Closed strings as single-valued open strings: A genus-zero derivation

Based on general mathematical assumptions we give an independent, elementary derivation of a theorem by Francis Brown and Cl\'ement Dupont which states that tree-level amplitudes of closed and open strings are related through the single-valued map `sv'. This relation can be traced back to the underlying moduli-space integrals over punctured Riemann surfaces of genus zero. The sphere integrals $J$ in closed-string amplitudes and the disk integrals $Z$ in open-string amplitudes are shown to obey $J = {\rm sv} \, Z$.


Introduction
The study of scattering amplitudes grew into a fertile and rapidly developing research area at the interface of particle physics, mathematics and string theory. A wealth of modern mathematical concepts including periods, motives and elliptic functions became a common theme in scattering amplitudes of quantum field theory and string theory: Field-theory amplitudes encounter various flavors of polylogarithms via Feynman integrals, and string amplitudes are formulated in terms of moduli-space integrals for punctured Riemann surfaces. In contrast to field theory, the infinite number of vibration modes in string spectra introduces transcendental numbers already into the tree level of string perturbation theory.
More specifically, the low-energy expansions of tree-level amplitudes of both open and closed strings involve multiple zeta values (MZVs), (1) ζ(k 1 , k 2 , . . . , k r ) = ∞ 0<l 1 <l 2 <...<lr . . . l −kr r , k 1 , k 2 , . . . , k r ∈ N , k r ≥ 2 , characterized by depth r and weight k 1 + k 2 + . . . + k r . MZVs are the periods of the moduli space M 0,n of n-punctured genus-zero surfaces [2]: For open strings, MZVs arise from iterated integrals over the boundary of a disk, and closed-string tree amplitudes are obtained from complex integration over punctures on a sphere. From the work of Kawai, Lewellen and Tye (KLT) in 1986 [3], the sphere integrals for closed strings are known to factorize into bilinears in disk integrals for open strings. However, the approach of KLT does not manifest whether the 'squaring procedure' for disk integrals induces any cancellations for certain classes of MZVs. From the observations of [4], only the so-called single-valued subclass of MZVs (see [5]) seems to persist in the final results for the sphere integrals in closed-string tree amplitudes. The purpose of this work is to give an elementary derivation for these conjectural selection rules. In fact, as will be detailed below, the closed-string amplitudes are tied to open-string amplitudes by the 'single-valued map'. While four-point tree-level scattering of open strings gives rise to all the Riemann zeta values ζ(m), 2 ≤ m ∈ N in the low-energy expansion, the analogous closed-string four-point function only involves odd zeta values ζ(2k+1), k ∈ N. Here, the cancellations of integer powers of π 2 can be tracked by the closed-form representation of the four-point amplitudes in terms of gamma functions of the kinematic data. In open-string amplitudes with n ≥ 5 external legs, in turn, the MZVs in the low-energy expansions include higher-depth instances and follow a more elaborate structure that can be understood in terms of motivic MZVs [4] and the Drinfeld associator [6,7] 1 . And it took until 2012 that an all-order conjecture for the selection rules on the MZVs in closed-string n-point amplitudes could be made [4], based on an experimental order-by-order inspection of the output of the KLT relations 2 .
According to the observations in [4], closed-string low-energy expansions are conjectured to follow from the single-valued map [5] of the MZVs in the disk integrals of open-string amplitudes [16,17]. A defining property of the resulting single-valued MZVs is their descent from singlevalued multiple polylogarithms at unit argument. The procedure of F. Brown [18] to eliminate the monodromies from harmonic polylogarithms induces a map on MZVs which is referred to as the single-valued map sv [19,5]. The MZVs in the low-energy expansion of n-point sphere integrals J can be obtained from specific disk integrals Z via the sv map: As will be detailed below, see (35), the relation conjectured by St. Stieberger and T. Taylor [17] (based on results of [4,16]) associates certain anti-meromorphic functions of the punctures on the sphere with cyclic orderings of the punctures on the disk boundary, following a Betti-deRham duality. An independent proof of (2) by F. Brown and C. Dupont was recently announced in [1]. The proof of Brown and Dupont relies on a 'motivic' version of the KLT formula (see section 2.4). This motivic KLT is proved to be closely related to the single-valued map (which is only proved to exists in the motivic setup). Finally, the authors define 'dihedral coordinates' to provide an explicit formula which handles the poles in the Laurent expansions of string tree-level amplitudes.
