aa r X i v : . [ h e p - t h ] M a y Periods and Superstring Amplitudes
S. Stieberger
Max–Planck–Institut f¨ur PhysikWerner–Heisenberg–Institut80805 M¨unchen, Germany
Abstract
Scattering amplitudes which describe the interaction of physical states play an im-portant role in determining physical observables. In string theory the physical states aregiven by vibrations of open and closed strings and their interactions are described (at theleading order in perturbation theory) by a world–sheet given by the topology of a disk orsphere, respectively.Formally, for scattering of N strings this leads to N − N − M ,N – the moduli space of Riemann spheres of N ordered marked points.The mathematical structure of these string amplitudes share many recent advances inarithmetic algebraic geometry and number theory like multiple zeta values, single–valuedmultiple zeta values, Drinfeld, Deligne associators, Hopf algebra and Lie algebra struc-tures related to Grothendiecks Galois theory. We review these results, with emphasis ona beautiful link between generalized hypergeometric functions describing the real iteratedintegrals on M ,N ( R ) and the decomposition of motivic multiple zeta values. Further-more, a relation expressing complex integrals on M ,N ( C ) as single–valued projection ofiterated real integrals on M ,N ( R ) is exhibited. MPP–2016–85 ontents M ,N
43 Volume form and period matrix on M ,N
64 Motivic and single–valued multiple zeta values 105 Motivic period matrix F m
156 Open and closed superstring amplitudes 217 Complex vs. iterated integrals 25 Introduction
During the last years a great deal of work has been addressed to the problem of revealingand understanding the hidden mathematical structures of scattering amplitudes in both field–and string theory. Particular emphasis on their underlying geometric structures seems to beespecially fruitful and might eventually yield an alternative way of constructing perturbativeamplitudes by methods residing in arithmetic algebraic geometry . In such a framework phys-ical quantities are given by periods (or more generally by L –functions) typically describingthe volume of some polytope or integrals of a discriminantal configuration. The mathematicalquantities which occur in string amplitude computations are periods which relate to fundamen-tal objects in number theory and algebraic geometry. A period is a complex number whosereal and imaginary parts are given by absolutely convergent integrals of rational functions withrational coefficients over domains in R n described by polynomial inequalities with rational coef-ficients. More generally, periods are values of integrals of algebraically defined differential formsover certain chains in algebraic varieties [5]. E.g. in quantum field theory the coefficients ofthe Laurent series in the parameter ǫ = (4 − D ) of dimensionally regulated Feynman integralsare numerical periods in the Euklidian region with all ratios of invariants and masses havingrational values [6]. Furthermore, the power series expansion in the inverse string tension α ′ of tree–level superstring amplitudes yields iterated integrals [7, 8, 9], which are periods of themoduli space M ,N of genus zero curves with N ordered marked points [10] and integrate to Q –linear combinations of multiple zeta values (MZVs) [11, 12]. Similar considerations [13] areexpected to hold at higher genus in string perturbation theory, cf. [14] for some fresh investi-gations at one–loop. At any rate, the analytic dependence on the inverse string tension α ′ ofstring amplitudes furnishes an extensive and rich mathematical structure, which is suited toexhibit and study modern developments in number theory and arithmetic algebraic geometry.The forms and chains entering the definition of periods may depend on parameters (mod-uli). As a consequence the periods satisfy linear differential equations with algebraic coeffi-cients. This type of differential equations is known as generalized Picard–Fuchs equations orGauss–Manin systems. A subclass of the latter describes the A –hypergeometric system orGelfand–Kapranov–Zelevinsky (GKZ) system relevant to tree–level string scattering. One no-torious example of periods are multivariate (multidimensional) or generalized hypergeometricfunctions . In the non–resonant case the solutions of the GKZ system can be represented bygeneralized Euler integrals [15], which appear as world–sheet integrals in superstring tree–levelamplitudes and integrate to multiple Gaussian hypergeometric functions [7]. Other occur-rences of periods as physical quantities are string compactifications on Calabi–Yau manifolds.According to Batyrev the period integrals of Calabi–Yau toric varieties also satisfy GKZ sys-tems. Therefore, the GKZ system is generic to functions describing physical effects in stringtheory as periods. In field–theory with N = 4 supersymmetry such methods have recently been pioneered by using tools inalgebraic geometry [1, 2] and arithmetic algebraic geometry [3, 4]. The initial data for a GKZ–system is an integer matrix A together with a parameter vector γ . For a givenmatrix A the structure of the GKZ–system depends on the properties of the vector γ defining non–resonantand resonant systems. More precisely, at an algebraic value of their argument their value is π P , with P being the set of periods. Periods on M ,N The object of interest is the moduli space M ,N of Riemann spheres (genus zero curves) of N ≥ P SL (2 , C ) on those points. The connectedmanifold M ,N is described by the set of N –tuples of distinct points ( z , . . . , z N ) modulo theaction P SL (2 , C ) on those points. As a consequence with the choice z = 0 , z N − = 1 , z N = ∞ (2.1)there is a unique representative( z , . . . , z N ) = (0 , t , . . . , t N − , , ∞ ) (2.2)of each equivalence class of M ,N M ,N ≃ { ( t , . . . , t N − ) ∈ (cid:0) P \{ , , ∞} (cid:1) N − | t i = t j for all i = q } , (2.3)and the dimension of M ,N ( C ) is N −
