A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes

Some aspects of Aomoto's generalized hypergeometric functions on Grassmannian spaces $Gr(k+1,n+1)$ are reviewed. Particularly, their integral representations in terms of twisted homology and cohomology are clarified with an example of the $Gr(2,4)$ case which corresponds to Gauss' hypergeometric functions. The cases of $Gr(2, n+1)$ in general lead to $(n+1)$-point solutions of the Knizhnik-Zamolodchikov (KZ) equation. We further analyze the Schechtman-Varchenko integral representations of the KZ solutions in relation to the $Gr(k+1, n+1)$ cases. We show that holonomy operators of the so-called KZ connections can be interpreted as hypergeometric-type integrals. This result leads to an improved description of a recently proposed holonomy formalism for gluon amplitudes. We also present a (co)homology interpretation of Grassmannian formulations for scattering amplitudes in ${\cal N} = 4$ super Yang-Mills theory.

Yangian symmetries. As mentioned in [26], the holonomy formalism also suggests a natural origin of the dual conformal symmetries. Towards the end of this note we would provide more rigorous mathematical foundations of the holonomy formalism and present an improved description of it. Lastly, we also consider the more familiar Grassmannian formulations of gluon amplitudes in the same framework. Namely, we analyze integral representations of the Grassmannian formulations and present a (co)homology interpretation of those integrals.
This note is organized as follows. In the next section we review some formal results of Aomoto's generalized hypergeometric functions on Gr(k + 1, n + 1), based on textbooks by Japanese mathematicians [15,27,28,29]. We present a review in a pedagogical fashion since these results are not familiar enough to many physicists. In section 3 we consider a particular case Gr(2, n+1) and present its general formulation. In section 4 we further study the case of Gr(2, 4) which reduces to Gauss' hypergeometric function. Imposing permutation invariance among branch points, we here obtain new realizations of the hypergeometric differential equation in a form of a first order Fuchsian differential equation.
In section 5 we apply Aomoto's results to the KZ equation. We first focus on four-point KZ solutions and obtain them in a form of the hypergeometric integral. We then show that (n + 1)-point KZ solutions in general can be represented by generalized hypergeometric functions on Gr(2, n+1). We further consider the Schechtman-Varchenko integral representations of the KZ solutions in this context. The (n + 1)-point KZ solutions can also be represented by the hypergeometric-type integrals on Gr(k + 1, n+ 1) but we find that there exist ambiguities in the construction of such integrals for k ≥ 2. In section 6 we review the construction of the holonomy operators of the KZ connections. We make a (co)homology interpretation of the holonomy operator and obtain a better understanding of analytic properties of the holonomy operator.
The holonomy operator gives a monodromy representation of the KZ equation, which turns out to be a linear representation of a braid group. This mathematical fact has been one of the essential ingredients in the holonomy formalism for gluon amplitudes. In section 7 we briefly review this holonomy formalism and present an improved description of it. In section 8 we also consider the Grassmannian formulations of gluon amplitudes. We observe that these formulations can also be interpreted in terms of the hypergeometric-type integrals. Lastly, we present a brief conclusion.
where k < n and the matrix elements are complex, z ij ∈ C (0 ≤ i ≤ k ; 0 ≤ j ≤ n). A function of Z, which we denote F (Z), is defined as a generalized hypergeometric function on Grassmannian space Gr(k + 1, n + 1) when it satisfies the following relations: where the parameters α j obey the non-integer conditions Integral representation of F (Z) and twisted cohomology The essence of Aomoto's generalized hypergeometric function [15] is that, by use of the so-called twisted de Rham cohomology, 1 F (Z) can be written in a form of integral: where Φ = n j=0 l j (τ ) α j (2.8) l j (τ ) = τ 0 z 0j + τ 1 z 1j + · · · + τ k z kj (0 ≤ j ≤ n) (2.9) The complex variables τ = (τ 0 , τ 1 , · · · , τ k ) are homogeneous coordinates of the complex projective space CP k , i.e., C k+1 − {0, 0, · · · , 0}. The multivalued function Φ is then defined in a space H j (2.11) where H j = {τ ∈ CP k ; l j (τ ) = 0} (2.12) We now consider the meaning of the integral path ∆. Since the integrand Φω is a multivalued k-form, simple choice of ∆ as a k-chain on X is not enough. Upon the choice of ∆, we need to implicitly specify branches of Φ on ∆ as well, otherwise we can not properly define the integral. In what follows we assume these implicit conditions.
Before considering further properties of ∆, we here notice that ω has an ambiguity in the evaluation of the integral (2.7). Suppose α is an arbitrary (k − 1)-form defined in X. Then an integral over the exact k-form d(Φα) vanishes: where ∇ can be interpreted as a covariant (exterior) derivative This means that ω ′ = ω+∇α is equivalent to ω in the definition of the integral (2.7). Namely, ω and ω ′ form an equivalent class, ω ∼ ω ′ . This equivalent class is called the cohomology class.
To study this cohomology class, we consider the differential equation l j (τ ) −α j (λ ∈ C × ) (2. 16) These local solutions are thus basically given by 1/Φ. The idea of locality is essential since even if 1/Φ is multivalued within a local patch it can be treated as a single-valued function. Analytic continuation of these solutions forms a fundamental homotopy group of a closed path in X (or 1/X to be precise but it can be regarded as X by flipping the non-integer powers α j in (2.8)). The representation of this fundamental group is called the monodromy representation. The monodromy representation determines the local system of the differential equation (2.15). The general solution f or 1/Φ gives a rank-1 local system in this sense 2 . We denote this rank-1 local system by L. The above cohomology class is then defined as an element of the k-th cohomology group of X over L, i.e., [ω] ∈ H k (X, L) (2.17) This cohomology group H k (X, L) is also called twisted cohomology group.

