Superstring amplitudes and the associator

Abstract

We investigate a pattern in the [alpha]' expansion of tree-level open superstring amplitudes which correlates the appearance of higher depth multiple zeta values with that of simple zeta values in a particular way. We rephrase this relationship in terms of the coaction on motivic multiple zeta values and show that the pattern takes a very simple form, which can be simply explained by relating the amplitudes to the Drinfel'd associator derived from the Knizhnik-Zamolodchikov equation. Given this correspondence we show that, at least in the simplest case of the four-point amplitude, the associator can be used to extract the form of the amplitude.

Published for SISSA by Springer

Received: June 19, 2013 Accepted: July 22, 2013

Published: August 29, 2013

JHEP08(2013)135

Superstring amplitudes and the associator

J.M. Drummonda,b and E. Ragoucyb

aPH-TH, CERN,

Case C01600, CH-1211 Geneva 23, Switzerland

bLAPTH, Universit de Savoie, CNRS,

B.P. 110, F-74941 Annecy-le-Vieux Cedex, France

E-mail: [email protected], mailto:[email protected]

Web End [email protected]

Abstract: We investigate a pattern in the expansion of tree-level open superstring amplitudes which correlates the appearance of higher depth multiple zeta values with that of simple zeta values in a particular way. We rephrase this relationship in terms of the coaction on motivic multiple zeta values and show that the pattern takes a very simple form, which can be simply explained by relating the amplitudes to the Drinfeld associator derived from the Knizhnik-Zamolodchikov equation. Given this correspondence we show that, at least in the simplest case of the four-point amplitude, the associator can be used to extract the form of the amplitude.

Keywords: Scattering Amplitudes, Brane Dynamics in Gauge Theories

ArXiv ePrint: 1301.0794

Open Access doi:http://dx.doi.org/10.1007/JHEP08(2013)135

Web End =10.1007/JHEP08(2013)135

Contents

1 Introduction and summary 1

2 Tree-level open superstring scattering amplitudes 3

3 Multi-zeta values and their motivic counterparts 5

4 Patterns in the expansion 7

5 Connection to the Drinfeld associator 9

6 The four-point amplitude from the associator 12

7 Closed strings and constrained multiple zeta values 13

A Isomorphisms between H and U 14

B Associator from the KZ equation 16

1 Introduction and summary

Much recent progress in understanding the structure of scattering amplitudes in eld theory has been based on understanding the analytic structure of loop corrections in terms of the Hopf structure underlying the iterated integrals dening multiple polylogarithms. Results along these lines have been obtained in [16], primarily based on using the symbol [79] or motivic coaction [1012] as a tool to perform analysis of the analytic structure in an algebraic manner. While loop corrections in eld theory quickly become rather complicated, there is a similar problem in which a transcendental structure appears in a much milder form, namely string theory scattering amplitudes at tree level. If one expands tree-level string amplitudes in , the inverse string tension, one nds coe cients which contain multiple zeta values, rather than multiple polylogarithms.

In [13] an intriguing pattern of multi-zeta values was found in the expansion of the open superstring amplitudes. The coe cients of all multiple zeta values of depth greater than one are xed in terms of those of depth one in a specic way, indicating that in the case of superstring amplitudes, the Hopf structure is not just a tool that can be used to analyse the results, but rather it is intrinsic to their structure.

The pattern among the multiple zeta values was expressed in [13] in terms of an auxiliary map which relates the (motivic) multi-zetas to a Hopf algebra of words endowed with the shu e product. The map was introduced in [11, 12] to help to explain the structure of the space of motivic multi-zetas and to produce an e cient algorithm for xing a basis

1

JHEP08(2013)135

and projecting a given multi-zeta value into that basis. As noted in [13] this pattern is closely related to a representation of the identity operator on the Hopf algebra of words, so that the pattern is something canonical. However the objects introduced to encode the structure do depend on the choice of basis of multiple zeta values. In particular, the isomorphism between the multi-zetas and the Hopf algebra of words is non-canonical in that it depends on the basis of multi-zetas chosen. One of the aims of the present work is to express this pattern purely in terms of the Hopf structure associated to multi-zeta values, without introducing a basis.

Indeed we will see that we can rephrase the pattern found in [13] purely in terms of the coaction on motivic multiple zetas, making the independence on the choice of basis manifest. Following [14, 15] we express the colour-ordered open superstring amplitudes in terms of a matrix R acting on a vector of independent colour-ordered super Yang-Mills amplitudes,

Aopen = RAYM . (1.1)

The matrix R = 11+O() encodes all the corrections to the super Yang-Mills amplitudes. In terms of the motivic coaction, the pattern found in [13] can be rephrased as follows,

Rm = Rm

Ra . (1.2)

The superscripts on R refer to the fact that all multi-zeta values should be replaced by their motivic versions (so that the coaction is dened) and moreover that in the right-hand factor we should work modulo m2. The equation (1.2) implies that all coe cients of multiple zetas of depth greater than one are xed in terms of those of depth one, in complete agreement with the structure presented in [13].

Moreover the equation (1.2) is similar to a property obeyed by another object, the Drinfeld associator [16, 17], dened in terms of the Knizhnik-Zamolodchikov equation [20]. It is a form of universal monodromy for solutions of the KZ equation. The associator can be written as a generating series for all (shu e regularised) multiple zeta values,

=

Xww (w) , (1.3)

where the w are words in two letters. obeys the relation

m = m

a , (1.4)

in complete analogy with (1.2), where is the Ihara action [19, 30, 31] on group-like series of words.