3. On the other hand, the real part of (2.3) describingthe space of points M ,N ( R ) := { (0 , t , . . . , t N − , , ∞ ) | t i ∈ R } (2.4)is not connected. Up to dihedral permutation each of its ( N − γ ) γ = ( z , z , . . . , z N ) (2.5)is completely described by the (real) ordering of the N marked points z < z < . . . < z N , (2.6)with: N [ i =1 { z i } = { , t , . . . , t N − , , ∞} . (2.7)In the compactification M ,N ( R ) the components γ become closed cells and give rise to as-sociahedra (Stasheff polytopes). The standard cell of M ,N is denoted by δ and given by theset of real marked points ( z , z , . . . , z N ) = (0 , t , t , . . . , t N − , , ∞ ) on M ,N subject to the(canonical) ordering (2.6), i.e.: δ = { t l ∈ R | < t < t < . . . < t N − < } . (2.8)A period on M ,N is defined to be a convergent integral [10] Z δ ω (2.9)over the standard cell (2.8) in M ,N ( R ) and ω ∈ H N − ( M ,N ) a regular algebraic ( N − δ and has no poles along δ . Every period on M ,N is a Q –linearcombination of MZVs [11]. Furthermore, every MZV can be written as (2.9).4o each cell γ a unique ( N − ω γ = N Y i =2 ( z i − z i − ) − dt ∧ . . . ∧ dt N − , (2.10)subject to (2.7) with z l = ∞ dismissed in the product. The form (2.10) is unique up toscalar multiplication, holomorphic on the interior of γ and having simple poles on all divisorson the boundary of that cell. To a cell (2.6) in M ,N ( R ) modulo rotations an oriented N –gon ( N –sided polygons) may be associated by labelling clockwise its sides with the markedpoints ( z , z , . . . , z N ). E.g. according to (2.6) the polygon with the cyclically labelled sides γ = (0 , , t , t , ∞ , t ) is identified with the cell 0 < < t < t < ∞ < t in M , ( R ) and thecorresponding cell form is: ω γ = ± dt dt dt ( − t ) ( t − t ) ( t − . The cell form (2.10) refers to the ordering (2.6). A cyclic structure γ corresponds to thecyclic ordering ( γ (1) , γ (2) , . . . , γ ( N )) of the elements { , , . . . , N } and refers to the standard N –gon (1 , , . . . , N ) modulo rotations. There is a unique ordering σ of the N marked points(2.2) as z σ (1) < z σ (2) < . . . < z σ ( N ) . (2.11)with σ ( N ) = N and compatible with the cyclic structure γ . The cell–form corresponding to γ is defined as [16] ω γ = N − Y i =2 (cid:0) z σ ( i ) − z σ ( i − (cid:1) − dt ∧ . . . ∧ dt N − . (2.12)E.g. for the cyclic structure (2 , , , , ,
3) the unique ordering σ compatible with the latterand with σ (6) = 6 is the ordering (4 , , , , , γ = ( t , t , t , , , ∞ ).In the following, we consider orderings (2.6) (01 cyclic structure γ ) of the set S Ni =1 { z i } = { , t , . . . , t N − , , ∞} with the elements z = 0 and z N − = 1 being consecutive, i.e. γ = (0 , , ρ )with ρ ∈ S N − some ordering of the N − { t , . . . , t N − , ∞} . The corresponding cell–function is given by ω ρ = z − ρ (2) N − Y i =3 (cid:0) z ρ ( i ) − z ρ ( i − (cid:1) − dt ∧ . . . ∧ dt N − , ρ ∈ S N − , (2.13)it is called 01 cell–function [16] and its associated N –gon, in which the edge referring to 0appears next to that referring to 1, is depicted in Fig. 1. The ( N − H N − ( M ,N ) of M ,N by constituting a basisof H N − ( M ,N , Q ), i.e. [16]: dim H N − ( M ,N , Q ) = ( N − . (2.14)As a consequence the cohomology group H N − ( M ,N ) is canonically isomorphic to the subspaceof polygons having the edge 0 adjacent to edge 1 [16].5 ρ (2) z z ρ ( N − z ρ ( N − z ρ ( N ) z N − z ρ (3) Figure 1: N –gon describing the 01 cyclic structure γ = (0 , , ρ ).Generically, in terms of cells a period (2.9) on M ,N may be defined as the integral [16] Z β ω γ (2.15)over the cell β in M ,N ( R ) and the cell–form ω γ with the pair ( β, γ ) referring to some polygonpair. Therefore, generically the cell–forms (2.10) integrated over cells (2.5) give rise to periodson M ,N , which are Q –linear combinations of MZV. By changing variables the period integral(2.15) can be brought into an integral over the standard cell δ parameterized in (2.8). Toobtain a convergent integral (2.15) in [16] certain linear combinations of 01 cell–forms (2.13)(called insertion forms) have been constructed with the properties of having no poles along theboundary of the standard cell δ and converging on the closure δ . E.g. in the case of M , thecell–form ω γ corresponding to the cell γ = (0 , , t , ∞ , t ) can be integrated over the compactstandard cell δ defined in (2.8): Z δ ω γ = Z ≤ t ≤ t ≤ dt dt (1 − t ) t = ζ (2) . (2.16) M ,N For a regular algebraic ( N − ω δ on M ,N conditions exist for the integral (2.9) overthe standard cell δ to converge. The set of all regular ( N − ω δ = dt ∧ . . . ∧ dt N − t ( t − t ) ( t − t ) · . . . · ( t N − − t N − ) (1 − t N − ) . (3.1)(Up to multiplication by Q + ) this form is the canonical volume form on M ,N ( R ) withoutzeros or poles along the standard cell (2.8). An algebraic volume form Ω on M ,N ( R ) may besupplemented by the P SL (2 , C ) invariant factor Q N − i 1, the integral (3.3)becomes I δ ( a ′ , b ′ , c ′ ) = N − Y i =1 Z dx i ! N − Y j =1 x a ′ j j (1 − x j ) b ′ j N − Y l = j +1 − l Y k = j x k ! c ′ jl , (3.4)with