Note that the proof of Brown and Dupont relies on the notion of 'motivic periods'. The motivic concept allows one to lift integrals from pure numbers (or functions) to objects in algebraic geometry. These objects contain the initial data of the integral (the form and the cycle) and provide a restricted set of transformations in algebraic cohomology. This bypasses notoriously difficult issues with transcendentality: While in many cases it is easy to see that certain numbers (such as MZVs) are related by equations, it is much harder to prove that a pair of numbers can never be related by a class of operations.
The notion of 'motivic periods' is a mathematically beautiful and deep construction which may not be readily accessible to physicists. The main claims relating motivic periods with pure numbers are: • Many properties of pure numbers can only be proved in the motivic setup (like e.g. the existence of a weight grading of MZVs). • The motivic setup is conjectured to be fully equivalent (isomorphic) to the pure number setup. • Any explicit relation which is derived within the motivic setup is also (proved to be) true in the pure number context. Because in (2) the objects J and Z are related by the single-valued map, the result can only be proved in the motivic context. Therefore, the mathematically beautiful proof of Brown and Dupont inevitably uses more advanced mathematics which may be somewhat less accessible to physicists.
In this work, we will deliver an elementary inductive derivation (a proof under general mathematical assumptions) that sphere integrals are single-valued versions of disk integrals and, equivalently, that closed-string tree-level amplitudes are single-valued open-string amplitudes.
The driving force for the derivation is the notion of single-valued integration [19] along with its properties that originate from motivic algebraic geometry [5,20]. The Betti-deRham duality between the anti-meromorphic factors in the integrands on the sphere and integration cycles on the disk boundary will arise naturally from the Stokes theorem.
The sv relations between disk and sphere integrals can be applied to closed strings in the supersymmetric, heterotic and bosonic theories [17] and triggered several directions of follow-up research. For instance, single-valued open-string amplitudes govern amplitude relations mixing gauge and gravitational states of the heterotic string [21] as well as the recent double-copy description of bosonic and heterotic strings [22]. Moreover, the appearance of single-valued MZVs in the sigma-model approach to effective gauge interactions of type-I and heterotic strings has been studied in [23]. The derivation in this work and the proof of Brown and Dupont will place these results on firm grounds without the need to rely on a conjectural status for the key relations (2) between sphere integrals and single-valued disk integrals.
The derivation in this article is not a proof in a full mathematical sense for the following three reasons: Mostly, one has to keep in mind that the single-valued map is only defined to exist in a 'motivic' framework. Because the singular divisors in the disk and sphere integrals are not normal crossing, it is a non-trivial step to set up a motivic theory for these objects. Alternatively, one can assume standard transcendentality conjectures for the related integrals.
Moreover, we use three natural properties of the single-valued map in section 3.1. These properties are thoroughly tested. Properties (i) and (ii) are proved or mostly proved in the stated literature. Property (iii) is proved in the text using a standard property of the 'falphabet' which may not be fully proved in the mathematical literature (the f -alphabet has not yet drawn much attention in mathematics).
Finally, we use the existence of a subtraction scheme whose existence we do not prove here. In tree-level string theory, the purpose of subtraction schemes is to capture the kinematic poles in disk and sphere integrals. These poles have already been investigated from various different perspectives [24,9,8]. Moreover, subtractions schemes are extensively studied in the much more complicated case of quantum field theory (see e.g. [25,26]). Further, note that in [1] Brown and Dupont prove the existence of this subtraction scheme in full mathematical rigor. So, we considered it more beneficial to provide the reader with explicit examples in appendix A and refer to [1] for the full proof.
In spite of these restrictions we informally use the word 'proof' in this article.

Reviewing the bases and relations of disk and sphere integrals
In this section, we review the classes of disk and sphere integrals that are related through the sv map. These moduli-space integrals encode the low-energy regime of string tree-level amplitudes through their series expansion in the dimensionless Mandelstam invariants where α ′ denotes the inverse string tension. The external momenta k i are Lorentz vectors referring to massless external states i = 1, 2, . . . , n of an n-point amplitude subject to k 2 i = 0 and momentum conservation n i=1 k i = 0. These kinematic constraints imply and involves all Riemann zeta values (while the Euler Mascheroni constant γ cancels).
