Single-Valued Integration and Superstring Amplitudes in Genus Zero

We study open and closed string amplitudes at tree-level in string perturbation theory using the methods of single-valued integration which were developed in the prequel to this paper (Brown and Dupont in Single-valued integration and double copy, 2020). Using dihedral coordinates on the moduli spaces of curves of genus zero with marked points, we define a canonical regularisation of both open and closed string perturbation amplitudes at tree level, and deduce that they admit a Laurent expansion in Mandelstam variables whose coefficients are multiple zeta values (resp. single-valued multiple zeta values). Furthermore, we prove the existence of a motivic Laurent expansion whose image under the period map is the open string expansion, and whose image under the single-valued period map is the closed string expansion. This proves the recent conjecture of Stieberger that closed string amplitudes are the single-valued projections of (motivic lifts of) open string amplitudes. Finally, applying a variant of the single-valued formalism for cohomology with coefficients yields the KLT formula expressing closed string amplitudes as quadratic expressions in open string amplitudes.


Introduction
1.1. The beta function. As motivation for our results, it is instructive to consider the special case of the Euler beta function (Veneziano amplitude [Ven68]) The integral converges for Re(s) > 0, Re(t) > 0. Less familiar is the complex beta function (Virasoro-Shapiro amplitude [Vir69,Sha70]), given by The integral converges in the region Re(s) > 0, Re(t) > 0, Re(s + t) < 1. The beta function admits the following Laurent expansion β(s, t) = 1 s and the complex beta function has a very similar expansion It is important to note that these Laurent expansions are taken at the point (s, t) = (0, 0) which lies outside the domain of convergence of the respective integrals. The coefficients in (4) can be expressed as 'single-valued' zeta values which satisfy: ζ sv (2n) = 0 and ζ sv (2n + 1) = 2 ζ(2n + 1) for n ≥ 1. The Laurent expansion (4) can thus be viewed as a 'single-valued' version of (3). To make this precise, we define a motivic beta function β m (s, t) which is a formal Laurent expansion in motivic zeta values: whose coefficients ζ m (n) are motivic periods of the cohomology of the moduli spaces of curves M 0,n+3 relative to certain boundary divisors. It has a de Rham version β m,dR (s, t), obtained from it by applying the de Rham projection term by term. One has β(s, t) = per (β m (s, t)) and β C (s, t) = s (β m,dR (s, t)) where s is the single-valued period map which is defined on de Rham motivic periods. We can therefore conclude that the Laurent expansions of β(s, t) and β C (s, t) are deduced from a single object, namely, the motivic beta function (5). The first objective of this paper is to generalise all of the above for general string perturbation amplitudes at tree-level.
Cohomology with coefficients There is another sense in which (1) is a single-valued version of (2) that does not involve expanding in s, t and uses cohomology with coefficients.
For generic values of s, t (i.e., s, t, s + t / ∈ Z), it is known how to interpret β(s, t) as a period of a canonical pairing between algebraic de Rham cohomology and locally finite Betti (singular) homology: and where X = P 1 \{0, 1, ∞}, ∇ s,t is the integrable connection on the rank one algebraic vector bundle O X , and L −s,−t is the rank one local system generated by x s (1 − x) t , which is a flat section of ∇ −s,−t = ∇ ∨ s,t (see Example 6.13).
An important feature of this situation is Poincaré duality which gives rise to de Rham and Betti pairings between (6) for (s, t) and for (−s, −t). Compatibility between these pairings amounts to the following functional equation for the beta function: sin(π s) sin(π t) sin(π(s + t)) , where the factor in brackets on the left-hand side is the de Rham pairing of dx x(1−x) with itself and the factor in brackets on the right-hand side is the inverse of the Betti pairing of (0, 1) As in the case of relative cohomology with constant coefficients studied in [BD20], there exists a single-valued formalism for cohomology with coefficients in this setting for which we give an integral formula (Theorem 7.8). This formula implies that β C (s, t) is a single-valued period of (6), which amounts to the equality and proves the second equality in (2). Applying the functional equation (7) we then get the following 'double copy formula' expressing a single-valued period as a quadratic expression in ordinary periods: sin(π(s + t)) β(s, t) 2 .
In conclusion, there are three different ways to deduce the complex beta function from the classical beta function: via (8) or the double copy formula (9), or by applying the single valued period map term by term in its Laurent expansion.
where ω is a meromorphic form with certain logarithmic singularities (see Sect. 3.2), and s = {s i j } are Mandelstam variables satisfying momentum conservation equations (30).
It turns out that one can write the closed string amplitudes in the form Later we shall rewrite the domain of integration as the complex points of the compactified moduli space of curves of genus 0 with N ordered marked points. Then, the form (t i+1 − t i ) −1 dt 1 ∧ · · · ∧ dt N −3 (t 0 = 0, t N −2 = 1) is logarithmic and has poles along the boundary of the domain of integration of the open string amplitude. It is in fact the image of the homology class of this domain under the map c ∨ 0 defined in [BD20]. The first task is to interpret the open and closed string amplitudes rigourously as integrals over the moduli space of curves M 0,N . An immediate problem is that the poles of the integrand lie along divisors which do not cross normally. Using a cohomological interpretation of the momentum conservation equations in Sect. 3.1, we show how to resolve the singularities of the integral by rewriting it in terms of dihedral coordinates. These are certain cross-ratios u c in the t i , indexed by chords c in an N -gon, whose zero loci form a normal crossing divisor. Thus, for example, we write in Sect. 3.3: are integrals over a domain X δ of a global regular form with logarithmic singularities. We can therefore interpret the previous integrals as motivic periods of universal moduli space motives, and hence define a motivic version of the string amplitude. The first statement has been used implicitly in [SS13,SS19] by assuming the period conjecture for multiple zeta values. The fact that the Laurent coefficients are multiple zeta values is folklore. A subtlety in the previous theorem is that the motivic lift I m (ω, s) is a priori not unique, as there are many possible ways to express the logarithms log(u c ) as integrals. We believe that one could fix these choices if one wished. In any case, the period conjecture suggests that the motivic amplitude I m (ω, s) is independent of these choices.
By applying the general theorems on single-valued integration proved in the prequel to this paper [BD20] we deduce that the closed string amplitude is the single-valued version of the motivic amplitude. Theorem 1.3. Let π m,dR denote the de Rham projection map from effective mixed Tate motivic periods to de Rham motivic periods (which maps ζ m to ζ m,dR ), and s the singlevalued period map (which maps ζ m,dR to ζ sv ). Then I closed (ω, s) = s π m,dR I m (ω, s).

It follows that the coefficients in the canonical Laurent expansion of the closed string amplitudes are single-valued multiple zeta values.
This theorem, for periods (i.e., assuming the period conjecture) was conjectured in [Sti14,ST14] and proved independently by a very different method from our own in [SS19]. Since the first draft of this paper was written, yet another approach to computing the closed string amplitudes appeared in [VZ18]. An interesting consequence of Theorem 1.3 is that it suggests that the space generated by closed string amplitudes might be closed under the action of the de Rham motivic Galois group. It is important to note that the proof of the previous theorem, in contrast to the approach sketched in [SS19], uses no prior knowledge of multiple zeta values or polylogarithms, and merely involves an application of our general results on single-valued integrals.
String amplitudes from the point of view of cohomology with coefficients and double copy formulae In the final parts of this paper Sect. 6, 7, we consider the open string amplitude as a period of the canonical pairing between algebraic de Rham cohomology with coefficients in a certain universal (Koba-Nielsen) algebraic vector bundle with connection, and locally finite homology with coefficients in its dual local system: As in the case of the beta function, Poincaré duality exchanges s and −s and leads to quadratic functional equations for open string amplitudes generalising (7). It is important to note that this interpretation of the open string amplitude, as a function of generic Mandelstam variables, is quite different from its interpretation as a Laurent series. After defining the single-valued period map, our main theorem (Theorem 7.8) provides an interpretation of the closed string amplitudes I closed (ω, s) as its singlevalued periods. Theorem 7.8 is in no way logically equivalent to the previous results since it is not obvious that the two notions of 'single-valuedness', namely as a function of the s i j , or term-by-term in their Laurent expansion, coincide. The paper [BD19] provides yet another connection between these two different cohomological points of view.
As a consequence of Theorem 7.8, we immediately deduce an identity relating closed and open string amplitudes which involves the period matrix, its inverse, and the de Rham pairing. By the compatibility between the de Rham and Betti pairings, it in turn implies a 'double copy formula' which generalizes (9). It expresses closed string amplitudes as quadratic expressions in open string amplitudes but this time using the Betti intersection pairing (Corollary 7.10). Since Mizera has recently shown [Miz17] that the inverse transpose matrix of Betti intersection numbers coincides with the matrix of KLT coefficients, our formula implies the KLT relations.
Because our results for genus zero string amplitudes are in fact instances of a more general mathematical theory [BD20], valid for all algebraic varieties, we expect that many of these results may carry through in some form to higher genera. It remains to be seen, in the light of [Wit12], if this has a chance of leading to a possible double copy formalism for higher genus string amplitudes.