some integers a ′ i , b ′ i , c ′ ij ∈ Z .Moreover, the form (3.2) can be generalized to the family of real period integrals R δ Ω on δ , with s ij ∈ R . Then, Taylor expanding (3.2) w.r.t. s ij at integral points s ij ∈ Z + yieldscoefficients representing period integrals of the form (3.3). Similar observations have beenmade in [7, 8] when computing α ′ –expansions of string amplitudes whose form–factors aredescribed by integrals of the type (3.4). In this setup the additional P SL (2 , C ) invariant factor Q N − i 1) (3.7)= Z π N − Y i =2 dz i ! N − Q i 2) ] = α ′ N − N − Y j =2 (cid:16) s ,j ρ + j − X k =2 θ ( j ρ , k ρ ) s j ρ ,k ρ (cid:17) , (3.11) The matrix S with entries S ρ,σ = S [ ρ | σ ] is defined as a ( N − × ( N − ρ ≡ { ρ (2) , . . . , ρ ( N − } and σ ≡ { σ (2) , . . . , σ ( N − } , respectively. The matrix S is symmetric, i.e. S t = S . j ρ = ρ ( j ) and θ ( j ρ , k ρ ) = 1 if the ordering of the legs j ρ , k ρ is the same in both orderings ρ (2 , . . . , N − 2) and σ (2 , . . . , N − S [ ρ | σ ] arepolynomials of the order N − N − × ( N − F πσ = ( − N − X ρ ∈ S N − Z π ( ρ ) S [ ρ | σ ] , (3.12)which according to (3.10) satisfies: F | α ′ − N = 1 . (3.13)The matrix F has rank rk( F ) = ( N − M ,N [21].In [22] it has been observed, that F can be written in the following way : F = P Q : exp (X n ≥ ζ n +1 M n +1 ) : . (3.14)This decomposition is guided by its organization w.r.t. MZVs as M n +1 = F | ζ n +1 ,P n = F | ζ n , (3.15)with: P = 1 + X n ≥ ζ n P n , (3.16) Q = 1 + X n ≥ Q n = 1 + 15 ζ , [ M , M ] + (cid:26) ζ + 114 ζ , (cid:27) [ M , M ]+ (cid:26) ζ ζ + 625 ζ ζ − ζ ζ + 15 ζ , , (cid:27) [ M , [ M , M ]] + . . . . (3.17)Hence, all the information is kept in the matrices P and M and the particular form of Q . Theentries of the matrices M n +1 are polynomials in s ij of degree 2 n + 1 (and hence of the order α ′ n +1 ), while the entries of the matrices P n are polynomials in s ij of degree 2 n (and hence ofthe order α ′ n ). E.g. for N = 5 we have P = α ′ (cid:18) − s s + s ( s − s ) s s s s ( s + s ) ( s + s ) − s s (cid:19) , (3.18)and M = α ′ (cid:18) m m m m (cid:19) , (3.19) The ordering colons : . . . : are defined such that matrices with larger subscript multiply matrices withsmaller subscript from the left, i.e. : M i M j := ( M i M j , i ≥ j ,M j M i , i < j . The generalization to iterated matrixproducts : M i M i . . . M i p : is straightforward. m = s [ − s ( s + 2 s + s ) + s s + s ] + s s ( s + s ) ,m = − s s ( s + s + s + s + s ) ,m = s s [ s + s + s − s + s ) ] ,m = ( s + s ) [ ( s + s )( s + s ) − s s ] − [ 2 s s − s + 2 s ( s + s ) ] s + s s . (3.20)As we shall see in section 5 the form (3.14) is bolstered by the algebraic structure of motivicMZVs. The form (3.14) exactly appears in F. Browns decomposition of motivic MZVs [23]. Insection 6 we shall demonstrate, that the period matrix F has also a physical meaning describingscattering amplitudes of open and closed strings. Multiple zeta values (MZVs) ζ n ,...,n r := ζ ( n , . . . , n r ) = X 14 [ e , [ e , [ e , e ]]] + [ e , [ e , [ e , e ]]] + 54 [ e , e ] (cid:17) + ζ ζ (cid:16) ([ e , [ e , e ]] − [ e , [ e , e ]) [ e , e ] + [ e , [ e , [ e , [ e , e ]]]] − [ e , [ e , [ e , [ e , e ]]]] (cid:17) + ζ (cid:16) [ e , [ e , [ e , [ e , e ]]]] − 12 [ e , [ e , [ e , [ e , e ]]]] − 32 [ e , [ e , [ e , [ e , e ]]]] + ( e ↔ e ) (cid:17) + . . . . (4.6)The set of integral linear combinations of MZVs (4.1) is a ring, since the product of any twovalues can be expressed by a (positive) integer linear combination of the other MZVs [26]. Thereare many relations over Q among MZVs. We define the (commutative) Q –algebra Z spannedby all MZVs over Q . The latter is the (conjecturally direct) sum over the Q –vector spaces Z N spanned by the set of MZVs (4.1) of total weight w = N , with n r ≥ 2, i.e. Z = L k ≥ Z k .For a given weight w ∈ N the dimension dim Q ( Z N ) of the space Z N is conjecturally given bydim Q ( Z N ) = d N , with d N = d N − + d N − , N ≥ d = 1 , d = 0 , d = 1 [26]. Startingat weight w = 8 MZVs of depth greater than one r > D w,r being the number of independent MZVs at weight w > r , which cannot be reduced to primitive MZVs of smaller depth and their products, it isbelieved, that D , = 1 , D , = 1 , D , = 1 , D , = 1 and D , = 1 [28]. For Z = Z > Z > ·Z > with Z > = ⊕ w> Z w the graded space of irreducible MZVs we have: dim ( Z w ) ≡ P r D w,r =1 , , , , , , , , , , , , , w = 3 , . . . , 16, respectively [28, 27].An important question is how to decompose a MZV of a certain weight w in terms of agiven basis of the same weight w . E.g. for the decomposition ζ , , = 43361925 ζ + 15 ζ ζ + 10 ζ ζ ζ − ζ − ζ ζ − ζ ζ , + ζ , (4.7)we wish to find a method to determine the rational coefficients. Clearly, this question cannotbe answered within the space of MZV Z as we do not know how to construct a basis of MZVsfor any weight. Eventually, we sick to answer the above question within the space H of motivicMZVs with the latter serving as some auxiliary objects for which we assume certain properties[23]. For this purpose the actual MZVs (4.1) are replaced by symbols (or motivic MZVs), whichare elements of a certain algebra. We lift the ordinary MZVs ζ to their motivic versions ζ m with the surjective projection (period map) [3, 29]:per : ζ m −→ ζ . (4.8) In [3, 29] motivic MZVs ζ m are defined as elements of a certain graded algebra H equipped with a periodhomomorphism (4.8). ζ m . In particular, H is a graded Hopf algebra H with a coproduct∆, i.e. H = M n ≥ H n , (4.9)and for each weight n the Zagier conjecture is assumed to be true, i.e. dim Q ( H n ) = d n . Toexplicitly describe the structure of the space H one introduces the (trivial) algebra–comodule: U = Q h f , f , . . . i ⊗ Q Q [ f ] . (4.10)The multiplication on U ′ = U (cid:14) f U = Q h f , f , . . . i (4.11)is given by the shuffle product (cid:1) f i . . . f i r (cid:1) f i r +1 . . . f i r + s = X σ ∈ Σ( r,s ) f i σ (1) . . . f i σ ( r + s ) , (4.12)Σ( r, s ) = { σ ∈ Σ( r + s ) | σ − (1) < . . . < σ − ( r ) ∩ σ − ( r + 1) < . . . < σ − ( r + s ) } . The Hopf–algebra U ′ is isomorphic to the space of non–commutative polynomials in f i +1 . The element f commutes with all f r +1 . Again, there is a grading U k on U , with dim( U k ) = d k . Then, thereexists a morphism φ of graded algebra–comodules φ : H −→ U , (4.13)normalized by: φ (cid:0) ζ mn (cid:1) = f n , n ≥ . (4.14)Furthermore, (4.13) respects the shuffle multiplication rule (4.12): φ ( x x ) = φ ( x ) (cid:1) φ ( x ) , x , x ∈ H . (4.15)The map (4.13) is defined recursively from lower weight and sends every motivic MZV ξ ∈ H N +1 of weight N + 1 to a non–commutative polynomial in the f i . The latter is given as seriesexpansion up to weight N + 1 w.r.t. the basis { f r +1 } φ ( ξ ) = c N +1 f N +1 + X ≤ r +1 ≤ N f r +1 ξ r +1 ∈ U N +1 , (4.16)with the coefficients ξ r +1 ∈ U N − r being of smaller weight than ξ and computed from thecoproduct as follows. The derivation D r : H m → A r ⊗ H m − r , with A = H /ζ H takes only asubset of the full coproduct, namely the weight ( r, m − r )–part. Hence, D r +1 ξ gives rise to aweight (2 r + 1 , N − r )–part x r +1 ⊗ y N − r ∈ A r +1 ⊗ H N − r and ξ r +1 := c φ r +1 ( x r +1 ) · φ ( y N − r ) A Hopf algebra is an algebra A with multiplication µ : A⊗A → A , i.e. µ ( x ⊗ x ) = x · x and associativity.At the same time it is also a coalgebra with coproduct ∆ : A → A ⊗ A and coassociativity such that the productand coproduct are compatible: ∆( x · x ) = ∆( x ) · ∆( x ), with x , x ∈ A . Note, that there is no canonical choice of φ and the latter depends on the choice of motivic generators of H . c φ r +1 ( x r +1 ), with x r +1 ∈ A r +1 determines the rational coefficient of f r +1 inthe monomial φ ( x r +1 ) ∈ U r +1 . Note, that the right hand side of ξ r +1 only involves elementsfrom H ≤ N for which φ has already been determined. On the other hand, the coefficient c N +1 cannot be determined by this method unless we specify a basis B and compute φ for thisbasis giving rise to the basis dependent map φ B . E.g. for the basis B = { ζ m ζ m , ζ m } we have φ B ( ζ m ζ m ) = f f and φ B ( ζ m ) = f , while φ B ( ζ m , ) = 3 f f + cf with c undetermined.To illustrate the procedure for computing the map (4.13) and determining the decompositionlet us consider the case of weight 10. First, we introduce a basis of motivic MZVs B = { ζ m , , ζ m ζ m , ( ζ m ) , ζ m , ζ m , ζ m ζ m ζ m , ( ζ m ) ( ζ m ) , ( ζ m ) } , (4.18)with dim( B ) = d . Then for each basis element we compute (4.13): φ B (cid:0) ζ m , (cid:1) = − f f − f f , φ B ( ζ m ζ m ) = f (cid:1) f ,φ B (cid:0) ( ζ m ) (cid:1) = f (cid:1) f , φ B (cid:0) ζ m , ζ m (cid:1) = − f f f ,φ B ( ζ m ζ m ζ m ) = f (cid:1) f f , φ B (cid:0) ( ζ m ) ( ζ m ) (cid:1) = f (cid:1) f f ,φ B (cid:0) ( ζ m ) (cid:1) = f . (4.19)The above construction allows to assign a Q –linear combination of monomials to everyelement ζ mn ,...,n r . The map (4.13) sends every motivic MZV of weight less or equal to N toa non–commutative polynomial in the f i ’s. Inverting the map φ gives the decomposition of ζ mn ,...,n r w.r.t. the basis B n of weight n , with n = P rl =1 n l . We construct operators actingon φ ( ξ ) ∈ U to detect elements in U and to decompose any motivic MZV ξ into a candidatebasis B . The derivation operators ∂ n +1 : U → U are defined as [23]: ∂ n +1 ( f i . . . f i r ) = ( f i . . . f i r , i = 2 n + 1 , , otherwise , (4.20)with ∂ n +1 f = 0. Furthermore, we have the product rule for the shuffle product: ∂ n +1 ( a (cid:1) b ) = ∂ n +1 a (cid:1) b + a (cid:1) ∂ n +1 b , a, b ∈ U ′ . (4.21)Finally, c n takes the coefficient of f n . By first determining the map (4.13) for a given basis B n we then can construct the motivic decomposition operator ξ n such that it acts trivially on thisbasis. This is established for the weight ten basis (4.19) in the following.With the differential operator (4.20) we may consider the following operator ξ = a ( ζ m ) + a ( ζ m ) ( ζ m ) + a ζ m ζ m ζ m + a ( ζ m ) + a ζ m ζ m , + a ζ m ζ m + a ζ m , (4.22) The choice of φ describes for each weight 2 r + 1 the motivic derivation operators ∂ φ r +1 acting on the spaceof motivic MZVs ∂ φ r +1 : H → H [23] as: ∂ φ r +1 = ( c φ r +1 ⊗ id ) ◦ D r +1 , (4.17)with the coefficient function c φ r +1 . a = c ∂ , a = c ∂ ∂ , a = ∂ + [ ∂ , ∂ ] a = c [ ∂ , ∂ ] , a = ∂ ∂ , a = [ ∂ , ∂ ] (4.23)acting on φ B ( ξ ). Clearly, for the basis (4.19) we exactly verify (4.22) to a be a decompositionoperator acting trivially on the basis elements.Let us now discuss a special class of MZVs (4.1) identified as single–valued MZVs (SVMZVs) ζ sv ( n , . . . , n r ) ∈ R (4.24)originating from single–valued multiple polylogarithms (SVMPs) at unity [30]. The latter aregeneralization of the Bloch–Wigner dilogarithm: D ( z ) = ℑ {L i ( z ) + ln | z | ln(1 − z ) } . (4.25)Thus, e.g.: ζ sv (2) = D (1) = 0 . (4.26)SVMZVs represent a subset of the MZVs (4.1) and they satisfy the same double shuffle andassociator relations than the usual MZVs