Indeed we will see in the simplest case (the four-point amplitude) that we can actually identify the associator with the matrix R appearing in the open superstring amplitude, which actually allows one to x also the coe cients of the zeta values of depth one.

The paper is organised as follows. We begin with a very brief review of superstring amplitudes in section 2. We introduce multiple zeta values and the motivic coaction in section 3. Then, in section 4 we describe the patterns in the expansion of the amplitudes

2

JHEP08(2013)135

found in [13] and describe how they can be rephrased as in eq. (1.2). In section 5 we introduce the associator and describe its properties under the motivic coaction. In section 6 we use the Knizhnik-Zamolodchikov equation to derive the form of the open superstring four-point amplitude. In section 7 we make some remarks on the structure of closed super-string amplitudes and the constraints on the multiple zeta values appearing in both closed and open superstring amplitudes.

2 Tree-level open superstring scattering amplitudes

Let us begin with tree-level scattering amplitudes in gauge theory. The n-gluon amplitude of a gauge theory can be decomposed as a sum over cyclic colour-ordered partial amplitudes,

Agauge =

XSn/Cn

Agauge((1), . . . , (n))Tr(T a(1) . . . T a(n)) . (2.1)

The tree-level open superstring amplitude Aopen of massless gauge bosons can similarly be decomposed in terms of colour-ordered partial amplitudes Aopen. The superstring amplitudes depend on , the inverse string tension and in the limit 0 one recovers the gauge theory amplitudes. All of these statements are independent of the number of uncompactied dimensions in which one studies the scattering process.

The colour-ordered partial amplitudes in gauge theory obey certain relations. The simplest among these are the cyclic and reection identities. Then one has the photon decoupling identity and Kleiss-Kuijf relations [21]. Finally there are the BCJ relations [22]. These relations among gauge theory partial amplitudes can be derived from the monodromy properties of the open string theory partial amplitudes [2325]. If one uses all these relations to reduce the set of colour-ordered partial amplitudes to a minimal set, then (n3)! partial amplitudes remain. One may choose these partial amplitudes to be those obtained from all permutations of the labels 2, . . . , n2, keeping 1, (n1) and n xed. We may arrange these remaining amplitudes into a vector which we denote by Agauge, or correspondingly, Aopen.

Thus we have a vector of independent colour-ordered open superstring amplitudes related to the corresponding eld theory amplitudes via a matrix R,

Aopen = RAgauge . (2.2)

The matrix R has an expansion R = 11 + O(). In general, terms of order ()m contain multi-zeta values of weight m. The detailed structure of the corrections has been studied in [14, 15]. As one might expect it is given in terms of (n 3)-fold integrals over the positions of the vertex operators not xed using conformal symmetry. Here we will outline the simplest cases n = 4, 5.

The simplest case is n = 4 where the matrix R is therefore a (1 1) matrix. In this case one has [26]

R = (1 s) (1 t)

(1 s t) . (2.3)

Here s = (p1 + p2)2 and t = (p2 + p3)2 are the two Mandelstam variables and we recall that the momenta satisfy p2i = 0 since we are considering the on-shell scattering of massless gauge bosons. If we expand the gamma functions as a series in s and t we nd zeta

3

JHEP08(2013)135

values. Specically, we can write R as the exponential of an innite series of contributions proportional to simple zeta values n = (n),

R = exp

Xn2nn [sn + tn (s + t)n] . (2.4)

If we wish we may decompose the exponential into the contributions of even and odd zeta values,

R = P E (2.5)

where

P = exp

Xn12n2n [s2n + t2n (s + t)2n] (2.6)

and

E = exp

Xn12n+12n + 1[s2n+1 + t2n+1 (s + t)2n+1] . (2.7)

For ve-point amplitudes, R is a (2 2) matrix,

R = F1 F2

~F2 ~F1

[parenrightBigg]

where we have

0 dyxs45ys12(1 x)s34(1 y)s23(1 xy)s241 . (2.9)

The ve independent Mandelstam variables are chosen to be s12, s23, s34, s45, s24 with sij =

(pi + pj)2. The functions ~F1 and ~F2 are given by ~Fi = Fi|p2p3. The integrals above can be expressed in terms of hypergeometric functions and gamma functions [15, 27].

F1 s12s34 =

F2 s13s24 =

JHEP08(2013)135

(2.8)

F1 = s12s34

F2 = s13s24

[integraldisplay]

1

1

0 dx [integraldisplay]

1

0 dyxs45ys121(1 x)s341(1 y)s23(1 xy)s24 ,

1

[integraldisplay]

(1 + s45) (s12) (s34) (1 + s23)

(1 + s45 + s34) (1 + s12 + s23) 3F2(1 + s45, s12, s24; 1 + s45 + s34, 1 + s12 + s23; 1) , (2.10)

0 dx [integraldisplay]

(1 + s12) (1 + s23) (1 + s34) (1 + s45)

(2 + s12 + s23) (2 + s34 + s45) , 3F2(1 + s12, 1 + s45, 1 s24; 2 + s12 + s23, 2 + s34 + s45; 1) . (2.11)

If we expand the above hypergeometric functions as a series in the Mandelstam variables we nd multiple zeta values. Before discussing the patterns in the expansion found in [13] we will give a short introduction to multiple zeta values and the Hopf structure of their motivic counterparts.

