Coaction and double-copy properties of configuration-space integrals at genus zero
Ruth Britto, Sebastian Mizera, Carlos Rodriguez, Oliver Schlotterer
PPrepared for submission to JHEP
UUITP–08/21
Coaction and double-copy properties of configuration-space integrals at genus zero
Ruth Britto, a,b
Sebastian Mizera, c Carlos Rodriguez, d Oliver Schlotterer da School of Mathematics and Hamilton Mathematics Institute, Trinity College,Dublin 2, Ireland b Institut de Physique Th´eorique, Universit´e Paris Saclay, CEA, CNRS,F-91191 Gif-sur-Yvette cedex, France c Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA d Department of Physics and Astronomy, Uppsala University, 75108 Uppsala, Sweden
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We investigate configuration-space integrals over punctured Riemannspheres from the viewpoint of the motivic Galois coaction and double-copy structuresgeneralizing the Kawai–Lewellen–Tye relations in string theory. For this purpose, ex-plicit bases of twisted cycles and cocycles are worked out whose orthonormality simpli-fies the coaction. We present methods to efficiently perform and organize the expan-sions of configuration-space integrals in the inverse string tension α (cid:48) or the dimensional-regularization parameter (cid:15) . Generating-function techniques open up a new perspectiveon the coaction of multiple polylogarithms in any number of variables and analyticcontinuations in the unintegrated punctures. We present a compact recursion for ageneralized KLT kernel and discuss its origin from intersection numbers of Stasheffpolytopes and its implications for correlation functions of two-dimensional conformalfield theories. We find a non-trivial example of correlation functions in ( p ,
2) minimalmodels, which can be normalized to become uniformly transcendental in the p → ∞ limit. a r X i v : . [ h e p - t h ] F e b ontents α (cid:48) -expansion 15 n, p ) = (5 ,
1) 183.3 Warm-up example ( n, p ) = (6 ,
1) 213.4 General result 24 F ( n,p ) ab and their building blocks 25 F ( n,p ) with p = n − F (5 , ( z ) 355.2 Warm-up example: Analytic continuation of F (6 , ( z , z ) 365.3 Initial values of F (6 , and their coaction 395.4 Analytic continuation of F ( n,p ) p = 1 526.5 Case p = 2 556.6 Recursion for general ( n, p ) 57– i – Implications for minimal models 59 p → ∞ limit 657.5 Example four-point correlators 67 α (cid:48) -expansion 77 A.1 Monodromy relations for F (5 , α (cid:48) -expansion of F (5 , P ( n, and M ( n, P (6 , and M (6 , B Braid group, monodromies and analytic continuation 88
B.1 Obtaining X ( n,p ) ( g ) for any g ∈ S n − p n, p ) = (5 ,
1) 89B.3 Example: Analytic continuation from two braidings 89B.4 Initial values in an alternative fibration basis of polylogarithms 90
Recent studies of scattering amplitudes revealed a wealth of mathematical structuresthat initiated a fruitful crosstalk between particle phenomenology, string theory, alge-braic geometry and number theory. Iterated integrals such as multiple polylogarithmsand multiple zeta values (MZVs) became a common theme of Feynman integrals andlow-energy expansions of string amplitudes. In a broad spectrum of physical settings,dramatic simplifications and striking connections between seemingly unrelated theo-ries have been found on the basis of the Hopf-algebra structures of polylogarithms andMZVs.Most prominently, amplitudes in a variety of theories were observed to exhibit uni-versal stability properties under the motivic Galois coaction of polylogarithms [1, 2].These observations support the coaction conjecture or coaction principle [3–6] whichstates that certain classes of amplitude building blocks close under the motivic Ga-lois coaction. So far, the coaction principle was found to apply to disk integrals in– 3 –pen-string tree-level amplitudes [7], periods in φ theory [5], the anomalous magneticmoment of the electron [6], six-point amplitudes in N = 4 super Yang–Mills theory[8], various families of Feynman integrals [9–14] and related hypergeometric functions[15, 16].The primary goal of this work is to extend the coaction principle in string tree-levelamplitudes to more general configuration-space integrals at genus zero where not all ofthe punctures on the Riemann sphere are integrated over. This relates to the incar-nation of the coaction principle in generalized hypergeometric functions through thesimilarity of their representations as Euler-type integrals amenable to the formalismof [17]. In the context of both string scattering [18, 19] and hypergeometric integrals(see for instance [20, 21] for earlier work on their connections), the underlying gener-alized disk integrals are dual pairings of twisted homologies and cohomologies. For agiven homology representative γ and cohomology representative ω in these spaces, thecoaction of the dual pairing given by the integral (cid:82) γ ω is conjectured to take the form[11, 12] ∆ (cid:90) γ ω = d (cid:88) a,b =1 c ab (cid:90) γ ω a ⊗ (cid:90) γ b ω , (1.1)where the { ω a } and { γ b } respectively generate the twisted (co-)homology group of di-mension d . The coefficients c ab are rational functions fixed by the choice of bases. Inthis paper, we will present a natural construction of such bases in the case of the gen-eralized disk integrals associated to tree-level string scattering, with the nice propertythat the coefficients c ab form the identity matrix.The master formula (1.1) can be viewed as a generating function of coaction iden-tities for polylogarithms and MZVs. In the string-theory incarnation of these integrals,the coaction acts order by order in the expansion with respect to the inverse stringtension α (cid:48) , or more precisely with respect to the dimensionless quantities 2 α (cid:48) k i · k j with lightlike momenta k i . For hypergeometric functions associated to dimensionally-regularized Feynman integrals, however, the analogous expansion is with respect tothe dimensional-regularization parameter (cid:15) . The formal analogy between α (cid:48) and (cid:15) hasalready been noticed by comparing differential equations of Feynman integrals andconfiguration-space integrals of string amplitudes at genus zero [22, 23] and at genusone [24–26], as well as in the context of twisted cohomology [27–33]. The discussionof this work only applies to the genus-zero case while leaving important extensions tonon-polylogarithmic integrals to the future.The main results in this work are: • To give explicit pairs of orthonormal bases { γ a } and { ω b } in (1.1) for generalizeddisk integrals over any number of punctures, while leaving an arbitrary number of– 4 –dditional punctures unintegrated. • To describe systematic methods of generating the uniformly transcendental α (cid:48) - or (cid:15) -expansions of the basis integrals (cid:82) γ a ω b in terms of multiple polylogarithms andMZVs. • To organize the multiple polylogarithms and MZVs contributing to the d × d matrix (cid:82) γ a ω b into matrix products (cid:90) γ a ω b ( z , z , . . . , z (cid:96) ) = d (cid:88) c ,c ,...,c (cid:96) =1 G (1) ac G ( z (cid:96) ) c c G ( z (cid:96) − ) c c . . . G ( z ) c (cid:96) − c (cid:96) G ( z ) c (cid:96) b (1.2)Each factor of G ( z j ) is by itself a matrix-valued series in α (cid:48) or (cid:15) , with polylogarithmsat the same argument z j in its coefficients (such that G (1) is a series of MZVs similarto those in open-string tree amplitudes [7]) and letters to be spelt out below. • To refine the coaction formula (1.1) to the individual factors in (1.2),∆ G ( z j ) = G ( z j ) × ad L (cid:0) G (1) G ( z (cid:96) ) G ( z (cid:96) − ) . . . G ( z j +1 ) (cid:1) G ( z j ) (1.3)where the operation ad L will be defined below and the contributions from MZVsobey the particularly simple special case ∆ G (1) = G (1) ⊗ G (1). • To explore the analytic continuation between configurations changing the order ofunintegrated punctures on the real axis. Such deformations can be compactly de-scribed by braid matrices acting on a vector of disk integrals and are relevant tothe study of monodromies and discontinuities of polylogarithmic Feynman integrals[9, 34–37].Another place in physics where identical integrals appear is in the context of con-formal field theories in the Coulomb gas formalism [38, 39]. On the one hand, theirconformal blocks are integrals of the type (cid:82) γ a ω b , where a subset of punctures are fixedwhile the remaining ones are integrated. On the other hand, the full correlation func-tions are given by sphere integrals, schematically (cid:82) C ( n,p ) ¯ ω a ω b . The integration domain C ( n,p ) is the configuration space of p punctures on a sphere with n − p points removed.We point out an interesting phenomenon in which correlation functions of ( p , p (cid:48) )minimal models in the p → ∞ limit (with p (cid:48) fixed and finite) behave as either the α (cid:48) → α (cid:48) → ∞ limit of string amplitudes, depending on whether charges of op-erators decay or grow in this limit. For ( p ,
2) models specifically, we find examples ofcorrelation functions exhibiting the uniform-transcendentality principle in the large- p – 5 –imit, familiar from the α (cid:48) → (cid:15) → N = 4 super Yang–Mills theory,it arises as a kinematic configuration space where the punctures are associated to themomenta of external scattering states. Motivated by this observation, amplitudes forarbitrary number of loops and legs are given in terms of single-valued multiple poly-logarithms [40–42]. Similar functional dependence can be seen in the high-energy limitof dijet scattering for generic gauge theories [43, 44].At this stage one may take inspiration from string theory, where the case of sphereintegrals with three unintegrated punctures form the backbone of closed-string tree-levelamplitudes. These sphere integrals are related to the disk integrals of open strings intwo complementary ways: • By the Kawai–Lewellen–Tye (KLT) relations [45], the sphere integrals (cid:82) C ( n,n − ¯ ω a ω b boil down to bilinears in disk integrals (cid:82) γ c ω a (cid:82) γ d ω b weighted by trigonometric func-tions of α (cid:48) built from inverse intersection numbers [18]. • At the level of the MZVs in their α (cid:48) -expansion, closed-string integrals (cid:82) C ( n,n − ¯ ω a ω b are single-valued images [3, 46] of disk integrals [7, 47–51] (cid:82) γ a ω b of open strings withsuitably chosen integration contours γ a .Another key achievement of this work is to generalize both the KLT relations andthe single-valued map between disk and sphere integrals to C ( n,p ) with p < n −
3, i.e.more than three unintegrated punctures. In these cases, the coefficients in the α (cid:48) -expansions augment single-valued MZVs by single-valued polylogarithms in one variable[52] ( p = n −
4) or multiple variables [41, 53] ( p ≤ n − p = n − p and unintegrated ones n − p , we willspell out the explicit form of the KLT-relations between C ( n,p ) -integrals and productsof generalized disk integrals and their complex conjugates. For a convenient choice ofbases for the twisted integration cycles of the disk integrals, we present an efficientrecursion for the generalized “KLT kernel” that determines the coefficients in theirbilinears. The generalized KLT kernel is again the inverse of an intersection matrixwith trigonometric functions in its entries which we derive from adjacency propertiesof Stasheff polytopes [55]. Our results furnish an explicit realization of several ofthe general mathematical concepts relating double copy, single-valued integration andstring amplitudes [51, 56]. Many all-multiplicity statements in this work are left as– 6 –onjectures, and we hope that the ideas of the references set the stage to find rigorousproofs.This work is organized as follows: The basic definitions of the configuration-spaceintegrals under investigation and the explicit form of their orthonormal bases of cy-cles { γ a } and forms { ω b } are given in section 2. We then discuss the structure of andpractical tool for the α (cid:48) -expansions of (cid:82) γ a ω b in section 3 and introduce their polyloga-rithmic building blocks G ( z j ) in (1.2). In section 4, the coaction (1.1) of the integralsis translated into that of the generating series G ( z j ) of polylogarithms, and we derivethe operation ad L in (1.3) in detail. Section 5 is dedicated to the analytic continuationof (cid:82) γ a ω b in the unintegrated punctures.In section 6, complex integrals (cid:82) C ( n,p ) ω a ω b are discussed from the perspectives ofthe single-valued map, intersection numbers and compact recursions for a KLT kernel.Finally, the implications for correlation functions of minimal models in the Coulomb-gas formalism can be found in section 7. Further details and examples of α (cid:48) -expansionsand analytic continuations are relegated to two appendices. In this section we introduce orthonormal bases of differential forms and integrationcycles. In order to do so, we start with reviewing the relevant notation and explainingwhy such bases are needed in the first place. We discuss the well-established caseof a single integration variable to set the stage for our general formula and verifyorthonormality using intersection theory.Let us consider a genus-zero Riemann surface, CP = C ∪ {∞} . The arena inwhich the integrals of our interest are defined is the configuration space of p points ona sphere with n − p punctures: C ( n,p ) = Conf p ( CP − { n − p points } ) . (2.1)In other words, out of the total n punctures, p are dynamical and are allowed to bemoved/integrated, while n − p are frozen in their positions. This space has p complexdimensions. We assume 1 ≤ p ≤ n − z i for i = 1 , , . . . , n . As the integrals of our interest are conformallyinvariant, we will work in the SL(2 , C )-frame with( z , z n − , z n ) = (0 , , ∞ ) . (2.2)– 7 –e will use the convention in which z , z , . . . , z p +1 are the integrated punctures. Inthese coordinates we can write explicitly C ( n,p ) = { ( z , z , . . . , z p +1 ) ∈ C p | z i (cid:54) = z , z i +1 , z i +2 , . . . , z n − for all i = 2 , , . . . , p +1 } , (2.3)since we fixed one puncture to infinity. We next introduce the generalized Koba–Nielsenfactor KN ( n,p ) = (cid:89) ≤ i ≤ p +1 (cid:32) | z i | s i (cid:89) i 3, where all but three punctures are integrated, (2.4) reduces tothe Koba–Nielsen factor in the integrand of string tree-level amplitudes. Note that ourdefinition omits the z ij for pairs of unintegrated punctures, i, j = 1 , p +2 , p +3 , . . . , n .We also assume that s ij are generic real numbers or formal variables. We are interested in the matrices of contour integrals F ( n,p ) ab , defined by F ( n,p ) ab = (cid:104) γ ( n,p ) a | ω ( n,p ) b (cid:105) = (cid:90) γ ( n,p ) a KN ( n,p ) ω ( n,p ) b , (2.7)where γ ( n,p ) a and ω ( n,p ) b denote integration cycles and holomorphic p -forms correspondingto bases of twisted homology and cohomology groups, respectively, for the twist 1-form given by d log KN ( n,p ) . Through γ ( n,p ) a and ω ( n,p ) b , the integrals F ( n,p ) ab dependon punctures or cross-ratios z p +2 , . . . , z n − and the Mandelstam invariants (2.6). The– 8 –ntegrals in (2.7) are of the form exhibited in the coaction formula (1.1), where in theintegrand we have now explicitly separated the twist factor KN ( n,p ) , and