and many more relations [31]. SVMZVs have recentlybeen studied by Brown in [31] from a mathematical point of view. They have been identifiedas the coefficients in an infinite series expansion of the Deligne associator [32] in two non–commutative variables. The latter is defined through the equation [31] W ( e , e ) = Z ( − e , − e ′ ) − Z ( e , e ) , (4.27)with the Drinfeld associator (4.6) and e ′ = W e W − . The equation (4.27) can systematicallybe worked out at each weight yielding [33]: W ( e , e ) = 1 + 2 ζ ([ e , [ e , e ]] − [ e , [ e , e ]) + 2 ζ (cid:16) [ e , [ e , [ e , [ e , e ]]]] − 12 [ e , [ e , [ e , [ e , e ]]]] − 32 [ e , [ e , [ e , [ e , e ]]]] + ( e ↔ e ) (cid:17) + . . . . (4.28)Strictly speaking, the numbers (4.24) are established in the Hopf algebra (4.9) of motivic MZVs ζ m . In analogy to the motivic version of the Drinfeld associator (4.6) Z m ( e , e ) = X w ∈{ e ,e } × ζ m ( w ) w (4.29)in Ref. [31] Brown has defined the motivic single–valued associator as a generating series W m ( e , e ) = X w ∈{ e ,e } × ζ m sv ( w ) w , (4.30)whose period map (4.8) gives the Deligne associator (4.27). Hence, for the motivic MZVs thereis a map from the motivic MZVs to SVMZVs furnished by the following homomorphismsv : H → H sv , (4.31)14ith: sv : ζ mn ,...,n r ζ m sv ( n , . . . , n r ) . (4.32)In the algebra H the homomorphism (4.31) together with ζ m sv (2) = 0 (4.33)can be constructed [31]. The motivic SVMZVs ζ m sv ( n , . . . , n r ) generate the subalgebra H sv ofthe Hopf algebra H and satisfy all motivic relations between MZVs.In practice, the map sv is constructed recursively in the (trivial) algebra–comodule (4.10)with the first factor (4.11) generated by all non–commutative words in the letters f i +1 . Wehave H ≃ U , in particular ζ m i +1 ≃ f i +1 . The homomorphismsv : U ′ −→ U ′ , (4.34)with w X uv = w u (cid:1) ˜ v , (4.35)and sv( f ) = 0 (4.36)maps the algebra of non–commutative words w ∈ U to the smaller subalgebra U sv , whichdescribes the space of SVMZVs [31]. In eq. (4.35) the word ˜ v is the reversal of the word v and (cid:1) is the shuffle product. For more details we refer the reader to the original reference [31] andsubsequent applications in [33]. With (4.35) the image of sv can be computed very easily, e.g.:sv( f i +1 ) = 2 f i +1 . (4.37)Eventually, the period map (4.8) implies the homomorphismsv : ζ n ,...,n r ζ sv ( n , . . . , n r ) , (4.38)and with (4.35) we find the following examples (cf. Ref. [33] for more examples):sv( ζ ) = ζ sv (2) = 0 , (4.39)sv( ζ n +1 ) = ζ sv (2 n + 1) = 2 ζ n +1 , n ≥ , (4.40)sv( ζ , ) = − ζ ζ , sv( ζ , ) = − ζ ζ − ζ , (4.41)sv( ζ , , ) = 2 ζ , , − ζ ζ + 90 ζ ζ + 125 ζ ζ − ζ ζ , . . . . (4.42) F m The motivic version F m of the period matrix (3.14) is given by passing from the MZVs ζ ∈ Z to their motivic versions ζ m ∈ H as F m = P m Q m : exp (X n ≥ ζ m n +1 M n +1 ) : , (5.1)15ith P m = P | ζ → ζ m , Q m = Q | ζ n ,...,nr → ζ mn ,...,nr , (5.2)and the matrices P, M and Q defined in (3.15) and (3.17), respectively. Extracting e.g. theweight w = 10 part of (5.1) F m | ζ m ζ m = M M ,F m | ζ m , = 114 [ M , M ] ,F m | ( ζ m ) = 12 M + 314 [ M , M ] ,F m | ζ m ζ m ζ m = P M M ,F m | ζ m ζ m , = 15 P [ M , M ] ,F m | ( ζ m ) ( ζ m ) = 12 P M ,F m | ( ζ m ) = P , (5.3)and comparing with the motivic decomposition operators (4.23) yields a striking exact matchin the coefficients and commutator structures by identifying the motivic derivation operatorswith the matrices (3.15) as: ∂ n +1 ≃ M n +1 , n ≥ ,c k ≃ P k , k ≥ . (5.4)This agreement has been shown to exist up to the weight w = 16 in [22] and extended throughweight w = 22 in [9]. Hence, at least up to the latter weight the decomposition of motivicMZVs w.r.t. to a basis of MZVs encapsulates the α ′ –expansion of the motivic period matrixwritten in terms of the same basis elements (5.1).In the following we shall demonstrate, that the isomorphism (4.13) encapsulates all therelevant information of the α ′ –expansion of the motivic period matrix (5.1) without furtherspecifying the latter explicitly in terms of motivic MZVs ζ m . In the sequel we shall applythe isomorphism φ to F m . The action (4.13) of φ on the motivic MZVs is explained in theprevious section. The first hint of a simplification under φ occurs by considering the weight w = 8 contribution to F m , where the commutator term [ M , M ] from Q m together with theprefactor ζ m , conspires into (with φ B ( ζ m , ) = − f f ): φ B ( ζ m ζ m M M + Q m ) = f f M M + f f M M . (5.5)The right hand side obviously treats the objects f , M and f , M in a democratic way. Theeffect of the map φ is, that in the Hopf algebra U , every non–commutative word of odd letters Note the useful relation φ B ( Q m ) = f f [ M , M ] for Q m = ζ m , [ M , M ]. k +1 multiplies the associated reverse product of matrices M k +1 . Powers f k of the commutinggenerator f are accompanied by P k , which multiplies all the operators M k +1 from the left.Most notably, in contrast to the representation in terms of motivic MZVs, the numerical factorsbecome unity, i.e. all the rational numbers in (3.17) drop out. Our explicit results confirm,that the beautiful structure with the combination of operators M i p . . . M i M i accompanyingthe word f i f i . . . f i p , continues to hold through at least weight w = 16. To this end, we obtainthe following striking and short form for the motivic period matrix F m [22]: φ B ( F m ) = ∞ X k =0 f k P k ! ∞ X p =0 X i ,...,ip ∈ N ++1 f i f i . . . f i p M i