4

3 Multi-zeta values and their motivic counterparts

Multi-zeta values can be dened in terms of nested sums. Here we will use an iterated integral representation. We consider iterated integrals of the following form,

I(a0; a1, . . . , an; an+1) =

Z dz1z1 a1 . . .dznzn an (3.1)

where is a path from a0 to an+1 avoiding the poles at a1, . . . , an, and the integral is performed so that a0, z1, . . . , zn, an+1 gives an ordering along the curve. If the poles coincide with the endpoints the integral may require regularisation.

Sometimes we will use the notation I(a0; w; a1) where w is the word a1 . . . an. The multi-zeta values are special cases where a0 = 0 and an+1 = 1, with being the path along the real axis, and ai {0, 1} for 1 i n. There is a also a sign depending on the number r of the ai equal to 1,

(p1, . . . , pr) = (1)rI(0; 10p11 . . . 10pr1; 1) . (3.2)

We take pr 2 to ensure convergence of the integral. Very often we write p1,...,pr to save a little space. Even more compactly we can write

(w) = (1)rI(0; w; 1) (3.3)

where w is a word built from the letters {0, 1}, beginning with a 1 and ending with a 0, and r is the number of ones in w as before. The simplest example of a zeta value is given by

2 = I(0; 10; 1) = [integraldisplay]

1

JHEP08(2013)135

dt1 t1 1

[integraldisplay]

dt2t2 . (3.4)

All even simple zeta values are rational multiples of powers of 2,

2n = bnn2 , bn = (1)n+1 12B2n

(24)n(2n)! , (3.5)

where B2n denotes the Bernoulli numbers.

The iterated integrals obey a shu e product relation,

I(a0; w1; an+1)I(a0; w2; an+1) = I(a0; w1 w2; an+1) , (3.6)

where we recall that the shu e product of two words w1 w2 is a sum over all words given by permutations of the elements of w1 and w2 which preserve the orderings within w1 and w2. For a0 = 0 and an+1 = 1 this implies the shu e relation for the multiple zeta values,

(w1)(w2) = (w1 w2) . (3.7)

We can use the above relation to dene a regularised version of the multi-zeta values whose words begin with 0 or end in a 1; we extract any leading zeros and trailing ones from a given zeta using the shu e relation and dene (0) = (1) = 0.

5

1

t1

0

ai2

ai3

ai1

ai4

an+1

JHEP08(2013)135

a0

Figure 1. A contribution to the motivic coaction for k = 4.

One may lift iterated integrals I to their motivic versions Im. These are dened as abstract elements of an algebra HMT , graded by weight and obeying the shu e product relation (3.6). The motivic iterated integrals satisfy all known algebraic relations satised by real iterated integrals and conjecturally capture all such relations. As special cases of the motivic iterated integrals one has the motivic multiple zeta values m(w) = Im(0; w; 1).

Since all even simple zeta values are known to be rational multiples of powers of 2,

the element m2 plays a special role. It is convenient to introduce the quotient AMT = HMT /m2, whose elements are denoted Ia (or a when specialising to motivic MZVs).

Then AMT is a Hopf algebra and HMT is a trivial comodule over AMT which can be non-canonically identied with Q[m2] AMT . As an algebra it is commutative, with the product being the shu e product (3.6). The coaction : HMT HMT AMT is

dened on the motivic iterated integrals as follows [1012].

Im(a0; a1, . . . , an; an+1) = (3.8)

=

Xk Xi0<i1...ik<ik+1Im(a0; ai1, . . . , aik; an+1)

[parenleftbigg]

k

Yp=0Ia(aip; aip+1, . . . , aip+11; aip+1)

.

The terms in the above formula can be associated with polygons inscribed into the semi-circle with (n + 1) marked points aj. Note we have reversed the order of the factors on the r.h.s. with respect to [11, 12]. The above coaction specialises in the obvious way to the algebra of motivic multiple zeta values, denoted by H.

The only primitive elements of H are the simple zeta values mn. Because of the fact that one should mod out by m2 on the right-hand factor, the coaction looks di erent for even and odd simple zetas,

m2n = m2n 1, m2n+1 = m2n+1 1 + 1 a2n+1 . (3.9)

6

4 Patterns in the expansion

In [13] the following pattern for R in the ve-point case was found up to weight 16,

R = P QE (4.1)

where

P = 1 +

X2nP2n , (4.2)

E =: exp

X2n+1M2n+1 : (4.3)

andQ = 1 +

Xn8Qn . (4.4)

The normal ordering symbol in E means that : MiMj := MjMi if i < j and : MiMj := MiMj otherwise. Alternatively one may write E as an ordered product of exponentials E = (. . . e 5M5e 3M3). The matrices Qn contain multiple zeta values of higher depth and are determined in terms of commutators of the M2n+1. Thus if all the Mn commute with each other, as is the case for the four-point amplitude, the matrix Q reduces to the identity. The structure is therefore a generalisation of the one for the four-point amplitude to non-commuting Mi.

Before we discuss in detail the structure from a Hopf algebraic point of view, let us exhibit the rst few terms in Q found in [13] in order to illustrate how the coefcients of multiple zetas of higher depth are xed in terms of those of depth 1. For this we need to make our basis of multi zeta values explicit. Up to weight 12 we take {2, 3, 5, 7, 3,5, 9, 3,7, 11, 3,3,5, 3,9, 1,1,4,6} and all possible products of these elements.