the remainingsingle-valued form is now denoted by ω ( n,p ) b .The indices a, b in (2.7) run from 1 to the dimensions d ( n,p ) of the associated twisted(co-)homologies [19, 57] d ( n,p ) = ( n − n − − p )! , (2.8)which, up to a sign, are the Euler characteristics of the configuration spaces C ( n,p ) .The twisted cycles γ ( n,p ) a can be taken to be regions of the real section of C ( n,p ) ,whose boundaries are contained in the union of hyperplanes { z ij = 0 } appearing in theKoba–Nielsen factor KN ( n,p ) . The unintegrated punctures z , z p +2 , z p +3 , . . . , z n − canbe assigned a fixed order on the real axis. We will always take0 = z < z p +2 < z p +3 < · · · < z n − < z n − = 1 , (2.9)except for the discussions of analytic continuations in section 5.Twisted cohomologies give a geometric description of the equivalence classes ofintegrands ω ( n,p ) b , up to total derivative terms: ω ( n,p ) b ∼ = ω ( n,p ) b + (d + d log KN ( n,p ) ∧ ) ξ (2.10)for any ( p − ξ . Both sides integrate to the same answer and hence can betreated as being equivalent. The representatives of the twisted cohomology classesare holomorphic p -forms with poles only at z i = z j . We will often strip the overalldifferential, so that the differential forms in (2.7) are written as ω ( n,p ) b = ˆ ω ( n,p ) b p +1 (cid:89) k =2 d z k , (2.11)where the functions ˆ ω ( n,p ) b are Laurent polynomials in the variables z ij . Let us see howthe equivalence relations (2.10) translate to these functions. The simplest case wouldbe to consider any closed form ξ (d ξ = 0), which can be written generally as ξ = p +1 (cid:88) i =2 ˆ ξ i p +1 (cid:89) k =2 k (cid:54) = i d z k with ∂ i ˆ ξ i = 0 ∀ i . (2.12) More generally, the Poincar´e polynomial of C ( n,p ) is given by P ( n,p ) ( t ) = (cid:81) n − k = n − p − (1 + kt ),which follows from a simple extension of the arguments given in [58]. The dimension of the onlynon-trivial p -th twisted cohomology is equal to ( − p P ( n,p ) ( − 1) = ( n − n − − p )! , which is smaller thanthat of the ordinary (untwisted) p -th cohomology, p ! ∂ pt P ( n,p ) (0) = ( n − n − − p )! , which in turn is evensmaller than the total number of possible real cycles (chambers in the real slice of C ( n,p ) ) [59] givenby P ( n,p ) (1) = ( n − n − − p )! . – 9 –ere we introduced the short-hand notation ∂ i = ∂/∂z i . Together with (2.10), it impliesthat any ˆ ω ( n,p ) b can be shifted by terms of the form( ∂ i log KN ( n,p ) ) ˆ ξ i = (cid:16) n − (cid:88) j =1 j (cid:54) = i s ij z ij (cid:17) ˆ ξ i (2.13)for any i . Throughout this work the symbol ∼ = will denote equality up to such equiva-lence relations (relations with d ξ (cid:54) = 0 will not be needed in our applications).We would like to choose bases of cycles γ ( n,p ) a and cocycles ω ( n,p ) b , for 1 ≤ a, b ≤ d ( n,p ) ,to yield orthonormal field-theory limitslim α (cid:48) → F ( n,p ) ab = δ ab . (2.14)If the condition (2.14) is satisfied, a coaction formula of the following form is claimed[16, 60]: ∆ F ( n,p ) ab = d ( n,p ) (cid:88) c =1 F ( n,p ) ac ⊗ F ( n,p ) cb , (2.15)consistent with the coaction of terms in the α (cid:48) -expansion. At p = n − 3, this specializesto the results of [7, 61] on the α (cid:48) -expansion of open-string tree-level amplitudes. As apractical advantage of orthonormal field-theory limits (2.14), they minimize the numberof terms in the coaction: One can identify (2.15) as a special case of the master formula(1.1) with c ab = δ ab and therefore d ( n,p ) in place of the ( d ( n,p ) ) summands that wouldarise for generic bases of γ ( n,p ) a and ω ( n,p ) b . Moreover, the (factorially growing) numbersof terms in the expressions below for ω ( n,p ) b are tailored to remove kinematic poles fromthe entire α (cid:48) -expansion of F ( n,p ) ab and to simplify the expressions at each order. Withthis motivation in mind, we now propose a pair of bases at general n and p satisfyingthe condition (2.14). As a warm-up, consider first the case of p = 1, where there is a single integrationvariable, z , and we have d ( n, = n − 3. The integrals F ( n, ab are then closely related toLauricella functions F n − D , for which a coaction was given in [15, 16]. By the ordering(2.9) of the unintegrated punctures on the real line, it is thus natural to choose thefollowing basis of integration contours, which are simply the intervals bounded byconsecutive finite punctures, γ ( n, = { < z < z } , γ ( n, n − = { z n − < z < } , (2.16) The difference is the absence of gamma-function prefactors in this work. The coaction for gammafunctions can easily be incorporated as desired according to the treatment in [16]. – 10 – ( n, a = { z a +1 < z < z a +2 } for 2 ≤ a ≤ n − . Now we would like to identify a set of forms ω ( n, b = d z ˆ ω ( n, b that are Laurent poly-nomials in the variables z i and satisfy the duality condition (2.14) with this set ofcontours. The functions ˆ ω ( n, b can be chosen to have only simple poles, as follows.ˆ ω ( n, = s z , ˆ ω ( n, n − = s z + n − (cid:88) j =3 s j z j , (2.17)ˆ ω ( n, b = s z + b +1 (cid:88) j =3 s j z j for 2 ≤ b ≤ n − . From the pole structure of these ω ( n, b , it is now easy to see that they are dual to the setof contours in (2.16). Contributions to the α (cid:48) → F ( n, ab arise onlywhen the poles coincide with the endpoints of integration. The logarithmic divergenceat such an endpoint, say z i , is regulated by the Koba–Nielsen factor, resulting in acontribution of s − i , cancelling the numerators in the differential forms. Thus thecontributions from the poles are either absent or cancel pairwise except when a = b .Adding a Koba–Nielsen derivative to (2.17) yields an alternative set of cohomologyrepresentatives, ˆ ω ( n, ∼ = n − (cid:88) j =3 s j z j , ˆ ω ( n, n − ∼ = s n − , z n − , , (2.18)ˆ ω ( n, b ∼ = n − (cid:88) j = b +2 s j z j for 2 ≤ b ≤ n − , which we will sometimes find more convenient in specific calculations below. For the general case ( n, p ) of (2.7), we select the basis of twisted cycles to correspondto regions labeled by distinct real orderings of the p integrated variables z i , z i , . . . , z i p among the ( n − p ) unintegrated variables in their fixed order (2.9). We write γ ( n,p ) (cid:126)A,(cid:126)i = (1 , A , i , A , i , A , . . . , A p , i p , A p +1 , n − , n ) (2.19)where (cid:126)A = ( A , A , . . . , A p +1 ) represents a partition of the ordered list of unintegratedvariables z p +2 , . . . , z n − into possibly empty parts A j . Each sequence . . . , A k , i k , A k +1 , . . . in (2.19) with A k = ( a k , a k , . . . , a k(cid:96) k ) translates into the range z a k(cid:96)k < z i k < z a k +1 , – 11 –or the associated integration variable z i k (with z i k − < z i k and z i k < z i k +1 in case of A k = ∅ and A k +1 = ∅ , respectively). Thus there are (cid:0) n − p (cid:1) values of (cid:126)A and p ! values of (cid:126)i = ( i , i , . . . , i p ) corresponding to permutations of (2 , , . . . , p +1). These cycles corre-spond to the bounded chambers of the hyperplane arrangement defined by { z ij = 0 } .The dual cocycle satisfying the condition of orthonormal field-theory limits (2.14),which can be understood as a recursive application of the case with p = 1 to successiveintegration variables, readsˆ ω ( n,p ) (cid:126)A,(cid:126)i = (cid:88) j ∈{ ,A } s i ,j z i ,j (cid:88) j ∈{ ,A ,i ,A } s i ,j z i ,j . . . (cid:88) jp ∈{ ,A ,i ,A ,......,Ap − ,ip − ,Ap } s i p ,j p z i p ,j p . (2.20)As in the p = 1 case, it is clear that the divergences contributed from endpoint sin-gularities of the integral result in the orthonormality required for the condition (2.14).Similar to (2.18), one can attain alternative cohomology representatives of (2.20) byadding Koba–Nielsen derivatives. The following p +1 choices without double poles fol-low from adding derivatives in z i k +1 , . . . , z i p with k = 0 , , . . . , p :ˆ ω ( n,p ) (cid:126)A,(cid:126)i ∼ = (cid:88) j ∈{ ,A } s i ,j z i ,j (cid:88) j ∈{ ,A ,i ,A } s i ,j z i ,j . . . (cid:88) jk ∈{ ,A ,i ,A ,......,Ak − ,ik − ,Ak } s i k ,j k z i k ,j k × (cid:88) jk +1 ∈{ Ak +2 ,ik +2 ,Ak +3 ,......,Ap,ip,Ap +1 ,n − } s j k +1 ,i k +1 z j k +1 ,i k +1 . . . (cid:88) j p ∈{ A p +1 ,n − } s j p ,i p z j p ,i p . (2.21)In case of double-integrals p = 2, the twisted cycles (2.19) and the dual functions (2.20)become γ ( n, A ,A ,A ) , ( i ,i ) = (1 , A , i , A , i , A , n − , n )ˆ ω ( n, A ,A ,A ) , ( i ,i ) = (cid:88) j ∈{ ,A } s i ,j z i ,j (cid:88) j ∈{ ,A ,i ,A } s i ,j z i ,j (2.22) ∼ = (cid:88) j ∈{ ,A } s i ,j z i ,j (cid:88) j ∈{ A ,n − } s j ,i z j ,i ∼ = (cid:88) j ∈{ A ,i ,A ,n − } s j ,i z j ,i (cid:88) j ∈{ A ,n − } s j ,i z j ,i , where the last two lines contain the alternative representatives (2.21) with k = 0 , More systematically, we can verify orthonormality (2.14) with the above cocycles usingintersection numbers. The α (cid:48) → F ( n,p ) ab is computed by intersection numbers– 12 –f twisted cocycles,lim α (cid:48) → F ( n,p ) (cid:126)A,(cid:126)i ; (cid:126)B,(cid:126)j = lim α (cid:48) → (cid:90) γ ( n,p ) (cid:126)A,(cid:126)i KN ( n,p ) ω ( n,p ) (cid:126)B,(cid:126)j = (cid:104) ν ( n,p ) (cid:126)A,(cid:126)i | ω ( n,p ) (cid:126)B,(cid:126)j (cid:105) , (2.23)since the forms constructed from the ˆ ω ( n,p ) (cid:126)B,(cid:126)j in (2.20) are logarithmic. Here the ν ( n,p ) (cid:126)A,(cid:126)i form a basis of dual cocycles that correspond to γ ( n,p ) (cid:126)A,(cid:126)i from (2.19), in the sense thateach ν ( n,p ) (cid:126)A,(cid:126)i has logarithmic singularities with unit residues along the boundaries of γ ( n,p ) (cid:126)A,(cid:126)i .In the terminology of [62], the ν ( n,p ) (cid:126)A,(cid:126)i are the canonical forms associated to the positivegeometries described by γ ( n,p ) (cid:126)A,(cid:126)i , and indeed any region bounded by hyperplanes is apositive geometry for which a canonical form exists. We can write out the latter as γ ( n,p ) (cid:126)A,(cid:126)i = { z b i < z i < z c i } × { z b i < z i < z c i } × · · · × { z b ip < z i p < z c ip } , (2.24)such that (cid:90) γ ( n,p ) (cid:126)A,(cid:126)i (cid:16) p +1 (cid:89) k =2 d z k (cid:17) = (cid:90) z ci z bi d z i (cid:90) z ci z bi d z i . . . (cid:90) z cip z bip d z i p (2.25)i.e. for each integrated puncture z i k , the indices b i k and c i k label the variables adjacentto it in the ordering (2.19). This gives a natural cocycle counterpart: ν ( n,p ) (cid:126)A,(cid:126)i = ˆ ν ( n,p ) (cid:126)A,(cid:126)i p +1 (cid:89) k =2 d z k (2.26)ˆ ν ( n,p ) (cid:126)A,(cid:126)i = (cid:32) z i ,b i − z i ,c i (cid:33) (cid:32) z i ,b i − z i ,c i (cid:33) · · · (cid:32) z i p ,b ip − z i p ,c ip (cid:33) . Since both bases ν ( n,p ) (cid:126)A,(cid:126)i and ω ( n,p ) (cid:126)A,(cid:126)i are logarithmic, the evaluation of intersection numberscan be carried out on the support of critical points of KN ( n,p ) [63] given by solutions ofthe equations: ∂ k log KN ( n,p ) = n − (cid:88) j =1 j (cid:54) = k s kj z kj = 0 , for k = 2 , , . . . , p +1 . (2.27) Note that in case of adjacent integration variables z , z bounded by z b < z < z < z c , only oneof z , z appears among the integration limits z b i , z c i , i.e. (cid:90) z b 7) that this formula gives rise tothe identity matrix, i.e., (cid:104) ν ( n,p ) (cid:126)A,(cid:126)i | ω ( n,p ) (cid:126)B,(cid:126)j (cid:105) = δ ( (cid:126)A,(cid:126)i ) , ( (cid:126)B,(cid:126)j ) , (2.30)which confirms (2.14). The largest checks required summing over d (10 , = 5040 criticalpoints for each entry of the 5040 × (cid:104) ν (10 , a | ω (10 , b (cid:105) . This high-multiplicitycomputation was made possible by following [64] to interpret log KN ( n,p ) as a log-likelihood function in algebraic statistics and extremizing it according to (2.27) usingthe Julia package HomotopyContinuation.jl [65]. For the maximum number p = n − ρ a , ρ b acting on 2 , , . . . , n − 2, i.e. a, b = 1 , , . . . , ( n − F ( n,n − ab = (cid:90) γ ( n,n − a (cid:16) n − (cid:89) j =2 d z j (cid:17) n − (cid:89) ≤ i 3, the samestatements are claimed to carry over to the MZV-dependent parts P ( n,p ) and M ( n,p ) of(3.7). We propose that P ( n,p ) = + φ − ∞ (cid:88) k =1 f k P ( n,p )2 k , (3.17) M ( n,p ) = φ − ∞ (cid:88) r =0 ∞ (cid:88) k ,k ,...,k r =1 f k +1 f k +1 . . . f k r +1 M ( n,p )2 k +1 M ( n,p )2 k +1 . . . M ( n,p )2 k r +1 , (3.18)where the entries of the d ( n,p ) × d ( n,p ) matrices P ( n,p ) w and M ( n,p ) w are again degree- w polynomials in the s ij with rational coefficients. Note that (3.17)–(3.18) is equivalentto ∆ P ( n,p ) = P ( n,p ) ⊗ , ∆ M ( n,p ) = M ( n,p ) ⊗ M ( n,p ) . (3.19)In the following, we will spell out examples of the P ( n,p ) w , M ( n,p ) w at p (cid:54) = n − P ( n ) w , M ( n ) w at n ≤ r ≥ φ -map in (3.18) dependson a choice of reference basis. We follow the conventions of [7, 78] to assign vanishingcoefficients of f w to the φ -image of those higher-depth MZVs at weight w in the (con-jectural) Q -bases of [75] (say ζ , , ζ , , ζ , , , . . . ). Still, the form of (3.15) to (3.18) doesnot depend on these choices, only the s ij -dependence in the entries of P ( n,p ) w and M ( n,p ) w depends on the reference bases for MZVs at weight w . ( n, p ) = (5 , G ( n,p ) { , ,... } ( z j ), we shall now give a detailed derivation of the α (cid:48) -expansion of F (5 , ab . Thetwo-dimensional bases of cocycles (2.17) and cycles (2.19) are γ (5 , = { < z < z } , γ (5 , = { z < z < } , (3.20)– 18 –s well as ˆ ω (5 , = s z ∼ = s z + s z , ˆ ω (5 , = s z + s z ∼ = s z . (3.21)We have discarded Koba–Nielsen derivatives ∂ KN (5 , = ( s z + s z + s z )KN (5 , inpassing between different representations of ˆ ω (5 , b in the twisted cohomology. Thesame integration-by-parts identities allow us to determine the 2 × e (5 , = (cid:18) s + s − s (cid:19) , e (5 , = (cid:18) − s s + s (cid:19) , (3.22)in the KZ equation (3.6) ∂ F (5 , ab = (cid:88) c =1 (cid:26) ( e (5 , ) bc z + ( e (5 , ) bc z (cid:27) F (5 , ac . (3.23)One can solve (3.23) through the generating series of polylogarithms G ( a k ∈ { , } ; z ) G (5 , { , } ( z ) = + (cid:88) a ∈{ , } G ( a ; z ) E (5 , a ,z + (cid:88) a ,a ∈{ , } G ( a , a ; z ) E (5 , a ,z E (5 , a ,z + O ( α (cid:48) )= ∞ (cid:88) r =0 (cid:88) a ,a ,......,ar ∈{ , } G ( a r , . . . , a , a ; z ) E (5 , a ,z E (5 , a ,z . . . E (5 , a r ,z . (3.24)with the transpose of the braid matrices (3.22) E (5 , ,z = ( e (5 , ) t = (cid:18) s + s − s (cid:19) , E (5 , ,z = ( e (5 , ) t = (cid:18) − s s + s (cid:19) , (3.25)which may multiply arbitrary z -independent matrices from the right. In order to tailorthese constant matrices to the target integrals F (5 , ab , we determine their asymptotics as z → z → F (5 , b ( z → 0) = δ b, | z | s + s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) , (3.26) Note that the soft limit s → e (5 , = e (5 , and e (5 , = e (5 , followed by relabelling s → s reproduces the four-point instances of the arguments of the 2 × z -derivatives of F (5 , ab have been simplified using partial fractions and integration by parts in order to attain the formon the right-hand side of (3.23) and to identify the expressions (3.22) for the braid matrices. While (3.26) follows from the rescaling z = xz of the integration variable with x ∈ (0 , z → − z in the derivation of (3.27). – 19 – (5 , b ( z → 1) = δ b, | − z | s + s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) . (3.27)Finally, it remains to expand the F (5 , b associated with the integration domain z ∈ ( z , z → × F (5 , ab in terms of polyloga-rithms with the same basepoint. In presence of the pole z − of ˆ ω , the α (cid:48) -expansion of F (5 , does not commute with the limit z → 0. Hence, as detailed in appendix A.1, weinstead infer the α (cid:48) -expansion via monodromy relations [83, 84] involving cycles wherethe α (cid:48) -expansions commute with the limit z → P (5 , M (5 , = (cid:32) Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) s s + s (cid:110) Γ(1+ s )Γ(1+ s + s )Γ(1+ s + s + s ) − Γ(1+ s )Γ(1 − s − s )Γ(1 − s ) (cid:111) Γ(1+ s + s )Γ(1+ s )Γ(1+ s + s + s ) (cid:33) . (3.28)Note that the factor of ( s + s ) − in the (2 , α (cid:48) = 0, s s + s (cid:26) Γ(1+ s )Γ(1+ s + s )Γ(1+ s + s + s ) − Γ(1+ s )Γ(1 − s − s )Γ(1 − s ) (cid:27) (3.29)= − ζ s ( s + s ) + ζ s ( s + s s + s s − s s ) + O ( α (cid:48) ) , which is consistent with the z → z → 0, the α (cid:48) -expansion of F (5 , ab at generic z ∈ (0 , 1) is obtained by right-multiplication with the series (3.24) in polylogarithms F (5 , ( z ) = P (5 , M (5 , G (5 , { , } ( z ) (3.30)with matrix multiplication between the three factors. The individual P (5 , k , M (5 , k +1 may be obtained from (3.28) by extracting the coefficients of ζ k , ζ k +1 in the Taylorexpansion of F (4 , ( s , s ) = Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) = exp (cid:16) ∞ (cid:88) k =2 ζ k k ( − k (cid:2) s k + s k − ( s + s ) k (cid:3)(cid:17) (3.31)i.e. they are determined by the single four-point integral ( d (4 , = 1). In (3.28) andlater expressions for initial values of F ( n,p ) , we already incorporate a central conjectureon the structure of the α (cid:48) -expansion by writing the left-hand side as a matrix productof P (5 , and M (5 , . Like this, the appearance of ζ is claimed to follow the expansionsin (3.17) and (3.18) which we have verified order by order in α (cid:48) . It would be interestingto find an all-order argument based on the right-hand side of (3.28).