalued function of the form

$$\displaystyle{ \varPhi =\prod _{1\leqslant i\leqslant m,1\leqslant \ell\leqslant n}(t_{i} - z_{\ell})^{\alpha _{\ell}}\prod _{1\leqslant i<j\leqslant m}(t_{i} - t_{j})^{2\gamma }. }$$

(10)

The local system on X _n, m is invariant under the action of the symmetric group $\mathfrak{S}_{m}$ and induces the local system on Y _n, m .

We have the following proposition.

Proposition 1

There is an open dense subset V in C ^ℓ+1 such that for (α ₁, ⋯α _ℓ , γ) ∈ V the associated local system on Y _n, m satisfies

and for any k ≠ m. Moreover, we have

(11)

where we use the same notation as in Eq. ( 4 ) for d _n, m .

For the purpose of describing the homology group and we introduce the following notation. We take the base point p = (1, ⋯ , n). For non-negative integers m ₁, ⋯ , m _n−1 satisfying

$$\displaystyle{ m_{1} + \cdots + m_{n-1} = m }$$

we define a bounded chamber $\varDelta _{m_{1},\cdots \,,m_{n-1}}$ in R ^m by

$$\displaystyle\begin{array}{rcl} & & 1 <t_{1} <\cdots <t_{m_{1}} <2 {}\\ & & 2 <t_{m_{1}+1} <\cdots <t_{m_{1}+m_{2}} <3 {}\\ & & \cdots {}\\ & & n - 1 <t_{m_{1}+\cdots +m_{n-2}+1} + \cdots + t_{m} <n. {}\\ \end{array}$$

We put M = (m ₁, ⋯ , m _n−1) and we write Δ _M for $\varDelta _{m_{1},\cdots \,,m_{n-1}}$. We denote by $\overline{\varDelta }_{M}$ the image of Δ _M by the projection map π _n, m . The bounded chamber Δ _M defines a homology class and its image $\overline{\varDelta }_{M}$ defines a homology class .

5 KZ Connection

Let $\mathfrak{g}$ be a complex semi-simple Lie algebra and {I _μ } be an orthonormal basis of $\mathfrak{g}$ with respect to the Cartan-Killing form. We set Ω = ∑ _μ I _μ ⊗ I _μ . Let $r_{i}: \mathfrak{g} \rightarrow \mathop{\mathrm{End}}\nolimits (V _{i})$, $1\leqslant i\leqslant n$, be representations of the Lie algebra $\mathfrak{g}$. We denote by Ω _ij the action of Ω on the i-th and j-th components of the tensor product V ₁ ⊗⋯ ⊗ V _n . It is known that the Casimir element c = ∑ _μ I _μ ⋅ I _μ lies in the center of the universal enveloping algebra $U\mathfrak{g}$. By means of this fact it can be shown that the infinitesimal pure braid relations

$$\displaystyle{ [\varOmega _{ik},\varOmega _{ij} +\varOmega _{jk}] = 0,\quad (i,j,k\ \ \mbox{ distinct}), }$$

(12)

$$\displaystyle{ [\varOmega _{ij},\varOmega _{k\ell}] = 0,\quad (i,j,k,\ell\ \ \mbox{ distinct}) }$$

(13)

hold.

We define the Knizhnik-Zamolodchikov (KZ) connection as the 1-form

$$\displaystyle{ \omega = \frac{1} {\kappa } \sum _{1\leqslant i<j\leqslant n}\varOmega _{ij}d\log (z_{i} - z_{j}) }$$