In this basis the Qn found in [13] take the form,

Q8 = 153,5[M5, M3] , (4.5)

Q9 = 0 , (4.6)

Q10 =[parenleftbigg]

3 1425 +

JHEP08(2013)135

1 143,7

[M7, M3] , (4.7)

Q11 =

929 + 625227 435325 +1 53,3,5

[M3, [M5, M3]] , (4.8)

Q12 =

2957 +1 273,9

[M9, M3]

+ 48

691

18353223 [phone] 72225 35223,5 323,7

1

1243

467 10857 +

799

72 39 +

2665

648 3,9 + 1,1,4,6

[parenrightbigg][parenleftBigg]

[M9, M3] 3[M7, M5]

[parenrightbig]

(4.9)

To be completely explicit, let us expand the matrix R up to order ()10,

R = 1 + 2P2 + 3M3 + 4P4 + 5M5 + 23P2M3 + 1223M3M3 + 6P6

7

+ 7M7 + 25P2M5 + 43P4M3 + 1

53,5[M5, M3] + 53M5M3

+ 12223P2M3M3 + 8P8 + 9M9 +

1633M3M3M3 + 27P2M7

+ 45P4M5 + 63P6M3 +

[parenleftbigg]

3 1425 +

1 143,7

[M7, M3] + 73M7M3

+ 1

225M5M5 +

1523,5P2[M5, M3] + 253P2M5M3

+ 1

2423P4M3M3 + 10P10 + . . . (4.10)

The fact that the coe cients of the higher depth multi zeta values are given in terms of those of the simple zeta values can be explained in terms of the Hopf algebra structure obeyed by the motivic multi zeta values. It is therefore useful to introduce the notation Rm to denote the matrix R with the zeta values replaced by their motivic versions and Ra to denote the same matrix modulo m2.

In [13], the structure underlying the matrix R was described by introducing an auxiliary Hopf algebra U, following [11, 12]. The Hopf algebra U is the commutative, graded Hopf algebra of all non-commutative words in generators of each odd degree d 3, denoted by {f3, f5, f7, f9, . . .}. These generators play the role of the simple zeta values. The product on U is the shu e product; the coproduct is given by deconcatenation. The even zeta values are taken into account by considering the trivial comodule

U = Q[f2] U , (4.11)

where f2 is a generator of degree 2. The references [11, 12] show how to construct isomorphisms B : H U, depending on the choice of basis B of the motivic multi-zeta values H. We describe the isomorphisms B in more detail in the appendix, describing their basis dependence.

Having introduced the isomorphisms B associated to a given basis of H, we can now state explicitly the structure found in [13] for the matrix R describing the open superstring scattering amplitudes. After replacing all zeta values with their motivic versions in the expansion of R, one nds the following simple pattern after applying the isomorphism B,

B(Rm) =

[parenleftbigg][summationdisplay]

fk2P2k

JHEP08(2013)135

[parenrightbigg] [summationdisplay]

p

Xi1,...,ipMi1 . . . Mipfi1 . . . fip . (4.12)

We remind the reader that both the matrices P and M and the map B depend on the choice of basis B for H, although we have not made explicit the dependence of Pr and Mr on B. The above structure (4.12) for the image of R under B, however, does not depend on B.

Indeed, it was noted in [13] that (4.12) has the same form as the canonical element of U U. Let us introduce the operators 2r+1 which act from the right on a word built from the fn so that 2r+1 removes the last letter of the word if it is f2r+1 or annihilates the word otherwise. Thus r1 . . . rn is the dual basis element to fr1 . . . frn. Similarly we can introduce operators 2r on the left which pick out the coe cient of fr2. Then replacing

P2k by 2k and M2r+1 by 2r+1, formula (4.12) turns into the canonical element of U U. Thus the matrices P2k and M2k+1 represent the operators r.

8

In fact one may state the above rather more simply by saying that the matrices P and M represent the duals of the primitive zeta values in a given basis B. Thus Rm is really representing the canonical element of H H. At this stage, no information is available on what the representation is, other than by directly expanding the result for the amplitude.

We now observe that the above structure can be restated as the following property of R,

Rm = Rm

Ra . (4.13)

Here the symbol

means that the two factors of the tensor product are multiplied as matrices. In addition to its compactness, the above expression makes manifest the fact that the structure found in [13] is independent of any choice of basis for H. The relation (4.13) for the coaction on Rm xes all but the primitive elements, i.e. contributions proportional to mn, corresponding to the matrices P2n and M2n+1 above. Another advantage to (4.13)

is that, in order to test it, one does not have to refer to the underlying algebra structure of U or H. One simply computes the motivic coaction on Rm and compares with the r.h.s.

The equation (4.13) for Rm is equivalent to the fact that Rm represents the canonical element R in H H. Indeed we have, in terms of a basis ei of H (andk of A) and the dual basis ei of H (andk of A),

R = [parenleftBig][summationdisplay]i

Xj(ej ej)

Pi fjkiei describes the action on the duals. We have checked explicitly that the expressions given in [13] do indeed satisfy the relation (4.13) to weight 15.

The consequence of (4.13) is that the amplitude is completely determined in terms of the matrices P2n and M2n+1. The matrices P2n and M2n+1 themselves are not determined.