Contents.
In §2 we review the geometry of the moduli spaces M 0,N , dihedral coordinates, and the forgetful maps which play a key role in the regularisation of singularities. In §3 we recall the definitions of tree-level string amplitudes, their interpretation as moduli space integrals, and discuss their convergence. Section 4 defines the 'renormalisation' of string amplitudes via the subtraction of counter-terms which uses the natural maps between moduli spaces. It uses in an essential way the fact that the zeros of dihedral coordinates are normal crossing. Lastly, in Sect. 5 we construct the motivic amplitude and prove the main theorems using [BD20]. The final sections Sect. 6, 7 treat cohomology with coefficients as discussed above. In an appendix, we prove a folklore result that the Parke-Taylor forms are a basis of cohomology with coefficients.

Dihedral Coordinates and Geometry of M 0,S
Let n ≥ 0 and let S be a set with n + 3 elements, which we frequently identify with {1, . . . , n + 3}. Let M 0,S denote the moduli space of curves of genus zero with marked points labelled by S. It is a smooth scheme over Z whose points correspond to sets of n + 3 distinct points p s ∈ P 1 , for s ∈ S, modulo the action of PGL 2 . Since this action is simply triply transitive, we can place p n+1 = 1, p n+2 = ∞, p n+3 = 0 and define the simplicial coordinates (t 1 , . . . , t n ) to be the remaining n points. In other words, they are defined for 1 ≤ i ≤ n as the cross-ratios Note that the indexing differs slightly from that in [Bro09]. These coordinates identify M 0,S as the hyperplane complement (P 1 \{0, 1, ∞}) n minus diagonals, and are widespread in the physics literature. We also use cubical coordinates: 2.1. Dihedral extensions of moduli spaces. A dihedral structure δ for S is an identification of S with the edges of an (n + 3)-gon (which we call (S, δ), or simply S when δ is fixed) modulo dihedral symmetries. When we identify S with {1, . . . , n + 3} we take δ to be the 'standard' dihedral structure that is compatible with the linear order on S. Let χ S,δ denote the set of chords of (S, δ). The dihedral extension M δ 0,S of M 0,S is a smooth affine scheme over Z of dimension n defined in [Bro09]. Its affine ring O(M δ 0,S ) is the ring over Z generated by 'dihedral coordinates' u c , for each chord c ∈ χ S,δ , modulo the ideal generated by the relations

Morphisms.
Given a subset T ⊂ S with |T | ≥ 3, let δ| T denote the dihedral structure on T induced by δ. There is a partially defined map f T : χ S,δ → χ T,δ |T induced by contracting all edges in (S, δ) not in T . Since some chords map to the outer edges of the polygon (T, δ| T ) under this operation, it is only defined on the complementary set of such chords in χ S,δ . It gives rise to a 'forgetful map' where c ∈ χ T,δ| T , and c ranges over its preimages in χ S,δ . The forgetful map restricts to a morphism f T : M 0,S → M 0,T between the open moduli spaces.
where T c is the set of four edges which meet the endpoints of c.

Strata.
Cutting along c ∈ χ S,δ breaks the polygon S into two smaller polygons, (S , δ ) and (S , δ ), with S = (S \{c}) (S \{c}) (see e.g. [Bro09, Figure 3]). There is a canonical isomorphism In particular, the restriction of a dihedral coordinate u c to the divisor D c , where c and c do not cross, is the dihedral coordinate u c on either (S , δ ) or (S , δ ), depending on which component c lies in.

Trivialisation maps.
A crucial ingredient in our 'renormalisation' of differential forms is to use dihedral coordinates to define a canonical trivialisation of the normal bundles of the divisors D c in a compatible manner. In order to define this, we shall fix a cyclic order γ on S, which is compatible with δ. Such a cyclic structure is simply a choice of orientation of the polygon (S, δ).

Definition 2.2.
Let c be a chord as above. Let T , T be the subsets of S consisting of the edges in S \{c}, S \{c}, respectively, together with the next edge in S with respect to the cyclic ordering γ . Let δ , δ be the induced dihedral structures on T , T . There are natural bijections S T and S T where in each case we identify the chord c with the next edge after S or S in the cyclic ordering. Consider the map T c is identified with A 1 . An illustration of this map is given in Fig. 2.
The first component f T c is simply the dihedral coordinate u c , and hence the restriction of (16) to D c induces the isomorphism (15). Note that (15) is canonical, but f γ c depends on the choice of cyclic structure γ .
Proof. Cutting along c 1 , c 2 decomposes the oriented polygon (S, γ ) into three smaller polygons (S 1 , γ 1 ), (S 12 , γ 12 ), (S 2 , γ 2 ), where S 1 has one edge labelled by c 1 , S 2 has one edge labelled by c 2 , and S 12 two edges labelled c 1 , c 2 . The graphs S 1 ∩ S and S 2 ∩ S each have one connected component and S 12 ∩ S has exactly two components (one of which may reduce to a single vertex). Extending each such component by the next edge in the cyclic order defines sets T 1 , T 12 , T 2 ⊂ S where |T 1 | = |S 1 | + 1, |T 2 | = |S 2 | + 1, and |T 12 | = |S 12 | + 2. One checks from the definitions that which is symmetric in c 1 , c 2 . The point is that the operation of 'extending by adding the next edge in the cyclic order' does not depend on the order in which one cuts along the chords c 1 , c 2 . When the cyclic ordering is fixed, we shall drop the γ from the notation.
2.5. Domains. Let X δ ⊂ M δ 0,S (R) be the subset defined by the positivity of the dihedral coordinates u c > 0, for all c ∈ χ S,δ . In simplicial coordinates (10) it is the open simplex {0 < t 1 < · · · < t n < 1}. In cubical coordinates (11) it is the open hypercube (0, 1) n . It serves as a domain of integration. On the domain X δ , every dihedral coordinate u c takes values in (0, 1) by (13). Given a cyclic ordering γ on S, (16) defines a homeomorphism More generally, for any set J of k non-crossing chords (Definition 2.1) set It follows by iterating the above that The closure X δ ⊂ M δ 0,S (R) for the analytic topology is a compact manifold with corners which has the structure of an associahedron. Note that the maps f γ J do not extend to homeomorphisms of the closed polytopes X δ .
2.6. Logarithmic differential forms. We define • S to be the graded Q-subalgebra of regular forms on M 0,S generated by the d log u c , c ∈ χ S,δ . These are functorial with respect to forgetful maps, i.e.
which follows from (14). One knows that all algebraic relations between the forms d log u c are generated by quadratic relations and furthermore, by Arnol'd-Brieskorn, that M 0,S is formal, i.e., the natural map is an isomorphism of Q-algebras. Consequently, one has [Bro09, §6.1] Finally, it follows from mixed Hodge theory [Del71] (see, e.g., [BD20,§4]), that Proof. This is simply the functoriality of the residue. It can also be checked explicitly using (14) and (21) which implies that where x ranges over chords in the preimage of c not equal to c. Thus the statement reduces to the equation , which is clear. For a form ω which does not have a pole along D c , we have Res D c ω = 0. One checks using (14) and the fact that f T (c) = c that Res D c f * T (ω) = 0. In the opposite direction, a cyclic structure γ defines maps . Lemma 2.6. We have and ) have no poles along D c , which proves the second equality.