(14)

with values in $\mathop{\mathrm{End}}\nolimits (V _{1} \otimes \cdots \otimes V _{n})$ for a non-zero complex parameter κ. We set ω _ij = dlog(z _i − z _j ), $1\leqslant i,j\leqslant n$. It follows from the above infinitesimal pure braid relations among Ω _ij together with Arnold’s relation

$$\displaystyle{ \omega _{ij} \wedge \omega _{jk} +\omega _{jk} \wedge \omega _{k\ell} +\omega _{k\ell} \wedge \omega _{ij} = 0 }$$

that ω ∧ω = 0 holds. This implies that ω defines a flat connection for a trivial vector bundle over the configuration space X _n = {(z ₁, ⋯ , z _n ) ∈ C ⁿ ;  z _i ≠ z _j  if i ≠ j} with fiber V ₁ ⊗⋯ ⊗ V _n . A horizontal section of the above flat bundle is a solution of the total differential equation dφ = ωφ for a function φ(z ₁, ⋯ , z _n ) with values in V ₁ ⊗⋯ ⊗ V _n . This total differential equation can be expressed as a system of partial differential equations

$$\displaystyle{ \frac{\partial \varphi } {\partial z_{i}} = \frac{1} {\kappa } \sum _{j,j\neq i} \frac{\varOmega _{ij}} {z_{i} - z_{j}}\,\varphi,\quad 1\leqslant i\leqslant n, }$$

(15)

which is called the KZ equation. The KZ equation was first introduced in [6] as the differential equation satisfied by n-point functions in Wess-Zumino-Witten conformal field theory.

Let ϕ(z ₁, ⋯ , z _n ) be the matrix whose columns are linearly independent solutions of the KZ equation. By considering the analytic continuation of the solutions with respect to a loop γ in X _n with base point x ₀ we obtain the matrix θ(γ) defined by

$$\displaystyle{ \phi (z_{1},\cdots \,,z_{n})\mapsto \phi (z_{1},\cdots \,,z_{n})\theta (\gamma ). }$$

Since the KZ connection ω is flat the matrix θ(γ) depends only on the homotopy class of γ. The fundamental group π ₁(X _n , x ₀) is the pure braid group P _n . As the above holonomy of the connection ω we have a one-parameter family of linear representations of the pure braid group

$$\displaystyle{ \theta: P_{n} \rightarrow \mathop{\mathrm{GL}}\nolimits (V _{1} \otimes \cdots \otimes V _{n}). }$$

The symmetric group $\mathfrak{S}_{n}$ acts on X _n by the permutations of coordinates. We denote the quotient space $X_{n}/\mathfrak{S}_{n}$ by Y _n . The fundamental group of Y _n is the braid group B _n . In the case V ₁ = ⋯ = V _n = V, the symmetric group $\mathfrak{S}_{n}$ acts diagonally on the trivial vector bundle over X _n with fiber V ^⊗n and the connection ω is invariant by this action. Thus we have one-parameter family of linear representations of the braid group $\theta: B_{n} \rightarrow \mathop{\mathrm{GL}}\nolimits (V ^{\otimes n}).$

6 Solutions of KZ Equation by Hypergeometric Integrals

In this section we describe solutions of the KZ equation for the case $\mathfrak{g} = sl_{2}(\mathbf{C})$ by means of hypergeometric integrals following Schechtman and Varchenko [14]. A description of the solutions of the KZ equation was also given by Date, Jimbo, Matsuo and Miwa [3]. We refer the reader to [1] and [13] for general treatments of hypergeometric integrals.

Let us recall basic facts about the Lie algebra sl ₂(C) and its Verma modules. As a complex vector space the Lie algebra sl ₂(C) has a basis H, E and F satisfying the relations:

$$\displaystyle{ [H,E] = 2E,\ \ [H,F] = -2F,\ \ [E,F] = H. }$$

(16)

For a complex number λ we denote by M _λ the Verma module of sl ₂(C) with highest weight λ. Namely, there is a non-zero vector v _λ ∈ M _λ called the highest weight vector satisfying

$$\displaystyle{ Hv_{\lambda } =\lambda v_{\lambda },\ Ev_{\lambda } = 0 }$$

and M _λ is spanned by F ^j v _λ , $j\geqslant 0$. It is known that if λ ∈ C is not a non-negative integer, then the Verma module M _λ is irreducible.