The simplest appearance of MZVs in a closed-string setup is the following complex integral in the four-point tree amplitude where z is the complex conjugate of z = x + iy and d 2 z := dx dy. The α ′ -expansion takes a particularly symmetric form in terms of s 13 = −s 12 − s 23 , see (3), and the first line of (8) manifests the cancellation of even Riemann-zeta values.
2.2. The integrals for n points. The above four-point integrals fall into the following general classes of n-point disk integrals Z(τ |ρ) and sphere integrals J(τ |ρ), where z i,j := z i − z j . Both types of integrals are indexed by two permutations ρ, τ ∈ S n of the legs {1, 2, . . . , n}. The absolute value in the integrand 1≤i<j≤n |z i,j | s ij of (9) ensures that only positive numbers are raised to the power of s ij , regardless of the integration domain characterized by z τ (i) < z τ (i+1) . The inverse factor of vol SL 2 (R) in the disk integrals (9) is implemented by dropping three integrations over any z i , z j , z k (with i, j, k ∈ {1, 2, . . . , n}), inserting |z i,j z i,k z j,k | and fixing (z i , z j , z k ) → (0, 1, ∞). Its analogue (vol SL 2 (C)) −1 in the sphere integral (10) instructs to insert |z i,j z i,k z j,k | 2 . The limit z k → ∞ is non-singular by the Mandelstam identity (4) and the choice of cyclic 'Parke-Taylor' denominators in (9) and (10).
Note that the four-point integrals (5) and (7) can be recovered from the general definition via after fixing (z 1 , z 3 , z 4 ) → (0, 1, ∞) and identifying z 2 → z. The low-energy expansions (6) and (8) of the four-point integrals generalize as follows to higher multiplicity: The n-point integrals where p k (τ |ρ) are Laurent polynomials in s i...j = α ′ (k i + . . . + k j ) 2 of homogeneity degree k with rational coefficients. The α ′ -expansion of J(τ |ρ) follows the same structure: Equation (2) translates the leading low-energy orders (12) of the disk integrals into cf. (25), with the same degree-k Laurent polynomials p k (τ |ρ) in s ij as seen in the α ′ -expansion (12) of the disk integrals.
By the results of refs. [27,9], the n-point tree-level amplitudes of open and closed superstrings are expressible in terms of the integrals (9) and (10), also see [28,22] for analogous statements on bosonic and heterotic strings.

2.3.
Relations of disk and sphere integrals. One can infer from the right-hand sides of (9) and (10) that the disk and sphere integrals Z(τ |ρ) and J(τ |ρ) only depend on the cyclic equivalence class of the permutations τ, ρ. The cyclic denominators manifest that and the same is true for the first entry of the sphere integrals [by reality J(τ |ρ) = J(ρ|τ )]. Also, the integration domain of the disk integrals (9) is cyclically invariant Still, the number (n−1)! of cyclically inequivalent permutations in S n overcounts the number of inequivalent disk and sphere integrals: Different choices of the cyclic denominators are related via integration-by-parts relations which lead to a basis of (n−3)! inequivalent permutations of (z 1,2 z 2,3 . . . z n,1 ) −1 . For disk integrals, dropping total derivatives w.r.t. the punctures yields [9] (16) n−1 j=2 k 1 ·(k 2 +k 3 + . . . +k j )Z(τ |2, 3, . . . , j, 1, j+1, . . . , n−1, n) = 0 ∀ τ ∈ S n , and the same relations hold for both entries of the sphere integrals. Since the first entry of the disk integrals (9) refers to an integration cycle −∞ ≤ z τ (1) ≤ z τ (2) ≤ . . . ≤ z τ (n) ≤ ∞ rather than a choice of integrand, i.e. Z(τ |ρ) = Z(ρ|τ ), monodromy properties of the Koba-Nielsen factor 1≤i<j≤n |z i,j | s ij yield [29,30] (17) The combinatorics of these monodromy relations follow the structure of (16) except for the promotion of the coefficients k 1 ·(k 2 +k 3 + . . . +k j ) to a trigonometric function. Hence, permutations of (17) leave (n−3)! independent integration cycles [29,30]. By combining permutations of (16) and (17), the moduli-space integrals Z(τ |ρ) and J(τ |ρ) can be expressed in a basis of (n−3)! × (n−3)! elements. For both entries, one can fix legs n−1, n, 1 in adjacent positions and take ρ = 1, β, n−1, n with permutations β ∈ S n−3 of {2, 3, . . . , n−2} as a convenient basis choice. These relations can be understood in the framework of intersection theory, where (n−3)! arises as the dimension of twisted homologies and cohomologies [31,32]. This is the simplest instance of the KLT relations [3] between sphere