This is already clear at the level of the four-point amplitude since any expression of the form

Rm = exp

[braceleftBig][summationdisplay]

Pj,k fjkiej k describes the coaction on H and (ej k) =

(4.15)

will obey Rm = Rm Ra. One may ask if there is any connection to the Hopf structure of zeta values which xes the coe cients cn to be precisely those appearing in the four-point amplitude.

5 Connection to the Drinfeld associator

By comparing formulas appearing in [13] and [31] one can see that the structure of the matrix R describing the open superstring amplitude mirrors closely the structure of another object, the Drinfeld associator [16, 17]. This is a form of universal monodromy for the Knizhnik-Zamolodchikov equation.

The associator can be written as a generating function for all multi-zeta values,

=

Xww (w) , (5.1)

9

ei ei

[parenrightBig]

=

Xi,j,kfjki(ej k) ei

=

JHEP08(2013)135

Xk(k k) = R ~R. (4.14)

Here ei =

cnmn

[bracerightBig]

where the sum is over all words w in the alphabet {e0, e1} and the multi-zeta values are regularised using the shu e product. The space of words on {e0, e1} can be identied with the universal enveloping algebra U(g) of the free Lie algebra on two generators g = Lie[e0, e1] .

Thus is an element of U(g) over the reals.

To express the associator it is useful to introduce derivations acting on U(g) as follows. Given an element y of g we dene (following [18, 19])

Dye0 = 0, Dye1 = [e1, y] , (5.2)

and extend Dy as a derivation to the whole of U(g). Given the derivations D above we can dene a right action of g on U(g) as follows,

x y = xy Dyx, y g, x U(g) . (5.3)

The Ihara bracket is dened as its antisymmetrisation after restricting U(g) to g in the natural way,

{x, y} = x y y x = [x, y] + Dxy Dyx , x, y g . (5.4)

Note that {x, y} is also an element of g (actually it is even an element of g = [g, g]). The Ihara bracket is the Poisson bracket which arises through commutation of the derivations D,

[Dx, Dy] = D{x,y} , (5.5)

and it therefore obeys the Jacobi identity.

By direct calculation one may compute the associator up to a given weight. We have explicitly computed it up to weight 13 and expressed the result in terms of the same zeta-values used in the previous section. Here we will display the expansion for the associator up to weight 10,

= 1 + 2p2 + 3w3 + 4p4 + 5w5 + 23(w2 w3) + 1223(w3 w3) + 6p6

+ 7w7 + 25(p2 w5) + 43(p4 w3) + 153,5{w5, w3} + 53(w5 w3)

+ 12223((p2 w3) w3) + 8p8 + 9w9 +

1633((w3 w3) w3) + 27(p2 w7)

1 143,7

1523,5(p2 {w5, w3}) + 253((p2 w5) w3)

+ 12423((p4 w3) w3) + 10p10 + . . . (5.6)

Comparing to equation (4.10) we see the obvious similarity. Going from (5.6) to (4.10), the words p2r are replaced by the matrices P2r, the words w2r+1 by the matrices M2r+1 and the product by the matrix product. The antisymmetrisation of the operation on two elements of g is the Ihara bracket which is therefore replaced by matrix commutators.

10

225(w5 w5) +

JHEP08(2013)135

+ 45(p4 w5) + 63(p6 w3) +

[parenleftbigg]

3 1425 +

{w7, w3} + 73(w7 w3)

+ 1

If we consider the coproduct g on all words in the alphabet {e0, e1} dened by demanding

g(ei) = ei 1 + 1 ei , (5.7)

g(w1w2) = g(w1) g(w2) , (5.8)

we nd is group-like, g = . (5.9)

Thus log is a Lie series. Moreover, since it contains no elements of length 1, it is a series in g = [g, g].

The words p2r and w2r+1 are not quite on the same footing. The reason is that all even simple zetas are related to powers of 2 via (3.5). Hence, while all the words w2r+1 are elements of g in accord with (5.9), this is not the case for the words p2r for r > 1. It is therefore helpful to dene the words w2r g via

w2 = p2 ,

w4 = p4 12b2 w2 w2 ,

w6 = p6 b2b3 w4 w2

and so on so that log is explicitly a Lie series,

log = 2w2 + 3w3 + 4w4 1222Dw2w2 + 5w5 + 23[parenleftbigg]

12[w4, w2] Dw2w4[parenrightbigg] 1632[parenleftbigg]12[Dw2w2, w2] Dw2Dw2w2 [parenrightbigg]

+ . . . (5.11)

To be more concrete we give the explicit form of the rst few words,

w2 = [e1, e0] (5.12)

w3 = [e0 e1, [e0, e1]] (5.13)

w4 = [e0, [e0, [e0, e1]]] 32[e1, [e0, [e1, e0]]] + [e1, [e1, [e1, e0]]] (5.14) w5 = [e0, [e0, [e0, [e0, e1]]]] 12[e0, [e0, [e1, [e0, e1]]]]

32[e1, [e0, [e0, [e0, e1]]]]

+ (e0 e1) . (5.15)

We now introduce an exponentiated version of the action , which we will denote by . Given group-like series A and B in U(g), we dene the action of B on A (following [19, 30, 31]),

A B = A(e0, Be1B1)B . (5.16)

11

JHEP08(2013)135

16b3 (w2 w2) w2 (5.10)