– 20 –iven that MZVs are recovered from polylogarithms at unit argument via ζ n ,n ,...,n r = ( − r G (0 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n r − , , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n r − − , , . . . , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) n − , 1; 1) (3.32)one can check that (3.30) is consistent with both (3.26) and (3.27), validating ourprocedure to determine the formal initial value of z = 0 from monodromy relations.The coaction properties of (3.30) extending our conjecture (3.19) for ∆ P ( n,p ) , ∆ M ( n,p ) are discussed in the later section 4, and the explicit form of the α (cid:48)≤ -orders can befound in appendix A.2. ( n, p ) = (6 , F (6 , ab . The bases of mastercontours γ (6 , = { < z < z } , γ (6 , = { z < z < z } , γ (6 , = { z < z < } (3.33)and dual cocycles (see (2.17))ˆ ω (6 , = s z ∼ = s z + s z + s z ˆ ω (6 , = s z + s z ∼ = s z + s z (3.34)ˆ ω (6 , = s z + s z + s z ∼ = s z give rise to the following 3 × e (6 , = s + s − s 00 0 00 0 0 , e (6 , = s s − s s s − s e (6 , = − s s s − s s s , e (6 , = − s s + s (3.35) e (6 , = − s s + s − s in the KZ equations (3.6) ∂ F (6 , ab = (cid:88) c =1 (cid:26) ( e (6 , ) bc z + ( e (6 , ) bc z + ( e (6 , ) bc z (cid:27) F (6 , ac (3.36)– 21 – F (6 , ab = (cid:88) c =1 (cid:26) ( e (6 , ) bc z + ( e (6 , ) bc z + ( e (6 , ) bc z (cid:27) F (6 , ac . (3.37)A convenient strategy is to focus on the differential equation (3.36) in z and to solveit in terms of polylogarithms G ( a j ∈ { , , z } ; z ), G (6 , { , ,z } ( z ) = ∞ (cid:88) r =0 (cid:88) a ,a ,...,ar ∈{ , ,z } G ( a r , . . . , a , a ; z ) E (6 , a ,z E (6 , a ,z . . . E (6 , a r ,z . (3.38)The formal initial value with respect to z = 0 multiplying G (6 , { , ,z } ( z ) from the left isstill a function of z which obeys the differential equation (3.37). The latter at z = 0is solved by G (6 , { , } ( z ) = ∞ (cid:88) r =0 (cid:88) a ,a ,......,ar ∈{ , } G ( a r , . . . , a , a ; z ) E (6 , a ,z E (6 , a ,z . . . E (6 , a r ,z (3.39)with a left-multiplicative factor that does not depend on z or z . Hence, the depen-dence of F (6 , ab on z , z stems from G (6 , { , } ( z ) G (6 , { , ,z } ( z ) multiplying a formal z , z → E (6 , ,z = ( e (6 , + e (6 , ) t = s s s + s + s − s − s − s (3.40) E (6 , ,z = ( e (6 , ) t = − s s + s , E (6 , ,z = ( e (6 , ) t = s + s − s E (6 , z ,z = ( e (6 , ) t = − s s + s − s , E (6 , ,z = ( e (6 , ) t = − s − s s s s s . The initial values are determined by the asymptotics F (6 , b ( z → , z ) = δ b, | z | s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) F (6 , b ( z , z → z ) = δ b, | z | s | − z | s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) (3.41) F (6 , b ( z , z → 1) = δ b, | − z | s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) , – 22 –nd one can again use monodromy relations as explained in appendix A.1 to alsoinfer the z → F (6 , b and F (6 , b (contours different from (3.33) arenecessary to ensure that the limit z → α (cid:48) -expansions). One arrivesat the formal limit F (6 , ab ( z → , z ) = | z | s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) s ˆ F (5 , s + s − K (6 , ˆ F (5 , ˆ F (5 , s ˆ F (5 , s + s ˆ F (5 , ˆ F (5 , ab , (3.42)where the hat notation stands for changes of arguments,ˆ F (5 , ab = F (5 , ab ( z ) (cid:12)(cid:12)(cid:12) s → s + s s → s s → s , (3.43)and the ( a, b ) = (2 , 1) entry of (3.42) involves K (6 , = sin( πs )sin( π ( s + s )) | z | s Γ(1+ s )Γ(1+ s )Γ(1+ s + s )= | z | s s s + s Γ(1+ s )Γ(1 − s − s )Γ(1 − s ) . (3.44)By importing the formal z → F (5 , ab from (3.28) with the above replacementrules for the s ij , we arrive at P (6 , M (6 , = Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) K (6 , s + s )Γ(1+ s )Γ(1+ s + s + s ) K (6 , ˆ K (6 , s + s + s )Γ(1+ s )Γ(1+ s + s + s + s ) (3.45)withˆ K (6 , = s s + s (cid:26) Γ(1+ s )Γ(1+ s + s )Γ(1+ s + s + s ) − Γ(1+ s )Γ(1 − s − s )Γ(1 − s ) (cid:27) , (3.46)ˆ K (6 , = s s + s + s (cid:26) Γ(1+ s )Γ(1+ s + s + s )Γ(1+ s + s + s + s ) − Γ(1+ s )Γ(1 − s − s − s )Γ(1 − s − s ) (cid:27) , ˆ K (6 , = s s + s + s (cid:26) Γ(1+ s )Γ(1+ s + s + s )Γ(1+ s + s + s + s ) − Γ(1+ s )Γ(1 − s − s − s )Γ(1 − s − s ) (cid:27) , i.e. the 3 × P (6 , k , M (6 , k +1 are again determined by the four-point integral(3.31). The factor of | z | s in (3.42) has been replaced by 1 in the formal z → | z | s = 1 + ∞ (cid:88) w =1 s w G (0 , , . . . , (cid:124) (cid:123)(cid:122) (cid:125) w ; z ) (3.47)– 23 –re later on generated by (3.39). The denominators ( s + s ) − and ( s + s + s ) − on the right-hand side of (3.46) are cancelled by the differences of Euler beta functionsas in (3.29) such that all entries of the matrices P (6 , k , M (6 , k +1 determined from (3.45)are indeed polynomials in s ij .By the above arguments, the α (cid:48) -expansion of F (6 , ab exhibits a matrix multiplicativestructure F (6 , ( z , z ) = P (6 , M (6 , G (6 , { , } ( z ) G (6 , { , ,z } ( z ) (3.48)similar to (3.30), where the building blocks are given by (3.38), (3.39), (3.45) and(3.46). This representation realizes the integration of the KZ form Ω (6 , in d F (6 , =Ω (6 , F (6 , along the path (0 , → (0 , z ) → ( z , z ), and the alternative choice of path(0 , → ( z , → ( z , z ) is discussed in section 5. The structural results (3.30) and (3.48) on the α (cid:48) -expansion of F (5 , and F (6 , can bereadily generalized to higher multiplicity: The KZ equations (3.6) can be solved by thematrix product (3.7), where the z j -dependent building blocks G ( n,p ) { , ,z j +1 ,z j +2 ,...,z n − } ( z j ) = ∞ (cid:88) r =0 (cid:88) a ,a ,...,ar ∈{ , ,zj +1 ,zj +2 ,...,zn − } G ( a r , . . . , a , a ; z j ) E ( n,p ) a ,z j E ( n,p ) a ,z j . . . E ( n,p ) a r ,z j (3.49)involve the following combinations of braid matrices E ( n,p ) z k ,z j = ( e ( n,p ) jk ) t ∀ k (cid:54) = 1 , E ( n,p )0 ,z j = ( e ( n,p ) j ) t + j − (cid:88) i = p +2 ( e ( n,p ) ij ) t . (3.50)The choice of fibration basis is adapted to the arrangement (3.1) of the unintegratedpunctures z p +2 , . . . , z n − on the real line and amounts to integrating the KZ form Ω ( n,p ) in d F ( n,p ) = Ω ( n,p ) F ( n,p ) along the path(0 , , . . . , → (0 , . . . , , z n − ) → (0 , . . . , , z n − , z n − ) → . . . (3.51) → . . . → (0 , z p +3 , . . . , z n − ) → ( z p +2 , z p +3 , . . . , z n − ) . The series (3.49) in polylogarithms act by right-multiplication on the z j -independentmatrices P ( n,p ) , M ( n,p ) in (3.7) that are claimed to carry the MZVs according to (3.18).As exemplified by (3.28), (3.46) and (A.18) for p = 1 and appendix A.4 for ( n, p ) =(6 , P ( n,p ) , M ( n,p ) are expected to be expressible in terms of the diskintegrals F ( k +3 ,k ) in string amplitudes with k ≤ p . Their compositions can be deter-mined via monodromy relations from the initial values z p +2 , . . . , z n − → α (cid:48) -expansions.– 24 – Coaction properties of F ( n,p ) ab and their building blocks The goal of this section is to investigate the coaction formula (2.15) of the F ( n,p ) ab atthe level of their factorized α (cid:48) -expansion (3.7). We will identify conjectural coactionproperties of the building blocks G ( n,p ) { , ,z j +1 ,z j +2 ,...,z n − } ( z j ) in (3.49) which imply (2.15)and mix different braid matrices and the matrices M ( n,p )2 k +1 accompanying the MZVs. Thesubsequent expressions for ∆ G ( n,p ) are generating functions for coactions of polyloga-rithms: Each contribution is already cast into a fibration basis, and they drasticallysimplify order-by-order tests of (2.15). The structures to be described in this section originate from the coproduct in the Hopfalgebra of multiple polylogarithms taken modulo their branch cuts, or equivalentlymodulo iπ [1, 2],∆ I ( a ; a , . . . , a n ; a n +1 ) (4.1)= (cid:88) i
3) = 0 and W (1 | 3) = − E E E E + 2 E E E E − E E E E + E E E E + E E E E − E E E E + 3 E E E E − E E E E (4.12)= [[[ E , E ] , E ] , E ] + [[[ E , E ] , E ] , E ] . In the remainder of this section, we specialize the abstract E , E to the braid matricesof various F ( n,p ) as for instance in (3.25) and find relations involving commutators ofmatrices and M ( n,p ) k . F ( n,p ) with p = n − p = n − 4, i.e.for functions in factorized form (3.7) that depend on one puncture z = z n − F ( n,n − ( z ) ad = P ( n,n − ab M ( n,n − bc G ( n,n − { , } ( z ) cd , (4.13)see (3.17) and (3.18) for the structure of P ( n,n − and M ( n,n − . We will find recursiverelations among the coefficients W ( (cid:126)u | (cid:126)k ) of the coaction in the second line of (4.9), andtheir solution can be resummed in terms of repeated adjoint actions in the generatingfunctions in (4.13).The conjectural coaction property for the full disk integrals is∆ F ( n,n − ( z ) ac = F ( n,n − ( z ) ab ⊗ F ( n,n − ( z ) bc , (4.14)– 27 –nd we start by investigating the regularized z → G ( n,n − { , } ( z ) → P ( n,n − ab M ( n,n − bd ) = P ( n,n − ab M ( n,n − bc ⊗ M ( n,n − cd . (4.15)This z → P ( n,n − and M ( n,n − which in turn follow from the expansion (3.17) and (3.18) in terms of matrices P ( n,n − w , M ( n,n − w of fixed polynomial degree w in s ij . In order for (4.14) to hold at nonzero z ,the series M ( n,n − and G ( n,n − { , } ( z ) need to be interrelated through the coaction,∆ (cid:0) M ( n,n − ac G ( n,n − { , } ( z ) ce (cid:1) = M ( n,n − ab G ( n,n − { , } ( z ) bc ⊗ M ( n,n − cd G ( n,n − { , } ( z ) de . (4.16)With the property ∆ M ( n,n − ac = M ( n,n − ab ⊗ M ( n,n − bc assumed in (3.19) and the ansatz(4.9) for the coaction of G ( n,n − { , } ( z ), the desired property (4.16) implies G ( n,n − { , } ( z ) ab ⊗ M ( n,n − bc − G ( n,n − { , } ( z ) bc ⊗ M ( n,n − ab (4.17)= (cid:88) (cid:126)u ∈{ , } × G ( (cid:126)u ; z ) ⊗ M ( n,n − ab (cid:1) (cid:88) (cid:126)k ∈ (2 N +1) × φ − ( f k f k . . . f k (cid:96) ) W ( (cid:126)u | (cid:126)k ) bc upon left- and right-multiplication with the inverses of M ( n,n − and G { , } ( z ). Theshuffle symbol in the second entry acts on the combinations of f k that are explicitin the second line of (4.17) and those in the expansion of M ( n,n − . The row- andcolumn indices a, b, . . . are spelt out since the order of matrix multiplication does notalways line up with the sequence of entries in the coaction as for instance for the term G ( n,n − { , } ( z ) bc ⊗ M ( n,n − ab .By isolating the coefficients of various G ( (cid:126)u ; z ) ⊗ f k f k . . . f k (cid:96) in (4.17), one obtains arecursion that relates W ( (cid:126)u | k , k , . . . , k (cid:96) ) associated with different numbers (cid:96) of letters f k . With the shorthand notation E ( (cid:126)u ) = E ( n,n − u w . . . E ( n,n − u E ( n,n − u (4.18)for the matrix product accompanying G ( u , u , . . . , u w ; z ) in (4.8) and suppressing thesuperscripts of M ( n,n − k , the coefficient equations at (cid:96) = 0 , , E ( (cid:126)u ) ab bc − E ( (cid:126)u ) bc ab = W ( (cid:126)u |∅ ) ac E ( (cid:126)u ) ab ( M k ) bc − E ( (cid:126)u ) bc ( M k ) ab = W ( (cid:126)u | k ) ac + ( M k ) ab W ( (cid:126)u |∅ ) bc , (4.19) E ( (cid:126)u ) ab ( M k M k ) bc − E ( (cid:126)u ) bc ( M k M k ) ab = W ( (cid:126)u | k , k ) ac + ( M k ) ab W ( (cid:126)u | k ) bc , + ( M k ) ab W ( (cid:126)u | k ) bc + ( M k M k ) ab W ( (cid:126)u |∅ ) bc . – 28 –t is easy to see from (4.17) that the generalization to coefficients of G ( (cid:126)u ; z ) ⊗ f k f k . . . f k (cid:96) at arbitrary (cid:96) is captured by the deshuffle (cid:80) (cid:126)p (cid:1) (cid:126)q = (cid:126)k on the right-hand side. The latterinstructs to sum over all pairs (cid:126)p = ( p , p , . . . , p i ) and (cid:126)q = ( q , q , . . . , q j ) of ordered setssuch that a given (cid:126)k = ( k , k , . . . , k (cid:96) ) with (cid:96) = i + j occurs in their shuffle product: (cid:2) E ( (cid:126)u ) , M k M k . . . M k (cid:96) (cid:3) ac = (cid:88) (cid:126)p (cid:1) (cid:126)q = (cid:126)k ( M p M p . . . M p i ) ab W ( (cid:126)u | (cid:126)q ) bc . (4.20)The recursion for the W ( (cid:126)u | k , k , . . . , k (cid:96) ) in (4.19) and (4.20) can be straightforwardlysolved in terms of nested matrix commutators such as W ( (cid:126)u |∅ ) = 0 ,W ( (cid:126)u | k ) = [ E ( (cid:126)u ) , M k ] , (4.21) W ( (cid:126)u | k , k ) = [[ E ( (cid:126)u ) , M k ] , M k ] , and more generally W ( (cid:126)u | k , k , . . . , k (cid:96) ) = [[ . . . [[ E ( (cid:126)u ) , M k ] , M k ] , . . . , M k (cid:96) − ] , M k (cid:96) ] . (4.22)For words (cid:126)u of length one, (4.21) relates the commutators [ E , M k +1 ] and [ E , M k +1 ]to products of braid matrices. One can for instance find[ E , M ] = 0 , [ E , M ] = 0[ E , M ] = [[[ E , E ] , E ] , E ] + [[[ E , E ] , E ] , E ][ E , M ] = [[[[[ E , E ] , E ] , E ] , E ] , E ] + 32 [[[[[ E , E ] , E ] , E ] , E ] , E ] (4.23)+ 12 [[[[[ E , E ] , E ] , E ] , E ] , E ] + 12 [[[[[ E , E ] , E ] , E ] , E ] , E ]+ 32 [[[[[ E , E ] , E ] , E ] , E ] , E ] + [[[[[ E , E ] , E ] , E ] , E ] , E ] , based on W (0 | 3) = W (0 | 5) = 0, and W (1 | 3) in (4.12) (with a similar expression for W (1 | E , the right-hand sides of (4.23) match thecoefficients of ζ and ζ in the Drinfeld associator Φ( E , E ) (when reducing the MZVsto the standard