Twisted homology and twisted cycles
Having defined the cohomology group H k (X, L), we can now define the dual of it, i.e., the k-th homology group H k (X, L ∨ ), known as the twisted homology group, where L ∨ is the rank-1 dual local system given by Φ. A differential equation corresponding to L ∨ can be written as We can easily check that the general solutions are given by Φ: As before, an element of H k (X, L ∨ ) gives an equivalent class called a homology class.
In the following, we show that the integral path ∆ forms an equivalent class and see that it coincides with the above homology class. Applying Stokes' theorem to (2.13), we find (2.20) where α is an arbitrary (k − 1)-form as before. The boundary operator ∂ is in principle determined from Φ (with information on branches). Denoting C p (X, L ∨ ) a p-dimensional chain group on X over L ∨ , we can express the boundary operator as ∂ : C p (X, L ∨ ) −→ C p−1 (X, L ∨ ). Since the relation (2.20) holds for an arbitrary α, we find that the k-chain ∆ vanishes by the action of ∂: ∂∆ = 0 (2.21) The k-chain ∆ satisfying above is generically called the k-cycle. In the current framework it is also called the twisted cycle. Since the boundary operator satisfies ∂ 2 = 0, the k-cycle has a redundancy in it. Namely, ∆ ′ = ∆ + ∂C (+1) also becomes the k-cycle where C (+1) is an arbitrary (k + 1)-chain or an element of C k+1 (X, L ∨ ). Thus ∆ and ∆ ′ form an equivalent class, ∆ ∼ ∆ ′ , and this is exactly the homology class defined by H k (X, L ∨ ), i.e., To summarize, the generalized hypergeometric function (2.7) is determined by the following bilinear form The condition l j (τ ) = 0 in (2.12) defines a hyperplane in (k + 1)-dimensional spaces. To avoid redundancy in configuration of hyperplanes, we assume the set of hyperplanes are nondegenerate, that is, we consider the hyperplanes in general position. This can be realized by demanding that any (k + 1)-dimensional minor determinants of the (k + 1) × (n + 1) matrix Z are nonzero. We then redefine X in (2.11) as X = {Z ∈ Mat k+1,n+1 (C)| any (k + 1)-dim minor determinants of Z are nonzero} (2.25) In what follows we implicitly demand this condition in Z. The configuration of n + 1 hyperplanes in CP k is determined by this matrix Z.
Apart from the concept of hyperplanes, we can also interpret that the above Z provides n + 1 distinct points in CP k . Since a homogeneous coordinate of CP k is given by C k+1 − {0, 0, · · · , 0}, we can consider each of the n + 1 column vectors of Z as a point in CP k ; the j-th column representing the j-th homogeneous coordinates of CP k (j = 0, 1, · · · , n).
The scale transformation, under which the CP k homogeneous coordinates are invariant, is realized by an action of H n+1 = {diag(h 0 , h 1 , · · · h n )|h j ∈ C × } from right on Z. The general linear transformation of the homogeneous coordinates, on the other hand, can be realized by an action of GL(k + 1, C) from left. These transformations are then given by Linear transformation: where g ∈ GL(k + 1, C) and h ∈ H n+1 . Under these transformations the integral F (Z) in (2.7) behaves as We now briefly show that the above relations lead to the defining equations of the generalized hypergeometric functions in (2.2) and (2.3), respectively. Let 1 n be the n-dimensional identity matrix 1 n = diag(1, 1, · · · , 1), and E (n) ij be an n × n matrix in which only the (i, j)-element is 1 and the others are zero. We consider g in a particular form of where ǫ is a parameter. Then gZ remains the same as Z except the p-th row which is replaced by (z p0 + ǫz i0 , z p1 + ǫz i1 , · · · , z pn + ǫz in ). Then the derivative of F (gZ) with respect to ǫ is expressed as On the other hand, using and (2.28), we find Evaluating the derivative at ǫ = 0 and ǫ = 1 for i = p and i = p, respectively, we then indeed find that (2.28) leads to the differential equation (2.2).
Similarly, parametrizing h as with 0 ≤ j ≤ n, we find that Zh has only one ǫ-dependent column corresponding to the j-th column, where in the last step we use the relation from (2.29): The same derivative can then be expressed as Setting ǫ = 0, we can therefore derive the equation ( This relation leads to which automatically derives the equation (2.4).
The integral F (Z) in (2.7) therefore indeed satisfies the defining equations (2.2)-(2.4) of the generalized hypergeometric functions on Gr(k + 1, n + 1). The Grassmannian space Gr(k+1, n+1) is defined as a set of (k+1)-dimensional linear subspaces in (n+1)-dimensional complex vector space C n+1 . It is defined as where Z is (k + 1) × (n + 1) complex matrices with rank Z = k + 1. Consider some matrix M and assume that there exists a nonzero r-dimensional minor determinant of M. Then the rank of M is in general defined by the largest number of such r's. Thus Z is not exactly same as Z defined in (2.25). Z is more relaxed since it allows some (k + 1)-dimensional minor determinants vanish, that is, Z ⊆ Z. In this sense F (Z) is conventionally called the generalized hypergeometric functions on Gr(k + 1, n + 1) and we follow this convention in the present note.
As before, from Φ we can define rank-1 local systems L, L ∨ on X, which lead to the k-th homology and cohomology groups, H k ( X, L ∨ ) and H k ( X, L). Then the integral over Φ ω is defined as In regard to the cohomology group H k ( X, L), Aomoto shows the following theorem 3 : Correspondingly, the homology group H k ( X, L ∨ ) has dimension n − 1 k and its basis can be formed finite regions bounded by H j . In terms of l j 's the basis of H k ( X, L) can also be chosen as [28]: 3 Generalized hypergeometric functions on Gr(2, n + 1) In this section we consider a particular case of Gr(2, n + 1). The corresponding configuration space is simply given by n + 1 distinct points in CP 1 . This can be represented by a 2 × (n + 1) matrix Z any of whose 2-dimensional minor determinants are nonzero. Allowing the freedom of coordinate transformations GL(2, C) from the right and scale transformations H 2 = diag(h 0 , h 1 ), we can uniquely parametrize Z as where z i = 0, 1, z j (i = j, 3 ≤ i, j ≤ n). Thus we can regard Z as The three other points (z 0 , z 1 , z 2 ) can be fixed at {0, 1, ∞} This agrees with the fact that the GL(2, C) invariance fixes three points out of the (n + 1) distinct points in CP 1 .
In application of the previous section, we can carry out a systematic formulation of the generalized hypergeometric functions on Gr(2, n+1) as follows. We begin with a multivalued function of a form As in (2.8, 2.9), the exponents obey the non-integer conditions As considered before, the latter condition applies to the expression (2.46), that is, when F (Z) is expressed as F (Z) = ∆ Φdt. The defining space of Φ is given by From Φ we can determine a rank-1 local system L on X and its dual local system L ∨ . Applying the result in (2.50), the basis of the cohomology group H 1 (X, L) is then given by In the present case the basis of the homology group H 1 (X, L ∨ ) can be specified by a set of paths connecting the branch points. For example, we can choose these by where ∆ pq denotes a path on CP 1 connecting branch points p and q. To summarize, for an element ∆ ∈ H 1 (X, L ∨ ) associated with Φ of (3.3), we can define a set of generalized hypergeometric functions on Gr(2, n + 1) as where 0 ≤ j ≤ n − 1. In the next section we consider the case of n = 3, the simplest case where only one variable exists, which corresponds to Gauss' hypergeometric function.
Reduction to Gauss' hypergeometric function 1: From defining equations From (4.4) we find the relevant 2 × 4 matrix in a form of The set of equations (2.2)-(2.4) then reduce to the followings: where ∂ ij = ∂ ∂z ij and ∂ 13 = − ∂ ∂z = −∂ z . The last relation (4.18) arises from (2.4); we here write down the one that is nontrivial and involves ∂ z . Since the sum of (4.10) and (4.11) equals to the sum of (4.14)-(4.17), we can easily find α 0 + · · · + α 3 = −2 in accord with (3.5). The second order equation (4.18) is then expressed as This can also be written as We can easily check that (4.20) identifies with the hypergeometric differential equation (4.3).
As seen in (3.1), there exist multiple complex variables for n > 3. In these cases reduction of the defining equations (2.2)-(2.4) can be carried out in principle but, unfortunately, is not as straightforward as the case of n = 3.
Reduction to Gauss' hypergeometric function 2: Use of twisted cohomology The hypergeometric equation (4.3) is a second order differential equation. Setting f 1 = F , f 2 = z b d dz F , we can express (4.3) in a form of a first order Fuchsian differential equation [15]: Using the results (3.3)-(3.9), we now obtain other first order representations of the hypergeometric function.
Let us start with a non-projected multivalued function Φ is defined on X = CP 1 − {0, 1, 1/z, ∞}. From these we can determine a rank-1 local system L and its dual L ∨ on X. Then, using (3.7), we can obtain a basis of the cohomology group H 1 (X, L) given by the following set Similarly, from (3.8) a basis of the homology group H 1 (X, L ∨ ) is given by In terms of these we can express Gauss' hypergeometric function as The derivative of f 01 (Z) = f 01 (z) with respect to z is written as where Thus the derivative comes down to the computation of ∇ z ϕ 01 ; notice that the choice of a twisted cycle ∆ is irrelevant as far as the derivative itself is concerned. In order to make sense of (4.31) we should require ∇ z ϕ 01 ∈ H 1 (X, L), that is, it should be represented by a linear combinations of (4.26)-(4.28). There is a caveat here, however. We know that an element of H 1 (X, L) forms an equivalent class as discussed earlier; see (2.13) and (2.14). In the present case (k = 1), α in (2.13) is a 0-form or a constant. So we can demand in the computation of ∇ z ϕ 01 . This means that the number of the base elements can be reduced from 3 to 2. Namely, any elements of H 1 (X, L) can be expressed by a combinations of an arbitrary pair in (4.26)-(4.28) under the condition (4.33). This explains the numbering discrepancies between (2.50) and (3.7) and agrees with the general result in the previous section that the dimension of the cohomology group is given by Choosing the pair of (ϕ 01 , ϕ ∞0 ), we find This leads to another first order differential equation Solving for f 01 , we can also check that (4.40) becomes Gauss' hypergeometric differential equation (4.3).
The representations (4.23) and (4.37) are obtained by Aomoto-Kita [15] and Haraoka [28], respectively. The last one (4.41) is not known in the literature as far as the author notices. Along the lines of the above derivation, we can also obtain the Aomoto-Kita representation (4.23) as follows. We introduce a new one-form The corresponding hypergeometric function is given by are equally well defined one-forms. We can then choose the pair (ϕ 01 , ϕ 1 1 z ) as a possible basis of the cohomology group. The derivatives ∇ z ϕ 01 , ∇ z ϕ 1 1 z are calculated as where we use the relations As before, ϕ 01 and ϕ 1 1 z have the same factor (1 − t) −1 . Thus, replacing c by c − 1, we obtain a first order differential equation We therefore reproduce the Aomoto-Kita representation (4.22), (4.23) by a systematic construction of first order representations of the hypergeometric differential equation.
Lastly, we note that ϕ ∞0 = dt t and ϕ 01 = dt t(1−t) have the same factor t −1 but we can not absorb this factor into Φ. This is because we can not obtain dt as a base element of H 1 (X, L) which is generically given in a form of d log l j+1 l j as discussed in (3.7).