1

2[w2, w3] Dw3w2

[parenrightbigg]

+ 6w6 + 24

It is clear that the innitesimal version of the action dened above is . Note that 1 A = A 1 = A and that is associative since

(A B) C = (A(e0, Be1B1)B) C= A(e0, B(e0, Ce1C1)Ce1C1B(e0, Ce1C1)1)B(e0, Ce1C1)C

= A(e0, (B C)e1(B C)1)(B C)

= A (B C) . (5.17)

Now we are in a position to compare to the relation obeyed by the matrix R in the open string amplitude. By replacing all zeta values in the expansion of with their motivic versions, we obtain an element of U(g) H which we call m. Calculating the coaction , one nds the following relation, completely analogous to (4.13),

m = m

indicates that the right-hand factor acts on the coe cient words of the left-hand factor via the action ,

m

a =

JHEP08(2013)135

a . (5.18)

Here the symbol

Xwm (w) w(e0, ae1( a)1) a . (5.19)

This is a series of words with coe cients in H A.

Thus we would like to identify the matrix R appearing in the open superstring amplitude with , with the matrix multiplication being a representation of the Ihara action . In the next section we will show that the associator can be used to reproduce the four-point amplitude by following this logic.

6 The four-point amplitude from the associator

The form (5.1) allows us to explicitly expand the associator, at least up to weights where (conjecturally) all relations between multiple zeta values are explicitly known [29]. We would like to ask if we can understand how to extract the matrices M2n+1 and P2n from the form of the words w2n+1 and p2n. We will examine the simplest case, namely the four-point amplitude, where we will in fact be able to identify P and M to all orders.

Since the matrix R is a (1 1) matrix in the four-point case, we need a commutative realisation of the Ihara action. In other words we need a representation where the Ihara brackets {wr, ws} vanish. This will guarantee that all multiple zeta values of depth greater than one will disappear.

Now we observe that for words wr and ws in g, the Ihara bracket {wr, ws} is an element of g = [g, g]. This can be seen as follows. First we note that if we work modulo g then all elements of g can be written as a linear combination of the basis elements

ukl = adk0adl1[e0, e1] , (6.1)

where adix = [ei, x] and the two adjoint actions commute. We recall that the Ihara bracket takes the form

{wr, ws} = [wr, ws] + Dwrws Dwswr . (6.2)

12

For wr, ws g the rst term on the r.h.s. is clearly in g. On the basis elements (6.1) we have Duklukl = uk+k+1,l+l+1 modulo g and so the nal two terms on the r.h.s. of (6.2)

cancel modulo g. Hence the Ihara brackets vanish in g/g as required.

In this approximation, the associator in has been calculated in [17]. For completeness we give a derivation in appendix B, starting from the Knizhnik-Zamolodchikov equation. The result is

[e0, e1], (6.3)

where u = ad0 and v = ad1 and again we work modulo g.

From the above result we deduce that

= 1 + 1

uv

log = 1

uv

(1 u) (1 v) (1 u v) 1

[e0, e1] mod (g)2. (6.4)

If we now think of acting on U(g)/(g)2 via the Ihara action, we nd that it is represented by the multiplicative operator (i.e. a (1 1) matrix),

R(u, v) = (1 u) (1 v)

(1 u v) , (6.5)

which, if we interpret the variables u and v as being the Mandelstam variables, is precisely the four-point amplitude R, with the coe cients Pn and Mn xed to their correct functional forms.

We emphasise here that we have derived only the simplest superstring amplitude, the four-point one, which can of course be derived simply from many approaches. Ultimately, given the form of the worldsheet integrals describing the amplitudes, it is not surprising that it can be related to monodromies of the Knizhnik-Zamolodchikov equation [32]. However the approach we have outlined shows that the relation between the associator and the amplitude guarantees that the relation (4.13) holds. Moreover, one can obtain more than just the fact that the coe cients of the multiple zetas of higher depth are xed in terms of the M2n+1 and P2n; one can also x the M2n+1 and P2n themselves.

7 Closed strings and constrained multiple zeta values

By employing the Kawai-Lewellen-Tye relations [33] one may obtain tree-level amplitudes for closed strings from the ordered partial amplitudes for open strings [25],

Aclosed = (Aopen)tSAopen . (7.1)

Here S is a (n 3)! (n 3)! matrix consisting of particular sin factors. It obeys the important relation [13],

P tSP = S0, (7.2)

where S0 is the 0 limit of S, a matrix of homogeneous rational functions of the Mandelstam variables.

The coe cients of odd zetas, the matrices Mn, obey (see also [13])

Mn = S10MtnS0 , (7.3)

13

(1 u) (1 v) (1 u v) 1

JHEP08(2013)135

implying that nested commutators Q(r) = [Ms1, [Ms2, [. . . [Msr1, Msr] . . .]]] obey

Q(r) = (1)r+1S10Qt(r)S0 . (7.4)

Since the nested commutators are the coe cients of specic multiple zeta values as dictated by (4.13), it follows that Aclosed gets contributions from only certain specic combinations of multiple zeta values. For example there are no linear contributions of multiple zeta values of even depth. This has important consequences for the closed IIB superstring e ective action. Since the IIB theory has an SL(2, Z) duality symmetry, the axion and dilaton appear in the e ective action through specic modular forms. At tree level these modular forms reproduce the zeta values. The restriction of the kinds of multiple zeta values appearing at tree level will have some interplay with the possible kinds of modular forms appearing. It would be interesting to investigate this point further.