conjectural Q -bases), see (5.6) below. Multiples of these expressionsalso feature as the nested brackets that define the elements D f and D f in the stablederivation algebra [86, 87]. Given that each W ( (cid:126)u | (cid:126)k ) in (4.22) involves the matrix product E ( (cid:126)u ) in (4.18), thesum over (cid:80) (cid:126)u ∈{ , } × G ( (cid:126)u ; z ) W ( (cid:126)u | (cid:126)k ) in (4.9) is expressible in terms of the generating For any element f ( x, y ) in a free Lie algebra with generators x, y , the derivation D f is defined by – 29 –eries G { , } ( z ) in (4.8): The coaction property (4.16) along with the ansatz (4.9) areequivalent to∆ G ( n,p ) { , } ( z ) = G ( n,p ) { , } ( z ) ⊗ G ( n,p ) { , } ( z ) + (cid:88) k ∈ N +1 (cid:2) G ( n,p ) { , } ( z ) , M ( n,p ) k (cid:3) ⊗ f k G ( n,p ) { , } ( z )+ (cid:88) k ,k ∈ N +1 (cid:2) (cid:2) G ( n,p ) { , } ( z ) , M ( n,p ) k (cid:3) , M ( n,p ) k (cid:3) ⊗ f k f k G ( n,p ) { , } ( z ) (4.24)+ (cid:88) k ,k ,k ∈ N +1 (cid:2) (cid:2) (cid:2) G ( n,p ) { , } ( z ) , M ( n,p ) k (cid:3) , M ( n,p ) k (cid:3) , M ( n,p ) k (cid:3) ⊗ f k f k f k G ( n,p ) { , } ( z ) + . . . = (cid:88) (cid:126)k ∈ (2 N +1) × (cid:2)(cid:2) . . . (cid:2)(cid:2) G ( n,p ) { , } ( z ) , M ( n,p ) k (cid:3) , M ( n,p ) k (cid:3) . . . , M ( n,p ) k (cid:96) − (cid:3) , M ( n,p ) k (cid:96) (cid:3) ⊗ f k f k . . . f k (cid:96) G ( n,p ) { , } ( z )with terms involving four or more f k in the ellipsis in the third line. In fact, thisderivation of (4.24) not only applies to p = n − p : Oneimposes the coaction properties of F ( n,p ) to hold for the matrix product (3.7) at generic z = z n − and vanishing z n − , z n − , . . . , z p +2 . Note that the pattern of f k i and M k i in(4.24) amounts to translating the matrix products in the expansion (3.18) of M ( n,p ) tothe adjoint representation: By introducing the formal operation X ⊗ ad L ( f k f k . . . f k r M k M k . . . M k r ) Y (4.25)= (cid:2)(cid:2) . . . [[ X, M k ] , M k ] , . . . , M k r − (cid:3) , M k r (cid:3) ⊗ f k f k . . . f k r Y that converts matrix products to nested commutators in the appropriate order and actslinearly ad L ( P + Q ) = ad L ( P ) + ad L ( Q ), one can compactly rewrite (4.24) as∆ G ( n,p ) { , } ( z ) = G ( n,p ) { , } ( z ) ⊗ ad L (cid:0) M ( n,p ) (cid:1) G ( n,p ) { , } ( z ) . (4.26)We emphasize that (4.24) is still conjectural and can be thought of as an economicreformulation of the coaction conjecture (2.15) for F ( n,p ) : We have started to decom-pose the coaction relation involving all the contributions G ( . . . ; z n − ) , . . . , G ( . . . ; z p +2 )and MZVs to the α (cid:48) -expansion of F ( n,p ) into simpler coaction formulae for the buildingblocks in (3.7). In the next section, this decomposition will be extended to polyloga-rithms in several variables. proposition 2 of [86]. The cases of f ( x, y ) relevant to (4.23) are [87] f = − [[ x, y ] , x ] − [[ x, y ] , y ] f = − x, y ] , x ] , x ] , x ] − x, y ] , x ] , x ] , y ] − [[[[ x, y ] , x ] , y ] , x ] − [[[[ x, y ] , y ] , x ] , y ] − x, y ] , y ] , y ] , x ] − x, y ] , y ] , y ] , y ]with x → E and y → E which are not to be confused with the f -alphabet description of MZVs. – 30 –e have tested (4.24) and (4.26) order by order in the α (cid:48) -expansion, namely upto and including α (cid:48) for ( n, p ) = (5 , 1) and α (cid:48) for ( n, p ) = (6 , M k +1 can be found in (3.25) and (3.28) for ( n, p ) = (5 , 1) aswell as appendix A.4 for ( n, p ) = (6 , α (cid:48) -order for ( n, p ) = (6 , 2) that are sensitive to the commutator structure of (cid:2) [ G ( n,p ) { , } ( z ) , M ( n,p ) k ] , M ( n,p ) k (cid:3) ⊗ f k f k along with ζ , , see appendix A.4.3 for details. Thesechecks go beyond the reach of ( n, p ) = (5 , 1) since [ M (5 , , M (5 , ] = 0 and therefore (cid:2) [ G (5 , { , } ( z ) , M (5 , ] , M (5 , (cid:3) = (cid:2) [ G (5 , { , } ( z ) , M (5 , ] , M (5 , (cid:3) . In preparation for the multivariable generalization of the expression (4.26) for ∆ G ( n,p ) { , } ( z ),we briefly repeat the analysis of the previous section in the two-variable case p = n − y = z n − and z = z n − , F ( n,n − ( z, y ) ae = P ( n,n − ab M ( n,n − bc G ( n,n − { , } ( y ) cd G ( n,n − { , ,y } ( z ) de (4.27)and study the coaction of G ( n,n − { , ,y } ( z ) de . We will arrive at a compact form of thegenerating function for coactions (4.7) of polylogarithms G ( (cid:126)u ; z ) with labels u i in thethree-letter alphabet { , , y } . Again, the general coaction formula (4.1) leads to thesimple class of terms from deconcatenation of (cid:126)u that are explicit in (4.7), and we willelaborate on the additional terms in the ellipsis with some G ( . . . ; u i ) , u i ∈ { , , y } intheir second entry. In terms of generating functions G { , ,y } ( z ) = (cid:88) (cid:126)u ∈{ , ,y } × G ( (cid:126)u t ; z ) E u ,z E u ,z . . . E u w ,z , (4.28)with unspecified matrices E ,z , E ,z , E y,z , it remains to determine the W ( (cid:126)u | (cid:126)k | (cid:126)m ) com-prising products of matrices with rational coefficients in∆ G { , ,y } ( z ) ab = G { , ,y } ( z ) ac ⊗ G { , ,y } ( z ) cb + (cid:88) (cid:126)u ∈{ , ,y } × G ( (cid:126)u ; z ) (4.29) ⊗ (cid:88) (cid:126)k ∈ (2 N +1) × φ − ( f k f k . . . f k (cid:96) ) (cid:88) (cid:126)m ∈{ , } × G ( (cid:126)m ; y ) W ( (cid:126)u | (cid:126)k | (cid:126)m ) ac G { , ,y } ( z ) cb . From coactions at weight two with G ( . . . ; y ) in their second entry such as∆ G (1 , y ; z ) = 1 ⊗ G (1 , y ; z ) + G ( y ; z ) ⊗ G (1; z ) + G (1 , y ; z ) ⊗ − G (1; z ) ⊗ G (0; y ) + G (1; z ) ⊗ G (1; y ) − G ( y ; z ) ⊗ G (1; y ) , – 31 –ne can for instance read off W (0 |∅| 0) = W (0 |∅| 1) = 0 W ( y |∅| 0) = [ E ,z , E y,z ] (4.31) W (1 |∅| 0) = − W (1 |∅| 1) = W ( y |∅| 1) = [ E ,z , E y,z ] . The matrix products W ( (cid:126)u | (cid:126)k | (cid:126)m ) in the coaction can again be determined by imposing(4.27) and furthermore assuming that ∆ P ( n,n − = P ( n,n − ⊗ M ( n,n − and G ( n,n − { , } . In this setting, the ansatz (4.29) for the coaction of interesthas to satisfy G ( n,n − { , ,y } ( z ) ab ⊗ M ( n,n − bc G ( n,n − { , } ( y ) cd − G ( n,n − { , ,y } ( z ) cd ⊗ M ( n,n − ab G ( n,n − { , } ( y ) bc (4.32)= (cid:88) (cid:126)u ∈{ , ,y } × G ( (cid:126)u ; z ) ⊗ M ( n,n − ab G ( n,n − { , } ( y ) bc (cid:1) (cid:88) (cid:126)k ∈ (2 N +1) × (cid:126)m ∈{ , }× φ − ( f k f k . . . f k (cid:96) ) G ( (cid:126)m ; y ) W ( (cid:126)u | (cid:126)k | (cid:126)m ) cd . By isolating the coefficients of G ( (cid:126)u ; z ) ⊗ f k . . . f k (cid:96) G ( m , . . . , m j ; y ), we obtain a recur-sion for W ( (cid:126)u | (cid:126)k | (cid:126)m ) in the total number of letters in (cid:126)k and (cid:126)m . With the shorthandnotation E z ( (cid:126)u ) = E u w ,z . . . E u ,z E u ,z (4.33)and an expansion of G ( n,n − { , } ( y ) in terms of G ( (cid:126)m ; y ) E m j ,y . . . E m ,y E m ,y , the simplestexamples are [ E z ( (cid:126)u ) , ] ac = W ( (cid:126)u |∅|∅ ) ac [ E z ( (cid:126)u ) , M k ] ac = W ( (cid:126)u | k |∅ ) ac + ( M k ) ab W ( (cid:126)u |∅|∅ ) bc [ E z ( (cid:126)u ) , E m ,y ] ac = W ( (cid:126)u |∅| m ) ac + ( E m ,y ) ab W ( (cid:126)u |∅|∅ ) bc [ E z ( (cid:126)u ) , M k M k ] ac = W ( (cid:126)u | k , k |∅ ) ac + ( M k ) ab W ( (cid:126)u | k |∅ ) bc + ( M k ) ab W ( (cid:126)u | k |∅ ) bc + ( M k M k ) ab W ( (cid:126)u |∅|∅ ) bc (4.34)[ E z ( (cid:126)u ) , E m E m ] ac = W ( (cid:126)u |∅| m , m ) ac + ( E m ,y ) ab W ( (cid:126)u |∅| m ) bc + ( E m ,y ) ab W ( (cid:126)u |∅| m ) bc + ( E m ,y E m ,y ) ab W ( (cid:126)u |∅|∅ ) bc [ E z ( (cid:126)u ) , M k E m ] ac = W ( (cid:126)u | k | m ) ac + ( E m ,y ) ab W ( (cid:126)u | k |∅ ) bc + ( M k ) ab W ( (cid:126)u |∅| m ) bc + ( M k E m ,y ) ab W ( (cid:126)u |∅|∅ ) bc . The general formula can again be written in terms of deshuffles similar to (4.20). Notethat the extraction of these identities from (4.32) hinges on the fact that all polyloga-rithms are already in a fibration basis. – 32 –imilar to (4.21) and (4.22), the solution to the recursion furnished by (4.34) andhigher-weight generalizations features nested commutators, starting with W ( (cid:126)u |∅|∅ ) = 0 , W ( (cid:126)u | k , k |∅ ) = [[ E z ( (cid:126)u ) , M k ] , M k ] W ( (cid:126)u | k |∅ ) = [ E z ( (cid:126)u ) , M k ] , W ( (cid:126)u |∅| m , m ) = [[ E z ( (cid:126)u ) , E m ,y ] , E m ,y ] (4.35) W ( (cid:126)u |∅| m ) = [ E z ( (cid:126)u ) , E m ,y ] , W ( (cid:126)u | k | m ) = [[ E z ( (cid:126)u ) , M k ] , E m ,y ]and more generally (note the reversal of the commutators of the E m i ,y ) W ( (cid:126)u | k , k , . . . , k (cid:96) | m , m , . . . , m j ) (4.36)= [[ . . . [[ . . . [[ E z ( (cid:126)u ) , M k ] , M k ] , . . . , M k (cid:96) ] , E m j ,y ] , . . . , E m ,y ] , E m ,y ] . As in the transition from (4.21) to (4.24), we recover the generating series G { , ,y } ( z )by summing the combinations of E z ( (cid:126)u ) (defined in (4.33) and coming from W ( (cid:126)u | (cid:126)k | (cid:126)m ))and G ( (cid:126)u ; z ) over (cid:126)u ∈ { , , y } × . In the context of the F ( n,p ) with y = z n − and z = z n − ,this yields∆ G ( n,p ) { , ,y } ( z ) = G ( n,p ) { , ,y } ( z ) ⊗ G ( n,p ) { , ,y } ( z ) + (cid:88) k ∈ N +1 (cid:2) G ( n,p ) { , ,y } ( z ) , M ( n,p ) k (cid:3) ⊗ f k G ( n,p ) { , ,y } ( z )+ (cid:88) m ∈{ , } (cid:2) G ( n,p ) { , ,y } ( z ) , E ( n,p ) m ,y (cid:3) ⊗ G ( m ; y ) G ( n,p ) { , ,y } ( z )+ (cid:88) k ,k ∈ N +1 (cid:2) (cid:2) G ( n,p ) { , ,y } ( z ) , M ( n,p ) k (cid:3) , M ( n,p ) k (cid:3) ⊗ f k f k G ( n,p ) { , ,y } ( z ) (4.37)+ (cid:88) m ,m ∈{ , } (cid:2) (cid:2) G ( n,p ) { , ,y } ( z ) , E ( n,p ) m ,y (cid:3) , E ( n,p ) m ,y (cid:3) ⊗ G ( m , m ; y ) G ( n,p ) { , ,y } ( z )+ (cid:88) m ∈{ , } k ∈ N +1 (cid:2) (cid:2) G ( n,p ) { , ,y } ( z ) , M ( n,p ) k (cid:3) , E ( n,p ) m ,y (cid:3) ⊗ f k G ( m ; y ) G ( n,p ) { , ,y } ( z ) + . . . = (cid:88) (cid:126)m ∈{ , }× (cid:126)k ∈ (2 N +1) × (cid:2)(cid:2) . . . (cid:2)(cid:2) . . . (cid:2)(cid:2) G ( n,p ) { , ,y } ( z ) , M ( n,p ) k (cid:3) , M ( n,p ) k (cid:3) . . . , M ( n,p ) k (cid:96) (cid:3) , E ( n,p ) m j ,y (cid:3) . . . , E ( n,p ) m ,y (cid:3) , E ( n,p ) m ,y (cid:3) ⊗ f k f k . . . f k (cid:96) G ( m , m , . . . , m j ; y ) G ( n,p ) { , ,y } ( z ) . The sum over (cid:126)m and (cid:126)k in the last line can be conveniently absorbed into a generalizationof the notation (4.25) to X ⊗ ad L (cid:0) f k f k . . . f k r M k M k . . . M k r G ( m , m , . . . , m j ; y ) E m j ,y . . . E m ,y E m ,y (cid:1) Y = (cid:2)(cid:2) . . . (cid:2) [ . . . [[ X, M k ] , M k ] , . . . , M k r ] , E m j ,y (cid:3) , . . . E m ,y (cid:3) , E m ,y (cid:3) (4.38) ⊗ f k f k . . . f k r G ( m , m , . . . , m j ; y ) Y , – 33 –amely ∆ G ( n,p ) { , ,y } ( z ) = G ( n,p ) { , ,y } ( z ) ⊗ ad L (cid:0) M ( n,p ) G ( n,p ) { , } ( y ) (cid:1) G ( n,p ) { , ,y } ( z ) . (4.39)We have explicitly verified this to be the case order by order in the α (cid:48) -expansion, namelyup to and including α (cid:48) for both ( n, p ) = (6 , 1) and ( n, p ) = (7 , G ( n,p ) of polylogarithms in the F ( n,p ) can be inductively extended to any number ofunintegrated punctures. With the obvious generalization of (4.38) to several speciesof braid matrices E ( n,p ) z k ,z j in (3.50) and polylogarithms in the fibration bases specifiedbelow, our conjecture for the coaction properties of the constituents of (3.7) is∆ G ( n,p ) { , ,z j +1 ,z j +2 ,...,z n − } ( z j ) = G ( n,p ) { , ,z j +1 ,z j +2 ,...,z n − } ( z j ) (4.40) ⊗ ad L (cid:16) M ( n,p ) G ( n,p ) { , } ( z n − ) G ( n,p ) { , ,z n − } ( z n − ) . . .. . . G ( n,p ) { , ,z j +2 ,z j +3 ,...,z n − } ( z j +1 ) (cid:17) G ( n,p ) { , ,z j +1 ,z j +2 ,...,z n − } ( z j ) . These coaction formulae at j = 3 , , . . . , n − α (cid:48) -expansion (3.7) to obey the coaction formula (2.15) ofthe F ( n,p ) . In an order-by-order check of the coaction properties of the α (cid:48) -expansion,the individual cases of (4.40) are considerably simpler to verify than dealing with thecomplete expressions for F ( n,p ) at once. The simplest examples of (4.40) with j = n − j = n − n, p ) = (5 , , (6 , , (6 , 1) and (7 , 1) to theorders of α (cid:48) , α (cid:48) , α (cid:48) and α (cid:48) , respectively. In this section we study the analytic continuation of the functions F ( n,p ) ab ( z p +2 , . . . , z n − )while keeping the orthonormal basis of forms and cycles fixed. Previously, we havedefined this family of functions with a specific branch choice in mind: the branchconsistent with 0 = z < z p +2 < z p +3 < . . . < z n − < z n − = 1 (5.1)when all the punctures sit on the real line. This branch choice is implicit in our selec-tion of cycles, γ ( n,p ) a ; the regularized initial values for these functions, P ( n,p ) M ( n,p ) ; andexplicit in the order of path-ordered integration from these initial values – schemati-cally shown in (3.51) – which induces a fibration basis on the multiple polylogarithmsappearing in F ( n,p ) ( z p +2 , . . . , z n − ). This is the usual branch choice for the polylogarithms appearing in F ( n,p ) ( z p +2 , . . . , z n − ). – 34 –he key nontrivial example to keep in mind for this section is F (6 , ( z p +2 , z n − ) = F (6 , ( z , z ), where we have assumed 0 < z < z < { z < z } has to be seen not asa permutation of z and z but rather as a braiding of these punctures. Fortunately,the theory of the KZ equations provides a representation of the braid group acting oncertain solutions to these equations [88]. In what follows and in appendix B, we spellout how this representation furnishes a group action on the solution space in which ourfunctions F ( n,p ) ( z p +2 , . . . , z n − ) live. F (5 , ( z )A monodromy is, of course, an example of analytic continuation. Because all the F ( n,p ) ( z p +2 , . . . , z n − ) are themselves defined to be holomorphic functions, the solutionsof our KZ equations have certain prescribed monodromies . In the case of F (5 , ( z ) = P (5 , M (5 , G (5 , { , } ( z ), the monodromy is determinedsolely from the generating series of polylogarithms G (5 , { , } ( z ) in (3.24). For example,the monodromies for z going anticlockwise around 0 and 1 are given by M ,z G (5 , { , } ( z ) = exp(2 πiE (5 , ) G (5 , { , } ( z ) (5.2) M ,z G (5 , { , } ( z ) = Φ( E (5 , , E (5 , ) exp(2 πiE (5 , )Φ( E (5 , , E (5 , ) − G (5 , { , } ( z ) , where the exponentials generalize the weight-one identities M ,z G (0; z ) = G (0; z ) + 2 πi , M ,z G (1; z ) = G (1; z ) + 2 πi . (5.3)Throughout this section, we shall use the shorthand E ( n,p ) ij = ( e ( n,p ) ij ) t (5.4)for transposed braid matrices, not to be confused with the special combinations E ( n,p ) i,z j or E ( n,p ) z i ,z j in (3.40) with z -variables appearing in the subscript. In the second line of(5.2), the expression Φ( E (5 , , E (5 , ) = G (5 , { , } ( z =1) (5.5)is a special case of the Drinfeld associator whose expansion in terms of MZVs andarbitrary non-commutative indeterminates E , E is given by [89]Φ( E , E ) = ∞ (cid:88) r =0 (cid:88) a ,a ,......,ar ∈{ , } G ( a r , . . . , a , a ; 1) E a E a . . . E a r One can build solutions to the KZ equations with no monodromy, by using certain non-holomorphicinitial values, see the discussion of sphere integrals and single-valued