Summary of structures.
With a view to generalisations we briefly list the geometric ingredients in our renormalisation procedure. We have a simple normal crossing divisor D ⊂ M δ 0,S whose induced stratification defines an operad structure (15). More precisely, this is a dihedral operad in the sense of [DV17]. We have spaces of global regular logarithmic forms equipped with • (Trivialisations, depending on a choice of cyclic structure on S) satisfying a certain number of compatibilities. In this paper, we also use the property u c | D c = 1 if D c and D c do not intersect, but we plan to return to the renormalisation of integrals in a more general context with a leaner set of axioms. Here, and in the next example, a subscript i j denotes the chord meeting the edges labelled {s i , s i+1 } and {s j , s j+1 } [Bro09, Figure 2]. The scheme M 0,4 is isomorphic, via the coordinate x, to P 1 \{0, 1, ∞} and its dihedral extension is M δ 0,4 = Spec Z[v 24 , v 13 ]/(v 24 + v 13 = 1) ∼ = A 1 . The domain X δ ⊂ R\{0, 1} is the open interval (0, 1). Its closure Let |S| = 5, and set S = {s 1 , s 2 , s 3 , s 4 , s 5 } with the natural dihedral structure δ. The five chords in the pentagon (S, δ) give rise to five dihedral coordinates which satisfy equations given in [Bro09,§2.2]. These equations define the affine scheme M δ 0,5 . The pair x = u 24 , y = u 25 are cubical coordinates (11), and embed (x, y) : M 0,5 −→ M 0,4 × M 0,4 ∼ = P 1 \{0, 1, ∞} × P 1 \{0, 1, ∞} Its image is the complement of the hyperbola x y = 1. We can write all other dihedral coordinates using (12) in terms of these two to give: The domain X δ maps to the open unit square {(x, y) : 0 < x, y < 1}. The first coordinate, x, is the forgetful map which forgets the edge s 5 : The induced map on affine rings satisfies π * (v 24 ) = u 24 , π * (v 13 ) = u 13 u 35 .

de Rham projection.
We now fix a dihedral structure δ on S and write S for (S, δ).
There is a volume form α S,δ on M δ 0,S which is canonical up to a sign [Bro09, §7.1]. A cyclic structure on S defines an orientation on the cell X δ and fixes the sign of α S,δ if we demand that its integral over X δ be positive. In simplicial coordinates it is given by [Bro09,(7.1)]: with the convention t 0 = 0, t n+1 = 1.
Definition 2.7. Writing u S,δ = c∈χ S,δ u c we define Note that the sign is the same as in [BD20, §4.5]. When the dihedral structure is clear from the context, we write ν S for ν S,δ .
Lemma 2.8. The form ν S,δ defines a meromorphic form on M 0,S with logarithmic singularities, and has simple poles only along those divisors which bound the cell X δ . In simplicial coordinates (10), and using the convention t 0 = 0, t n+1 = 1, Proof. Using the equation 1 − u c = c ∈A u c , where A is the set of chords which cross c , which as an instance of (12), we deduce that Using the definition of dihedral coordinates as cross-ratios [Bro09, (2.6) and §2.1], Using the fact that u S,δ is positive on X δ one obtains After passing to cubical coordinates one obtains the more symmetric expression The following proposition follows from the computations in [BD20,§4].
Working in cubical coordinates and using (28) we get the following compatibility between the ν S,δ and the maps f The sign is compatible with the single-valued Fubini theorem discussed in [BD20, Let J = { j 1 , . . . , j k } be a set of k non-crossing chords. With the notation of Definition 2.1 we may define We have the compatibility with a sign that is compatible with the single-valued Fubini theorem.

String Amplitudes in Genus 0
We give a self-contained account of open and closed string amplitudes in genus 0, recast them in terms of dihedral coordinates, and discuss their convergence. The results in this section are standard in the physics literature, which is very extensive. The seminal references are [GSW12], [KLT86]. More recent work, including [SS13], [Sti14], [ST14], [BSST14], served as the main inspiration for the results below.

Momentum conservation.
Let N = n + 3 ≥ 3 and let s i j ∈ C for 1 ≤ i, j ≤ N satisfying s i j = s ji and s ii = 0. Let (x i : y i ) denote homogeneous coordinates on P 1 for 1 ≤ i ≤ N . Consider the functions The former is multi-valued, the latter is single-valued.
In this case, they are automatically PGL 2 -invariant and define (multi-valued, in the case of f s ) functions on the moduli space M 0,N (C).
Proof. The functions f s , g s are invariant under scalar transformations (30) holds. The first part of the statement follows. For the second, observe that GL 2 acts by left matrix multiplication on Since each term x j y i − x i y j is minus the determinant of the matrix formed from the columns i, j, GL 2 acts via scalar multiplication. We have already established that scalar invariance is equivalent to (30), and hence proves the second part. The last part follows since the moduli space M 0,N is the quotient of the configuration space of N distinct points in P 1 modulo the action of PGL 2 .
We call (30), together with s i j = s ji and s ii = 0 the momentum conservation equations. The solutions to these equations form a vector space (scheme) V N .
When they hold, denote the above functions simply by where p i = x i /y i . The former has a canonical branch on the locus where the points p i are located on the circle P 1 (R) in the natural order, which corresponds to the domain X δ ⊂ M 0,N (R). When (30) holds, the differential 1-form defines a logarithmic 1-form on (M 0,N , ∂M 0,N ). We therefore obtain a linear map for any field K of characteristic zero.

Lemma 3.2. The map (32) is an isomorphism.
Proof. It is injective: if ω s were to vanish then its residue along p i = p j , viewed as a divisor in the configuration space of N distinct points on the projective line, would vanish. Hence all s i j = 0. Next observe that V N −1 ⊂ V N , and that V N /V N −1 is generated by s i N = s Ni , for 1 ≤ i ≤ N − 1, subject to the single relation (21), and so (32) is an isomorphism.

String amplitudes in simplicial coordinates.
It is customary in the physics literature to write the open and closed string amplitudes in simplicial coordinates (10). We use the coordinate system on V N consisting of the s i j for 1 ≤ i < j ≤ n along with the s i,n+1 and s i,n+3 for 1 ≤ i ≤ n. We use the notation s 0,i = s i,n+3 . Let |S| = N = n + 3 and let ω ∈ n S be a global logarithmic form. Let s i j ∈ C be a solution to the momentum conservation equations (30). The associated open string amplitude is formally written as the integral with the convention t 0 = 0, t n+1 = 1. In the literature (see Theorem 6.10 below), ω is typically of the form for some permutation σ of {0, . . . , n + 1}. Closed string amplitudes are written in the form where ν S was given in Definition 2.7. For ω of the form (34), we can rewrite (35) as with the notation d 2 z = dRe(z) ∧ dIm(z). Note that the apparently complicated sign in the definition of ν S is such that all signs cancel in the previous formula, in agreement with the conventions in the physics literature. Convergence of these integrals is discussed below. As we shall see, a huge amount is gained by first rewriting them in dihedral coordinates.

String amplitudes in dihedral coordinates.
Let S = (S, δ) be a set of cardinality N = n + 3 ≥ 3 and fix a dihedral structure. Suppose that s i j are solutions to the momentum-conservation equations. It follows from Lemma 3.2 and (21) that we can uniquely write where the s c are linear combinations of the s i j indexed by each chord in (S, δ). Thus the s c form a natural system of coordinates for the space V N . More precisely: Proof. See [Bro09, (6.14) and (6.17)].
The coordinates s c are better suited than the s i j for studying (33) and (35). By (36), we have on appropriate branches (e.g., on X δ ) the equation: Define the closed string amplitude, when it converges, to be: These definitions are equivalent to (33) and (35), respectively, after passing to simplicial coordinates. For the closed string case, one can change its domain using the fact that M 0,S (C) ⊃ M 0,S (C) ⊂ C n differ by sets of Lebesgue measure zero.
In the physics literature, one usually wants to expand string amplitudes in the Mandelstam variables s. However, the integrals (37) and (38) generally do not converge if s is close to zero, as the following propositions show.