Reduction to Gauss' hypergeometric function 3: Permutation invariance
The choice of twisted cycles or ∆'s is irrelevant in the above derivations of the first order Fuchsian differential equations. The hypergeometric differential equation is therefore satisfied by a more general integral form, rather than (4.30), i.e., where (p, q) represents an arbitrary pair among the four branch points p, q ∈ {0, 1, 1/z, ∞}. This means that we can impose permutation invariance on the branch points. ∆ pq is then given by the following set of twisted cycles: so that the number of elements becomes 4 2 = 6. Correspondingly, the base elements of the cohomology group also include (For derivations and details of these relations, see [28].) The relevant configuration space represented by Z is given by Gr(2, 4)/S 4 where S 4 denotes the rank-4 symmetry group. The permutation invariance can also be confirmed by deriving another set of the first order differential equations with the choice of ϕ 01 and one of (4.51)- (4.53). This is what we will present in the following.
For the choice of (ϕ 01 , ϕ 1∞ ) we find The corresponding differential equation is then expressed as d dz Solving for f 01 , we can check that (4.63) indeed becomes Gauss' hypergeometric differential equation (4.3).
Similarly, for (ϕ 01 , ϕ 1 z ∞ ) we find The first order differential equation is then expressed as d dz We can check that (4.67) reduces to the hypergeometric differential equation for f 01 .
Lastly, for (ϕ 01 , ϕ 0 1 z ) we find The corresponding differential equation becomes d dz We can check that (4.71) reduces to the hypergeometric differential equation for f 01 as well.
As in the case of (4.42), it is tempting to think of ϕ 1 z ∞ = z−1 z ϕ 1 z ∞ = (z−1) dt 1−zt . But, with ϕ 01 and ϕ 1 z ∞ , it is not feasible to obtain a first order differential equation in the form of (4.67) which leads to the hypergeometric differential equation. This is because, if expanded in ϕ 01 and ϕ 1 z ∞ , the z-dependence of the derivatives ∇ z ϕ 01 and ∇ z ϕ 1 z ∞ , can not be written in terms of 1 z or 1 z−1 .

Summary
In this section we carry out a systematic derivation of first order representations of the hypergeometric differential equation by use of twisted cohomology as the simplest reduction of Aomoto's generalized hypergeometric function. The first order equations are generically expressed as d dz where (pq) denotes a pair of four branch points A list of the (2 × 2) matrices A (pq) 01 obtained in this section is given by the following: where we include the Aomoto-Kita representation A (1 1 z ) 01 . As far as the author notices, these expressions except (4.74, 4.76) are new for the description of the hypergeometric differential equation. A common feature among these matrices is that the determinant is identical: In terms of the first order differential equation (4.73), this means that the action of the derivative on the basis f 01 f pq of the cohomology group H 1 (X, L) can be represented by a generator of the SL(2, C) algebra. In other words, a change of the bases is governed by the SL(2, C) symmetry. The SL(2, C) invariance corresponds to the global conformal symmetry for holomorphic functions on CP 1 . In the present case we start from the holomorphic multivalued function Φ in (4.24) which is defined on X = CP 1 − {0, 1, 1/z, ∞}. The result (4.80) is thus natural in concept but nontrivial in practice because the equivalence condition d log Φ ≡ 0 in (4.33) is implicitly embedded into the expressions (4.74)-(4.79).

Integral representations of the KZ solutions
Having investigated thoroughly reduction of Aomoto's generalized hypergeometric function to Gauss' original hypergeometric function, we now consider a physical problem in relation to the above results. As mentioned in the introduction, solutions of the Knizhnik-Zamolodchikov (KZ) equation are known to be related to the generalized hypergeometric functions. In this section we shed light on this relation along the lines of arguments on cohomology and homology.
The KZ equation We first review basic properties of the Knizhnik-Zamolodchikov (KZ) equation, following the description in [21]. The KZ equation is defined by where Ψ = Ψ(z 1 , z 2 , · · · , z n ) is a function of n complex variables, z i (i = 1, 2, · · · , n), corresponding to a correlation function in the Wess-Zumino-Witten (WZW) model. κ is parametrized as where l is the level number and h ∨ is the dual Coexter number of the Lie algebra g associated to the WZW model. The function Ψ is defined on a space where H ij is denotes a hyperplane in C n defined by z i − z j = 0: Quantum theoretically, the function Ψ should be evaluated as a vacuum expectation value of operators acting on the Hilbert space where V i (i = 1, 2, · · · , n) denotes a Fock space for a particle labeled by i. Ω ij 's in the KZ equation are bialgebraic operators acting on the (i, j) entries of the Hilbert space V ⊗n . These operators satisfy the infinitesimal braid relations: It is well-known that the KZ equation is invariant under the global SL(2, C) symmetry or the conformal transformations. It is then natural to consider each variable z i on CP 1 rather than on C. In practice, this means that we can add an extra variable z 0 = ∞ in the definitions of Ψ, that is, Ψ(z 1 , z 2 , · · · , z n ) −→ Ψ(z 0 , z 1 , z 2 , · · · , z n ). In general, the solutions of the KZ equation (5.1) give (n + 1)-point correlation functions on CP 1 in the WZW model, which we call (n + 1)-point KZ solutions. Fixing (z 1 , z 2 ) = (0, 1), we can then identify the configuration space (5.3) as the 2 × (n + 1) matrix Z in (3.1) defined for the generalized hypergeometric function on Gr(2, n + 1).
In order to relate Ψ to the generalized hypergeometric functions, we need to determine a multivalued function analogous to Φ in (2.8). From the defining space (5.3) we find that the relevant multivalued function is given by In order to interpret Φ 0 as a function, we need to specify the Lie algebra g for Ω ij and relevant actions to the vacuum state. We shall not specify these algebraic properties here and interpret Ω ij as Ω ij , the vacuum expectation value of Ω ij , for the moment. We will specify the algebraic structure to SL(2, C) in the next section. In analogy to (2.15) or (2.18), we now consider the following covariant derivatives General solutions of Df = 0 are given by As before, analytic continuation of these solutions forms a fundamental homotopy group of a closed path in X n . The representation of Π 1 (X n ) gives the monodromy representation of the differential equation Df = 0.
We now notice that in a differential form the KZ equation (5.1) can be expressed as Notice that from the result in section 2 we find that ω ij 's form a basis of the cohomology group of X n . These satisfy the identity From (5.6), (5.7) and (5.14) we can show the flatness of Ω, i.e., Ω is called the KZ connection one-form.
Imposing permutation invariance on X n , we can in fact express the defining space as where S n is the rank-n symmetric group. The fundamental homotopy group of C is given by the the braid group The braid group B n has generators, b 1 , b 2 , · · · , b n−1 , and they satisfy the following relations where we identify b n with b 1 . One of the main results in the KZ equation is that the monodromy representation of the KZ equation can be given by the braid group.