polylogarithms in section 6 andfor instance [41, 46]. – 35 – 1 + ζ [ E , E ] − ζ [ E + E , [ E , E ]] + . . . , (5.6)in lines with (3.24). Its inverse can be written in two different ways:Φ( E , E ) − = ∞ (cid:88) r =0 ( − r (cid:88) a ,a ,......,ar ∈{ , } G ( a , a , . . . , a r ; 1) E a E a . . . E a r = Φ( E , E ) . (5.7)Thus, the monodromy of F (5 , ( z ) is given by [74] M ,z P (5 , M (5 , G (5 , { , } ( z ) = P (5 , M (5 , exp(2 πiE (5 , ) G (5 , { , } ( z ) , (5.8)with a similar expression for M ,z P (5 , M (5 , G (5 , { , } ( z ). Also for F ( n,p ) ( z p +2 , . . . , z n − )at more general n, p , the monodromies are clearly determined by the generating seriesof polylogarithms, i.e. the G ( n,p ) in (3.49). The monodromies of generating functionsof multiple polylogarithms, as studied in this work, have already been spelled out indetail in [74]. From now on we will focus on the analytic continuation of the func-tions F ( n,p ) ( z p +2 , . . . , z n − ) from z i < z i +1 to branches with z i +1 < z i , which are notmonodromies. F (6 , ( z , z )We shall now study the analytic continuation of F ( n,p ) ( z p +2 , . . . , z n − ) from 0 < z p +2 1. Further examples of analytic continuations to one of z , z being < > n, p ) = (5 , 1) example in appendix B.2. F (6 , and their coaction As a side effect of (5.21), it determines a formal initial value for F (6 , adapted to path-ordered integration in the order shown in (5.12). On top of the Q -linear combinationsof MZVs seen in the α (cid:48) -expansion (3.18), (3.17) of the initial values P (6 , M (6 , , the α (cid:48) -expansion of the initial value in (5.21) involves powers of iπ . Hence, we reorganize P (6 , M (6 , (cid:2) X (6 , ( σ , ) (cid:3) − = ˜ P (6 , ˜ M (6 , (5.23)with an expansion of ˜ P (6 , , ˜ M (6 , in terms of alternative 3 × P (6 , w , ˜ M (6 , w whose entries are still degree- w polynomials in s ij with rational coefficients:˜ P (6 , = + iπ ˜ P (6 , + ζ ˜ P (6 , + iπζ ˜ P (6 , + ζ ˜ P (6 , + iπζ ˜ P (6 , + ζ ˜ P (6 , + O ( s ij ) In particular, (5.20) is consistent with the numerics of PolyLogTools if arg( z ) > arg( z ). – 39 – + ∞ (cid:88) k =1 ( iπζ k − ˜ P (6 , k − + ζ k ˜ P (6 , k ) , (5.24)˜ M (6 , = φ − ∞ (cid:88) r =0 ∞ (cid:88) k ,k ,...,k r =1 f k +1 f k +1 . . . f k r +1 ˜ M (6 , k +1 ˜ M (6 , k +1 . . . ˜ M (6 , k r +1 . The ˜ P (6 , w -matrices associated with odd w do not have any counterparts in the ex-pansion of P (6 , . Moreover, the ˜ P (6 , k and ˜ M (6 , k +1 differ from the P (6 , k and M (6 , k +1 determined by (3.45) and (3.46) as exemplified in appendix B.4.Still, the coefficients of the MZVs and their products with iπ in (5.23) are expectedto be compatible with the coaction principle in the sense that∆(˜ P (6 , ab ˜ M (6 , bd ) = ˜ P (6 , ab ˜ M (6 , bc ⊗ ˜ M (6 , cd , (5.25)which we have tested up to and including the order of α (cid:48) . At the level of the MZVs thatsolely arise from words in f k +1 , the coaction (5.25) is again equivalent to an expansion˜ M (6 , = + ζ ˜ M (6 , + ζ ˜ M (6 , + 12 ζ ˜ M (6 , ˜ M (6 , + ζ ˜ M (6 , + ζ ζ ˜ M (6 , ˜ M (6 , + 15 ζ , [ ˜ M (6 , , ˜ M (6 , ] + O ( s ij ) (5.26)as in (3.10), where the commutator [ ˜ M (6 , , ˜ M (6 , ] vanishes just like [ M (6 , , M (6 , ] = 0.In other words, ζ , drops out from ˜ M (6 , in the same way as it does from M (6 , . Infact, we have checked that all irreducible MZVs of depth ≥ ≤ 11 alreadycancel from the individual Drinfeld associators in (5.19).Moreover, already the matrix-multiplicative structure on the right-hand side of(5.23) is not manifest on its left-hand side. Hence, the fact that the coefficients of iπζ , iπζ and iπζ ζ in (5.23) are given by matrix products ˜ P (6 , ˜ M (6 , , ˜ P (6 , ˜ M (6 , and˜ P (6 , ˜ M (6 , , respectively, can be viewed as non-trivial checks of the coaction principle. F ( n,p ) The examples in section 5.2 have set the stage to describe the analytic continuation of F ( n,p ) ( z p +2 , z p +3 , . . . , z n − ). The simplest analytic continuation of these functions wasdescribed in (5.18) and (5.19) as a group action of certain generators σ , . The groupin question is B N , the braid group of N strands, acting on the N = n − p unintegrated Doing a complete turn around SL(2 , C )-fixed punctures performs a monodromy as for instancein (5.2). These operations can also be described as part of a braid group. See Appendix B for moredetails. – 40 –unctures. The braid group B N can be defined as the non-commutative group withgenerators σ i := σ i,i +1 , where 1 ≤ i ≤ N − 1, that satisfy the relations [92] σ i σ j = σ j σ i for | i − j | ≥ , (5.27) σ i σ i +1 σ i = σ i +1 σ i σ i +1 for 1 ≤ i ≤ N − . For convenience, we will label the generators according to the punctures, i.e. σ i,i +1 denotes the generator that interchanges punctures z i and z i +1 via braiding, with z i +1 going around z i counterclockwise.We will now describe the group action of a generator of the braid group σ i,i +1 on F ( n,p ) ( z p +2 , z p +3 , . . . , z n − ). This corresponds to performing a change of branch fromthe branch consistent with0 = z < z p +2 < z p +3 < · · · < z i < z i +1 < · · · < z n − < z n − = 1 (5.28)when all the punctures lie on the real line, into a branch consistent with0 = z < z p +2 < z p +3 < · · · < z i +1 < z i < · · · < z n − < z n − = 1 . (5.29)Now, the analytic continuation of F ( n,p ) = P ( n,p ) M ( n,p ) G ( n,p ) ( z p +2 , . . . , z i , z i +1 , . . . , z n − )with G ( n,p ) ( . . . ) comprising all the factors of G ( n,p ) in (3.7) is given by a matrix actingon the generating series of polylogarithms , σ i,i +1 F ( n,p ) = P ( n,p ) M ( n,p ) X ( n,p ) ( σ i,i +1 ) G ( n,p ) ( z p +2 , . . . , z i , z i +1 , . . . , z n − ) (5.30)=: P ( n,p ) M ( n,p ) ˜ G ( n,p ) ( z p +2 , . . . , z i +1 , z i , . . . , z n − ) , where X ( n,p ) ( σ i,i +1 ) is given by X ( n,p ) ( σ i,i +1 ) = Φ (cid:32) E ( n,p )1 ,i +1 + i − (cid:88) j = p +2 E ( n,p ) j,i +1 , E ( n,p ) i,i +1 (cid:33) exp (cid:16) iπE ( n,p ) i,i +1 (cid:17) × Φ (cid:32) E ( n,p ) i,i +1 , E ( n,p )1 ,i + i − (cid:88) j = p +2 E ( n,p ) j,i (cid:33) . (5.31)The ( n, p ) = (6 , 1) cases of these expressions for σ i,i +1 F ( n,p ) and X ( n,p ) ( σ i,i +1 ) can befound in (5.22) and (5.19), respectively. Before the analytic continuation in (5.30), one While this is a known formula in the literature, it is not usually written down explicitly. Anexplicit version of it can be found in Proposition 5.1 of [93] for the genus 1 case, which apparently hasthe same formula. – 41 –an translate the rewriting F ( n,p ) = P ( n,p ) M ( n,p ) [ X ( n,p ) ( σ i,i +1 )] − ˜ G ( n,p ) into a modifiedinitial value ˜ P ( n,p ) ˜ M ( n,p ) = P ( n,p ) M ( n,p ) [ X ( n,p ) ( σ i,i +1 )] − (5.32)as done in section 5.3 at ( n, p ) = (6 , P ( n,p ) ˜ M ( n,p ) to inherit the coactionproperties of F ( n,p ) , i.e. to generalize (5.25) to arbitrary n and p . Accordingly, the α (cid:48) -expansion of ˜ M ( n,p ) will share the structure of the leading-order terms in (5.26), andthe α (cid:48) -expansion of ˜ P ( n,p ) will involve odd powers of iπ as in (5.24).The image ˜ G ( n,p ) ( z p +2 , . . . , z i +1 , z i , . . . , z n − ) under σ i,i +1 describes the path-orderedintegration of the KZ form Ω ( n,p ) in d F ( n,p ) = Ω ( n,p ) F ( n,p ) , with an initial value equalto the identity, and along the path(0 , , . . . , → (0 , . . . , , z n − ) → (0 , . . . , , z n − , z n − ) → . . . → (0 , . . . , , , z i +2 , . . . , z n − ) → (0 , . . . , , z i , , z i +2 , . . . , z n − ) (5.33) → (0 , . . . , , z i , z i +1 , z i +2 , . . . , z n − ) → (0 , . . . , , z i − , z i , z i +1 , z i +2 , . . . , z n − ) → . . . → ( z p +2 , z p +1 , . . . z n − ) . Both the matrices that enter into the definition of ˜ G ( n,p ) ( z p +2 , . . . , z i +1 , z i , . . . , z n − ) andthe fibration basis of its component polylogarithms respect this integration order above.Equivalently, we can define ˜ G ( n,p ) ( z p +2 , . . . , z i +1 , z i , . . . , z n − ) to be given by:˜ G ( n,p ) ( z p +2 , . . . , z i +1 , z i , . . . , z n − ) = G ( n,p ) ( z p +2 , . . . , z i , z i +1 , . . . , z n − ) (cid:12)(cid:12) i ↔ i +1 , (5.34)where i ↔ i +1 means to interchange z i with z i +1 and E i,j with E i +1 ,j everywhere, butwithout modifying the Mandelstam variables in their entries. In the case of ( n, p ) =(6 , n, p ) = (7 , 1) cases of (5.30) and (5.31) for σ , and σ , up to and including α (cid:48) . For these explicit checks, changes of fibration bases wereperformed via PolyLogTools [90], with the sign of iπ as in the last term of (5.10) andthe analogous identity with ( z , z ) → ( z , z ) to take the orientation of the braidinginto account.In conclusion, the key achievement in this section is to spell out the action of theelementary braid σ i,i +1 involving neighboring punctures on F ( n,p ) . Since the braid groupof N strands, B N , is generated by these σ i,i +1 , the results of this section determine theanalytic continuation due to arbitrary braiding of the punctures. Further examples ofanalytic continuation can be found in Appendix B.– 42 – Sphere integrals This section is dedicated to sphere integrals over the forms ω ( n,p ) a of section 2 and theircomplex conjugates. When interpreting the F ( n,p ) ab as open-string integrals with a subsetof the vertex-operator insertions integrated out, the sphere integrals in this section canbe viewed are their closed-string counterparts. Moreover, they are directly applicableto computations of correlation functions in two-dimensional conformal field theories aswill be further elaborated on in section 7.We will express the α (cid:48) -expansion of the sphere integrals in this section both assingle-valued maps of the F ( n,p ) ab and as Kawai–Lewellen–Tye (KLT) formulae involvingproducts of open-string integrals and their complex conjugates. We will propose twoprescriptions for computing the entries of the KLT matrix and its inverse. The latterwill be given in terms of combinatorial rules describing adjacency properties of Stasheffpolytopes associated to each integration cycle, while the former will be an explicitexpression in terms of polynomials of trigonometric functions. The sphere integrals of interest in this section take the form (cid:90) C ( n,p ) (cid:16) p +1 (cid:89) k =2 d z k (cid:17) | KN ( n,p ) | ˆ ω ( n,p ) a ˆ ω ( n,p ) b , (6.1)where d z = i d z ∧ d¯ z and a, b = 1 , , . . . , d ( n,p ) independently run over the bases offorms ω ( n,p ) a = ˆ ω ( n,p ) a (cid:81) p +1 j =2 d z j specified in section 2.3. For p = n − 3, we recover thesphere integrals of closed-string amplitudes which are known to be single-valued mapsof open-string integrals F ( n,n − ab if ˆ ω ( n,p ) a are replaced by suitably chosen Parke–Taylorforms [7, 47–51, 56]:sv F ( n,n − ab = 1 π n − (cid:90) C ( n,n − (cid:16) n − (cid:89) k =2 d z k (cid:17) | KN ( n,n − | ˆ ν ( n,n − a ˆ ω ( n,n − b , (6.2)with ˆ ν ( n,n − a = ( − n − z ,n − ρ a ( z , z , . . . z n − ,n − z n − ,n − ) . (6.3)The ˆ ν ( n,n − a are SL(2 , C )-fixed antiholomorphic Parke–Taylor factors, furnish the Betti–de Rham duals [56, 94, 95] to disk orderings of the F ( n,n − ab and are indexed by permu-tations ρ a ∈ S n − of the labels { , , . . . , n − } in lexicographic ordering.– 43 –ne of the goals of this section is to extend (6.2) to generic p , i.e., to spell out theforms ˆ ν ( n,p ) a that generalize (6.3) to the Betti–de Rham dual of the cycles γ ( n,p ) a withan arbitrary number of integrated and unintegrated punctures. For each collection ofadjacent integrated punctures z i , z i , . . . , z i k located between unintegrated ones z b , z c ,the forms ˆ ν ( n,p ) a pick up a factor as on the right-hand side of (6.3), i.e., { z b < z i < z i < . . . < z i k < z c } ↔ ( − k z b,c z b,i z i ,i z i ,i . . . z i k − ,i k z i k ,c . (6.4)After combining the contributions from all integrated and unintegrated punctures, oneobtains the basis of ν ( n,p ) a given in (2.26) which reduces to (6.3) in the special case of p = n − 3. This will be further illustrated through the examples at various ( n, p ) in thenext subsections.Given the d ( n,p ) forms ν ( n,p ) a defined in this way, we claim that a basis of sphereintegrals (6.1) can be computed from the single-valued map acting on the MZVs andpolylogarithms in the α (cid:48) -expansion of F ( n,p ) ab : sv F ( n,p ) ab = 1 π p (cid:90) C ( n,p ) (cid:16) p +1 (cid:89) k =2 d z k (cid:17) | KN ( n,p ) | ˆ ν ( n,p ) a ˆ ω ( n,p ) b = (cid:104) ν ( n,p ) a | ω ( n,p ) b (cid:105) . (6.5)The single-valued map is compatible with the product of MZVs and polylogarithmsand can be evaluated separately for each factor in the α (cid:48) -expansion of F ( n,p ) ab in (3.7).For example, single-valued MZVs relevant for sv P ( n,p ) = 1 and sv M ( n,p ) have beenintroduced in [3, 46]: Their simplest cases include sv ζ k +1 = 2 ζ k +1 , sv ζ k = 0 , sv ζ , = − ζ ζ , (6.6)and the f -alphabet admits the closed formula (with i , . . . , i r ∈ N +1) sv f N f i f i . . . f i r = δ N, r (cid:88) j =0 f i j . . . f i f i (cid:1) f i j +1 f i j +2 . . . f i r . (6.7) The notation (cid:104) ν | ω (cid:105) with an antiholomorphic form ν will always refer to (6.5) as opposed to theright-hand side of (2.23), which takes two holomorphic forms instead. Strictly speaking, the single-valued map is only well-defined in the setting of motivic MZVs. Wewill informally drop the superscript of ζ m n ,n ,...,n r in (6.6) and use the same notation sv for the single-valued map of motivic MZVs and their images in the f -alphabet in (6.7). The conventions of this work differ from those of [3, 46] by A ⊗ B → B ⊗ A and therefore by areversal f k +1 f k +1 . . . f k r +1 (cid:55)→ f k r +1 . . . f k +1 f k +1 . Accordingly, (6.7) features a reversal in thefirst part f i j . . . f i f i of the deconcatenated word f i f i . . . f i r on the right-hand side and not in thesecond part f i j +1 f i j +2 . . . f i r as seen in the references. – 44 –he expansion coefficients of sv G ( n,p ) { , } ( z n − ) are single-valued polylogarithms in onevariable [52] that includesv G (0; z ) = sv log z = log | z | , (6.8)sv G (1; z ) = sv log(1 − z ) = log | − z | , as well as sv G (0 , z ) = − sv Li ( z )= G (0 , z ) + G (0; z ) G (1; z ) + G (1 , z )= − Li ( z ) + Li (¯ z ) + log(1 − ¯ z ) log | z | (6.9)sv G (1 , z ) = sv G (0; z ) sv G (1; z ) − sv G (0 , z )sv G ( a, a ; z ) = 12 sv G ( a ; z ) , andsv G (0 , , , z ) = G (0 , , , z ) + G (0 , , z ) G (1; z ) + G (0 , z ) G (1 , z ) (6.10)+ G (0; z ) G (1 , , z ) + G (1 , , , z ) + 2 ζ G (1; z ) . Single-valued polylogarithms in multiple variables that enter the remaining sv G ( n,p ) ... can be found in [41, 53].It should be possible to derive (6.5) from the inductive techniques of [49, section3.3]. However, this is not a fully rigorous proof since the use of the single-valued maprelies on transcendentality conjectures on MZVs. The techniques of Brown and Dupont[51, 56] in turn should allow for a proof without any such assumptions. An second goal of this section is to write the sphere integrals (6.5) as bilinears inthe open-string integrals F ( n,p ) ab and their complex conjugates, following the Kawai–Lewellen–Tye (KLT) formula for the case p = n − p = n − all ( n − / ( n − p − C ( n,p ) , including those outside of the d ( n,p ) basis. To be precise, introducing J ( n,p ) ab = (cid:90) γ ( n,p ) a (cid:16) p +1 (cid:89) k =2 d z k (cid:17) KN ( n,p ) ˆ ν ( n,p ) b , (6.11)we have sv F ( n,p ) ab = (cid:18) − πi (cid:19) p ( n − n − p − (cid:88) c,d =1 e iπφ cd J ( n,p ) da F ( n,p ) cb . (6.12)– 45 –oth of c, d run over the ( n − n − p − cycles γ ( n,p ) c that impose the ordering (2.9) of theunintegrated punctures z , z p +2 , . . . , z n − but allow the integrated ones z , z , . . . , z p +1 to be in ( −∞ , 0) or (1 , + ∞ ) besides the standard interval (0 , 1) of the d ( n,p ) -elementbasis. The only subtlety in (6.12) comes from the fact that each integral on the right-hand side comes with a specific phase of the Koba–Nielsen factor prescribed in (2.4).This is corrected by the explicit phase factor e iπφ cd , where φ cd = (cid:88) ≤ i 3) and (2 , n = 5, is always held fixed. ( b ) Illustration of theindependence of the phase (in this case s ij + s jk + s ik ) on the way of drawing straightlines, as long as they only intersect pairwise. For practical purposes it is beneficial to eliminate redundant terms from (6.12) toinvolve only a sum over the minimal d ( n,p ) basis. To this end we construct dual cycles β ( n,p ) a such that I ( n,p ) ab = (cid:90) β ( n,p ) a (cid:16) p +1 (cid:89) k =2 d z k (cid:17) KN ( n,p ) ˆ ν ( n,p ) b (6.14)reduce to δ ab in the α (cid:48) → I ( n,p ) ab in a basis of ˜ J ( n,p ) cb ,which are the integrals from (6.11) but with a shifted basis of cycles ˜ γ ( n,p ) a (defined– 46 –elow in (6.19)) instead of γ ( n,p ) a : I ( n,p ) ab = d ( n,p ) (cid:88) c =1 S α (cid:48) ( ρ a | ρ c ) ˜ J ( n,p ) cb , (6.15)for example by the use of monodromy relations. We will describe two distinct prescrip-tions for deriving the coefficients S α (cid:48) ( ρ a | ρ c ), which we will refer to as the generalizedKLT kernel. As is known from the p = n − 3, the inverse of the matrix S α (cid:48) ( ρ a | ρ c )are intersection numbers of cycles γ ( n,p ) a [18, 98, 99]. In fact, this is a general featureof complex integrals (see [100] and [101, Section 6]), which allows us to extend thisprescription to all other values of p . Intersection numbers are given by combinatorialrules describing how the cycles γ ( n,p ) a intersect one another in the moduli space. Basedon this computation and direct manipulations using monodromy relations, we proposean explicit recursive expression for the KLT matrix S α (cid:48) ( ρ a | ρ c ) and verify its correctnessup to n = 8 with any p .Putting everything together, the resulting expression is the second major claim ofthis section: sv F ( n,p ) ab = 1 π p d ( n,p ) (cid:88) c =1 I ( n,p ) ca F ( n,p ) cb = 1 π p d ( n,p ) (cid:88) c,d =1 ˜ J ( n,p ) da S α (cid:48) ( ρ c | ρ d ) F ( n,p ) cb , (6.16)which is the generalization of the KLT formula to arbitrary ( n, p ). In this subsection we describe combinatorial rules for computing the intersection num-bers of twisted cycles H ( n,p ) ab = (cid:104) γ ( n,p ) a | γ ( n,p ) b (cid:105) (6.17)in terms of adjacency properties of Stasheff polytopes (or associahedra) [55] tiling thereal slice of the configuration space Re C ( n,p ) . In fact, it will prove rewarding to constructthe d ( n,p ) × d ( n,p ) matrix ˜ H ( n,p ) ab = (cid:104) γ ( n,p ) a | ˜ γ ( n,p ) b (cid:105) (6.18) Our terminology is not to be confused with the generalized KLT kernel in [97]. This referencegeneralizes the field-theory version of the KLT kernel at p = n − n − × ( n − n − × ( n − p (cid:54) = n − 3, and it would beinteresting to also derive the recursion relations (6.62) for its entries from the S-bracket of [97]. Wewould like to thank Carlos Mafra for discussions on this point. – 47 –ith an alternative basis of cycles ˜ γ ( n,p ) b in the second entry, where some of the punctures j i are integrated over subsets of ( −∞ , γ ( n,p ) (cid:126)B,(cid:126)j ↔ ρ (cid:126)B,(cid:126)j = ( B , j , B , j , . . . , B p , j p , B p +1 , n − , n − , n ) . (6.19)In this setup, the above KLT matrix is given by S α (cid:48) ( ρ a | ρ b ) = ( ˜ H ( n,p ) ) − ba . (6.20)For a given ( n, p ), the cycles γ ( n,p ) a are in bijection to n -gons with edges labelledaccording to the given ordering ρ a of labels { , , . . . , n } . We will consider all possiblepermutations ρ a where the unintegrated (fixed) labels (1 , p +2 , p +3 , . . . , n ) always ap-pear in this specific order. By an extension of the combinatorics describing the modulispace M ,n [74, 102], adjacency properties on Re C ( n,p ) can be described by drawingtessellations of decorated n -gons. A flip move corresponds to drawing a single chord c and reflecting one side of the chord as illustrated in figure 3. A given chord is ad- k k +1 k +2 n k k +1 k +2 nc c flip ←→ Figure 3 : Example of an admissible flip by a chord c between n -gons labelled by(1 , , . . . , k, k +1 , k +2 , . . . , n ) and ( k, . . . , , , k +1 , k +2 , . . . , n ). Edges corresponding tounintegrated punctures are indicated in red. Flipped side of the n -gon contains only onered edge, which makes the flip admissible. missible only if the result of the flip leaves the fixed labels (1 , p +2 , p +3 , . . . , n ) in thesame order. In other words, the side of the n -gon we flip has to have exactly 0 or1 edges corresponding to fixed punctures. (In particular, for p = n − c we associate the Mandelstam variable s c = α (cid:48) (cid:32)(cid:88) i ∈ F c k i (cid:33) , (6.21)where F c is the set of edges being flipped (2 ≤ | F c | ≤ n − s c = s ...k . If a chord is admissible, it labels an element of the boundary More precisely, here and in the following, whenever we talk about boundaries of cycles, we mean ∂π − (( γ ( n,p ) a ) o ), the boundary of the closure of the interior of γ ( n,p ) a after the resolution of exceptionaldivisors of the configuration space C ( n,p ) by a blowup map π − (see [103]). – 48 –f γ ( n,p ) a and the whole boundary structure is governed by how these chords fit intotessellations. The cycles γ ( n,p ) a are combinatorially isomorphic to Stasheff polytopesand their direct products. A given tessellation T a associated to γ ( n,p ) a is admissible if it includes only admissiblechords { c (cid:96) } (cid:96) =1 , ,..., | T a | ∈ T a , where | T a | ∈ { , , . . . , p } is the number of chords used, andone imposes that these chords do not cross. Following [104, 105] we find the followingformula for self-intersection numbers: (cid:104) γ ( n,p ) a | γ ( n,p ) a (cid:105) = (2 i ) p (cid:88) T a (cid:89) c (cid:96) ∈ T a e πis c(cid:96) − , (6.22)where the sum goes over all admissible tessellations (for | T a | = 0 the set of chordsis empty and the term contributing to the sum is 1) and we introduce the followingshorthand t ij... = e πis ij... − . (6.23)The geometric understanding of this formula is that a given tessellation T a with | T a | chords labels the codimension- | T a | boundary of γ ( n,p ) a . For example, the terms withmaximum number of chords, max | T a | , label its vertices, and those with a single chordlabel its facets. Tessellations describe the combinatorics of how these elements of theboundary fit together.For example, at p = 1: (cid:104) γ ( n, ··· n | γ ( n, ··· n (cid:105) = 2 i (cid:18) t + 1 t (cid:19) = sin( π ( s + s ))sin( πs ) sin( πs ) . (6.24)Here and below, to make the connection with n -gons easier to follow, we label the cycles γ ( n,p ) a directly by their permutation ρ a and underline the integrated (unfixed) labels.For p = 2 the answer depends on the number of fixed punctures separating the twounfixed ones: (cid:104) γ ( n, ··· n | γ ( n, ··· n (cid:105) = − (cid:18) t + 1 t + 1 t + 1 t + 1 t (6.25)+ 1 t t + 1 t t + 1 t t + 1 t t + 1 t t (cid:19) , (cid:104) γ ( n, ··· n | γ ( n, ··· n (cid:105) = − (cid:18) t + 1 t + 1 t + 1 t + 1 t (6.26)– 49 – 1 t t + 1 t t + 1 t t + 1 t t + 1 t t (cid:19) , and (cid:104) γ ( n, ...k k +1 ··· n | γ ( n, ··· n (cid:105) = − (cid:18) t + 1 t (cid:19) (cid:18) t k + 1 t ,k +1 (cid:19) (6.27)for k ≥ 5. Factorization of the final example reflects the fact that the correspondingchamber is combinatorially a square (a product of two one-dimensional Stasheff poly-topes), while the first two were two-dimensional Stasheff polytopes, combinatoriallypentagons. A more interesting case is the intersection number of distinct cycles, which geometricallydescribes the boundary of their intersection in the moduli space. If two n -gons cannotbe transformed into one another with a series of admissible flips, their intersectionnumber is zero. Otherwise, associated to γ ( n,p ) a and γ ( n,p ) b , there exists a unique set ofchords T ab that flips one into another in the minimal number of steps, as illustratedin figure 4. The resulting n -gon is tessellated into a number of smaller polygons P ab . c ←→ c c c ( a ) ( b )12 3 4 56 c c Figure 4 : ( a ) Example of the tessellation T ab by admissible chords c , c for γ (6 , and γ (6 , . We have s c = s and s c = s . ( b ) The set of chords can be determined,following [98, 106], by embedding the second n -gon (blue) inside the first one by connectingmidpoints of its edges in the order ρ b . Provided this can be done without self-overlaps, thechords are determined by places where the second n -gon folds over (admissibility criterianeed to be checked separately). For each P ab we can define the set of admissible tessellations T P ab that have chordsonly within P ab (admissibility is determined with respect to the original n -gon). Theformula for the intersection number becomes (cid:104) γ ( n,p ) a | γ ( n,p ) b (cid:105) = ( − w ( ρ a | ρ b )+1 (cid:32) (cid:89) c (cid:96) ∈ T ab πs c (cid:96) ) (cid:33) (cid:89) P ab (2 i ) max | T Pab | (cid:88) T Pab (cid:89) c (cid:96) ∈ T Pab e πis c(cid:96) − , (6.28)– 50 –here w ( ρ a | ρ b ) is the relative winding number of the two permutations as defined in[98, Appendix A]. The proof of this formula is analogous to those in [18, 104]. Thedefinition collapses to (6.22) when a = b because T ab is the empty set and P ab is simplythe original n -gon, so T P ab = T ab and max | T P ab | = p .For example, let us compute the intersection number of γ (6 , and γ (6 , , whichwe already know is non-zero from figure 4. We found two chords defining T ab , whichdissects the original 6-gon into three polygons we will call { P , P , P } (cid:51) P ab below.For each polygon we find that there is exactly one admissible tessellation, see figure 5.In this case the winding number is w (123456 | P P P T P = T P = T P = , , Figure 5 : Example polygon decomposition needed for the computation of the intersectionnumber in (6.29). Out of the three polygons, P and P are triangles and hence admitonly one admissible tessellation. The last one, P , has three possible tessellation, but thelast two are not admissible as they separate two red edges on either side of the additionalchord. answer: (cid:104) γ (6 , | γ (6 , (cid:105) = ( − sin( πs ) sin( πs ) × × × . (6.29)As another example, we can consider the intersection of γ (6 , and γ (6 , , whichonly differs from (6.29) by the fact that the label 4 is now integrated. Hence, all thecomputations are identical, except for the fact that the final two tessellations in T P offigure 5 are now admissible. We therefore find (cid:104) γ (6 , | γ (6 , (cid:105) = ( − sin( πs ) sin( πs ) × × × i (cid:18) t + 1 t (cid:19) = − sin( π ( s + s ))sin( πs ) sin( πs ) sin( πs ) sin( πs ) . (6.30)Another way of stating this result is that the intersection of the two cycles is a one-dimensional Stasheff polytope, while in (6.29) it was a zero-dimensional one (a point).We will provide more examples in the following subsections. Alternative prescrip-tions for computing intersection numbers (cid:104) γ ( n,p ) a | γ ( n,p ) b (cid:105) were given in [107, 108]. The– 51 –dvantage of our approach is that it provides combinatorial insight in terms of tessel-lations of n -gons (or equivalently planar trees). p = 1Let us start with an instructive case of p = 1, which will inspire the choice of bases ofcycles for the p > d ( n, = n − γ ( n, = { z ∈ R | z < z < z < · · · < z n } , (6.31) γ ( n, a = { z ∈ R | z < · · · < z a +1 < z < z a +2 < · · · < z n } for 2 ≤ a ≤ n − , or in the notation introduced above γ ( n, a = (cid:16) γ ( n, ...n , γ ( n, ...n , . . . , γ ( n, ...n − , ,n − ,n (cid:17) a . (6.32) Let us compute the intersection matrix H ( n, ab in (6.17) explicitly. For the n -gon associ-ated to γ ( n, ...j k... , only two chords c s j , c s k are admissible: precisely those correspondingto the Mandelstam invariants s j , s k . Hence, we conclude that elements in the basis(6.32) which are more than one element apart have no common chords and therebyzero intersection number.It remains to consider the other two cases. For the self-intersection number, wealready computed the answer in (6.24), which after relabeling and expressing in termsof trigonometric functions gives (cid:104) γ ( n, ...j k... | γ ( n, ...j k... (cid:105) = sin( π ( s j + s k ))sin( πs j ) sin( πs k ) = cot( πs j ) + cot( πs k ) . (6.33)Two adjacent cycles γ ( n, ...j k,k +1 ,... and γ ( n, ...jk ,k +1 ,... share a single chord c s k , which decom-poses the n -gon into a triangle P and an ( n − P . Both of these have only oneadmissible empty tessellation, which is given by the polytope itself, T P i = P i . Togetherwith the fact that the relative winding number of the two permutations is 2, we have (cid:104) γ ( n, ...j k,k +1 ,... | γ ( n, ...jk ,k +1 ,... (cid:105) = − πs k ) = − csc( πs k ) (6.34)and the same result for (cid:104) γ ( n, ...jk ,k +1 ,... | γ ( n, ...j k,k +1 ,... (cid:105) by hermitian symmetry of the intersec-tion product. Organizing these results into an ( n − × ( n − 3) symmetric tridiagonal– 52 –atrix we obtain H ( n, = cot( πs )+ cot( πs ) − csc( πs ) 0 · · ·− csc( πs ) cot( πs )+ cot( πs ) − csc( πs ) · · · − csc( πs ) cot( πs )+ cot( πs ) · · · ... ... ... . . . . (6.35)We find that these matrices have the inverse with entries (recall that s ii = 0):( H ( n, ) − ab = sin( π (cid:80) min( a,b )+1 i =1 s i ) sin( π (cid:80) n − i =max( a,b )+2 s i )sin( π (cid:80) n − i =1 s i ) . (6.36) In spite of the appeal of a symmetric basis choice for the entries of ˜ H ( n,p ) ab , we found amore convenient choice of bases that simplifies the entries of the KLT matrix. Let usdenote the corresponding intersection matrix by˜ H ( n, ab = (cid:104) γ ( n, a | ˜ γ ( n, b (cid:105) , (6.37)where the right basis is now taken to be˜ γ ( n, a = (cid:16) γ ( n, ...n , γ ( n, ...n , . . . , γ ( n, ...n − , ,n − ,n − ,n (cid:17) a . (6.38)We simply “shifted” the position of 2 by one slot to the left compared to (6.32) whicheffectively moves the diagonals of the intersection matrix and leads to the new form˜ H ( n, = − csc( πs ) cot( πs )+ cot( πs ) − csc( πs ) · · · − csc( πs ) cot( πs )+ cot( πs ) · · · − csc( πs ) · · · ... ... ... . . . . (6.39)This fact is crucial in simplifying the computation of the inverse, which can easily seento take the upper-triangular form( ˜ H ( n, ) − = − sin( πs ) sin( π ( s + s )) sin( π ( s + s + s )) · · · πs ) sin( π ( s + s )) · · · πs ) · · · ... ... ... . . . , (6.40)or more explicitly ( ˜ H ( n, ) − ab = − sin (cid:18) π b +1 (cid:88) i = a +1 − δ a s i (cid:19) . (6.41)– 53 –he entries vanish for b < a and are polynomial in terms of the sines, i.e. do