integrals and bilinears in disk integrals which can be derived by suitable deformations of the complex integration contours. Their generalizations to n-points do not depend on the Parke-Taylor denominators in the integrand of J(τ |ρ) and may be described in terms of a (n−3) The KLT matrix S α ′ (σ|β) 1 is indexed by permutations σ, β ∈ S n−3 of {2, 3, . . . , n−2} and admits a recursive definition [34,35] Here, we are employing the notation A = a 1 a 2 . . . a p and B = b 1 b 2 . . . b q for words of length p, q ≥ 0 composed of external-state labels a i and b j as their letters. We also use k B = q j=1 k b j for the overall momentum associated with the word B = b 1 b 2 . . . b q . The recursive step in (20) removes the last leg j in the first entry of S α ′ (·|·) 1 which is not necessarily in the last position in the second entry. The subscript of S α ′ (σ|β) 1 indicates that the entries in (20) depend on both k 1 and the momenta k 2 , k 3 . . . , k n−2 associated with the permutations σ, β.
Similar to the integration-by-parts and monodromy relations (16) and (17), the KLT relations (19) can be elegantly understood in terms of intersection theory [31] where they follow from the twisted period relations [36].
The permutations 1, σ, n, n−1 and 1, β, n−1, n in (19) reflect a particular basis choice of twisted homologies that is tailored to simplify the KLT matrix (20): The three legs 1, n−1, n are kept in adjacent positions, and the sets of integration cycles for Z(1, σ, n, n−1|τ ) and Z(1, β, n−1, n|ρ) in (19) are related through the transposition n−1 ↔ n. With this choice of bases, the entries of S α ′ (σ|β) 1 do not depend on k n−1 or k n .
At n ≥ 5 points, similar cancellations have been observed [4] by inserting explicit α ′ -expansions of disk integrals into KLT formulae equivalent to (19). In the following we will not use the KLT relations. We rather give a general proof that the observed patterns of MZVs in sphere integrals are governed by the single-valued map.
Note that, in contrast to our approach, the proof of (2) in [1] uses a motivic version of the KLT relations.

The main result
3.1. Single-valued iterated integrals and single-valued MZVs. The notion of singlevalued (motivic 3 ) MZVs is based on the representation of generic MZVs (1) in terms of multiple (harmonic) polylogarithms at unit argument (see [37] for the general definition of iterated integrals I) where z ∈ C. For each choice of a 1 , a 2 , . . . , a w ∈ {0, 1}, a construction by F. Brown [18] provides a unique single-valued iterated integral I (0, a 1 a 2 . . . a w , z). The latter can be considered as iteratively performing 'single-valued integrations' from the base point 0 to z in complete analogy to the analytic integration in (21).
In the f -alphabet (motivic) iterated integrals become words in some alphabet which reflects the number-theoretical contents of the iterated integral. The f -alphabet exists for arbitrary a 1 , . . . , a w ∈ C, in which case the iterated integrals (21) are hyperlogarithms. (Single) logarithms are primitive, i.e. they are represented by a single letter (of weight one). The product becomes shuffle x , and there is some admixture of polynomial type from pure periods (integrals without boundary which in the case of hyperlogarithms are polynomials in 2πi).
Iterated integrals in several analytic variables are represented by words with purely analytic letters 4 . In an f -alphabet with purely analytic letters the sv map on a word w is given by (27) sv whereũ is u in reversed order (and • is complex conjugation). Moreover, sv 2πi = 0. In physical terminology the f -alphabet can be considered as a complete symbol [41]. In particular, the conversion into the f -alphabet has trivial kernel, so that no information is lost when one uses the f -alphabet.
In pure mathematics the sv map exists as evaluation of 'deRham' periods in a very general motivic context. Here, we only use sv as the map (28) sv: I(0, a 1 a 2 . . . a w , z) → I(0, a 1 a 2 . . . a w , z) (which is consistent with (24)). In general, there exist relations between iterated integrals (e.g. for MZVs). A priori it is unclear (surprising even) that the map sv is well-defined (i.e. it is consistent with all relations). However, the sv-map on I(0, a 1 a 2 . . . a w , z) can be proved to have the following three natural properties: (i) The sv-map is well-defined.