Relation to 4-point KZ solutions
For n = 3 we can check that the KZ equation (5.1) has a solution of the form [21]: where G(z) satisfies the differential equation This equation is equivalent in structure to the first order differential equation (4.73). The solution G(z) thus corresponds to Gauss' hypergeometric function if Ω 12 and Ω 23 are represented by 2 × 2 matrices. In what follows we shall clarify this statement by constructing hypergeometric-type integral representations of the 4-point KZ solutions.
As before, G(z) has singular points at 0, 1, ∞. Thus, evaluated on the Riemann sphere CP 1 , the solution (5.20) can be interpreted as a 4-point solution. Thanks to the SL(2, C) invariance, without losing generality, we can fix the three points (0, 1, ∞). For example, choosing (z 0 , z 1 , z 2 , z 3 ) → (∞, 0, z, 1), we find where we omit the z-independent factor (z 3 − z 1 ) 1 κ (Ω 12 +Ω 13 +Ω 23 ) → e i2πm 1 κ (Ω 12 +Ω 13 +Ω 23 ) with m ∈ Z. When evaluated as a vacuum expectation value (vev), this factor enters nontrivially into the solution Ψ. If the exponent is evaluated as an integer we can not properly define Φ or Φ 0 since these function can be multiplied by 1 = e i2πm for any times. Thus we can naturally demand non-integer conditions for the vev of these exponents: These conditions are analogous to the non-integer conditions we have considered for the exponents α j , say, in (2.44). As mentioned earlier, we will omit the brackets in the following expressions. In terms of the covariant derivative (5.9), we find In this case the KZ equation D z Ψ = 0 directly reduces to the differential equation (5.21). In general, the reduction is not so simple but, as expected, we may relate the KZ equation D z Φ = 0 to the equation (5.21). For example, take the previous choice (z 0 , z 1 , z 2 , z 3 ) → (∞, 0, 1, 1/z), which leads to the following parametrization: The last relation arises from the equivalent relations that are associated to the cohomology group H 1 (X 3 , L 0 ) where L 0 is a rank-1 local system determined by Φ 0 . Considering in one-form, we can express D z Ψdz as The KZ equation D z Ψdz = 0 then reduces to the differential equation for G(z) in (5.21). In the above analysis the concept of cohomology plays an essential role to replace the covariant derivative D z by the ordinary derivative ∂ z in (5.28) and (5.30). In the language of gauge theory the replacement can be implemented by taking a pure gauge of the KZ connection one-form.
Essence of the integral representation lies in the choice of multivalued function. The choice of Φ 0 in (5.26) is, however, not appropriate to derive Gauss' hypergeometric function as Φ 0 does not contain the integral parameter t. To obtain Gauss' hypergeometric function, we need to incorporate the following multivalued function which is defined on X = CP 1 − {∞, 0, 1, 1/z}. Note that the choice of X 3 automatically determines X. As discussed earlier, X n is represented by distinct (n + 1) points in CP 1 . The physical configuration space C = X n /S n imposes permutation (or bosonic) invariance on these so that it gives rise to distinct ordered (n + 1) points in CP 1 .
Another issue on Φ 0 is that it is an algebraic function because of the exponent 1 κ Ω ij . A rank-1 local system determined by Φ 0 is therefore an algebraic one. To incorporate Φ t into Φ 0 we then need to demand that the exponents α i of Φ t be algebraic. To be concrete, we set where Ω i acts on the Fock space V i of V ⊗n in (5.5). We assume non-integer conditions on these, α i ∈ Z, as well. The multivalued function of interest becomes As usually Φ determines a rank-1 local system which we denote L g to indicate the algebraic nature of Φ. The covariant derivative associated with Φ is expressed as where D z is the same as (5.27): The covariant derivative (5.34) is essentially the same as the one in (4.32) except that the derivative ∂ z is now covariantized as D z . We should emphasize that, owing to the equivalent relation (5.29), the action of D z on a one-form F (z)dz reduces to D z F (z)dz ≡ ∂ z F (z)dz where F (z) is an arbitrary function (or 0-form) of z. The negative sign in the exponents (5.32) is chosen such that (5.34) and (4.32) are compatible.
A basis of the cohomology group H 1 (X, L g ) can be given by a pair (ϕ 01 , ϕ pq ) where concrete expressions of these elements are shown in (4.26)-(4.28) and (4.51)-(4.53). Similarly, elements of the homology group H 1 (X, L ∨ g ) can also be obtained from (4.50).
For simplicity, we now choose the pair (ϕ 01 , ϕ ∞0 ). The hypergeometric-type integrals of interest are expressed as where ϕ 01 = dt t(1−t) , ϕ ∞0 = dt t and ∆ ∈ H 1 (X, L ∨ g ). The covariant derivative D z of F 01 with respect to z is calculated as The calculation of ∇ z ϕ 01 should be executed under the equivalent condition The calculations of ∇ z ϕ 01 and ∇ z ϕ ∞0 are therefore exactly the same as those of (4.34) and (4.35), respectively, except that the exponents are now replaced by (a, c − a, −b) → ( 1 κ Ω 1 , 1 κ Ω 2 , 1 κ Ω 3 ). To be concrete, we obtain the following result: Note that we have used notation Ω i for the vacuum expectation value Ω i in the above expressions. The differential equation (5.40) is first order in the covariant derivative D z . Using the results in the previous section, we then find that F 01 (z) satisfies the following covariantized hypergeometric differential equation Notice that, due to the equivalent relation D z F 01 (z)dz ≡ ∂ z F 01 (z)dz, the covariantized equation (5.42) becomes the ordinary hypergeometric differential equation (4.3) when evaluated in the integrand over the physical configuration space C for n = 3.
An integral representation of the 4-point KZ solution can be constructed as From this representation we can easily compute D z Ψ as , respectively. As mentioned earlier, the basis of the cohomology group H 1 (X, L g ) can be given by a pair (ϕ 01 , ϕ pq ) where in the previous section, with the replacements in (5.43). In general, Ω 12 and Ω 23 should be related to these B 1 , respectively. So far, we argue the 4-point KZ solutions in terms of the parametrization (z 0 , z 1 , z 2 , z 3 ) = (∞, 0, 1, 1/z) so that we can utilize the results in the previous section. Now that the integral representation of the solutions become clear, we consider the simplest parametrization (z 0 , z 1 , z 2 , z 3 ) = (∞, 0, z, 1) again. As discussed in (5.10), general solutions of the KZ equation is expressed as λΦ 0 (λ ∈ C × ) for any n. Thus the essential part of solving the KZ equation is to find λ, in the present case, in a form of integrals. From the above analyses we find that for n = 3 we have two independent solutions and these can be expressed in terms of the hypergeometric-type integrals. The key ingredient of the integral is the elements of cohomology and homology groups H 1 (X, L g ), H 1 (X, L ∨ g ) where X = CP 1 − {∞, 0, z, 1} and the rank-1 local system L g and its dual L ∨ g are determined by Φ t . We now rewrite Φ t as The element of the homology group ∆ = H 1 (X, L ∨ g ) or the twisted cycle defines the integral path over X. The possible twisted cycles are given by ∆ pq where (p, q) denotes distinct pairs of {∞, 0, z, 1}. The basis of the cohomology group, on the other hand, defines a one-form to be integrated apart from the factor of the multivalued function Φ = Φ 0 Φ t . Owing to the equivalent relation d log Φ t ≡ 0, the basis consists of two elements of H 1 (X, L ∨ g ). As considered earlier, there are six different choices for the bases. From (5.46) we can easily find the one of these can be given by { dt t , dt t−z }. The two independent 4-point KZ solutions can then be expressed as κ Ω 23 and Φ t is given by (5.46). As discussed below (5.22), the solution allows a phase factor arising from (z 3 − z 1 ) 1 κ (Ω 12 +Ω 13 +Ω 23 ) . To summarize, the 4-point KZ solutions can be given by the following generalized hypergeometric functions on Gr(4, 2): where ∆ ∈ H 1 (X, L ∨ g ) and ϕ j ∈ H 1 (X, L g ). From our study on the generalized hypergeometric functions on Gr(4, 2), we find the elements ϕ j are given by the following set The 4-point KZ solutions have two independent solutions. These are obtained by choosing two elements from the above. The simplest choice may be { dt t−z 1 , dt t−z 2 }. This corresponds to the solutions in (5.47).