For Λ = (λ ₁, ⋯ , λ _n ) ∈ C ⁿ we put | Λ | = λ ₁ + ⋯ + λ _n and consider the tensor product $M_{\lambda _{1}} \otimes \cdots \otimes M_{\lambda _{n}}$. For a non-negative integer m we define the space of weight vectors with weight | Λ | − 2m by

$$\displaystyle{ W[\vert \varLambda \vert - 2m] =\{ x \in M_{\lambda _{1}} \otimes \cdots \otimes M_{\lambda _{n}}\;\ Hx = (\vert \varLambda \vert - 2m)x\} }$$

and consider the space of null vectors defined by

$$\displaystyle{ N[\vert \varLambda \vert - 2m] =\{ x \in W[\vert \varLambda \vert - 2m]\;\ Ex = 0\}. }$$

The KZ connection ω commutes with the diagonal action of $\mathfrak{g}$ on $V _{\lambda _{1}} \otimes \cdots \otimes V _{\lambda _{n}}$, hence it acts on the space of null vectors N[ | Λ | − 2m].

For parameters κ and λ we consider the multi-valued function

$$\displaystyle{ \varPhi _{n,m} =\prod _{1\leqslant i<j\leqslant n}(z_{i} - z_{j})^{\frac{\lambda _{i}\lambda _{j}} {2\kappa } }\prod _{1\leqslant i\leqslant m,1\leqslant \ell\leqslant n}(t_{i} - z_{\ell})^{-\frac{\lambda _{\ell}} {\kappa }}\prod _{1\leqslant i<j\leqslant m}(t_{i} - t_{j})^{\frac{2} {\kappa } } }$$

(17)

defined over X _n+m . The function Φ _n, m is called the master function. Let denote the local system associated to the multi-valued function Φ _n, m .

The symmetric group $\mathfrak{S}_{m}$ acts on X _n, m by the permutations of the coordinate functions t ₁, ⋯ , t _m . The function Φ _n, m is invariant by the action of $\mathfrak{S}_{m}$. The local system over X _n, m defines a local system on Y _n, m , which we denote by . The local system dual to is denoted by .

We put $v = v_{\lambda _{1}} \otimes \cdots \otimes v_{\lambda _{n}}$ and for J = (j ₁, ⋯ , j _n ) set $F^{J}v = F^{j_{1}}v_{\lambda _{ 1}} \otimes \cdots \otimes F^{j_{n}}v_{\lambda _{ n}}$, where j ₁, ⋯ , j _n are non-negative integers. The weight space W[ | Λ | − 2m] has a basis F ^J v for each J with | J | = j ₁ + ⋯ + j _n = m. For the sequence of integers $(i_{1},\cdots \,,i_{m}) = (\mathop{\underbrace{1,\cdots \,,1}}\limits _{j_{1}},\cdots \,,\mathop{\underbrace{n,\cdots \,,n}}\limits _{j_{n}})$ we set

$$\displaystyle{ S_{J}(z,t) = \frac{1} {(t_{1} - z_{i_{1}})\cdots (t_{m} - z_{i_{m}})} }$$

and define the rational function R _J (z, t) by

$$\displaystyle{ R_{J}(z,t) = \frac{1} {j_{1}!\cdots j_{n}!}\sum _{\sigma \in \mathfrak{S}_{m}}S_{J}(z_{1},\cdots \,,z_{n},t_{\sigma (1)},\cdots \,,t_{\sigma (m)}). }$$

Since π _n, m : X _m+n → X _n is a fiber bundle with fiber X _n, m the fundamental group of the base space X _n acts naturally on the homology group . Thus we obtain a representation of the pure braid group which defines a local system on X _n denoted by . In the case λ ₁ = ⋯ = λ _n there is a representation of the braid group which defines a local system on Y _n, m .