(ii) The sv-map commutes with evaluation.
(iii) The sv-map extends to several (analytic) variables. I.e. I(0, a 1 a 2 . . . a w , z) is singlevalued in all variables a 1 , a 2 , . . . , a w , z of its letters. 4 In the context of quantum field theory, iterated integrals with non-analytic letters also play a prominent role [26,40]. Handling these objects is more complicated. Here, we only need the straightforward analytic case.
These results have a deep origin in motivic algebraic geometry. The Ihara action [42] plays a major role in the proof of property (i) for iterated integrals. Property (i) is theorem 1.1 in [5] and property (ii) in the context of multiple polylogarithms is corollary 5.4 in [5]. More on the evaluation of hyperlogarithms at special values of the arguments can be found in [20]. Property (iii) can be proved in the f -alphabet [39]: Proof of (iii). In the general hyperlogarithmic context, monodromies can be expressed in terms of an 'infinitesimal' object, M = exp(m), Here, m can be considered as picking the part of the monodromy which is proportional to 2πi. Note that m is a derivative (i.e. it obeys the Leibniz rule). In the f -alphabet, m is obtained from the first letter on the Betti side (here, the left-hand side) [41], where a is a letter and w is a word. Expressions with trivial monodromy lie in the kernel of m. For hyperlogarithms, the only functions represented by single letters in the f -alphabet are logarithms (all logarithms are 'primitives' of weight one). Hence, only words with logarithms (like I(0, a 1 , z) = log(1 − z/a 1 )) as first letters contribute to the differential monodromy m. For such a logarithm the differential monodromy around z = a 1 is 2πi. The complex conjugate letter log(1 − z/a 1 ) has differential monodromy −2πi around z = a 1 (this also remains true if one considers the monodromy of the variable a 1 around a fixed value of z). From this we conclude that in the f -alphabet for hyperlogarithms single-valuedness means that all words not beginning in constants come in pairs with complex conjugate first letters.
For a letter a we define ∂ a aw = w (clipping off the first Betti letter) and ∂ a bw = 0 if b = a. Note that ∂ a is a differential with respect to the shuffle product. Because of the monodromy property of the f -alphabet, the proof of property (iii) reduces to showing that (30) ∂ a sv w = ∂ a sv w for all words w and all letters a (with complex conjugate a). From (27) we have Likewise, Both expressions on the right-hand sides are equivalent to (33) w=uavũ x v which completes the proof.
Note that property (iii) means that single-valued integration with respect to any variable of a single-valued iterated integral is single-valued in all variables. A priori, this property of 'single-valued integration' is as mysterious as properties (i) and (ii). Single-valued integration was originally introduced by F. Brown using generating functions [5]. In practice, it is more convenient to use a bootstrap algorithm first defined in [19]. A practical and fully general approach uses a commutative hexagon [26,40].
Also note that property (iii) relates I(0, a 1 a 2 . . . a w , z) to the single-valued multiple polylogarithms in more than one variable constructed in [43,44]. where sv is understood on the expansion in the parameters s ij . By ζ sv (2k) = 0, the sv map rationalizes the trigonometric functions sin(πs ij ) = πs ij exp −2 ∞ k=1 ζ(2k) 2k s 2k ij in the monodromy relations (17), sv sin(πs ij )/π = s ij . Hence, the observation (34) extends to all four-point disk and sphere integrals of the general form (9) and (10).
The general conjecture of St. Stieberger and T. Taylor we want to prove in this work concerns the striking connection between n-point disk and sphere integrals in (9) and (10) via [17] This relation identifies sphere integrals J as single-valued disk integrals sv Z, where the antimeromorphic part (z τ (1),τ (2) . . . z τ (n),τ (1) ) −1 of the sphere integrand reflects the ordering of the integration cycle −∞ ≤ z τ (1) ≤ z τ (2) ≤ . . . ≤ z τ (n) ≤ ∞ on the disk boundary. The conjecture (35) of [17] is based on equivalent conjectures on an (n−3)! × (n−3)! basis of disk and sphere integrals that have been made in [4,16]. The latter conjectures are based on an experimental order-by-order inspection of the output of the KLT relations (19), e.g. up to transcendental weight 18 at five points or weight 9 at six points: The MZVs in the α ′ -expansions of sphere integrals were observed to realize the representation (27) of the single-valued map in the f -alphabet when comparing with the dependence of disk integrals on s ij [4,16]. Assuming that (35) holds for said (n−3)! × (n−3)! bases of disk and sphere integrals, integration-by-parts and monodromy relations (16) and (17) imply its general validity for arbitrary pairs of permutations τ, ρ ∈ S n [17]. As emphasized in the reference, this argument relies on the action of the sv map on the trigonometric functions sv sin(πs ij )/π = s ij in the monodromy relations.