In fact there are also restrictions on the kinds of multiple zeta values that can enter the open superstring amplitude. For a given number of external particles there are relations among the di erent commutators of the Mn simply due to the fact that they are nite matrices. A very strong restriction appears at four points where, since the matrices are (11), they all commute, i.e. [Mr, Ms] = 0. This is linked to the fact that no multiple zeta values appear in the four-particle amplitude. At higher points the restrictions become successively weaker. For example, at ve points the matrices are (2 2) and according to (7.3) above they are conjugate to symmetric matrices. This implies that the commutators [Mr, Ms] are

conjugate to antisymmetric matrices, but since there is a unique (22) antisymmetric matrix up to rescaling, the commutators commute with each other [[Mr, Ms], [Mt, Mu]] = 0.

Note that these constraints imply [Mr, [Ms, [Mt, Mu]]] = [Ms, [Mr, [Mt, Mu]]] and that such relations were useful in [13] in establishing that the matrix Q takes an exponential form up to weight 18 in the case of the ve point amplitude. Relations such as these imply that only specic linear combinations of multiple zeta values appear in the n-point amplitude for xed n. The constraints become weaker and weaker in the sense that more and more linearly independent combinations of multiple zeta values appear as n grows.

Acknowledgments

JMD would like to thank the organisers of the workshop Amplitudes and periods held at IHES in December 2012 for the opportunity to present this work and the participants for many interesting and helpful comments. We would also like to thank Francis Brown for some helpful suggestions and insightful comments, particularly regarding the role of the associator.

A Isomorphisms between H and U

In [13], the structure underlying the matrix R was described by introducing an auxiliary Hopf algebra U, following [11, 12]. This Hopf algebra is the commutative Hopf algebra of all words constructed from the alphabet {f3, f5, f7, f9 . . .}, i.e. the alphabet with a generator of every odd degree, r 3. The Hopf algebra product is the shu e product and

14

JHEP08(2013)135

the coproduct is given by deconcatenation,

fa1, . . . , fan =

Now one may construct many isomorphisms B between H and U as described in [11]. The map B depends on the choice of basis of H and is constructed as follows. First one introduces an innitesimal version of the coproduct (3.8) encapsulated by operators Dr : H H L for r odd and r 3 and where L is H modulo m2 and modulo all non-trivial products. Explicitly we have

DrIm(a0; a1, . . . , an; an+1) =

=

nr

Xifa1 . . . fai fai+1 . . . fan . (A.1)

By analogy with the structure of H above we dene a trivial comodule over U by

U = Q[f2] U , (A.2)

where f2 is of degree 2. We can think of f2 as a letter which commutes with all of the f2r+1. For convenience we will introduce the symbols f2n in analogy with (3.5) for the even zeta values,

f2n = bnfn2 . (A.3)

JHEP08(2013)135

Xp=0

Im(a0; a1, . . . , ap, ap+r+1, . . . , an; an+1) IL(ap; ap+1, . . . , ap+r; ap+r+1) . (A.4)

This is the coproduct , restricted so that the right-hand factor is taken modulo non-trivial products.

Now we construct the map B by assuming that our basis contains m2 and all the m2n+1 and their products and imposing

B(mn) = fn, n = 2, 3, 5, 7, 9, . . . (A.5)

Furthermore we impose that it is a homomorphism for the shu e product on U, i.e.

B(xy) = B(x) B(y) . (A.6)

Then for all non-trivial elements in the basis, i.e. the multi-zetas of higher depth, we impose recursively that

B(x) =

Xr(B 2r+1 B)D2r+1(x) , (A.7)

where 2r+1 is the projection in U2r+1 onto the word f2r+1 of length 1 and is simply concatenation (not the shu e) of words in U, treating f2 as commutative.

Let us look at weight 8 as an example. A basis of words in U at weight 8 is given by {f42, f2f3 f3, f3 f5, f3f5}. The rst three elements are the result of applying B to the products 42, 223, 35. To obtain the remaining word at weight 8 we can, for example, include 3,5 in the basis B. Applying the operators D3 and D5 to 3,5 one nds

D33,5 = 0 , D53,5 = 53 5 (A.8)

15

and hence B(3,5) = 5f3f5. Having xed the basis at a given weight, one may express all multi-zetas of that weight in terms of the basis. To decompose a particular multi-zeta value one applies the same recursive algorithm to obtain its image under B, except that now one allows an arbitrary amount of the unique primitive element fn of the given weight in the result. For example one nds

B(5,3) = 6f3f5 + f5f3 + af8 . (A.9)

Thus we conclude that5,3 = 3,5 + 35 + a8 . (A.10)

By numerical evaluation (or in this case simply application of the stu e relation (p)(q) = (p, q) + (q, p) + (p + q)) one concludes that a = 1 . Had we chosen instead a di erent basis B where 5,3 was included as a basis element we would have found

B (5,3) = 6f3f5 + f5f3, B (3,5) = 5f3f5 + f8 , (A.11)

Thus B and B dene di erent isomorphisms between H and U. Note that the rst time one has an ambiguity involving an odd primitive zeta value is at weight 11 where di erent choices of the depth 3 element to be included in the basis result in di erent coe cients of f11 in the application of B to a given multi-zeta value. Many more examples on the denition and application of B are given in [11, 13].