Proposition 3.5. The integral I open (ω, s) of (37) converges absolutely for s c ∈ C satisfying
Proof. Let J ⊂ χ S be a set of non-crossing chords. The set J can be extended to a maximal set J ⊂ J ⊂ χ S of non-crossing chords. The u j for j ∈ J form a system of local coordinates on M δ 0,S [Bro09, §2.4]. For any ε > 0, consider the set The sets S J ε , for varying J , cover X δ for sufficiently small ε. This follows because the latter is defined by the domain u c ≥ 0 for all c ∈ χ S . Since u c and u c can only vanish simultaneously if c and c do not cross by (13), it follows that This implies the covering property by compactness of X δ . It suffices to show that the integrand is absolutely convergent on each S J ε . In the local coordinates u j , the normal crossing property means that we can write the integrand of (37) as where p c = −ord D c ω is the order of the pole of ω along u c = 0, and ω 0 has no poles on S J ε . Since x α dx is integrable on [0, ε) for Re α > −1, the condition Re (s c − p c ) > −1 for all c ∈ J guarantees absolute convergence over S J ε . Note that the region of convergence does not permit a Taylor expansion at s c = 0. Proposition 3.6. Let N = |S|. The integral I closed (ω, s) of (38) converges absolutely for s c ∈ C satisfying Proof. Let denote the integrand of (38). Let D ⊂ M 0,S be an irreducible boundary divisor. Supose first that D is a component of ∂M δ 0,S and is therefore defined by u c = 0 for some c ∈ χ S . By Lemma 2.8, ν S has a simple pole along D. In the local coordinate z = u c , has at worst poles of the form: In polar coordinates z = ρ e iθ , the left-hand term is proportional to ρ 2s c −1 dρ dθ and hence integrable for Re(s c ) > 0, the right-hand term to ρ 2s c dρ dθ and hence integrable for Re(s c ) > −1/2. Now consider a boundary divisor D which is a component of M 0,S \M 0,S but which is not a component of ∂M δ 0,S (at 'infinite distance'). It is defined by a local coordinate z = 0 (which is a dihedral coordinate with respect to some other dihedral structure on S). By Lemma 2.8, ν S has no pole along z = 0. Since ω has logarithmic singularities, is locally at worst of the form where p is a linear form in the s c . Since any cross-ratio u c has at most a simple zero or pole along D, it follows that p = c∈χ S a c s c where a c ∈ {0, ±1} (an explicit formula for p in terms of s c is given in [Bro09, §7.3]). By passing to polar coordinates one sees that the integrability condition reads 2 Re( p) > −1. Assuming (39) one gets the inequality and we are done.
Put differently, for any s c satisfying the assumptions (39), the integrand of (38) is polar-smooth on (M 0,S , ∂M 0,S ) in the sense of Definition 3.7 of [BD20].
This is the classical beta function β(s, t), which converges for Re(s) > 0, Re(t) > 0. For the closed string amplitude we get . This is the complex beta function β C (s, t), which converges for Re(s) > 0, Re(t) > 0, Re(s + t) < 1.

Formal moduli space integrands.
Let us fix S = (S, δ) as above. We shall interpret the integrands of string amplitudes as formal symbols in dihedral coordinates, with a view to either taking a Taylor expansion in the variables s c , or specialising to complex numbers in the case when the integrals are convergent. This will furthermore enable us to treat the open and closed string integrands simultaneously. To this end, consider a fixed commutative monoid (M, +) which is free with finitely many generators. The main example will be M S = c∈χ S N s c , the monoid of non-negative integer linear combinations of the symbols s c . Similarly, if c is a chord, let F c (M) be the Q-algebra generated by u m c for m ∈ M modulo the above relations. Let us write We write the elements of A S (M) without the tensor product, as linear combinations Definition 4.2. Let J ⊂ χ S be any subset of non-crossing chords as in Definition 2.1.   (19) we get a morphism We can realise the formal moduli space integrands as differential forms as follows.
Definition 4.5. Given an additive map α : It is single-valued since u α c = exp(α log(u c )) and log(u c ) has a canonical branch on X δ , which is the region 0 < u c < 1. In a similar manner, we can define

Infinitesimal behaviour.
We define a kind of residue of formal differential forms along boundary divisors which encodes the infinitesimal behaviour of functions in the neighbourhood of the divisor. We first define the evaluation map as the morphism sending a formal symbol u c to 1 if c crosses c, and all other symbols to identically named symbols.
Definition 4.6. For any c ∈ χ S we define the map to be the tensor product of the evaluation map ev c and the map of logarithmic differential Proof. The commutativity for the evaluation maps is clear.
be a subset of non-crossing chords as in Definition 2.1. By the previous lemma one can compute the iterated residue R J = R j 1 • · · · • R j k in any order, which provides a linear map

Trivialisation maps.
Fix a cyclic ordering γ on S which is compatible with δ. Using the morphisms (40) define for each chord c ∈ χ S a trivialisation map Proof. Use the notations of lemma 2.3. We wish to show the following diagram commutes, where the horizontal maps are induced by forgetful morphisms: The commutativity of this diagram on the level of formal symbols is clear, and the commutativity on the level of logarithmic forms is a consequence of Lemma 2.5.
Note that (44) has to be understood via the Koszul sign rule.
We can simply write it in the unambiguous form R c 2 • f * c 1 = f * c 1 • R c 2 since the source and target of a map f * c or R c is uniquely determined by the data of c. The following lemma will not be needed in our renormalisation procedure, but will play a role in the analysis of the convergence of string amplitudes.

Lemma 4.12. For
∈ A S (M) and a chord c ∈ χ S , the difference − f * c R c is a linear combination of elements: , for some chord c crossing c and some m ∈ M.
Proof. This is a consequence of Lemma 4.8. for some m in the abelian group generated by M, by the same argument as in the proof of Proposition 3.6. This is integrable around z = 0 for Re α( m) > − 1 2 . Since there are finitely many such divisors, the latter condition is implied by the hypotheses (2) for sufficiently small ε.

Integrability and residues.
Remark 4.14. In the case of closed string amplitudes, an integrand ρ closed α ( ) satisfying the assumptions of Proposition 4.13 (2) is polar-smooth on (M 0,S , ∂M 0,S ) in the sense of Definition 3.7 of [BD20].

Definition 4.15. Define a renormalisation map
where J ranges over all sets of non-crossing chords in χ S .
The reason for calling this map the renormalisation map, even though it does not agree with the notion of renormalisation in the strict physical sense, is that it is mathematically very close to the renormalisation procedure given in [BK13].
Each summand is of the form By the first part of Lemma 4.10, By the commutation relation (44), this is R c f * K R K , and therefore the previous expression vanishes. We extend the renormalisation map to tensor products of forms by defining it be the identity on every A c (M). For |J | = k it acts upon via id ⊗k ⊗ ren ⊗k+1 , and is denoted also by ren.
where the sum is over non-crossing sets of chords in χ S .
Proof. We prove formula (46) by induction on |S|. Suppose it is true for all sets S with < N elements, and let |S| = N . Then applying the formula (46) to each component of R K , for K = ∅, we obtain Now, substituting into the definition of ren , we obtain Via the binomial formula, Rearranging gives (46) and completes the induction step. The initial case with |S| = 3 is trivial, since = ren .