Relation to (n + 1)-point KZ solutions
At the present stage, it is straightforward to generalize the above results to (n + 1)-point solutions of the KZ equation. These are obtained as generalized hypergeometric functions on Gr(2, n + 1). Following the representation (5.48)-(5.51), we can write down the (n + 1)-point KZ solutions as where Φ t is now defined on X = CP 1 − {z 0 , z 1 , · · · , z n }. The construction of F j is essentially the same as the one considered in (3.9). From Φ t we can determine the homology group H 1 (X, L ∨ g ) and the cohomology group H 1 (X, L g ). The twisted cycle ∆ and the one-form ϕ j are elements of these, respectively, i.e., ∆ ∈ H 1 (X, L ∨ g ) and ϕ j ∈ H 1 (X, L g ). The number of elements for the basis of the cohomology group is n − 1 and such a basis can be chosen as There are n − 1 solutions and these correspond to n − 1 independent solutions of the (n + 1)point KZ equation. Namely, we can express the (n + 1)-point KZ solution as As in the n = 3 case, it is known that this Ψ satisfy the differential equation dΨ = BΨ where B is called the Gauss-Manin connection and represented by an (n + 1) × (n + 1) matrix. This Gauss-Manin connection is associated to the definition of the generalized hypergeometric functions on Gr(2, n + 1). In general, such a Gauss-Manin connection can be constructed in association with the generalized hypergeometric functions on Gr(k + 1, n + 1), where the dimension of B is given by n − 1 k . The study of the Gauss-Manin connection is beyond the scope of this note. Interested readers are advised to see mathematical literature, e.g., Section 3.8 in [15]. In this context, the KZ connection (5.12) can be interpreted as the Gauss-Manin connection for the Gr(2, n + 1)-type generalized hypergeometric functions.
So far, we consider the case of k = 1. This is natural because it clarifies the relation between Gauss' hypergeometric function and the 4-point KZ solutions. For general (n + 1)point KZ solutions, however, there are no particular reasons to choose k = 1 except it leads to the simplest hypergeometric integrals. In principle, we can consider k ≥ 2 cases and relate the KZ solutions to generalized hypergeometric functions on Gr(k + 1, n + 1). In fact, there is a remarkable result or a theorem by Schechtman and Varchenko [17,18] that, with g being the SL(2, C) algebra, the KZ solutions can be expressed by the following hypergeometric-type integral In this expression a multivalued function relevant to Φ t in (5.54) is given by This multivalued function Φ t is defined on the space Namely, X represents a coordinate on C k , eliminating n distinct points (t 1 , t 2 , · · · , t k ) = (z j , z j , · · · , z j ) for 1 ≤ j ≤ n. This space is analogous to the one in (2.47) except that we have a different l j (t) here. From Φ t we can then define the k-th homology and cohomology groups, H k ( X, L ∨ g ) and H k ( X, L g ). In the integral (5.57), ∆ denotes an element of H k ( X, L ∨ g ) or a twisted k-cycle.
According to Schechtman and Varchenko [17,18], R J (t, z) in (5.57) is a rational function of t s and z j and is expressed as follows (see also recent reviews [30,31]). Let J be a set of n non-negative integers J = (j 1 , j 2 , · · · , j n ) under the condition |J| = j 1 +j 2 +· · ·+j n = k. Note that there is no particular maximum limit for k (such as k < n) in the original derivation [17,18]. (In an alternative derivation with a free field OPE method [20], k corresponds to the number of insertions in an (n+ 1)-point correlators.) Thus, for j i ∈ Z ≥0 (i = 1, 2, · · · , n), we have n + k − 1 n − 1 different elements in J. This corresponds to the number of possible n-partitions of integer k, allowing an empty set. To be concrete, for each choice of J = (j 1 , j 2 , · · · , j n ), we can define an n-partition of the sequence of k integers (s 1 , s 2 , · · · , s k ) such that (s 1 , s 2 , · · · , s k ) = 1, · · · , 1 j 1 , 2, · · · , 2 j 2 , · · · , n, · · · , n jn (5.62) Accordingly, we can define a rational function S J (z 1 , · · · , z n , t 1 , · · · , t k ) = The rational function R J (t, z) is then defined as R J (t, z) = 1 j 1 !j 2 ! · · · j n ! σ∈S k S J (z 1 , · · · , z n , t σ 1 , · · · , t σ k ) (5.64) where the summation of S k is taken over the permutations σ = 1 2 · · · k σ 1 σ 2 · · · σ k . For simple cases, R J (t, z) are written down as In what follows we show that R J (t, z)dt 1 ∧ · · · ∧ dt k can be interpreted as an element of the k-th cohomology group H k ( X, L g ) for k < n and j i ∈ {0, 1} (i = 1, 2, · · · , n). The condition k < n is necessary to relate the solutions to generalized hypergeometric functions on Gr(k + 1, n + 1). The other condition j i ∈ {0, 1} (i = 1, 2, · · · , n) arises from the fact that the number of the basis for H k ( X, L g ) is given by n − 1 k as mentioned in (2.49) and below.
Under these conditions the label J can be replaced by a set of k integers {j 1 , j 2 , · · · , j k }, satisfying 1 ≤ j 1 < j 2 < · · · < j k ≤ n − 1. The basis of H k ( X, L g ) is then given by In comparison with (5.64), we find that ϕ j 1 ,j 2 ,··· ,j k are equivalent to R I (t, z) where the elements of I = (i 1 , i 2 , · · · , i n ) are set to i l = 1 for l = j 1 , j 2 , · · · , j k and i l = 0 otherwise. This illustrates a direct relation of the (n + 1)-point KZ solutions to the hypergeometric integrals of a form where l j (t) is defined by (5.59). Notice that the basis ϕ j 1 j 2 ···j k in (5.67) is not the unique choice. We can choose different bases which lead to alternative parametrization of the solutions F j 1 j 2 ···j k For example, using the result in (2.50), we can also express the basis of H k ( X, L g ) as In general, the basis can be chosen as ϕ j 1 ,j 2 ,··· ,j k = d log Lastly, we notice that l j (t) in (5.59) is linear in terms of the elements of t = (t 1 , t 2 , · · · , t k ). This enable us to determine the rank-1 local systems L g , L ∨ g in association to Φ t and make it straightforward to construct the hypergeometric integrals (5.68). However, if we consider rather than Φ t , as a multivalued function of interests, we can not properly determine rank-1 local systems out of it since it can not be factorized into functions linear in t. In order to circumvent this issue, one may regard the above multivalued function as a function on C n+k . But this leads to mixture of variables in t s and z i and a resulting hypergeometric integral may be regarded as that on Gr(1, n + k + 1). Thus it is not appropriate to think of (5.72) as a multivalued function of interest when we interpret F j 1 j 2 ···j k as hypergeometric functions.
One may still wonder why Φ t instead of (5.72) should be extracted as the defining multivalued function. The author do not have a satisfying answer to it; this could be an ambiguity in the construction of F j 1 j 2 ···j k . This issue arises from the fact that F j 1 j 2 ···j k is not exactly defined as generalized hypergeometric function on Gr(k + 1, n + 1). As discussed in the beginning of this section, the configuration space of the KZ solutions Ψ(z 0 , z 1 , z 2 , · · · , z n ) is equivalent to that of generalized hypergeometric functions on Gr(2, n + 1), which is represented by n + 1 distinct points in CP 1 . If we relate the (n + 1)-point KZ solutions to generalized hypergeometric functions on Gr(k + 1, n + 1) in a rigorous manner, we need to expand the configuration space such that it is represented by n + 1 distinct points in CP k but this brings about ambiguities with the actual/physical configuration space mentioned above.
In conclusion, we can express solutions of the KZ equation in terms of hypergeometrictype integrals. The (n + 1)-point KZ solutions in general can be represented by generalized hypergeometric functions on Gr(2, n + 1) as shown in (5.52)-(5.54). We can generalize this expression to represent the (n + 1)-point KZ solution as hypergeometric-type integrals on Gr(k + 1, n + 1), as shown in (5.68)-(5.70), but there exists a subtle ambiguity in rigorous construction of the integrals for k ≥ 2.