The twisted de Rham complex (Ω ^∗(X _n, m ), ∇) is a complex with differential ∇: Ω ^j(X _n, m ) → Ω ^j+1(X _n, m ) defined by

$$\displaystyle{\nabla \omega = d\omega + d\log \varPhi _{n,m} \wedge \omega.}$$

for ω ∈ Ω ^j(X _n, m ). We define a map

$$\displaystyle{\rho: W[\lambda -2m] \rightarrow \varOmega ^{m}(X_{ n,m})}$$

given by

$$\displaystyle{\rho (w) = R_{J}(t,z)dt_{1} \wedge \cdots \wedge dt_{m}}$$

for w = F ^J v using the rational function R _J (t, z). It turns out that ρ induces a map to the cohomology of the twisted de Rham complex

$$\displaystyle{N[\lambda -2m]\longrightarrow H^{m}(\varOmega ^{{\ast}}(X_{ n,m}),\nabla ).}$$

By this construction we obtain a map

defined by

$$\displaystyle{\langle \phi (c),w\rangle =\int _{c}\varPhi _{n,m}\rho (w).}$$

A lot of works have been done on the expression of the solutions of the KZ equation by means of hypergeometric type integrals (see [3] and [14]). For any horizontal section c(z) of the local system we have the following.

Theorem 3 (Schechtman and Varchenko [14])

The integral

$$\displaystyle{\sum _{\vert J\vert =m}\left (\int _{c(z)}\varPhi _{n,m}R_{J}(z,t)\ dt_{1} \wedge \cdots \wedge dt_{m}\right )F^{J}v}$$

lies in the space of null vectors N[ | Λ | − 2m] and is a solution of the KZ equation.

7 Relation Between Homological Representation and KZ Connection

We fix a complex number λ and consider the space of null vectors

$$\displaystyle{N[n\lambda - 2m] \subset M_{\lambda }^{\otimes n}}$$

by putting λ ₁ = ⋯ = λ _n = λ in the definition of Sect. 6. As the monodromy of the KZ connection

$$\displaystyle{\omega = \frac{1} {\kappa } \sum _{1\leqslant i<j\leqslant n}\varOmega _{ij}d\log (z_{i} - z_{j})}$$

with values in N[nλ − 2m] we obtain the linear representation of the braid group

$$\displaystyle{\theta _{\lambda,\kappa }: B_{n}\longrightarrow \mathop{ \mathrm{Aut}}\nolimits N[n\lambda - 2m].}$$

We put

$$\displaystyle{F(z_{1},\cdots \,,z_{n}) =\prod _{1\leqslant i<j\leqslant n}(z_{i} - z_{j})^{\frac{\lambda ^{2}} {2\kappa } }.}$$

The multivalued function F gives an abelian representation of the braid group

$$\displaystyle{a_{n}: B_{n}\longrightarrow \mathbf{C}^{{\ast}}}$$

and the representation θ _λ, κ is expressed in the form $a_{n} \otimes \tilde{\theta }_{\lambda,\kappa }$.

By comparing the action of the braid group on and the monodromy representations of the KZ connection by means of an explicit description of the horizontal sections by hypergeometric integrals we obtain the following theorem.

Theorem 4 ([9, 10])

There exists an open dense subset U in (C ^∗)² such that for (λ, κ) ∈ U the homological representation ρ _n, m with the specialization

$$\displaystyle{q = e^{-2\pi \sqrt{-1}\lambda /\kappa },\quad t = e^{2\pi \sqrt{-1}/\kappa }}$$

is equivalent to the monodromy representation of the KZ connection $\tilde{\theta }_{\lambda,\kappa }$ with values in the space of null vectors N[nλ − 2m] ⊂ M _λ ^⊗n. 