Laurent expansion of open string integrals. Let
where Res D J denotes the iterated residue along irreducible components of D J and χ J denotes the set of chords in χ S \J which do not cross any element of J . Let The integral of over X δ can be canonically renormalised as follows.
Theorem 4.20. For all s c ∈ C satisfying the assumptions of Proposition 4.13 (1), where the sum in the right-hand side is over all subsets of non-crossing chords (including the empty set). The integrals on the right-hand side converge for Proof. For any subset J of non-crossing chords, we have By similar arguments to those in the proof of theorem 4.20, we have Proposition 4.17 is stated for a form ∈ A S (M) but holds more generally for a tensor products of forms in A S (M) ⊗ A S (M), where S , S are the polygons obtained by cutting S along c. This is because the maps f * , R and ren are all compatible with tensor products. Therefore writing R c = ω ⊗ω (Sweedler's notation) with ω ∈ A S (M), We therefore deduce that where the last equality follows from the same arguments as in the proof of theorem 4.20.
We have therefore shown that both sides of (51) coincide for all values of s c such that the integrals converge. Note that since the left-hand side admits a Laurent expansion, the same is true of the right-hand side.
Proof. Since d|z| 2s = s|z| 2s dz z + dz z , the integrand equals For ε > 0 small enough, let U ε be the open subset of P 1 (C) given by the complement of three open discs of radius ε around 0, 1, ∞ (in the local coordinates z, 1 − z, z −1 ). By Stokes' theorem, where the boundary ∂U is a union of three negatively oriented circles around 0, 1, ∞. By using Cauchy's theorem, we see that all integrals in the right-hand side are bounded as ε → 0, and that the only one which is non-vanishing in the limit as ε → 0 is around the point 1, giving Let be as in (47). We set The closed string amplitudes can be canonically renormalised as follows.
where cutting along c decomposes S into S , S . The proof is similar to the proof of (51).

Motivic String Perturbation Amplitudes
Having performed a Laurent expansion of string amplitudes, we now turn to their interpretation as periods of mixed Tate motives. where I ranges over all subsets of χ , and

Decomposition of convergent forms. Let
Proof. First observe that the case where χ = {c} is a single chord follows from Lemma 4.12, since R c = 0 implies that = − f * c R c , and therefore Although the sum is not direct, the decomposition into two parts can be made canonical.

Logarithmic expansions.
For each chord c, let c be a formal symbol which we think of as corresponding to a logarithm of u c .
There is a continuous homomorphism of algebras defined on generators by This extends to a map (resp. c → log |u c | 2 ).

Definition 5.2. A convergent monomial is one of the form
where for every c ∈ χ S such that Res D c ω = 0, there exists another chord c ∈ χ S which crosses c such that k c ≥ 1. for some α, β, γ . By applying this to every chord c for which Res D c ω = 0, we can rewrite the above integrals as linear combinations of where F, G have at most logarithmic singularties near boundary divisors, χ ⊂ χ S is a subset of chords, and ω has no poles along D c , for all c ∈ χ S . Convergence in both cases follows from a very small modification of propositions 3.5, 3.6 to allow for possible logarithmic divergences. The latter do not affect the convergence since | log(z)| k z s tends to zero as z → 0 for any Re s > 0.

Proposition 5.4. Let ∈ A S (M) such that R c = 0 for all chords c. Then admits a canonical expansion which only involves convergent monomials (57).
Proof. Apply (55) to each term in (I ) χ in the decomposition (54).

Corollary 5.5. Renormalised amplitudes, where they converge, can be canonically written as infinite sums of integrals of convergent monomials in logarithms:
where a K , a K lie in the Q-subalgebra of C generated by α(M). Each integral on the right-hand side converges.
Equivalently, if we treat the elements of M as formal variables then the open and closed string amplitudes admit expansions in C[[M]] whose coefficients are canonically expressible as Q-linear combinations of integrals of convergent monomials as above.
Example 5.6. We apply the above recipe to Example 4.18. By abuse of notation we identify s, t with their images under a realisation α : Ns ⊕ Nt → C, and write Note that log(x) vanishes at x = 1, and log(1 − x) at x = 0, so the integrals on the right-hand side are convergent. In the closed case we get s m m! t n n! P 1 (C) log m |z| 2 log n |1 − z| 2 d 2 z |z| 2 (1 − z) Again, the integrals on the right-hand side are convergent.

Removing a logarithm.
We can now replace each logarithm with an integral one by one. It suffices to do this once and for all for |S| = 5. The projection restricts to the real domains X 5 → X 4 whose fibers are identified with (0, 1) with respect to the coordinate y. Then In this manner we shall inductively replace all logarithms of dihedral coordinates with algebraic integrals. Note that it is not possible to express the logarithmic dihedral coordinate log v 24 = log x as an integral of another logarithmic dihedral coordinate over the fiber in y with respect to the same dihedral structure. It is precisely this subtlety that complicates the following arguments.
From now on, we fix a dihedral structure (S, δ), and consider a differential form of degree |S| − 3 of the following type: where ω 0 ∈ |S|−3 (M 0,S ). Suppose that it is convergent, i.e., for every chord c such that Res D c ω 0 = 0, there exists a c ∈ I which crosses c with n c ≥ 1. Define We can remove one logarithm at a time as follows.

Consider the diagram
It commutes since forgetful maps are functorial. Let β ∈ 1 (M 0,5 ) denote the form d log u 13 of example 5.7 whose integral in the fiber yields log(v 13 ) and set .
and an isomorphism f : X δ c ∼ = X δ × (0, 1). Since = f * ( ∧ β), we find by changing variables along the map f that The second integral takes place on the fiber product M 0,S × M 0,4 M 0,5 and is computed using (61). This proves equation (63).
We now check that is of the required shape (62) with respect to M δ c 0,S c and convergent. First of all, observe that for any forgetful morphism f : S → S and any chords a, b in S which cross, we have by (14) where a , b range over chords in S in the preimage of a and b respectively. Every pair a , b crosses. It remains to check the convergence condition along the poles of the 1-form β. For this, denote the two chords in S c lying above the chord c by c 1 , c 2 . The chord c 1 corresponds to edges {t 5 , t 1 ; t 3 , t 4 } and c 2 to {t 1 , t 2 ; t 3 , t 4 }. By (14), we have f * S (u c ) = u c 1 u c 2 , and by example 5.7 f * T β = du c 2 u c 2 , (β corresponds to d log u 13 in example 2.9). We therefore check that where c crosses c. In the sum, c ranges over the preimages of c under f S , and necessarily crosses both c 1 and c 2 . It follows that is a sum of convergent monomials in logarithms. The statement about the weights is clear.
Remark 5.9. Note that S c depends on the choice of where to insert the new edge we called t 5 . Similarly, the computation in example 5.7 also involves a choice: we could instead have used Thus there are two different ways in which we can remove each logarithm. One can presumably make these choices in a canonical way.
Corollary 5.10. Let be of the form (62) and convergent. Then the integral I of over X δ is an absolutely convergent integral where S ⊃ S is a set with dihedral structure δ compatible with δ, and ω ∈ S a logarithmic algebraic differential form with no poles along the boundary of M δ 0,S . Furthermore, |S | = weight( ) + 3.
Proof. Apply the previous lemma inductively to remove the logarithms log(u c ) one at a time. At each stage the total degree of the logarithms decreases by one. One obtains a Z-linear combination of convergent integrals of the form (64). Add the integrands together to obtain a single integral of the required form.

Motivic versions of open string amplitude coefficients.
Let MT (Z) denote the tannakian category of mixed Tate motives over Z with rational coefficients [DG05]. An object H ∈ MT (Z) has two underlying Q-vector spaces H dR (the de Rham realisation) and H B (the Betti realisation) together with a comparison isomorphism comp : Remark 5.12. Note that the motivic lift I m of I depends on some choices which go into Lemma 5.8. One expects, from the period conjecture, that it is independent of these choices. One can possibly make the lift canonical by fixing choices in the application of Lemma 5.8.
We deduce a number of consequences: Proof. Use the fact that the periods of universal moduli space motives are multiple zeta values [Bro09]. One can obtain the statement about the weights either by modifying the argument of loc. cit or as a corollary of the next theorem (using the fact that a real motivic period of weight n of an effective mixed Tate motive over Z is a Q-linear combination of motivic multiple zeta values of weight n).
Theorem 5.13 is well-known in this field using results scattered throughout the literature, but until now lacked a completely rigorous proof from start to finish.
Theorem 5.14. The above expansion admits a (non-canonical) motivic lift where ζ m n is a Q-linear combination of motivic multiple zeta values of weight N +|n|−3, whose period is ζ n .
Proof. Apply the Laurent expansions (58) to each renormalised integrand (50) and invoke Corollary 5.11. This expresses the terms in the Laurent expansion as linear combinations of products of motivic periods of the required type.
Remark 5.15. The existence of a motivic lift is a pre-requisite for the computations of Schlotterer and Stieberger [SS13], in which the motivic periods are decomposed into an ' f -alphabet' (rephrased in a different language, that paper and related literature studies the action of the motivic Galois group on I m (ω, s)). In [SS19], this is achieved by assuming the period conjecture. The computation of universal moduli space periods in terms of multiple zeta values can be carried out algorithmically [Bro09], [Pan15], [Bog16]. This type of analytic argument (or [Ter02], [BSST14]) is used in the literature to deduce a theorem of the form 5.13, but it is not capable of proving the much stronger statement 5.14.
Example 5.16. ('Motivic' beta function). We now treat the case of the beta function (Example 4.18) by using the expansion (59) of the renormalised part. We can remove all logarithms at once and write, for We thus get the following expansion: which can be rewritten as The above argument yields a 'motivic' beta function whose period, applied termwise, gives back (66). Note that Ohno and Zagier observed in [OZ01] that (66) agrees with the more classical expansion of the beta function (3). Likewise, one can verify using motivic-Galois theoretic techniques that (67) indeed coincides with the definition (5).
Definition 5.17. Given a choice of motivic lift (65), define its de Rham projection to be its image after applying π m,dR term-by-term: This makes sense since ζ m n is effective. Likewise, define its single-valued version It is a Laurent series whose coefficients are Q-linear combinations of single-valued multiple zeta values.
Since sv •π m,dR = s • π m,dR , we could equivalently have applied the map sv, which is specific to the mixed Tate situation (see [BD20, §2.6]). We now compute I sv (ω, s).
Proof. This follows from the computations of [BD20, §6.3] after a change of coordinates.
Theorem 5.19. Consider an integral of the form where the integrand is convergent of the form (62). Let I m denote a choice of motivic lift (Corollary 5.11). Its single-valued period I sv = s π m,dR (I m ) is and in particular does not depend on the choice of motivic lift.
Proof. Repeated application of Lemma 5.8 (which may involve a choice at each stage), gives rise to a dihedral structure (S , δ ), a morphism and a differential form ω ∈ S with no poles along ∂M δ 0,S such that The form ω satisfies ω = f * (ω 0 ∧ β) where and x 1 , . . . , x k denote the coordinates on A k and correspond to the 1 − u c , with multiplicity n c , taken in some order. By [BD20, Theorem 3.16] and Corollary 2.9, Since ω , ν S are logarithmic with singularities along distinct divisors, the integral converges. By repeated application of Lemma 2.10, we obtain that ν S is, up to a sign, the pullback by f of the form the sign being such that after changing coordinates via f we obtain: Formula (68) follows on applying Lemma 5.18.
Theorem 5.20. We have

In other words, the coefficients in the canonical Laurent expansion of the closed string amplitudes (52) are the images of the single-valued projection of the coefficients in any motivic lift of the expansion coefficients of open string amplitudes.
Proof. By (50), we write the open string amplitude as By Corollary 5.5, each integrand on the right-hand side admits a Taylor expansion, whose coefficients are products of integrals over moduli spaces, each of which can be lifted to motivic periods by Corollary 5.11. Thus By abuse of notation, we may express the previous theorem as the formula which is equivalent to the form conjectured in [Sti14].
be the subfields of C generated by the exp(2πis kl ) and s kl respectively.
Definition 6.1. Let O S denote the structure sheaf on M 0,S × Q Q dR s . The Koba-Nielsen connection [KN69] is the logarithmic connection on O S defined by and ω s was defined in (31). The Koba-Nielsen local system is the Q B s -local system of rank one on M 0,S (C) defined by Since ω s is a closed one-form, the connection ∇ s is integrable. The horizontal sections of the analytification (O an S , ∇ an s ) define a rank one local system over the complex numbers that is naturally isomorphic to the complexification of L s : We will also consider the dual of the Koba-Nielsen local system Let (S, δ) be a dihedral structure. In dihedral coordinates, for every subset I ⊂ S with |I | ≥ 2, and |S\I | ≥ 2. Remark 6.4. The formal one-form ω defines a logarithmic connection on the universal enveloping algebra of the braid Lie algebra. It is the universal connection on the affine ring of the unipotent de Rham fundamental group π dR 1 (M 0,S ): The Koba-Nielsen connection (viewed as a connection over the field Q dR s , i.e., for the universal solution of the momentum-conservation equations) is its abelianisation. Given any particular complex solution to the moment conservation equations, the latter specialises to a connection over C.