Holonomy operators of KZ connections
In the present section we review the construction of holonomy operators of the Knizhnik-Zamolodchikov (KZ) connection, following [21,25], and consider the holonomy operators in relation to cohomology and homology of the physical configuration space C in (5.16). We first reconsider the KZ equation. We rewrite the KZ equation (5.1) as a differential form (5.11), DΨ = (d − Ω)Ψ = 0, where the KZ connection Ω is defined as In doing so, we implicitly use the condition Ω ij = Ω ji . The KZ equation is originally derived from the application of a Ward identity to the current correlators. Action of the operator Ω ij on the Hilbert space V ⊗n = V 1 ⊗ V 2 ⊗ · · · ⊗ V n is then defined as where I µ (µ = 1, 2, · · · , dimg) are elements of the Lie algebra g and ρ denotes its representation. Thus, by definition, Ω ij satisfies Ω ij = Ω ji .
For example, in the conventional choice of g being the SL(2, C) algebra Ω ij can be defined as where the operators a (±,0) i act on the i-th Fock space V i and forms the SL(2, C) algebra: where δ ij denotes Kronecker's delta. Note that in the case of i = j, Ω ii becomes the quadratic Casimir of SL(2, C) algebra which acts on V i . This defines the operator Ω i that we have introduced in (5.32).
The resultant KZ solutions in an integral form show that the solutions can be described in terms of Ω ij where 1 ≤ i < j ≤ n. This is a natural consequence of our setting that the physical configuration space of the KZ solution C can be represented by ordered distinct n + 1 points in CP 1 . We have already considered such a space C = X n /S n in (5. 16) and see that the monodromy representation of the KZ equation is given by the braid group B n = Π 1 (C). From these perspectives we can discard the operators Ω ji (i < j) and begin with DΨ = (d − Ω)Ψ = 0 as the defining KZ equation. The study of a KZ system can then be attributed to the classification of the KZ connections which satisfy the infinitesimal braid relations (5.6) and (5.7). As shown in (5.15), such a KZ connection becomes a flat connection, DΩ = 0. Thus, a general solution of the KZ equation can be given by a holonomy of Ω. In the language of gauge theory the holonomy is given by a Wilson loop operator of a gauge field in question. According to Kohno [21], the holonomy of Ω provides a general linear representation of the braid group on the Hilbert space V ⊗n .

Holonomy operators of the KZ connections: a review
The holonomy of Ω can be defined as [21]: where γ represents a closed path on C r = X r /S r where X r is defined in (5.3). In the following we shall denote the physical configuration space, with the the number of dimensions being explicit. Since the integrand in (6.6) is an r-form, the corresponding integral is taken over the r-dimensional complex space C r . Formally, the above integral can be evaluated as an iterated integral of K. -T. Chen [22]. Let the path γ in C r be represented by Denoting the pull-back γ * ω ij as we can explicitly express Θ γ as an iterated integral where (i < j) means that the set of indices (i 1 , j 1 , · · · , i r , j r ) are ordered such that 1 ≤ i l < j l ≤ r for l = 1, 2, · · · , r.
Let the initial point of γ(t) be z i (0) = z i for i = 1, 2, · · · , r and the final point be (z 1 (1), z j (1)) = (z 1 , z σ j ) for j = 2, 3, · · · , r and σ = 2 3 · · · r σ 2 σ 3 · · · σ r . Since C r = X r /S r is permutation invariant, the initial and final points are identical and we can naturally interpret γ as a closed path on C r . By definition γ represents an element of the braid group: To make Θ γ permutation invariant explicitly, we now redefine the holonomy operator of Ω as an analog of the Wilson loop operator in gauge theory [25]: The meanings of the symbol P and the trace Tr γ are clarified below. As in (6.9), the exponent in (6.10) can be expanded as Action of the symbol P on the above integral imposes the ordering conditions 1 ≤ i 1 < i 2 < · · · < i r ≤ r and 2 ≤ j 1 < j 2 < · · · < j r ≤ r + 1, with r + 1 ≡ 1 (mod r). Thus we have P γ Ω ∧ · · · ∧ Ω r = 1 κ r γ Ω 12 Ω 23 · · · Ω r1 ω 12 ∧ ω 23 ∧ · · · ∧ ω r1 (6.13) The trace Tr γ in (6.11) is carried out by summing over permutations of the indices: This can be interpreted as a trace over generators of the braid group B r and is called a braid trace.

Homology and cohomology interpretations of the integral I r
The holonomy operator (6.11) is essentially calculated by the above integral I r . We now consider I r in terms of cohomology and homology of C r . The cohomology part is relatively straightforward. Since the KZ connection Ω is a flat connection DΩ = 0, it is a closed form with respect to the covariant derivative D = d−Ω. But, by definition, it is not an exact form, that is, Ω = Df for any function f of (z 1 , · · · , z r ). Thus it is an element of the cohomology group H 1 (C r ) whose coefficients are given by g ⊗ g, To generalize, the integrand of I r can be considered as where g ⊗r denotes a set of operators acting on the Hilbert space V ⊗r .
Since γ is defined as a closed path in C r , the algebraic coefficients Ω i 1 j 1 · · · Ω irjr can be extracted out of the integrand. In order to make sense of Θ γ as an integral, we need to regard γ as an element of the r-th homology group H r (C r , R), with the coefficients being real number: γ ∈ H r (C r , R) (6.17) The element γ can be considered as a path in C r connecting r hyperplanes H ij defined in (5.4). Since the KZ equation has branch points at H ij (i < j), (6.17) is in accord with a general concept of the homology group by use of boundary operators as discussed in (2.20)-(2.22). Note that the homology and cohomology considered here are not the twisted ones as before. Thus we can not directly relate I r to the bilinear construction of hypergeometrictype integrals in (2.23) and (2.24). In the present note, however, we have been equipped with a certain level of understanding of the hypergeometric integrals in terms of homology and cohomology. Namely, we learned that a (co)homology interpretation provides a systematic treatments of analytic continuation or monodromy representation of the solutions to a differential equation of interests. In what follows, we further consider these aspects of the integral I r .
Let LC r be a loop space in C r . It is known that the fundamental homotopy group of C r is isomorphic to the 0-dimensional homology group of LC r [32]: With the result (6.10), we then find that γ can also be an element of H 0 (LC r ), with the coefficients being the real number: This suggests that the integrand of I r can and should be interpreted as an element of the 0-dimensional cohomology group of the loop space LC r . We can then naturally assume In the following, we briefly argue that this assumption is a favorable one. Remember that the basis of H r−1 ( X, L g ) considered in (5.67), with n = r and k = r − 1, can be expressed as Adding the parameters (t 1 , z 1 ), we can redefine the above basis as By definition (see e.g., (2.45)), the dimension-0 cohomology means there are no extra parameters except {z 1 , z 2 , · · · , z r } ∈ LC r . The basis of the dimension-0 cohomology group H 0 (LC r ) can then be obtained by identifying {t 1 , t 2 , · · · , t r } with {z 1 , z 2 , · · · , z r } in the above. To avoid divergence, we need to require t i = z i in the denominator; since z i 's are permutation invariant we can choose t i 's arbitrarily such that ϕ 1,2,··· ,r becomes finite. One of the simplest nontrivial results can be obtained by setting t i = z σ i+1 in the denominator of (6.22), with t r = z σ r+1 = z 1 . We then have Apart from the algebraic part involving Ω ij , the above factor is identical to the integral I r at the level of integrand. Thus, as far as the integrand is concerned, the fact that the basis of H 0 (LC r ) is given by (6.23) leads to a favorable confirmation of the assertion (6.20).
As discussed earlier, we need to determine a multivalued function to make a (co)homology interpretation of the integral I r . The above analyses show that a suitable choice is given by which is defined on C r . As studied in the previous sections, elements of homology groups satisfy the equivalent relation d log Φ 0 ≡ 0. In particular, from (5.30) we see that this enable us to replace the ordinary derivative ∂ z by the covariant derivative D z . In a sense this equivalent relation can be thought of as an origin of the minimal coupling principle in gauge theories.
In conclusion, the holonomy operator of the KZ connection is essentially given by the integral I r . Motivated by the bilinear construction of the generalized hypergeometric functions (which we have reviewed in the previous sections), we analyze I r in terms of (non-twisted) homology and cohomology group of the relevant physical configuration space C r = X r /S r or its loop space LC r . Following the notation in (2.23), the construction can be written as The two interpretations of the integral I r suggest that an unambiguous constituent of I r is given by its integrand I r (z), i.e., The expressions (6.28) and (6.29) (or (6.14)) correspond to the interpretations (6.26) and (6.25), respectively. Since the equation between (6.28) and (6.29) is mathematically subtle, we need to further consider the meaning of this. Obviously, the equation holds when the r-form α r σ : is equivalent to dz 1 ∧ · · · ∧ dz r , regardless choices of σ. Since z i 's are coordinates of C r they satisfy the permutation invariance and z i − z j = 0 (i = j). Thus natural non-vanishing coordinates on C r can be taken by (z 1 − z σ 2 , z σ 2 − z σ 3 , · · · , z σr − z 1 ). These are the coordinates on C r , that is, these are supposed to be independent of the permutation σ. If we impose the condition z i = 0, we can then identify the above non-vanishing coordinates as (z 1 , · · · , z r ). This means that z i 's are the coordinates on CP r−1 rather than C r ; remember that C r = X r /S r and we have defined This means that as far as we consider the projected complex spaces we can equate the above r-form α r σ to the simplest form dz 1 ∧ · · · ∧ dz r , and the above relations (6.28) and (6.29) hold.
In terms of I r (z) integrand part of the holonomy operator Θ γ can be written as Θ γ (z) = exp r≥2 I r (z) (6.30) Simplification of the algebraic part Having considered analytic aspects of the holonomy operators, we now consider simplification of the algebraic structure of I r . As emphasized earlier in this section, the algebra of the KZ connections is determined by the infinitesimal braid relations (5.6) and (5.7). Using the SL(2, C) algebra in (6.5), we introduce a bialgebraic operator [25]: A ij satisfy the infinitesimal braid relations (5.6), (5.7). Thus, the operator This relation guarantees the existence of the holonomy operator for A.
The bialgebraic structures of Ω and A are different but the constituents of these remain the same, i.e., they are given by a i . Thus, we can use the same Hilbert space V ⊗n and physical configuration C for both Ω and A. The KZ equation of A is then given by where Ψ is a function of a set of complex variables (z 1 , z 2 , · · · z n ). In analogy with (6.11) the holonomy operator of A can be defined as The algebraic part of Θ (A) γ can be simplified as follows. We first note that the commutator [A 12 , A 23 ] can be calculated as An analog of (6.13) is then expressed as P γ A ∧ · · · ∧ A r = γ A 12 A 23 · · · A r1 ω 12 ∧ ω 23 ∧ · · · ∧ ω r1 (6.36) ⊗ · · · ⊗ a (hr) r γ ω 12 ∧ · · · ∧ ω r1 (6.37) where h i denotes h i = ± = ±1 (i = 1, 2, · · · , r). In the above expression, we define a The braid trace over (6.37) is expressed as where σ µ = (1, σ i ) where σ i (i = 1, 2, 3) denotes the (2 × 2) Pauli matrices and 1 is the (2 × 2) identity matrix. Requiring that p µ be real, we can takeūȦ as a conjugate of u A , i.e., uȦ = (u A ) * .
Lorentz transformations of u A are given by where g ∈ SL(2, C) is a (2 × 2)-matrix representation of SL(2, C); the complex conjugate of this relation leads to Lorentz transformations ofūȦ. Four-dimensional Lorentz transformations are realized by a combination of these, that is, the four-dimensional Lorentz symmetry is given by SL(2, C) × SL(2, C). Scalar products of u A 's orūȦ's, which are invariant under the corresponding SL(2, C), are expressed as where ǫ AB is the rank-2 Levi-Civita tensor. This can be used to raise or lower the indices, e.g., u B = ǫ AB u A . Notice that these products are zero when i and j are identical.
For a theory with conformal invariance, such as Maxwell's electromagnetic theory and N = 4 super Yang-Mills theory, we can impose scale invariance on the spinor momentum, i.e., where λ is non-zero complex number. With this identification, we can regard the spinor momentum u A as a homogeneous coordinate of the complex projective space CP 1 . The local coordinate of CP 1 is represented by a complex variable z ∈ C − {∞}. This can be related to u A by the following parametrization: The local complex coordinate of CP 1 can be taken as z = β/α except in the vicinity of α = 0, where we can instead use 1/z = α/β as the local coordinate.
A helicity of a massless particle is generally determined by the so-called Pauli-Lubanski spin vector. In the spinor-momenta formalism, we can define an analog of this spin vector. This enables us to define a helicity operator of massless particles as This means that the helicity of the particle is determined by the degree of homogeneity in u.