8 Space of Conformal Blocks

First, we recall the definition of the space of conformal blocks. We refer the reader to [8] for an introductory treatment of this subject. Let $\mathfrak{g}$ be a complex simple Lie algebra and $\mathfrak{h}$ be its Cartan subalgebra. We denote by Δ the set of all roots and for a root α set

$$\displaystyle{\mathfrak{g}_{\alpha } =\{ v \in \mathfrak{g}\;\ [h,v] =\alpha (h)v\ \mbox{ for any}\ h \in \mathfrak{h}\}.}$$

We have the root space decomposition

$$\displaystyle{\mathfrak{g} = \mathfrak{h} \oplus \bigoplus _{\alpha \in \varDelta }\mathfrak{g}_{\alpha }.}$$

We denote by $\mathfrak{g}_{+}$ the direct sum of $\mathfrak{g}_{\alpha }$ for all positive roots and $\mathfrak{g}_{-}$ the direct sum of $\mathfrak{g}_{\alpha }$ for all negative roots. We have a decomposition of Lie algebras

$$\displaystyle{\mathfrak{g} = \mathfrak{g}_{+} \oplus \mathfrak{h} \oplus \mathfrak{g}_{-}.}$$

Let θ be the longest root and 〈⋅ , ⋅ 〉 be the invariant bilinear form normalized by 〈θ, θ〉 = 2. We take α ₁, ⋯ , α _r a system of simple roots and $\rho = \frac{1} {2}\sum _{\alpha>0}\alpha$ the half sum of positive roots.

We fix a positive integer K called the level. Let $\lambda \in \mathfrak{h}^{{\ast}}$ be a dominant integral weight satisfying $\langle \lambda,\theta \rangle \leqslant K$. We call such weight a dominant integral weight of level K. We denote by V _λ the irreducible representation of $\mathfrak{g}$ with highest weight λ.

Let us recall the notion of affine Lie algebras. We start from the loop algebra $\mathfrak{g} \otimes \mathbf{C}((\xi ))$, where C((ξ)) denotes the ring of Laurent series. We consider the central extension $\widehat{\mathfrak{g}} = \mathfrak{g} \otimes \mathbf{C}((\xi )) \oplus \mathbf{C}c$ defined by

$$\displaystyle{[X \otimes f,Y \otimes g] = [X,Y ] \otimes fg +\langle X,Y \rangle \ \mathop{\mathrm{Res}}\nolimits _{\xi =0}(df\,g)}$$

We call $\widehat{\mathfrak{g}}$ the affine Lie algebra. We denote by A ₊ the subalgebra of C((ξ)) consisting of the series with only positive powers. Similarly, A ₋ denotes the subalgebra consisting of the series with only negative powers. We define Lie subalgebras N ₊, N ₀, N ₋ by

$$\displaystyle{N_{+} = [\mathfrak{g} \otimes A_{+}] \oplus \mathfrak{g}_{+},\quad N_{0} = \mathfrak{h} \oplus \mathbf{C}c,\quad N_{+} = [\mathfrak{g} \otimes A_{-}] \oplus \mathfrak{g}_{-}.}$$

We have a direct sum decomposition

$$\displaystyle{\widehat{\mathfrak{g}} = N_{+} \oplus N_{0} \oplus N_{-}}$$

as Lie algebras.

Let λ be a dominant integral weight of level K. We start from the finite dimensional irreducible representation V _λ of $\mathfrak{g}$. We consider the Verma module defined as satisfying N ₊ V _λ = 0, where the action of U(N ₋) is free and the central elements c acts as the multiplication by K. It turns out that the Verma module contains a null vector, which means that there exists a non-zero vector such that N ₊ χ = 0. We consider the quotient module

and it can be shown that is an irreducible $\widehat{\mathfrak{g}}$-module. We call the integral highest weight module of $\widehat{\mathfrak{g}}$ with highest weight λ of level K.

We regard the Riemann sphere C P ¹ as the one point compactification C ∪{∞} and fix an affine coordinate function z for C. We take local coordinates around p _j , $1\leqslant j\leqslant n$ as ξ _j = z − z(p _j ) and take ξ _n+1 = 1∕t as a local coordinate around p _n+1 = ∞. We associate to the points p ₁, ⋯ , p _n+1 dominant integral weights λ ₁, ⋯ , λ _n , λ _n+1 of level K.