B Associator from the KZ equation

Here we provide a derivation of the form of the associator used in section 6. The associator can be obtained as a regularised limit as z 1 of the solution of the Knizhnik-Zamolodchikov equation,ddz L(z) = L(z)

given by a formal sum over all harmonic polylogarithms,

L(z) =

XwwH(; z) , (B.2)

where is the word w (with e0 treated as 0 and e1 as 1) reversed. This reversal is needed simply because of a di erence of ordering conventions between [11, 12] and [28]. The solution L(z) is the unique one obeying the boundary condition,

L(z) ze0 as z 0 . (B.3)

One can likewise dene the unique solution obeying

L1(z) (1 z)e1 as z 1 . (B.4)

The Drinfeld associator can be identied with the connection relating the two solutions,

L1(z) = L(z) . (B.5)

16

JHEP08(2013)135

e0z +e11 z

, (B.1)

From the fact that both L and L1 are solutions of the KZ equation and are invertible one concludes that above is a constant series.

The solution L(z) is divergent at z = 0 and z = 1 as can be seen by expanding,

L(z) = 1 + e0H0(z) + e1H1(z) + . . . = 1 + e0 log z e1 log(1 z) + . . . . (B.6)

We regularise to obtain a quantity nite at these points,

L(z) = ze0L(z)(1 z)e1 . (B.7)

The regularised solution obeys the following equation

d dz

L(z) = L(z)(1 z)e1 e0z (1 z)e1

JHEP08(2013)135

L(z) . (B.8)

We can rewrite the rst term on the r.h.s. in terms of the adjoint action of e1 on e0 leading to

z d

dz

e0 z

L(z) = L(z)(1 z)ad1(e0) e0 L(z) , (B.9)

where adi(x) = [ei, x].

Now since L(z) is group-like,

eL(z) = L(z) L(z) , (B.10)

and L(z) di ers from L(z) only by multiplication of group-like elements (1 z)e1 and

ze0 then L(z) is also group-like. This means that it is the exponential of a Lie series in g = Lie[e0, e1]. In fact the regularised solution L(z) actually contains no words of length 1 (they were removed by the regularisation) so we actually know that it is the exponential of a Lie series in g = [g, g],

L(z) = expL(z), L(z) g . (B.11)

In the rst instance we are looking for a representation of the Ihara action which is given by (1 1) matrices of multiplicative operators. To this end we will simplify the problem and work modulo any products in g so that we can actually write

L(z) = 1 + L(z) modulo products. (B.12)

This approximation has the result that the words wn corresponding to the coe cients of the primitive elements n survive (they are elements of g) but that any Ihara brackets are killed since they are all actually elements of g = [g, g]. Thus we already know that no

multiple zeta values will survive in this approximation.

The di erential equation (B.9) above is not quite suitable for simplication since the unit term in L(z) means that the whole of the factor (1z)ad1(e0) contributes, even modulo products. However we can improve the situation by taking a second derivative,

[parenleftbigg]

ddz +z

d2 dz2

L(z) = ddzL(z)(1z)ad1(e0)+ L(z) ad11 z (1z)ad1(e0)e0ddzL(z) . (B.13)

17

In the rst term on the r.h.s. we can now replace (1 z)ad1(e0) by e0 if we work modulo

products in g. This term then combines neatly with the last term to ad0 L(z). In the second term on the r.h.s. we can pull out a total adjoint action ad1 and subtract the action on L(z) and then use the rst-order equation (B.9) to rewrite the result in terms of the derivative L(z),

L(z)ad1(1 z)ad1(e0) = ad1[L(z)(1 z)ad1(e0)] ad1 L(z)(1 z)ad1(e0) , (B.14)

= ad1

z ddzL(z) + e0 L(z)

ad1 L(z)(1 z)ad1(e0) . (B.15)

As above, we can replace the factor (1 z)ad1(e0) by e0 and combine the nal two terms into an adjoint action of e0 on ad1 L(z) plus a term [e1, e0]. Finally combining everything we have

(1 z)z d2

dz2

L(z) +

(1 z)(1 + ad0) zad1[bracketrightbig]ddzL(z) ad0ad1 L(z) + [e0, e1] = 0 . (B.16)

Now, working modulo products in g means replace the non-commuting variable L(z) = L(z) 1 with a function of two commuting variables ad0 = u and ad1 = v,

L(z) = L(u, v, ; z)[e0, e1] (B.17)

We can represent a general word in U(g) modulo (g)2 as a function of u and v as follows (see also the discussion around eq. (6.1)),

w = c0 +

Xckladk0adl1[e0, e1] c0 +

Xckluk+1vl+1 . (B.18)

In this representation the second order equation above becomes the hypergeometric equation with a constant inhomogenous term,

(1 z)z d2

dz2 L(z; u, v) +

JHEP08(2013)135

(1 z)(1 u) zv[bracketrightbig]ddz L(z; u, v) + uvL(z; u, v) + 1 = 0 . (B.19)

The solution obeying the relevant boundary conditions is

L(z; u, v) = 2F1(u, v, 1 u; z) 1

uv . (B.20)

The logarithm of the associator is this solution evaluated at z = 1 (applied to [e0, e1])

log = 1

uv

(1 u) (1 v) (1 u v) 1

[e0, e1] mod g. (B.21)

Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

18

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