p . . . M i M i . (5.6)In (5.6) the sum over the combinations f i f i . . . f i p M i p . . . M i M i includes all possible non–commutative words f i f i . . . f i p with coefficients M i p . . . M i M i graded by their length p . Ma-trices P k associated with the powers f k always act by left multiplication. The commutativenature of f w.r.t. the odd generators f k +1 ties in with the fact that in the matrix orderingthe matrices P k have the well–defined place left of all matrices M k +1 . Alternatively, we maywrite (5.6) in terms of a geometric series: φ B ( F m ) = ∞ X k =0 f k P k ! − ∞ X k =1 f k +1 M k +1 ! − . (5.7)Thus, under the map φ the motivic period matrix F m takes a very simple structure φ B ( F m ) interms of the Hopf–algebra.After replacing in (5.6) the matrices (3.15) by the operators as in (5.4) the operator (5.6)becomes the canonical element in U ⊗ U ∗ , which maps any non–commutative word in U toitself. In this representation (5.6) gives rise to a group like action on U . Hence, the operators ∂ n +1 and c k are dual to the letters f n +1 and f k and have the matrix representations M n +1 and P k , respectively. By mapping the motivic MZVs ζ m of the period matrix F m to elements φ B ( ζ m ) of the Hopf algebra U the map φ endows F m with its motivic structure: it maps thelatter into a very short and intriguing form in terms of the non–commutative Hopf algebra U .In particular, the various relations among different MZVs become simple algebraic identitiesin the Hopf algebra U . Moreover, in this writing the final result (5.6) for period matrix doesnot depend on the choice of a specific set of MZVs as basis elements. In fact, this featurefollows from the basis–independent statement in terms of the motivic coaction (subject tomatrix multiplication) [34] ∆ F m = F a ⊗ F m , (5.8)with the superscripts a and m referring to the algebras A and H , respectively. Furthermorewith [35] ∂ n +1 F m = F m M n +1 (5.9)one can explicitly prove (5.6). For instance instead of taking a basis containing the depth one elements ζ m n +1 one also could choose the setof Lyndon words in the Hoffman elements ζ mn ,...,n r , with n i = 2 , 17t has been pointed out in [36] that the simplification occurring in (5.6) can be interpretedas a compatibility between the motivic period matrix and the action of the Galois group ofperiods. Let us introduce the free graded Lie algebra F over Q , which is freely generated bythe symbols τ n +1 of degree 2 n + 1. Ihara has studied this algebra to relate the Galois Liealgebra G of the Galois group G to the more tractable object F [37]. The dimension dim( F m )can explicitly given by [38]dim( F m ) = X d | m d µ ( d ) X ⌈ m d ⌉≤ n ≤⌊ m − d d ⌋ n (cid:18) n md − n (cid:19) , (5.10)with the M¨obius function µ . Alternatively, we have [37]dim( F m ) = 1 m X d | m µ (cid:16) md (cid:17) X i =1 α di − − ( − d ! , (5.11)with α i being the three roots of the cubic equation α − α − Z = Z > Z > ·Z > with Z > = ⊕ w> Z w is isomorphic to the dual of F , i.e. dim( Z m ) = dim( F m ) [39, 40]. This property relates linearly independent elements F in the α ′ –expansion of (3.14) or (5.1) to primitive MZVs. The linearly independent algebraelements of F and irreducible (primitive) MZVs (in lines of [27]) at each weight m are displayedin Tables 1 and 2 through weight m = 22. m dim( F m ) linearly independent elements at α ′ m irreducible MZVs1 0 − − − − τ ζ − − τ ζ − − τ ζ τ , τ ] ζ , τ ζ 10 1 [ τ , τ ] ζ , 11 2 τ , [ τ , [ τ , τ ]] ζ , ζ , , 12 2 [ τ , τ ] , [ τ , τ ] ζ , , ζ , , , 13 3 τ , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] ζ , ζ , , , ζ , , 14 3 [ τ , τ ] , [ τ , τ ] , [ τ , [ τ , [ τ , τ ]]] ζ , , ζ , , ζ , , , 15 4 τ , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] ζ , ζ , , , ζ , , , ζ , , , , 16 5 [ τ , τ ] , [ τ , τ ] , [ τ , τ ] , ζ , , ζ , , ζ , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] ζ , , , , ζ , , , 17 7 τ , [ τ , [ τ , [ τ , [ τ , τ ]]]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , ζ , ζ , , , , , ζ , , , , , ζ , , , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] ζ , , , ζ , , , ζ , , Table 1: Linearly independent elements in L m and primitive MZVs for m = 1 , . . . , dim( F m ) linearly independent elements at α ′ m irreducible MZVs18 8 [ τ , τ ] , [ τ , τ ] , [ τ , τ ] , ζ , , ζ , , ζ , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , ζ , , , , ζ , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] ζ , , , , ζ , , , , ζ , , , , , 19 11 τ , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , ζ , ζ , , , ζ , , , ζ , , , , , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , ζ , , , ζ , , , ζ , , , ζ , , , , , [ τ , [ τ , [ τ , [ τ , τ ]]]] , [ τ , [ τ , [ τ , [ τ , τ ]]]] , ζ , , , , , ζ , , , , , [ τ , [ τ , [ τ , [ τ , τ ]]]] ζ , , , , 20 13 [ τ , τ ] , [ τ , τ ] , [ τ , τ ] , [ τ , τ ] , ζ , , ζ , , ζ , , ζ , , , , [ τ , [ τ , [ τ , [ τ , [ τ , τ ]]]]] , ζ , , , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , ζ , , , , ζ , , , , ζ , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , ζ , , , , ζ , , , , ζ , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] ζ , , , , , , ζ , , , , , , 21 17 τ , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , ζ , ζ , , , ζ , , , , , ζ , , , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , ζ , , , ζ , , , , , ζ , , , , , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , [ τ , [ τ , τ ]] , ζ , , , ζ , , , ζ , , , [ τ , [ τ , [ τ , [ τ , τ ]]]] , [ τ , [ τ , [ τ , [ τ , τ ]]]] , ζ , , , , , ζ , , , , , [ τ , [ τ , [ τ , [ τ , τ ]]]] , [ τ , [ τ , [ τ , [ τ , τ ]]]] , ζ , , , , , ζ , , , , , [ τ , [ τ , [ τ , [ τ , τ ]]]] , [ τ , [ τ , [ τ , [ τ , τ ]]]] , ζ , , , , , ζ , , , , , , , [ τ , [ τ , [ τ , [ τ , τ ]]]] ζ , , , , 22 21 [ τ , τ ] , [ τ , τ ] , [ τ , τ ] , [ τ , τ ] ζ , , ζ , , ζ , , ζ , , , [ τ , [ τ , [ τ , τ ]]] ζ , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] ζ , , , , ζ , , , , ζ , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]] ζ , , , , ζ , , , , , , ζ , , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , τ ]]]] , [ τ , [ τ , [ τ , τ ]]]] ζ , , , , ζ , , , , ζ , , , , , [ τ , [ τ , [ τ , τ ]]] , [ τ , [ τ , [ τ , [ τ ]]]]] , [ τ , [ τ , [ τ , τ ]]] ζ , , , , ζ , , , , ζ , , , [ τ , [ τ , [ τ , τ ]]] ζ , , , , , [ τ , [ τ , [ τ , [ τ , [ τ , τ ]]]]] , [ τ , [ τ , [ τ , [ τ , [ τ , τ ]]]]] ζ , , , , , , ζ , , , , , [ τ , [ τ , [ τ , [ τ , [ τ , τ ]]]]] ζ , , , , , Table 2: Linearly independent elements in L m and primitive MZVs for m = 18 , . . . , M n +1 defined in (3.15) are represented as ( N − × ( N − M n , [ M n , . . . , [ M n r , M n ]] . . . ] (5.12)in the expansion of (3.14) or (5.1). These structures can be related to a graded Lie algebraover Q L = M r ≥ L r , (5.13)which is generated by the symbols M n +1 with the Lie bracket ( M i , M j ) [ M i , M j ]. Thegrading is defined by assigning M n +1 the degree 2 n + 1. More precisely, the algebra L is19enerated by the following elements: M , M , M , [ M , M ] , M , [ M , M ] , M , [ M [ M , M ]] , [ M , M ] , [ M , M ] , . . . . (5.14)However, this Lie algebra L is not free for generic matrix representations M i +1 referring to any N ≥ 5. Hence, generically L 6≃ F . In fact, for N = 5 at weight w = 18 we find the relation[ M , [ M , [ M , M ]]] = [ M , [ M , [ M , M ]]] leading to dim( L ) = 7 in contrast to dim( F ) = 8.For a given multiplicity N the generators M n +1 , which are represented as ( N − × ( N − M tl by a similarity (conjugacy) transformation S S − M tl S = M l , (5.15)i.e. M l and M tl are similar (conjugate) to each other. The matrix S is symmetric and hasbeen introduced in [22]. The relation (5.15) implies, that the matrices M l are conjugate tosymmetric matrices. An immediate consequence is the set of relations S Q ( r ) + ( − r Q t ( r ) S = 0 , (5.16)for any nested commutator of generic depth r Q ( r ) = [ M n , [ M n , . . . , [ M n r , M n ]] . . . ] , r ≥ . As a consequence any commutator Q r is similar to an anti–symmetric matrix and any commu-tator Q r +1 is similar to a symmetric and traceless matrix. Depending on the multiplicity N the relations (5.16) impose constraints on the number of independent generators at a givenweight m given in the Tables 1 and 2. E.g. for N = 5 the constraints (5.16) imply : r + r ∈ Z + : [ Q ( r ) , ˜ Q ( r ) ] = 0 , (5.17) r + r ∈ Z + + 1 : { Q ( r ) , ˜ Q ( r ) } = 0 . (5.18)As a consequence, for N = 5 the number of independent elements at a given weight m doesnot agree with the formulae (5.10) nor (5.11) starting at weight w = 18. The actual number ofindependent commutator structures at weight w is depicted in Table 5. m dim( F m ) dim( L (5) m ) irreducible MZVs18 8 7 719 11 11 1120 13 11 1121 17 16 1622 21 16 1623 28 25 25Table 3: Linearly independent elements in F , L (5) m and primitive MZVs for m = 18 , . . . , 23 for N = 5. The relation (5.16) implies, that any commutator Q (2) is similar to an anti–symmetric matrix, and hence(5.17) implies [ [ M a , M b ] , [ M c , M d ] ] = 0, which in turn as a result of the Jacobi relation yields the followingidentity: [ M a , [ M b , [ M c , M d ]]] − [ M b , [ M a , [ M c , M d ]]] = [ [ M a , M b ] , [ M c , M d ] ] = 0. Furthermore, (5.16) impliesthat the commutator Q (3) is similar to a symmetric and traceless matrix. As a consequence from (5.18), weobtain the following anti–commutation relation: { [ M a , M b ] , [ M c , [ M d , M e ] } = 0. Relations for N = 5 betweendifferent matrices M i +1 have also been discussed in [41]. N = 5 an other algebra L (5) rather than F is relevant for describing the expansionof (3.14) or (5.1). For N ≥ F m ) = dim( L m ) to show up at higherweights m . This way, for each N ≥ L ( N ) , which is not free.However, we speculate that for N large enough, the matrices M k +1 should give rise to the freeLie algebra F , i.e.: lim N →∞ L ( N ) ≃ F . (5.19) The world–sheet describing the tree–level scattering of N open strings is depicted in Fig. 3.Asymptotic scattering of strings yields the string S –matrix defined by the emission and ab-sorption of strings at space–time infinity, i.e. the open strings are incoming and outgoing atinfinity. In this case the world–sheet can conformally be mapped to the half–sphere with theemission and absorption of strings taking place at the boundary through some vertex opera-tors. Source boundaries representing the emission and absorption of strings at infinity becomepoints accounting for the vertex operator insertions along the boundary of the half–sphere(disk). After projection onto the upper half plane C + the strings are created at the N posi-tions z i , i = 1 , . . . , N along the (compactified) real axis RP . By this there appears a naturalordering Π ∈ S N of open string vertex operator insertions z i along the boundary of the diskgiven by z Π(1) < . . . < z Π( N ) . C+ Diskz z1 2 zNzN−1 12 N conformaltransformation Figure 3: World–sheet describing the scattering of N open strings.21o conclude, the topology of the string world–sheet describing tree–level scattering