We denote by the set of meromorphic functions on C P ¹ with poles of any order at most at p ₁, ⋯ , p _n+1. Then has a structure of a Lie algebra and acts diagonally on the tensor product by means of the Laurent expansions of a meromorphic function at the points p ₁, ⋯ , p _n+1 ∈ C P ¹ with respect to the above local coordinates. Here we notice that this action is well-defined since the affine Lie algebra is defined by means of a central extension given by a 2-cocycle expressed by the residue of a 1-form and the sum of the residues is zero on C P ¹.

The space of conformal blocks is defined as the space of coinvariants

There is also a dual formulation as follows. We define as the space of invariant multilinear forms by

where the action of on C is supposed to be trivial. This means that the dual space of conformal blocks is defined as the space of invariant multilinear forms by the action of .

It is a basic result in conformal field theory that the spaces of conformal blocks form a vector bundle over the configuration space X _n equipped with a flat connection. This connection is explicitly give by the KZ connection.

$$\displaystyle{\omega = \frac{1} {\kappa } \sum _{i<j}\varOmega _{ij}d\log (z_{i} - z_{j}),\quad \kappa = K + h}$$

where h is the dual Coxeter number. Therefore, the pure braid group P _n acts on the space of conformal blocks by means of the holonomy of this connection.

It turns out that there is a surjective map

and the kernel is described by some algebraic equations coming from the definition of the space of conformal blocks. The reason that the above map is not an isomorphism is as follows. The integrable highest module is not a Verma module and there exists a null vector in . The existence of such null vectors yields the above algebraic equations.

By generalizing the master function (17) for sl ₂(C) we introduce the following function for the Lie algebra $\mathfrak{g}$ and the dominant integral weights λ ₁, ⋯ , λ _n , λ _n+1. We write

$$\displaystyle{\sum _{i=1}^{n}\lambda _{ i} -\lambda _{n+1} =\sum _{ j=1}^{r}k_{ j}\alpha _{j}}$$

and put m = ∑ _{j = 1} ^r k _j . We set

$$\displaystyle\begin{array}{rcl} \varPhi _{n,m}^{\mathfrak{g}}& =& \prod _{ 1\leqslant i<j\leqslant n}(z_{i} - z_{j})^{\frac{\langle \lambda _{i},\lambda _{j}\rangle } {\kappa } }\prod _{1\leqslant i\leqslant m,1\leqslant \ell\leqslant n}(t_{i} - z_{\ell})^{-\frac{\langle \alpha _{i},\lambda _{\ell}\rangle } {\kappa } }\prod _{1\leqslant i<j\leqslant m}(t_{i} - t_{j})^{\frac{\langle \alpha _{i},\alpha _{j}\rangle } {\kappa } } {}\\ \end{array}$$

and denote by the associated local system on the configuration space X _n, m .

As is shown in [5] one can construct horizontal sections of the KZ connections by means of hypergeometric integrals of the form

$$\displaystyle{\int _{\varDelta }\varPhi _{n,m}^{\mathfrak{g}}R_{ w}(z,t)dt_{1} \wedge \cdots \wedge dt_{m}}$$

with some rational function R _w (z, t) for $w \in (V _{\lambda _{1}} \otimes \cdots \otimes V _{\lambda _{n}} \otimes V _{\lambda _{n+1}}^{{\ast}})/\mathfrak{g}$. Here Δ is a cycle in . There is an action of the symmetric group $\mathfrak{S}_{m}$ on and we define the subspace by

It follows from [5] that we have a well-defined period map

defined by

$$\displaystyle{\langle \phi (c),w\rangle =\int _{c}\varPhi _{n,m}^{\mathfrak{g}}R_{ w}(z,t)dt_{1} \wedge \cdots \wedge dt_{m}.}$$

Combining with results by A. Varchenko [16], we can summarize the properties of the period map ϕ as follows.

Theorem 5

The period map

is equivariant with respect to the action of the pure braid group P _n . If K is sufficiently large relative to λ ₁, ⋯ , λ _n the period map ϕ gives an isomorphism.