not haveany analogue of the denominator in (6.36). This choice of bases will inform the choicesfor general ( n, p ).The matrix entries in (6.41) can be used to compute β ( n, a cycles in terms of thebasis ˜ γ ( n, a needed in the definition of the integrals I ( n, ab given in (6.14). Using (6.15)and (6.20) we have, for instance, β (4 , = − sin( πs ) γ (4 , , (6.42)as well as β (5 , = − sin( πs ) γ (5 , ,β (5 , = − sin( π ( s + s )) γ (5 , − sin( πs ) γ (5 , , (6.43)and β (6 , = − sin( πs ) γ (6 , ,β (6 , = − sin( π ( s + s )) γ (6 , − sin( πs ) γ (6 , , (6.44) β (6 , = − sin( π ( s + s + s )) γ (6 , − sin( π ( s + s )) γ (6 , − sin( πs ) γ (6 , . Before looking at p = 2 examples, let us see how the same results could have beenobtained from the overcomplete (but extremely simple) form of the KLT relations from(6.12). We can write it with an ( n − × ( n − 1) kernel matrix Φ with entries Φ ab = i e iπφ ab given by (6.13). More explicitly, we haveΦ = i e iπs e iπ ( s + s ) · · · e iπ (cid:80) n − i =1 s i e iπs e iπs · · · e iπ (cid:80) n − i =3 s i e iπ ( s + s ) e iπs · · · e iπ (cid:80) n − i =4 s i ... ... ... . . . ... e iπ (cid:80) n − i =1 s i e iπ (cid:80) n − i =3 s i e iπ (cid:80) n − i =4 s i · · · , (6.45)where the columns and rows are labelled by all cycles in R \ { z = z j } ,Γ = (cid:16) γ ( n, ...n , γ ( n, ...n , . . . , γ ( n, ...n − , ,n (cid:17) t . (6.46)In order to reduce this to the ( n − × ( n − 3) form, one makes use of the fact thatonly n − (cid:32) e iπs e iπ ( s + s ) · · · e iπ (cid:80) n − i =1 s i e − iπs e − iπ ( s + s ) · · · e − iπ (cid:80) n − i =1 s i (cid:33) Γ = (cid:18) (cid:19) , (6.47)– 54 –here s = 0 and the first (second) row comes from considering a contour right above(below) the real z -axis and deforming it to a point in the upper-half (lower-half) plane,also see appendix A.1. Let us invert these relations to construct projectors onto thetwo bases (6.32), (6.38) we considered in this subsection. We can write (Γ , Γ n − ) P = Γwith P = − (cid:32) e iπ (cid:80) n − i =1 s i e − iπ (cid:80) n − i =1 s i (cid:33) − (cid:32) e iπs e iπ ( s + s ) · · · e iπ (cid:80) n − i =1 s i e − iπs e − iπ ( s + s ) · · · e − iπ (cid:80) n − i =1 s i (cid:33) = 1sin( πs n ) (cid:18) sin( π (cid:80) n − i =3 s i ) sin( π (cid:80) n − i =4 s i ) · · · sin( πs ,n − )sin( πs ) sin( π ( s + s )) · · · sin( π (cid:80) n − i =1 s i ) (cid:19) , (6.48)which expresses the first and last elements of Γ in terms of the basis γ ( n, a . Similarly,eliminating the second-last and last elements we have a projector onto the ˜ γ ( n, a basis:˜ P = − (cid:32) e iπ (cid:80) n − i =1 s i e iπ (cid:80) n − i =1 s i e − iπ (cid:80) n − i =1 s i e − iπ (cid:80) n − i =1 s i (cid:33) − (cid:32) e iπs e iπ ( s + s ) · · · e iπ (cid:80) n − i =1 s i e − iπs e − iπ ( s + s ) · · · e − iπ (cid:80) n − i =1 s i (cid:33) = 1sin( πs ,n − ) (cid:18) − sin( π (cid:80) n − i =1 s i ) − sin( π (cid:80) n − i =2 s i ) · · · − sin( π ( s ,n − + s ,n − ))sin( π (cid:80) n − i =1 s i ) sin( π (cid:80) n − i =3 s i ) · · · sin( πs ,n − ) (cid:19) . (6.49)With these computations in place, we can simply apply the projectors to the relevantcolumns and rows of the overcomplete KLT matrix to obtain: P n − P t Φ P n − P = ( H ( n, ) − , n − ˜ P ˜ P t Φ P n − P = ( ˜ H ( n, ) − , (6.50)reproducing the results from (6.36) and (6.41). p = 2Recall that for p = 2 the basis γ ( n, a consists of all cycles where z < z < z < · · · 4) elements in the basis.Motivated by the simplicity of the results for p = 1, we will also introduce a secondbasis ˜ γ ( n, a where z , z are placed between z n = −∞ and z n − (or z for p = n − n = 5 , 6. – 55 – .5.1 Example ( n, p ) = (5 , γ (5 , a = (cid:16) γ (5 , , γ (5 , (cid:17) a , ˜ γ (5 , b = (cid:16) γ (5 , , γ (5 , (cid:17) b . (6.51)Computing the intersection matrix according to the above combinatorial rules we find˜ H (5 , = csc( πs ) (cid:18) − csc( πs ) cot( πs ) + cot( πs )cot( πs ) + cot( πs ) − csc( πs ) (cid:19) , (6.52)where the tilde refers to the asymmetric basis choice ˜ H (5 , ab = (cid:104) γ (5 , a | ˜ γ (5 , b (cid:105) analogous to(6.37). Inverting the matrix we obtain the well-known result for the local KLT matrix( ˜ H (5 , ) − = (cid:18) sin( πs ) sin( πs ) sin( πs ) sin( π ( s + s ))sin( πs ) sin( π ( s + s )) sin( πs ) sin( πs ) (cid:19) , (6.53)in agreement with momentum-kernel techniques [54]. With this result the basis cycles β (5 , a in (6.14) and (6.16) read as follows, β (5 , = sin( πs ) (cid:16) sin( πs ) γ (5 , + sin( π ( s + s )) γ (5 , (cid:17) ,β (5 , = sin( πs ) (cid:16) sin( π ( s + s )) γ (5 , + sin( πs ) γ (5 , (cid:17) . (6.54) ( n, p ) = (6 , γ (6 , a = (cid:16) γ (6 , , γ (6 , , γ (6 , , γ (6 , , γ (6 , , γ (6 , (cid:17) a , (6.55)˜ γ (6 , b = (cid:16) γ (6 , , γ (6 , , γ (6 , , γ (6 , , γ (6 , , γ (6 , (cid:17) b . (6.56)Among the intersection numbers ˜ H (6 , ab = (cid:104) γ (6 , a | ˜ γ (6 , b (cid:105) in the asymmetric basis choice,we already computed one example in the entry ˜ H (6 , in (6.29). Due to space limitationswe do not present the full intersection matrix here. Its 6 × H (6 , ) − = sin( πs ) sin( πs ) sin( πs ) sin( πs , ) sin( πs ) sin( πs , )0 sin( πs ) sin( πs ) 00 0 0sin( πs ) sin( πs , ) sin( πs ) sin( πs , ) sin( πs ) sin( πs )0 0 00 0 0 (6.57)– 56 –in( πs ) sin( πs , ) sin( πs , ) sin( πs , ) sin( πs , ) sin( πs , )0 sin( πs ) sin( πs , ) sin( πs ) sin( πs , )0 sin( πs ) sin( πs ) sin( πs ) sin( πs , )sin( πs ) sin( πs , ) sin( πs , ) sin( πs , ) sin( πs , ) sin( πs , )sin( πs ) sin( πs ) sin( πs ) sin( πs , ) sin( πs ) sin( πs , )0 sin( πs ) sin( πs , ) sin( πs ) sin( πs ) , where we use the notation s i i ...,j = s i j + s i j + · · · . In terms of the β (6 , a cycles from(6.14), this translates to β (6 , = sin( πs ) (cid:16) sin( πs ) γ (6 , + sin( πs , ) γ (6 , (cid:17) ,β (6 , = sin( πs ) (cid:16) sin( πs , ) γ (6 , + sin( πs ) γ (6 , + sin( πs , ) γ (6 , (cid:17) ,β (6 , = sin( πs ) (cid:16) sin( πs , ) γ (6 , + sin( πs ) γ (6 , (cid:17) , (6.58) β (6 , = sin( πs ) (cid:16) sin( πs , ) γ (6 , + sin( πs , ) γ (6 , + sin( πs ) γ (6 , (cid:17) ,β (6 , = sin( πs , ) (cid:16) sin( πs , ) γ (6 , + sin( πs ) γ (6 , + sin( πs , ) γ (6 , (cid:17) + sin( πs ) (cid:16) sin( πs ) γ (6 , + sin( πs , ) γ (6 , + sin( πs , ) γ (6 , (cid:17) ,β (6 , = sin( πs ) (cid:16) sin( πs , ) γ (6 , + sin( πs , ) γ (6 , + sin( πs ) γ (6 , (cid:17) + sin( πs , ) (cid:16) sin( πs , ) γ (6 , + sin( πs , ) γ (6 , + sin( πs ) γ (6 , (cid:17) . This example already illustrates the general rule: for each integrated puncture i ∈ { , } we have a sine factor in the generalized KLT kernel S α (cid:48) in (6.20). Thearguments of the sine functions are given by the overlap between labels to the left of i in γ (6 , a which are also to the right of i in ˜ γ (6 , b . We make this observation moreconcrete in the following. ( n, p )The goal of this subsection is to find the explicit form of the cycles β ( n,p ) in (6.14)which need to be field-theory orthonormal with respect to the forms ν ( n,p ) that are theBetti–de Rham duals of the integration cycles γ ( n,p ) (cid:126)A,(cid:126)i ↔ ρ (cid:126)A,(cid:126)i = (1 , A , i , A , i , . . . , A p , i p , A p +1 , n − , n ) (6.59)in order to avoid inconsistency in the α (cid:48) → z , z n − , z n ) =(0 , , ∞ ) in a vector (cid:126)A of words A , A , . . . each of which gathers (possibly zero) ad-jacent unintegrated punctures. The i , i , . . . , i p in turn are a permutation of the p integrated punctures z , z , . . . , z p +1 . – 57 –he examples of the β ( n,p ) in the earlier subsections have orthonormal intersectionnumbers with the γ ( n,p ) in (6.59) in the sense that (cid:104) γ ( n,p ) (cid:126)A,(cid:126)i | β ( n,p ) (cid:126)B,(cid:126)j (cid:105) = δ (cid:126)A, (cid:126)B δ (cid:126)i,(cid:126)j . (6.60)At general n and p , the β ( n,p ) with this property are conjecturally given by β ( n,p ) (cid:126)A,(cid:126)i = (cid:88) (cid:126)B,(cid:126)j S α (cid:48) (1 , (cid:126)A,(cid:126)i | (cid:126)B,(cid:126)j, n − 2) ˜ γ ( n,p ) (cid:126)B,(cid:126)j , (6.61)where we employ the alternative basis (6.19) of d ( n,p ) cycles ˜ γ ( n,p ) (cid:126)B,(cid:126)j instead of (6.59),in order to obtain a local expression for the generalized KLT kernel S α (cid:48) . Since thefinal two labels are always the same, we suppress them in (6.61) and below for clarity,i.e., S α (cid:48) ( X | Y ) = S α (cid:48) ( X, n − , n | Y, n − , n ). The latter is claimed to obey the followingrecursion in the number of integrated punctures i k S α (cid:48) (1 , A , i , A , . . . , A p , i p , A p +1 | X, i p , Y ) (6.62)= − sin(2 πα (cid:48) k i p · (cid:80) (cid:96) ∈ Y ∩ (1 ,A ,i ,...,i p − ,A p ) k (cid:96) ) S α (cid:48) (1 , A , i , A , . . . , i p − , A p , A p +1 | X, Y ) . This step may only be applied to remove the rightmost integrated puncture i p in thefirst entry, and the recursion terminates with S α (cid:48) (1 , (cid:126)A | (cid:126)B, n − 2) = δ (1 , (cid:126)A ) , ( (cid:126)B,n − (6.63)when there are no more integrated punctures left. This has been verified up to andincluding n = 8 for any value of p ≤ n − S α (cid:48) is indeed the inverse of˜ H ( n,p ) obtained with combinatorial rules. More generally, the recursion (6.62) can berewritten as S α (cid:48) ( P, i, Q | X, i, Y ) = − sin(2 πα (cid:48) k i · k P ∩ Y ) S α (cid:48) ( P, Q | X, Y ) , (6.64)where Q has no integrated punctures, i.e., the momenta in the sine functions are deter-mined by the punctures that appear on opposite sides of i in the two entries of S α (cid:48) ( ·|· ).The structure of the recursion (6.64) resonates with the momentum-kernel formalism[54] and its generalization to the KLT formulae for p = n − Throughout this subsection we have assumed that p (cid:54) = n − 3: otherwise, the basis (6.19) of ˜ γ ( n,p ) would be of the form ( . . . , , n − , n ) rather than ( . . . , n − , n − , n ) when maintaining the recursionfor S α (cid:48) . Since local representations of the KLT formula for p = n − p < n − – 58 –ith the expansion (6.61), the desired orthonormality property (6.60) takes theform (with collective indices a, b, c taking the role of (cid:126)A,(cid:126)i ), (cid:104) γ ( n,p ) a | β ( n,p ) b (cid:105) = d ( n,p ) (cid:88) c =1 (cid:104) γ ( n,p ) a | ˜ γ ( n,p ) c (cid:105) ( S α (cid:48) ) bc = δ ab , (6.65)In order to deduce the desired orthonormality of β ( n,p ) a and ν ( n,p ) b in the α (cid:48) → γ ( n,p ) c and cocycles ω ( n,p ) d ,lim α (cid:48) → (cid:104) β ( n,p ) a | ν ( n,p ) b (cid:105) = lim α (cid:48) → d ( n,p ) (cid:88) c,d =1 (cid:104) β ( n,p ) a | γ ( n,p ) c (cid:105) ( F ( n,p ) ) − cd (cid:104) ω ( n,p ) d | ν ( n,p ) b (cid:105) = d ( n,p ) (cid:88) c,d =1 δ ac δ cd δ db = δ ab , (6.66)where F ( n,p ) cd = (cid:104) ω ( n,p ) d | γ ( n,p ) c (cid:105) = (cid:104) γ ( n,p ) c | ω ( n,p ) d (cid:105) as in (2.7). The final two Kroneckerdeltas in passing to the last line stem from the fact that the ω ( n,p ) d are engineered tobe field-theory orthonormal to both γ ( n,p ) c and ν ( n,p ) b . The first Kronecker delta arisesfrom the conjectural orthonormality (6.65). It would be interesting to find a rigorousall-multiplicity proof that (6.61) together with the recursion (6.62) indeed leads toorthonormal intersection numbers. In this section we give an interpretation of our results in terms of correlation functionsin two-dimensional conformal field theories (CFTs). We will focus on the family oftheories known as the minimal models whose spectrum can been completely classifiedand solved in terms of irreducible representations of the Virasoro algebra. We startwith a lightning review of these models, where we focus only on the parts necessary tomake connections with the rest of this paper. For more comprehensive expositions werefer the reader to [111–113]. In general, intersection numbers satisfy (cid:104) γ | ˜ γ (cid:105) = (cid:104) ˜ γ | γ (cid:105) , but in our normalizations they are purelyreal, which is why the equality (6.65) also implies (cid:104) β ( n,p ) a | γ ( n,p ) c (cid:105) = δ ac . Saying that minimal models are solved by no means implies that their correlation functions havebeen computed, or are easy to compute, in general. – 59 – .1 Lightning review of the Coulomb gas formalism Our starting point is the action of a free boson φ ( x ) coupled linearly to the scalarcurvature R of the genus-zero surface: S p , p (cid:48) = (cid:90) CP d x √ g (cid:18) ∂ µ φ∂ µ φ + i √ Q p , p (cid:48) φR (cid:19) . (7.1)Here the strength of the coupling is given by the background charge Q p , p (cid:48) , which makesthe U(1) symmetry anomalous. Since the action is complex, it does not automaticallygive rise to a unitary theory. In fact, families of unitary models written in this way areheavily constrained and can be classified by a pair of co-prime integers ( p , p (cid:48) ), in termsof which Q p , p (cid:48) = p − p (cid:48) √ pp (cid:48) . (7.2)The central charge is c = 1 − Q p , p (cid:48) and we take p > p (cid:48) by convention. These arethe minimal models. For example ( p , p (cid:48) ) = (4 , 3) gives the critical Ising model with Q , = √ and c = , while ( p , p (cid:48) ) = (5 , 2) is the Yang–Lee edge singularity with Q , = √ and c = − .Conformal primary operators O q ( r,s ) in the ( p , p (cid:48) ) minimal model are classified bytwo integers ( r, s ) such that1 ≤ r ≤ p (cid:48) − , ≤ s ≤ p − . (7.3)Charges q ( r,s ) and conformal dimensions h ( r,s ) of these operators are given by q ( r,s ) = p (1 − r ) − p (cid:48) (1 − s )2 √ pp (cid:48) , h ( r,s ) = ( r p − s p (cid:48) ) − ( p − p (cid:48) ) pp (cid:48) . (7.4)Notice that operators O q and O Q p , p (cid:48) − q share the same conformal dimension and they areindistinguishable at the level of correlation functions. In other words, we can identifyoperators with ( r, s ) and ( p (cid:48) − r, p − s ). For instance, in the case of the critical Isingmodel we have the following Kac table: s = 1 s = 2 s = 3 r = 1 O = O √ = σ O √ = ε r = 2 O − √ = ε O − √ = σ O √ = (7.5)Here , σ , and ε are the usual identity, spin, and energy operators of conformal weight0, and , respectively. – 60 –e will be interested in computing the correlation function of N such operators.For readability we will simply label the j -th vertex operator O q j ( x j ) = e i √ q j φ ( x j ) by itscharge q j : (cid:104)O q ( x ) O q ( x ) · · · O q N ( x N ) (cid:105) p , p (cid:48) . (7.6)Such a computation might not seem approachable, because we deal with a strongly-interacting system. However, one can simplify this problem conceptually using the Coulomb gas formalism [38, 39], which is the idea that correlation functions in inter-acting theories with background charge can be equivalently represented as those in afree theory with insertions of p charged operators integrated over the whole surface. Asa result, the correlation functions (7.6) can be represented as (cid:90) ( CP ) p p +1 (cid:89) i =2 d z i (cid:104)O q ( x ) O q ( x ) · · · O q N ( x N ) p +1 (cid:89) i =2 O q ± ( z i ) (cid:105) free (7.7)up to a constant. The additional operators are called screening charges and theircharges can only take two values, q + and q − , given by q + = (cid:112) p / p (cid:48) , q − = − (cid:112) p (cid:48) / p , (7.8)such that q + + q − = Q p , p (cid:48) . We will