Singular (co)homology.
Denote the (singular) homology, locally finite (Borel-Moore) homology, cohomology, and cohomology with compact supports of M 0,S with coefficients in L s by They are finite-dimensional Q B s -vector spaces. The second is the cohomology of the complex of formal infinite sums of cochains with coefficients in L s whose restriction to any compact subset have only finitely many non-zero terms.
Because of (74), duality between homology and cohomology gives rises to canonical isomorphisms of Q where the right-hand side denotes the cohomology of the complex of global smooth differential forms on M 0,S (C) with differential ∇ s . Recall from [BD20, §3] the notation A • Proof. This is a smooth version of [Del70, Proposition 3.13]. The assumptions of [loc.
cit.] are implied by (75) and one can check that its proof can be copied in the smooth setting.
By a classical argument due to Esnault-Schechtman-Viehweg [ESV92], we can replace global logarithmic smooth forms with global algebraic smooth forms and the cohomology group H k (M 0,S , ∇ s ) is given, under the assumptions (75), by the cohomology of the complex ( • S ⊗ Q dR s , ω s ∧ −). In particular, H n (M 0,S , ∇ s ) is simply the quotient of n S ⊗ Q dR s by the subspace spanned by the elements ω s ∧ ϕ for ϕ ∈ n−1 S . The following theorem gives a basis of that quotient. Theorem 6.9. Assume that the s i j are generic in the sense of (75). A basis of H n (M 0,S , ∇ s ) is provided by the classes of the differential forms for the tuples (i 1 , . . . , i n ) with 0 ≤ i k ≤ k − 1 and where we set t 0 = 0.
Proof. This is a special case of [Aom87, Theorem 1].
The following more symmetric basis is more prevalent in the string theory literature. Its elements are called Parke-Taylor factors [PT86]. Therefore we shall refer to it as the Parke-Taylor basis, as opposed to the Aomoto basis of Theorem 6.9. Although the following theorem is frequently referred to in the literature, we could not find a complete proof for it and therefore provide one in "Appendix 7.4". Theorem 6.10. Assume that the s i j are generic in the sense of (75). A basis of H n (M 0,S , ∇ s ) is provided by the classes of the differential forms for permutations σ ∈ n , where we set t σ (0) = 0 and t σ (n+1) = 1.
Other bases can be found in the literature, e.g., the βnbc bases of Falk-Terao [FT97]. of Q dR s -vector spaces. This is easily checked after extending the scalars to C by working with smooth de Rham complexes. The only part to check is that this pairing is algebraic, i.e., defined over Q dR s . Indeed, it can be defined algebraically (see [CM95] for the general case of curves), and computed explicitly for hyperplane arrangements [Mat98], which contains the present situation as a special case.
If ω, ν ∈ Q dR s ⊗ n S are logarithmic n-forms, let ν be a smooth ∇ s -closed n-form on M 0,S (C) which represents [ν] and has compact support. Then Our normalisation differs from the one in the literature by the factor of (2πi) −n . 6.6. Periods. Since algebraic de Rham cohomology is defined over Q dR s , we can meaningfully speak of periods. Using (73) we see that integration induces a perfect pairing of complex vector spaces: which is well-defined by Stokes' theorem. By (78) it induces an isomorphism: We will use the notation comp B,dR (s) when we want to make the dependence on s explicit. If we choose a Q dR s -basis of the left-hand vector space, and a Q B s -basis of the right-hand vector space, the isomorphism comp B,dR (s) can be expressed as a matrix P s , and we will sometimes abusively use the notation P s instead of comp B,dR (s).
Theorem 6.11. (Twisted period relations [KY94a,CM95]) Assume that the s i j are generic in the sense of (75). Let ω, ν ∈ Q dR s ⊗ n S be logarithmic n-forms giving rise to classes in H n (M 0,S , ∇ −s ) and H n (M 0,S , ∇ s ) respectively. We have the equality: where the cohomological Betti pairing is naturally extended by C-linearity.
Proof. This follows from the fact that the (iso)morphisms (76) and (77) and Poincaré-Verdier duality are compatible with the comparison isomorphisms.
The reason for the factor (2πi) n in the formula (80) is because of our insistence that the de Rham intersection pairing I dR be algebraic and have entries in Q dR s .
Proposition 6.12. Assume that the s i j are generic in the sense of (75). Let δ be a dihedral structure on S and let ω ∈ Q dR s ⊗ n S be a regular logarithmic form on M 0,S of top degree. If the inequalities of Proposition 3.5 hold then we have Proof. 1) We first prove that the formula holds for any algebraic n-form ω on M δ 0,S with logarithmic singularities along ∂M δ 0,S , provided Re(s c ) > 0 for every chord c ∈ χ S,δ . By definition we have for every s the formula: where ω is a global section of A n M 0,S (log ∂M 0,S ) with compact support which is cohomologous to ω, i.e., such that ω − ω = ∇ s φ, with φ a global section of A n−1 M 0,S (log ∂M 0,S ). Thus, we need to prove that the integral of f s ∇ s φ = d( f s φ) on X δ vanishes if Re(s c ) > 0 for all chords c ∈ χ S,δ . We note that in general f s φ has singularities along the boundary of X δ unless Re(s c ) > 1 for all chords c ∈ χ S,δ , so that we cannot apply Stokes' theorem directly. We can write where the sum is over subsets of chords J ⊂ χ S,δ and φ J extends to a smooth form on M δ 0,n (i.e., has no poles along the boundary of X δ ). By properties of dihedral coordinates, we can furthermore assume that φ J = 0 if J contains two crossing chords. Indeed, a form c∈J du c u c extends to a regular form on M δ 0,n if every chord in J is crossed by another chord in J by (13). It is therefore sufficient to consider a single term given by a set J ⊂ χ S,δ consisting of chords that do not cross. We write φ = φ J and set f s = c / ∈J u s c c so that we have The forgetful maps (18) give rise to a diffeomorphism X δ (0, 1) k × X J with k = |J | and X J = X δ 0 × · · · × X δ k . We can thus write where the x i are the coordinates on (0, 1) k , corresponding to the dihedral coordinates The period matrix P s is the 1 × 1 matrix whose entry is the beta function: For any small ε > 0, a representative for the regularisation of where S i (ε) denotes the small circle of radius ε winding positively around i. From this one easily deduces the intersection product with the class of σ ⊗ sin(π(s + t)) sin(πs) sin(πt) · See, e.g., [CM95], [KY94a,§2], or [MY03,§2]. Dually: On the other hand, the de Rham intersection pairing [Mat98] is In this case, equation (80) reads using (82) and (83), as observed in [CM95], or in terms of the gamma function: This can easily be deduced from the well-known functional equation for the gamma function (s) (−s) = − π s sin(π s) , and is in fact equivalent to it (set t = −s/2). 6.7. Self-duality. It is convenient to reformulate the above relations as a statement about self-duality. Consider the object and denote the comparison The results in the previous section can be summarised by saying that the triple of objects (M dR , M B , P) is self-dual. In other words, the Betti and de Rham pairings induce isomorphisms

Single-Valued Periods for Cohomology with Coefficients
We fix a solution (s i j ) of the momentum conservation equations over the complex numbers.
7.1. Complex conjugation and the single-valued period map. We can define and compute a period pairing on de Rham cohomology classes by transporting complex conjugation which is the anti-holomorphic diffeomorphism: Since it reverses the orientation of simple closed loops, and since a rank one local system on M 0,S (C) is determined by a representation of the abelian group H 1 (M 0,S (C)) we see that we have an isomorphism of local systems: We thus get a morphism of local systems on M 0,S (C): which at the level of cohomology induces a morphism of Q B s -vector spaces We call F ∞ the real Frobenius or Frobenius at the infinite prime. We will use the notation F ∞ (s) when we want to make dependence on s explicit. One checks that the Frobenius is involutive: Remark 7.1. The isomorphism (88) is induced by the trivialisation of the tensor product conj * L s ⊗ L s given by the section Thus, the action of real Frobenius on homology is given by the formula Remark 7.2. A morphism similar to F ∞ was considered in [HY99] and leads to similar formulae but has a different definition. Our definition only uses the action of complex conjugation on the complex points of M 0,S , whereas the definition in [loc. cit.] conjugates the field of coefficients of the local systems. Note that our definition does not require the s i j to be real.
Definition 7.3. The single-valued period map is the C-linear isomorphism In other words, it is defined by the following commutative diagram: The single-valued period matrix (the matrix of s) is then the product This formula is often impractical because one needs to compute all the entries of the period matrix in order to compute any single entry of the single-valued period matrix.
Example 7.4. With the notation of Example 6.13 we have σ = σ since (0, 1) is real, and the single-valued period matrix is  Proof. We have where the first equality is the definition of the single-valued period map, and the second equality follows from Theorem 6.11. The second formula follows from the definition of P s and Remark 7.1.
Example 7.6. Following up on Example 7.4 and using (83) we see that we have given on the level of sections by Proof. Recall the notation g s = i< j | p j − p i | 2s i j . We first check that s an is a morphism of complexes: On the level of horizontal sections, we compute: Thus, s an induces the isomorphism L s → conj * L −s and the result follows.
We can now give an explicit formula for single-valued periods in the case of forms with logarithmic singularities.
2) We have p ≤ r . Without loss of generality, let p = r . Then φ has the form 7.4. Double copy formula. By equating the two expressions for the single-valued period given in Proposition 7.5 and Theorem 7.8 we obtain an equality that expresses a volume integral as a quadratic expression in ordinary period integrals.
Corollary 7.10. Under the assumptions of Theorem 7.8 we have the equality: This formula bears very close similarity to the KLT formula [KLT86], and makes it apparent that the 'KLT kernel' should coincide with the Betti intersection pairing on twisted cohomology, which is the inverse transpose of the intersection pairing on twisted homology. Indeed, Mizera has shown in [Miz17] that the KLT kernel indeed coincides with the inverse transpose matrix of the intersection pairing.
Example 7.11. Examples 7.6 and 7.9 give rise to the equality which is an instance of Corollary 7.10.
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Appendix A: The Parke-Taylor basis
In this appendix we prove the following theorem, which is Theorem 6.10 from the main body of the paper. for permutations σ ∈ n , where we set t σ (0) = 0 and t σ (n+1) = 1.