of openstrings is a disk or upper half plane C + . On the other hand, the tree–level scattering of closedstrings is characterized by a complex sphere P with vertex operator insertions on it.At the N positions z i massless strings carrying the external four–momenta k i , i = 1 , . . . , N and other quantum numbers are created, subject to momentum conservation k + . . . + k N = 0.Due to conformal invariance one has to integrate over all vertex operator positions z i in anyamplitude computation. Therefore, for a given ordering Π open string amplitudes A o (Π) areexpressed by integrals along the boundary of the world–sheet disk (real projective line) asiterated (real) integrals on RP giving rise to multi–dimensional integrals on the space M ,N ( R )defined in (2.4). The N external four–momenta k i constitute the kinematic invariants of thescattering process: s ij = ( k i + k j ) = 2 k i k j . (6.1)Out of (6.1) there are N ( N − 3) independent kinematic invariants involving N external mo-menta k i , i = 1 , . . . , N . Any amplitude analytically depends on those independent kinematicinvariants s ij .A priori there are N ! orderings Π of the vertex operator positions z i along the boundary.However, string world–sheet symmetries like cyclicity, reflection and parity give relations be-tween different orderings. In fact, by using monodromy properties on the world–sheet furtherrelations are found and any superstring amplitude A o (Π) of a given ordering Π can be expressedin terms of a minimal basis of ( N − A o ( σ ) := A o (1 , σ (2 , . . . , N − , N − , N ) , σ ∈ S N − . (6.2)The amplitudes (6.2) are functions of the string tension α ′ . Power series expansion in α ′ yieldsiterated integrals (3.3) multiplied by some polynomials in the parameter (6.1).On the other hand, closed string amplitudes are given by integrals over the complex world–sheet sphere P as iterated integrals integrated independently on all choices of paths. Whilein the α ′ –expansion of open superstring tree–level amplitudes generically the whole space ofMZVs (4.1) enters [7, 43, 22], closed superstring tree–level amplitudes exhibit only a subsetof MZVs appearing in their α ′ –expansion [43, 22]. This subclass can be identified [33] as thesingle–valued multiple zeta values (SVMZVs) (4.24).The open superstring N –gluon tree–level amplitude A oN in type I superstring theory decom-poses into a sum A oN = ( g oY M ) N − X Π ∈ S N / Z Tr( T a Π(1) . . . T a Π( N ) ) A o (Π(1) , . . . , Π( N )) (6.3)over color ordered subamplitudes A o (Π(1) , . . . , Π( N )) supplemented by a group trace overmatrices T a in the fundamental representation. Above, the YM coupling is denoted by g oY M ,which in type I superstring theory is given by g oY M ∼ e Φ / with the dilaton field Φ. The sumruns over all permutations S N of labels i = 1 , . . . , N modulo cyclic permutations Z , whichpreserve the group trace. The α ′ → N –gluon scattering amplitude of super Yang–Mills (SYM): A o (Π(1) , . . . , Π( N )) | α ′ =0 = A (Π(1) , . . . , Π( N )) . (6.4)22s a consequence from (6.2) also in SYM one has a minimal basis of ( N − A ( σ ) := A (1 , σ (2 , . . . , N − , N − , N ) , σ ∈ S N − . (6.5)Hence, for the open superstring amplitude we may consider a vector A o with its entries A oσ = A o ( σ ) describing the ( N − N –point superstring subamplitudes (6.2), whilefor SYM we have an other vector A with entries A σ = A ( σ ): A o = ( N − all independent superstring subamplitudes A oσ = A o ( σ ) , σ ∈ S N − ,A = ( N − all independent SYM subamplitudes A σ = A ( σ ) , σ ∈ S N − . The two linear independent ( N − A and A are related by a non–singularmatrix of rank ( N − A o = F A , (6.6)with the period matrix F given in (3.12). Note, that with (3.13) the Ansatz (6.6) matches thecondition (6.4). In components the relation (6.6) reads: A o ( π ) = X σ ∈ S N − F πσ A ( σ ) , π ∈ S N − . (6.7)In fact, an explicit string computation proves the relation (6.6) [45, 46].Let us now move on to the scattering of closed strings. In heterotic string vacua gluonsare described by massless closed strings. Therefore, we shall consider the closed superstring N –gluon tree–level amplitude A cN in heterotic superstring theory. The string world–sheet de-scribing the tree–level scattering of N closed strings has the topology of a complex sphere with N insertions of vertex operators. The closed string has holomorphic and anti–holomorphicfields. The anti–holomorphic part is similar to the open string case and describes the space–time (or superstring) part. On the other hand, the holomorphic part accounts for the gaugedegrees of freedom through current insertions on the world–sheet. As in the open string case(6.3), the single trace part decomposes into the sum A cN, s . t . = ( g HET Y M ) N − X Π ∈ S N / Z tr( T a Π(1) . . . T a Π( N ) ) A c (Π(1) , . . . , Π( N )) (6.8)over partial subamplitudes A c (Π) times a group trace over matrices T a in the vector represen-tation. In the α ′ → N –gluon scattering subamplitudes of SYM A c (Π(1) , . . . , Π( N )) | α ′ =0 = A (Π(1) , . . . , Π( N )) , (6.9)similarly to open string case (6.4). Again, the partial subamplitudes A c (Π) can be expressed interms of a minimal basis of ( N − A c ( ρ ) , ρ ∈ S N − . The latter have been computedin [47] and are given by A c ( ρ ) = ( − N − X σ ∈ S N − X ρ ∈ S N − J [ ρ | ρ ] S [ ρ | σ ] A ( σ ) , ρ ∈ S N − , (6.10)23ith the complex sphere integral J [ ρ | ρ ] := V − N Y j =1 Z z j ∈ C d z j N Y i 5. 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