Twistor and supertwistor spaces
Twistor space is defined by a four-component spinor W I = (π A , vȦ) (I = 1, 2, 3, 4) where π A and vȦ are two-component complex spinors. From this definition, it is easily understood that twistor space is represented by the complex homogeneous coordinates of CP 3 . Thus, W I correspond to homogeneous coordinates of CP 3 and satisfy the following relation.
In twistor space, the relation between π A and vȦ is defined as where xȦ A represent the local coordinates on S 4 . This can be understood from the fact that CP 3 is a CP 1 -bundle over S 4 . We consider that the S 4 describes a four-dimensional compact spacetime. Notice that in twistor space the spacetime coordinates xȦ A are emergent quantities. Four-dimensional diffeomorphisms, i.e., general coordinate transformations, is therefore realized by u A → u ′A (7.10) The key idea of the spinor-helicity formalism in twistor space is the identification of the CP 1 fibre of twistor space with the CP 1 on which the spinor momenta are defined. In other words, we identify π A with the spinor momenta u A so that we can essentially describe fourdimensional physics in terms of the coordinates of CP 1 . In the spinor-momenta formalism, the twistor-space condition vȦ = xȦ A π A is then expressed as vȦ = xȦ A u A (7.11) In what follows we use the spinor momenta u A for the role of π A in twistor space.

Holonomy formalism for MHV amplitudes
Having reviewed the spinor-helicity formalism in twistor space, we now define creation operators of gluons. As mentioned in (7.7), the helicity of a particle is determined by the degree of homogeneity in u. In accordance with (7.7), we define the gluon operators a (±) i of helicity ± and their superpartners as where i = 1, 2, · · · , n andĥ i = (0, ± 1 2 , ±) respectively denote the numbering index and the helicity of the particle. The color factor of gluon can be attached to each of the physical operators: a where t c i 's are given by the generators of the SU(N) gauge group.
In the coordinate-space (or superspace) representation, the physical operators can be expressed as where dµ(p) denotes the Nair measure: The maximally helicity violating (MHV) amplitudes are the scattering amplitudes of (n − 2) positive-helicity gluons and 2 negative-helicity gluons. In a momentum-space representation, the MHV tree amplitudes are expressed as Tr(t c 1 t cσ 2 t cσ 3 · · · t cσ n ) (u r u s ) 4 (u 1 u σ 2 )(u σ 2 u σ 3 ) · · · (u σn u 1 ) (7.22) where r and s denote the numbering indices of the two negative-helicity gluons out of the total n gluons. g is the coupling constant of gluon interactions.
Now it is straightforward to construct an S-matrix functional for the MHV amplitudes by use of the integrand part of the holonomy operator in (6.41). We first replace the operators a (7.19). We further use the spinor momenta u i 's for the complex variables z i 's on CP 1 . This leads to a supersymmetric versions of I (A) r (z), i.e., where we define the coupling constant g by In terms of the supersymmetric version of Θ where a (h)c 's denote the gluon creation operators in the momentum-space representation, which are treated as source functions here. In the above derivation we use the fact that the Grassmann integral over θ's vanishes unless we have the following integrand: This relation guarantees that only the MHV amplitudes are picked up upon the execution of the Grassmann integral.
Notice that A (r − s − ) MHV (u,ū) contains the delta function (2π) 4 δ (4) (p 1 + · · · + p n ) = d 4 x e i(p 1 +···+pn)x (7.29) Thus the resultant expression in (7.27) can be proportional to the products of δ (4) (x)'s. Physically, this is obvious because n gluons are supposed to be scattering at a single point. But, mathematically, we need to be careful about the treatment of the products of the delta functions since there are no rigorous definitions for such products. The problem may be solved if we consider in the momentum representation where the holomorphic MHV amplitudes A (r − s − ) MHV (u) in (7.22) can be generated as δ where a (h)c (x)'s play the same role as a (h)c in (7.27) except that they are now given by x-space representation of the gluon creation operators: As studied in [26], however, the expression (7.27) turns out to be more useful for the computation of one-loop amplitudes.