A.1. Working with the Configuration Space of Points in C
We consider the configuration space and its (rational algebraic de Rham) cohomology algebra By a classical result of Arnol'd [Arn69], it is generated by the (classes of the) forms The natural (diagonal) C * -action on Conf(n+2, C) induces a linear map in cohomology: Compatibility with the cup-product and the Koszul sign rule implies that it is a graded derivation, i.e., that it satisfies the Leibniz rule ∂(ab) = ∂(a)b + (−1) |b| a ∂(b) for b homogeneous of degree |b|. It is uniquely determined by ∂(ω i, j ) = 1 for all i, j and is given more generally by the formula: ∂(ω i 1 , j 1 ∧ · · · ∧ ω i r , j r ) = r k=1 (−1) k−1 ω i 1 , j 1 ∧ · · · ∧ ω i k , j k ∧ · · · ∧ ω i r , j r .
We will make use of the following relations.
Proof. The second relation follows from the first, which is already satisfied at the level of differential forms.
A special case is the classical Arnol'd relations ∂(ω i, j ∧ω i,k ∧ω j,k ) = 0, which generate all the relations among the generators ω i, j in the algebra A • [Arn69]. The (homological) complex (A • , ∂) is contractible. A contracting homotopy is given, for instance, by multiplication by the generator ω 0,1 .
There is a natural quotient morphism (by the affine group C C * ): Conf(n + 2, C) M 0,n+3 , (z 0 , . . . , z n+1 ) → (z 1 , . . . , z n+1 , ∞, z 0 ), defined so that the simplicial coordinates t 0 = 0, t 1 , . . . , t n , t n+1 = 1 on M 0,n+3 are related to the coordinates z i by the formula The pullback by this quotient identifies • = H • dR (M 0,n+3 ) with the subalgebra ker(∂) ⊂ A • . Since (A • , ∂) is contractible we have ker(∂) = Im(∂) and we get a short exact sequence: We now move to cohomology with coefficients. A solution s to the momentum conservation equations is completely determined by the complex numbers s i, j for 0 ≤ i < j ≤ n + 1, where we set s 0, j := s j,n+3 as in Sect. 3.2, subject to the single relation: 0≤i< j≤n+1 Denote the pullback of the Koba-Nielsen form to Conf(n + 2, C) by the same symbol: In the remainder of this appendix we extend the scalars from Q to the field Q dR s = Q(s i, j ) generated by the s i, j inside C. In order to keep the notations simple, we continue to write A • and • for A • ⊗ Q Q dR s and • ⊗ Q Q dR s , respectively. The short exact sequence (92) induces a short exact sequence of complexes : This is because ∂(ω s ) = 0, which follows from equation (93). Assume that s is generic (75). Then H k ( • , ∧ω s ) = 0 for k = n (Remark 6.6) and the long exact sequence in cohomology shows that the morphism induced by ∂ in top degree cohomology is an isomorphism. An easy computation (see, e.g., [Miz17,Claim 3.1]) shows that the Parke-Taylor form (91) from Theorem A.1 is, up to the sign sgn(σ ), the image by ∂ of the form ω σ = ω 0,σ (1) ∧ ω σ (1),σ (2) ∧ · · · ∧ ω σ (n−1),σ (n) ∧ ω σ (n),n+1 ∈ A n+1 , for σ ∈ n . Thus, a restatement of Theorem A.1 is as follows: Theorem A.3. Assume that the s i, j are generic in the sense of (75). Then a basis of H n+1 (A • , ∧ω s ) = A n+1 /(A n ∧ ω s ) is provided by the classes of the forms ω σ , for permutations σ ∈ n .
The rest of this appendix is devoted to the proof of this theorem.
Proof. The dimension of A n+1 is (n + 1)! by [Arn69], so it is enough to prove that the ω σ are linearly independent in A n+1 . For 1 ≤ i ≤ n + 1 we have a residue morphism along {z 0 = z i }: Res i : H n+1 dR (Conf(n + 2, C)) −→ H n dR (Conf(n + 1, C)), where in the target Conf(n + 1, C) consists of tuples (z 0 , . . . , z i , . . . , z n+1 ). It satisfies: where in the second case we implicitly use the natural bijection The result follows by induction on n, since the case n = 0 is trivial.
Lemma A.5. Any element of A n+1 which is the exterior product of a form with an element of the following type ω 0,i 1 ∧ ω i 1 ,i 2 ∧ · · · ∧ ω i k−2 ,i k−1 ∧ ω i k−1 ,n+1 lies in F k A n+1 .
Proof. To a graph with set of vertices {0, . . . , n + 1} and (n + 1) edges we associate a monomial ω (well-defined up to a sign) obtained by multiplying the generators ω i, j together for {i, j} an edge of . What we need to prove is that ω ∈ F k A n+1 if contains a path of length k between the vertices 0 and n + 1. If contains a cycle then ω = 0 by Lemma A.2 and there is nothing to prove. Since has n + 2 vertices and n + 1 edges we can thus assume that it is a tree. If we choose 0 to be the root of then all edges inherit a preferred orientation (from the root to the leaves). We let δ (i) denote the distance between the vertices 0 and i in and set We have δ ≤ 1+2+· · ·+(n +1) and the equality holds exactly when is linear, i.e., has only one leaf. We prove by decreasing induction on δ the statement: ω ∈ F k A n+1 if δ (n +1) ≥ k. If is linear then ω is one of the basis elements ω σ for some σ ∈ n+1 such that σ −1 (n + 1) ≥ k and thus ω ∈ F k A n+1 by definition. Now assume that is not linear and let a be a vertex of with two children b 1 and b 2 (which means that there is an edge from a to b 1 and an edge from a to b 2 in ). Then we can use the Arnol'd relation ω a,b 1 ∧ ω a,b 2 = ω a,b 1 ∧ ω b 1 ,b 2 − ω a,b 2 ∧ ω b 1 ,b 2 and deduce a relation ω = ± ω 2 ± ω 1 where s is obtained from by deleting the edge {a, b s } and adding the edge {b 1 , b 2 }, for s ∈ {1, 2}. One easily sees that we have, for all i ∈ {1, . . . , n + 1}, δ s (i) ≥ δ (i), and δ s > δ , for s ∈ {1, 2}. One can thus apply the induction hypothesis to 1 and 2 , which completes the induction step and the proof.
We have the relation, in H n+1 (A • , ω s ): We note that if {i, j} = {i a , i b } for 0 ≤ a < b ≤ k then we have X ∧ ω i, j = 0 by Lemma A.2. Let P denote the set of remaining pairs of indices. We then get the relation: We now claim that the term X ∧ ∂(ω i, j ∧ Y ) is in F k+1 for all {i, j} ∈ P. There is one easy case: if {i, j} = {i a , i b } for k ≤ a < b ≤ n + 1 then we have ∂(ω i j ∧ Y ) = 0 by Lemma A.2. Thus, we only have to treat the case where {i, j} = {i a , i b } for 0 ≤ a ≤ k and k + 1 ≤ b ≤ n + 1. We proceed by decreasing induction on a. If a = k then the first case applies and we are done. If a < k then the Leibniz rule implies: We now set X = ω i 0 ,i 1 ∧ · · · ∧ ω i a−1 ,i a ∧ ω i a+1 ,i a+2 ∧ · · · ∧ ω i k−1 ,n+1 so that we have X = ±ω i a ,i a+1 ∧ X . We thus get where we have used the Arnol'd relation ∂(ω i a ,i a+1 ∧ ω i a ,i b ∧ ω i a+1 ,i b ) = 0 in the form: ω i a ,i a+1 ∧ ∂(ω i a ,i b ∧ ω i a+1 ,i b ) = ±ω i a ,i b ∧ ω i a+1 ,i b . Now Lemma A.5 applied to X ∧ ω i a ,i b ∧ ω i a+1 ,i b ∧ ∂(Y ) and the induction hypothesis respectively imply that are in F k+1 H n+1 (A • , ω s ), which implies that it is also the case for X ∧ ∂(ω i a ,i b ∧ Y ). This concludes the induction step and the proof by induction. Returning to (94) we see that we have S ω σ ∈ F k+1 for S = {i, j}∈P where we have used equation (93). Thus S = 0 by the genericity assumption, and ω σ ∈ F k+1 , which finishes the proof of the proposition.
We can now conclude with the proof of Theorem A.3. By Proposition A.6 we have which is spanned by the ω σ for σ ∈ n . Since the dimension of H n+1 (A • , ω s ) is n! if the s i, j are generic, this implies that the ω σ are a basis, and Theorem A.3 is proved. Theorem A.1 follows as explained in Sect. A.1.