Reducing the sphere integral J(τ |ρ) to a single-valued disk integral has both a conceptual and a practical advantage over the KLT formula: The low-energy expansion of (35) bypasses the spurious appearance of MZVs beyond ζ sv (n 1 , . . . , n r ), and the summation over (n−3)! × (n − 3)! terms 5 on the right-hand side of (19) is replaced by a single term sv Z(τ |ρ). The implications of (35) on the leading low-energy orders of disk and sphere integrals are spelled out in (12) and (13). 3.3. The proof. As the main result of this work, this section is dedicated to a proof of (35). We emphasize again that the proof is subject to the restrictions detailed at the end of the introduction.
does not affect the subsequent arguments. The valuesz 1 = 0 andz n−1 = 1 are meant to be inserted in the denominator of (37) and subsequent expressions. The integrals Z (open string) and J (closed string) are connected by a Betti-deRham duality [45,46]: In (36) the chain of integration is bounded by the identities z i = z i+1 for i = 1, . . . , n−2. Likewise, the integrand in (37) has the anti-meromorphic singular divisor ∪ n−2 i=1 {z i = z i+1 } which is the deRham version of the chain of integration in (36). Accordingly, J(τ |ρ) becomes the deRham analogue of Z(τ |ρ). It is explained in [5] that single-valued MZVs are evaluations of deRham periods (after a projection from motivic periods into deRham periods which suppresses 2πi, see also [20]). So, it is natural that J is the image of Z under the single-valued map. These statements, however, do not have the status of a theorem so that we need a proof of the result (35).
We will show by induction that Note that the absolute values in the numerators of (39) and (40) play completely different roles in both cases. In Z 1 there exist no complex conjugate variables and |z a,b | with a < b is −z a,b . In fact, the only motivation for employing the absolute values for disk integrals stems from (9), where the integrand does not need any explicit reference to the permutation τ of the integration cycle. We consider the numerator as a generating series of logarithms with the expansion parameters s ab . In J 1 the numerator is a generating series of logarithms in |z a,b | 2 = z a,b z a,b . Because (42) sv log( for any complex numbers or variables x, y, equation (41) holds for i = 1. Now, assume (41) holds for i. In the calculation of Z i+1 , the integrand may have a singularity at z i+1 = z i+2 or at z i+1 = 0. In these cases, one has to subtract the asymptotic expansion at the singular locus which will be exemplified in appendix A.
Note that F. Brown and C. Dupont give a full mathematical proof in [1] that the subtraction of singularities is always possible.
The subtraction at z i+1 = z i+2 is of the form c|z i+1,i+2 | s−1 for some c = c(z i+2 , . . . , z n−2 ) which is constant in z i+1 but may depend on the integration variables z i+2 , . . . , z n−2 of later steps. The exponent in |z i+1,i+2 | s−1 refers to a sum s = s ab for some pairs a, b that are determined by previous integration steps. Assuming that 6 s > 0, the subtraction can trivially be integrated from 0 = z 1 to z i+2 yielding − c s ·|z 1,i+2 | s (providing a pole in s = 0). The analogous result holds for a singularity at z i+1 = 0.
The systematics of the kinematic poles of disk integrals generated in this way have been discussed in the literature from various perspectives [24,9,8]. Note that for the present proof, we only need the existence of such a subtraction scheme, i.e. the four-and five-point examples in appendix A are merely displayed for illustrative purposes. The closed-string analogues of the disk integrals with kinematic poles can be addressed with almost identical subtraction schemes, where the primitives involve factors of |z i+1,i+2 | 2s rather than |z i+1,i+2 | s . All the intermediate steps of the open-string and closed-string subtraction scheme are related through the sv map as one can see from the Taylor expansions of |z i+1,i+2 | 2s and |z i+1,i+2 | s .