CSW rules and non-MHV amplitudes in holonomy formalism
General amplitudes, the so-called non-MHV amplitudes, can be expressed in terms of the MHV amplitudes A (r − s − ) M HV (u) at tree level. Prescription for these expressions is called the Cachazo-Svrcek-Witten (CSW) rules [33]. For the next-to-MHV (NMHV) amplitudes, which contain three negative-helicity gluons, the CSW rules can be expressed as where the sum is taken over all possible choices for (i, j) that satisfy the ordering i < r < s < j < t. The momentum transfer q ij between the two MHV vertices is given by q ij = p i + p i+1 + · · · + p r + · · · + p s + · · · + p j (7.33) where p's denote four-momenta of gluons as before. General non-MHV amplitudes are then obtained by an iterative use of the relation (7.32).
In terms of the expression (7.27) the CSW rules can be implemented by a contraction operator where we take the limit y → x with x 0 − y 0 → 0 + (keeping the time ordering x 0 > y 0 ). q denotes a momentum transfer of a virtual gluon. As explicitly parametrized in (7.33), this is off-shell quantity q 2 = 0. Its on-shell partner can be defined as where η µ is a reference null vector (η 2 = 0) and w is a real parameter. In (4.29) a (±) p denote the creation operators of a pair of virtual gluons at ends of a propagator or at MHV vertices. In the calculation of (4.29) we also use the well-known identity where ǫ is a positive infinitesimal.
Using the above contraction operator, we can define an S-matrix functional for general non-MHV amplitudes as Generalization of the expression (7.27) can be written as (u,ū) denotes a non-MHV version of the gluon amplitudes in the form of (7.21). This is called N k−2 MHV amplitudes where h i = ± denotes the helicity of the i-th gluon, with the total number of negative helicities being k. (k = 2, 3, · · · , n − 2) Note that the expression (7.38) is not necessarily limited to tree-level amplitudes; for details on applications of the CSW rules to one-loop amplitudes in the holonomy formalism, see [26]. The above formulation illustrates that the holonomy operator of the KZ connection plays an essential role in the construction of the S-matrix functional F a (h)c for gluon amplitudes at least at tree level. One of the main purposes for the present note is to study mathematical foundations of the holonomy operator (which we have carried out in the previous section) so as to give an improved description of the holonomy formalism.

Grassmannian formulations of gluon amplitudes
In the following we briefly review more powerful formulations of gluon amplitudes, known as Grassmannian formulations [1]- [6]. These are powerful, in particular, in computation of higher loop amplitudes but we here consider basic tree-level formulations.
The supertwistor conditions as the momentum conservation laws We first review various representations of gluon amplitudes, following [25]. The twistor space condition vȦ = xȦ A u A in (7.11) naturally leads to the relations vȦūȦ = xȦ A p AȦ = 2(x 0 p 0 − x 1 p 1 − x 2 p 2 − x 3 p 3 ) = 2x µ p µ (8.1) where we use the rules of scalar products (7.4) between the spinor momenta. We also parametrize xȦ A in terms of the Minkowski coordinates as The product (8.1) suggests that the physical phase space can be spanned by (vȦ,ūȦ), rather than (x µ , p µ ), in twistor space. We can then relate a function of (u,ū) to a function of (u, v) by Fourier transform integrals Similarly, by taking conjugates of these, we have These integrals are referred to as Fourier transforms in twistor space.
What is remarkable about the use of supertwistor space in gluon amplitudes is that the supertwistor conditions (7.14) automatically arise from the momentum conservation. We shall review this point in the following. We start from n-point N k−2 MHV amplitudes of the form A n,k (u,ū) = ig n−2 (2π) 4 δ (4) n i=1 p i A n,k (u,ū) (8.5) This is a generalized version of A (r − s − ) MHV (u,ū) in (7.21). Notice that A n,k (u,ū) is no more holomorphic to u A 's for k ≥ 1. The momentum conservation is realized by the delta function: We now introduce a fermionic partner of the momentum conservation: whereη α =ūȦθȦ α as defined in (7.16). Adding this fermionic delta function, the amplitudes can be represented by (u,ū,η): A n,k (u,ū,η) = δ (8) i u A iη iα A n,k (u,ū) (8.8) The amplitudes can of course be expressed in terms of the supertwistor variables WÎ = (u A , vȦ, ξ α ) in (7.12). Using Fourier transforms in supertwistor space, we can obtain such representations as where we use (8.6) and (8.7). The inverse transformation is given by Notice that the supertwistor conditions (7.14) are indeed embedded in A n,k (u, v, ξ). This illustrates intimate connections between supertwistor space and massless gauge bosons. In the holonomy formalism we consider the physical operators in the coordinate-space representation, see (7.19) or (7.31), and put the supertwistor conditions by hand. This formulation is suitable as far as we rely on the CSW rules where the amplitudes factorize into the MHV vertices A m,1 (u,ū) (m ≤ n) which are holomorphic to u A 's, apart from the contributions from the momentum conservation. To describe the non-MHV amplitudes in a more democratic manner, we need to handle the non-holomorphic part of the amplitudes properly. This can not be done in the holonomy formalism. The most promising formulation is given by the Grassmannian formulations.

Grassmannian formulations of gluon amplitudes
In the Grassmannian formulations the gluon amplitudes are originally considered in terms of the so-called dual supertwistor variables WÎ = (v A ,ūȦ,η α ) (8.11) The relevant amplitudes A n,k (v,ū,η) can be Fourier transformed into A n,k (u,ū,η) in (8.10) as A n,k (v,ū) = δ n j=1 k i=1 z ij ρ A iv jA A n,k (ū) (8.15) where z ij (i = 1, 2, · · · , k ; j = 1, 2, · · · , n ; k = 2, 3, · · · , n − 2) denotes an element of (k × n) complex matrix Z. The delta function is analogous to the momentum-conservation delta functions but it is qualitatively different from them as it involves mixing of the numbering indices. In (8.15) ρ A i 's are considered as another set of the holomorphic spinors (which are not necessarily spinor momenta). Performing thev-integral, A n,k (u,ū) are then expressed as Together with (8.13), this means that on top of the supertwistor conditions we further impose the relation u A j = z 1j ρ A 1 + z 2j ρ A 2 + · · · + z kj ρ A k (j = 1, 2, · · · , n) (8.17) As mentioned in (7.10), the four-dimensional spacetime is an emergent concept and the spinor momenta are more fundamental quantities in twistor space. In a multigluon system there is a possibility to extend our perspective on the spinor momenta; we may express one entry of the spinor momenta in terms of another set of spinor momenta which are given by two-component holomorphic spinors. The extra condition (8.17) exactly realizes this possibility. One of the remarkable features in the Grassmannian formalism is that the number of linearly independent ρ A i 's is related to the number of negative helicity gluons in the amplitudes. This makes it possible to describe non-MHV amplitudes in a universal fashion.
Integrating over ρ A i 's, we see that the above expressions lead to the Grassmannian formulations of the gluon amplitudes. Apart from the color factor, the n-point N k−2 MHV gluon amplitudes in the Grassmannian formulation are conjectured in a form of [1]: z ij WÎ j (12 · · · k)(23 · · · k + 1) · · · (n n + 1 · · · n + k − 1) (8.18) where (j 1 j 2 · · · j k )'s denote k-dimensional minor determinants of Z which consist of the j 1 -th to j k -th columns. As explicitly shown in (8.18), j's are ordered such that (j 1 j 2 · · · j k ) are cyclic to (12 · · · k) in mod n. Thus the above minor determinant is an ordered k-dimensional minor determinant of the k × n matrix Z, which can be computed as (j 1 j 2 · · · j k ) := σ∈S k 1 2 · · · k σ 1 σ 2 · · · σ k z σ 1 j 1 z σ 2 j 2 · · · z σ k j k (8.19) description for a multi-particle systems. Since the monodromy representation of the KZ equation is given by the braid group, the KZ system is advantageous especially to analyze the multi-particle system by use of braid groups. The monodromy representation is also given by the holonomy of the KZ connection, which can be expressed in terms of the iterated integral. In section 6 we review the definition of the holonomy operator of the KZ connection. We find that the integral representation of the holonomy operator can be constructed by a set of homology and cohomology groups in two different ways. This interpretation is schematically summarized in (6.25) and (6.26).
Equipped with this mathematical construction of the holonomy operator, in section 7, we present an improved review of the holonomy formalism for gluon amplitudes. We also carry out a similar analysis on the Grassmannian formulations of gluon amplitudes in section 8. After a detailed introduction to the dual supertwistor variables WÎ , we review the definition of the integral representation of the amplitudes L n,k (W) in (8.18). We find that the integral can also be constructed in bilinear forms in terms of homology and cohomology group of the relevant physical configuration space; see (8.26) and (8.27). From these analyses we observe that the integral contours of L n,k (W) are given by elements of B n , the braid group on n strands.