After the subtraction, the integrands of (39) and (40) have an integrable expansion at s ij = 0 and we can consider the integrand as a generating series in the s ij . With this prescription we define the primitive F i of Z i with respect to z i+1 and obtain In general, the right-hand side of (43) is a series of Laurent type whose coefficients are iterated integrals in the letters 0, 1, z k for k = i+2, . . . , n−2.
By the inductive assumption we have We calculate the integral with the residue theorem of section 2.8 in [19]. To do so we need a single-valued primitive of the integrand with respect to the holomorphic variable z i+1 . By single-valued integration -see property (iii) in section 3.1 -this primitive is Because the denominator of F i is of degree two in z i+1 , its anti-residue at infinity (the residue with respect to the anti-holomorphic variable z i+1 ) vanishes. Moreover, F i has simple poles at z i+1 = z 1 = 0 and at z i+1 = z i+2 whose anti-residues are obtained by substitution. From the residue theorem in [19] (using Stokes' theorem) 7 we obtain Because the evaluation of F i commutes with the sv-map -see property (ii) in section 3.1 -this reproduces the shifted form i → i+1 of the inductive assumption (41) and therefore completes the induction.
The proof confirms the result of [47,2] that the Laurent series of Z has MZV coefficients and provides a method to calculate them which closely follows the lines of [9,8]. At the same time, it clarifies that the coefficients of J are single-valued MZVs which can be inferred from open-string results on Z without any reference to KLT relations (19). 7 Schematically, after using Stokes' theorem we use the residue theorem in the following way in passing to the second line of (46) where the function f is regular at z b = za, zc. Note that the 'boundary' of C \ {za, zc} has negative orientation. A proof of this identity is in [19], see theorem 2.29.

Conclusions
In this work, we have proved that the moduli-space integrals in n-point tree-level amplitudes of open and closed strings are related by the sv map, confirming the conjectures of [4,16,17]. More precisely, sphere integrals are expressed as single-valued disk integrals, where the singular parts of the anti-meromorphic sphere integrand are traded for an integration cycle on the disk boundary related by Betti-deRham duality. Our proof puts an intriguing web of connections between low-energy interactions of gauge-and gravity states in different string theories [17,21,22] on firm grounds. These results go beyond the reach of the KLT relations (19) as well as the known string dualities [48,49,50] and call for various directions of follow-up research.
In the same way as the notion of a single-valued map applies to a variety of periods [20], the sv relations between string tree-level amplitudes should have an echo at loop level. At genus one, this gives rise to expect a relation between elliptic multiple zeta values [51] in open-string α ′ -expansions [52,53] and modular graph functions in closed-string expansions [54,55,56,57] Single-valued polylogarithms and MZVs were found to play a key role in one-loop amplitudes of closed superstrings [60,57]. Moreover, first explicit connections between open-and closed-string results at genus one were established in [61], along with an empirically motivated conjecture for the form of an elliptic single-valued map. Since the proof of this work only relies on general properties of the genus-zero integrals -such as singularities of the integrands and the existence of suitable primitives -it is conceivable that similar methods can be applied to timely research problems at genus one and beyond.
At higher genus, the α ′ -expansion of moduli-space integrals of closed strings was pioneered in [62,63], and the last months witnessed tremendous progress in understanding their systematics and degenerations [64,65]. However, a higher-genus framework of elliptic multiple zeta values is still lacking, so the knowledge of open-string low-energy expansions is very limited. We hope that the ideas of the proof in this work are helpful to identify a language for loop-level integrals in open-and closed-string amplitudes that is tailored to expose their relations.
amplitudes of string theories, the singularities are logarithmic once the disk and sphere integrals are brought into the form of Z(τ |ρ) and J(τ |ρ) via integration by parts. Since there is no need for dimensional regularization in string tree-level amplitudes, the analogue of Conjecture 4.12 in [26] becomes a lemma that follows from blowing up all singular loci in the integrand. See [66] for the application of the concept of blowing up singularities in the context of quantum field theory.
The subtractions on the right-hand side are tailored to isolate the field-theory limits The evaluation of (viii) is completely analogous to (vi) and yields −1/s 23 . Note that, following the proof in section 3.3, the meromorphic parts of the primitives in (54) and (56) are identical.