denote the number of screening charges O q ± by p ± , such that p + + p − = p . These numbers can be determined by imposing theneutrality condition (Ward identity), i.e. requiring that the sum of charges equals tothe background charge, N (cid:88) i =1 q i + p + q + + p − q − = Q p , p (cid:48) . (7.9)As a heuristic, for sufficiently generic ( p , p (cid:48) ), reading off the coefficients of the irrationalnumbers (cid:112) p / p (cid:48) and (cid:112) p (cid:48) / p translates to the following condition for the integers ( r i , s i )labeling every operator: p + = 12 (cid:32) N (cid:88) i =1 r i − N + 2 (cid:33) , p − = 12 (cid:32) N (cid:88) i =1 s i − N + 2 (cid:33) . (7.10)For instance, the four-point correlation function of O = O q (2 , operators requires p + = 3and p − = 1 and hence can be written as (cid:104)O ( x ) O ( x ) O ( x ) O ( x ) (cid:105) p , p (cid:48) = (cid:90) ( CP ) (cid:89) i =2 d z i (cid:104)O ( x ) O ( x ) O ( x ) O ( x ) (7.11) O q + ( z ) O q + ( z ) O q + ( z ) O q − ( z ) (cid:105) free – 61 –ince in this case the neutrality condition reads4 q (2 , + 3 q + + q − = Q p , p (cid:48) . (7.12)However, this representation is not unique. For example, since we can dually representone of O as (cid:101) O = O Q p , p (cid:48) − q (2 , , we find a simpler representation (cid:104)O ( x ) O ( x ) O ( x ) (cid:101) O ( x ) (cid:105) p , p (cid:48) = (cid:90) CP d z (cid:104)O ( x ) O ( x ) O ( x ) (cid:101) O ( x ) O q + ( z ) (cid:105) free , (7.13)given that 3 q (2 , + ( Q p , p (cid:48) − q (2 , ) + q + = Q p , p (cid:48) . (7.14)We will return to this example in section 7.5 once we establish the connection tothe results of this paper. (Note that when we use the dual description (cid:101) O , ( r , s ) =( p (cid:48) − , p − 1) are p , p (cid:48) -dependent and we can no longer use (7.10), which would otherwisepredict p ± ≥ 1. The neutrality condition (7.9) always holds.)Of course, the free-theory correlator inside of the integrand of (7.7) can be writtendown explicitly, giving us the explicit formula (cid:104)O q ( x ) O q ( x ) · · · O q N ( x N ) (cid:105) p , p (cid:48) = (cid:90) ( CP ) p p +1 (cid:89) i =2 d z i e W + W , (7.15)where W = 2 N (cid:88) ≤ i 2) minimal models (with p odd). Vertexoperators are labeled by (1 , s i ) with 1 ≤ s i ≤ p − q (1 ,s i ) = s i − √ p . (7.30)In this situation the background and screening charges are given by Q p , = p − √ p , q + = (cid:112) p / , q − = − (cid:112) / p . (7.31)Hence, if we can avoid using the screening charge q + , all the charge pairings ˜ q i ˜ q j wouldscale as 1 / p , and the correlation function in the limit p → ∞ would be on the samefooting as string-theory amplitudes in the low-energy approximation, α (cid:48) → 0. This cancertainly be done. Let us consider an N-pt function of operators O q (1 ,si ) and representthe N-th one via its dual O q (1 , p − si ) , i.e., (cid:104)O q (1 ,s ( x ) O q (1 ,s ( x ) · · · O q (1 ,s N − ( x N − ) O q (1 ,s N) ( x N ) (cid:105) p , = (cid:104)O q (1 ,s ( x ) O q (1 ,s ( x ) · · · O q (1 ,s N − ( x N − ) O q (1 , p − s N) ( x N ) (cid:105) p , . (7.32)We use the second representation in the Coulomb gas formalism. Here the neutralitycondition is satisfied if p + = 0 , p − = 12 (cid:32) N − (cid:88) i =1 s i − s N − N + 2 (cid:33) , (7.33)and if p − is an integer. This leads to a potential W proportional to 1 / p : W = 1 p (cid:18) N − (cid:88) ≤ i 1) log( x i − x j ) − N − (cid:88) i =1 p − +1 (cid:88) j =2 ( s i − 1) log( x i − z j ) + 16 p − +1 (cid:88) ≤ i 2) models coupled to Liouvilletheory has been recently conjectured to be describing Jackiw–Teitelboim gravity [121],which adds further physical motivation for studying such correlation functions.– 66 –he individual conformal blocks, once expressed in terms of F (N+ p,p ) cb , satisfy all themonodromy properties described in previous section as well as the coaction formulafrom (2.15). In addition, once expressed in this basis, the correlation function can beexpressed as a single-valued map of a single conformal block:lim x N →∞ | x N | q N ( q N − Q p , p (cid:48) ) π p N − (cid:89) ≤ i 2) minimal modelsin the p → ∞ limit below. In order to illustrate the above formulae on concrete examples we will consider thefour-point functions G ( x, x ) = lim x →∞ | x | q ( q − Q p , p (cid:48) ) π p | x | − q q | − x | − q q (cid:104)O q (0) O q ( x ) O q (1) O q ( x ) (cid:105) p , p (cid:48) , (7.36)where to avoid clutter we expressed it in terms of the cross-ratio x . In the intermediatesteps we will restrict to x ∈ (0 , , 1) and (1 , 2) operators, which both involve a single screeningcharge, p = 1. In the case (7.13), we have a single screening charge q + and(I) : q = q = q = q (2 , = − (cid:112) p / p (cid:48) , q = Q p , p (cid:48) − q (2 , = 3 p − p (cid:48) √ pp (cid:48) , q + = (cid:114) pp (cid:48) . (7.37)On the other hand, we can consider a special case of (7.32) with p (cid:48) = 2 and(II) : q = q = q = q (1 , = (cid:112) p (cid:48) / p , q = Q p , p (cid:48) − q (1 , = − p (cid:48) − p √ pp (cid:48) , q − = − (cid:114) p (cid:48) p , (7.38)as well as a single screening charge q − . This example of course can be considered also for p (cid:48) (cid:54) = 2. We can compute these different correlation functions using the same formulaeprovided we treat q , q , q , and q ± as abstract variables and plug in their values onlyat the end.Explicitly, G ( x, x ) is given by the integral G ( x, x ) = 1 π (cid:90) C \{ ,x, } d z | z | q q ± | z − x | q q ± | z − | q q ± , (7.39)– 67 –hich we can easily express in terms of the following contour integrals obtained byplacing the screening charge between the external operators in all possible ways: F ( x ) F ( x ) F ( x ) F ( x ) = (cid:90) −∞ d z ( − z ) q q ± ( x − z ) q q ± (1 − z ) q q ± (cid:90) x d z z q q ± ( x − z ) q q ± (1 − z ) q q ± (cid:90) x d z z q q ± ( z − x ) q q ± (1 − z ) q q ± (cid:90) ∞ d z z q q ± ( z − x ) q q ± ( z − q q ± , (7.40)where the phase of W is chosen such that each F a is real and agrees with the conventionwith absolute values for the Koba–Nielsen factor in (2.4). The overcomplete KLTrelation (7.22) then reads: G ( x, x ) = i F ( x ) F ( x ) F ( x ) F ( x ) t e πiq q ± e πi ( q + q ) q ± e πi ( q + q + q ) q ± e πiq q ± e πiq q ± e πi ( q + q ) q ± e πi ( q + q ) q ± e πiq q ± e πiq q ± e πi ( q + q + q ) q ± e πi ( q + q ) q ± e πiq q ± F ( x ) F ( x ) F ( x ) F ( x ) . (7.41)It is however beneficial to express it in terms of a minimal basis, which according to(7.23) is | χ ( C (5 , ) | = 2. To minimize the number of computations let us pick F and F for both holomorphic and antiholomorphic blocks. It leads to a simplificationbecause the two contours do not intersect and hence the intersection matrix is diagonal(another natural choice would be F and F ), cf. (6.35). We therefore immediately get G ( x, x ) = sin(2 πq q ± ) sin(2 πq q ± )sin(2 π ( q + q ) q ± ) |F ( x ) | + sin(2 πq q ± ) sin(2 π ( q + q + q ) q ± )sin(2 π ( q + q ) q ± ) |F ( x ) | . (7.42)Computation of the relevant conformal blocks explicitly gives F ( x ) = x q + q ) q ± B(1+2 q q ± , q q ± ) F ( − q q ± , q q ± ; 2+2( q + q ) q ± ; x ) , F ( x ) = − q q ± B( − q + q + q ) q ± , q q ± )1 + 2( q + q + q ) q ± (7.43) × F ( − q q ± , − − q + q + q ) q ± ; − q + q ) q ± ; x ) , The case (7.37) with p (cid:48) = 3 is special because it leads to integrands which are not branched atinfinity, given that e W | (7.37) , p (cid:48) =3 = [ z ( z − x )( z − − p / → z − p as z → ∞ . Because of this fact the sizeof the basis decreases to 1. It is the same problem as sitting on a factorization channel s = 0 in stringtheory amplitudes, or considering Feynman integrals in integer dimensions, see [27, Section 4]. In thosesituations one needs to correct KLT relations using the framework of relative twisted cohomologies[122]. While in this subsection we ignore this problem to retain generality, we will return to it insections 7.5.1 and 7.5.2. – 68 –here B( a, b ) = Γ( a )Γ( b )Γ( a + b ) is the Euler beta function. This result is in agreement with[38, 39, 50, 107].Next we analyze the p → ∞ behavior of these correlation functions for p (cid:48) fixedand finite. This limit can be qualitatively different, depending on whether chargesbecome small, such as in the case (7.38) where q i q − → 0, or large, as is the case in theexample (7.37) where q i q + → ∞ (recall that q does not enter the expressions directly).While the first case is fairly easy to analyze and leads to interesting connections withtranscendentality, the second is more subtle due to the presence of Stokes phenomenasimilar to those appearing in the α (cid:48) → ∞ limit of the Veneziano amplitude [31]. Weconsider examples of these limits below. Before doing so, we give an explicit examplewhere correlation functions can be expressed in terms of elementary functions. In ordernot to confuse the two cases (7.37) and (7.38), we will label the correlation functionand conformal blocks evaluated on the two sets of charges with superscripts I and II,respectively. Let us consider the example of the critical Ising model with ( p , p (cid:48) ) = (4 , r, s ) = (2 , F I4 does not contribute sinceits prefactor in (7.42) is proportional tosin(2 π ( q + q + q ) q ± ) (cid:12)(cid:12) (7.37) = − sin(3 π p / p (cid:48) ) (cid:12)(cid:12) p (cid:48) =3 = 0 . (7.44)It is consistent with the size of the basis dropping to 1 in the case (7.37) with p (cid:48) = 3,although the fact that the limit p (cid:48) → F I2 ( x ) = x − p / p (cid:48) B(1 − p / p (cid:48) , − p / p (cid:48) ) F (1 − p / p (cid:48) , p / p (cid:48) ; 2 − p / p (cid:48) ; x ) (cid:12)(cid:12) ( p , p (cid:48) )=(4 , = Γ( − ) Γ( − ) ( x − x + 1)(1 − x ) / x / . (7.45)Plugging back into (7.36) we findlim x →∞ (cid:104) ε (0) ε ( x ) ε (1) ε ( x ) (cid:105) Ising = c | x | (cid:12)(cid:12)(cid:12)(cid:12) x − x − − (cid:12)(cid:12)(cid:12)(cid:12) , (7.46)– 69 –here c = − √ π Γ( − ) / Γ( − ) . Restoring the original coordinates one finds (cid:104) ε ( x ) ε ( x ) ε ( x ) ε ( x ) (cid:105) Ising = c (cid:12)(cid:12)(cid:12)(cid:12) Pf (cid:18) x i − x j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (7.47)where Pf( · · · ) denotes the Pfaffian of the antisymmetric matrix with entries labelledby i, j = 1 , , , 4. This is the correct result.Let us move on to the four-point function of spin operators with ( r, s ) = (1 , F II2 ( x ) = x − p (cid:48) / p B(1 − p (cid:48) / p , − p (cid:48) / p ) F ( p (cid:48) / p , − p (cid:48) / p ; 2 − p (cid:48) / p ; x ) (cid:12)(cid:12) ( p , p (cid:48) )=(4 , = Γ( ) √ π (cid:112) √ − x (cid:112) x (1 − x ) , (7.48)as well as F II4 ( x ) = p (cid:48) p − p (cid:48) B(3 p (cid:48) / p , − p (cid:48) / p ) F ( p (cid:48) / p , p (cid:48) / p − 1; 2 p (cid:48) / p ; x ) (cid:12)(cid:12) ( p , p (cid:48) )=(4 , = Γ( ) √ π (cid:112) √ x − (cid:112) − √ x (cid:112) x (1 − x ) . (7.49)Putting everything together according to (7.42) (with coefficients in front of the twofactors) and restoring all the coordinates x i , one finds agreement with the free-fermioncomputation (cid:104) σ ( x ) σ ( x ) σ ( x ) σ ( x ) (cid:105) = c (cid:88) e i = ± (cid:80) i e i =0 (cid:89) ≤ i 16 in the domain x ∈ (0 , p limit for (2 , four-point correlators We now consider the p → ∞ limit of the four-point functions of (2 , 1) operators withcharges given in case I in (7.37) and p (cid:48) ≥ W .(One cannot easily apply saddle-point analysis directly to the correlator because it isnot written in terms of a holomorphic integrand.) There is a large number of criticalpoints located on different sheets of the Riemann surface of z . On the first sheet we Since in this case we have e W | (7.37) = [ z ( z − x )( z − − p / p (cid:48) (7.51) – 70 –ave ∂ z W ( z ∗ ) = 2 q + (cid:18) q z ∗ + q z ∗ − x + q z ∗ − (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) (7.37) = − pp (cid:48) z ∗ − x ) z ∗ + xz ∗ ( z ∗ − x )( z ∗ − 1) = 0 . (7.52)Explicitly, it gives two solutions which we denote by z ±∗ , z ±∗ = 13 (cid:16) x ± √ x − x + 1 (cid:17) . (7.53)It is clear that the positions of these critical points depend on the cross-ratio x . Thisis the source of the Stokes phenomenon: the large- p asymptotics depends on the valueof x .Here we focus on the case x ∈ (0 , z −∗ ∈ (0 , x ) and exactly one z + ∗ ∈ ( x, q i ˜ q j have the same sign and x i ’s are ordered, the problem of com-puting critical points is equivalent to that of finding stable configurations of mutually-repelling charges on a line. There are exactly (N+ p − / (N − C (N+ p,p ) .While there might be a large number of critical points (with two per sheet), theyall give same-magnitude contributions to the large- p asymptotics and only differ inthe complex phase. These phases typically resum to trigonometric functions. We canexploit the KLT formula with a judicious choice of bases to drastically simplify thecomputation. In the example at hand, the contours (0 , x ) and ( x, 1) are already pathsof steepest descent (also known as Lefschetz thimbles) for the potential W at x ∈ (0 , z −∗ and z + ∗ , respectively. Hence using F I2 and F I3 as bases we cancompute the asymptotic behavior with only two saddle points from the first sheet, onefor each conformal block. For p (cid:48) > 3, using the intersection numbers computed from(6.36) and plugging in (7.37) we have G I ( x, x ) (cid:12)(cid:12) p (cid:48) > = − sin( π p / p (cid:48) ) sin(3 π p / p (cid:48) ) (cid:18) F I2 ( x ) F I3 ( x ) + F I3 ( x ) F I2 ( x )+ 2 cos( π p / p (cid:48) ) (cid:0) |F I2 ( x ) | + |F I3 ( x ) | (cid:1) (cid:19) . (7.54) with a finite p (cid:48) co-prime to p , the number of sheets is p (cid:48) . This is because the corresponding Riemannsurface of z is p (cid:48) -branched around the three points 0, x , 1 (monodromies around infinity are notindependent). Each sheet can be labelled by a point in a Z p (cid:48) lattice counting how many times z winded around each of the branch points. This situation is different to string theory, where s ij aregeneric non-rational variables and hence the numbers of sheets and critical points are infinite. – 71 –ote that there are no poles or zeros due to the sine factors because p and p (cid:48) are co-prime and p (cid:48) > 3. The case p (cid:48) = 3 is simpler for the same reason as in the case of theenergy correlator (7.47) in the critical Ising model. Namely, even in the basis F I2 , F I3 the coefficient of F I3 is zero, as in (7.44), as hence we have a simplified result G I ( x, x ) (cid:12)(cid:12) p (cid:48) =3 = − sin( π p / sin(2 π p / |F I2 ( x ) | , (7.55)which means it only receives contributions from the single critical point z −∗ . The phys-ical reason for this simplification is that four (2 , 1) operators can only exchange anidentity operator when p (cid:48) = 3.At any rate, the asymptotics of the blocks F I2 and F I3 can now be easily computed.The Hessian evaluated at the two critical points is J ± = ∂ z W ( z ±∗ ) = pp (cid:48) (cid:18) z ±∗ ) + 1( z ±∗ − x ) + 1( z ±∗ − (cid:19) (7.56)and is positive for x ∈ (0 , p →∞ F I2 ( x ) = 1 √ πJ − [ z −∗ ( x − z −∗ )(1 − z −∗ )] − p / p (cid:48) , (7.57)lim p →∞ F I3 ( x ) = 1 √ πJ + [ z + ∗ ( z + ∗ − x )(1 − z + ∗ )] − p / p (cid:48) . (7.58)They together give the asymptotics of (7.54) and (7.55) in the case x ∈ (0 , p → ∞ . p limit for (1 , four-point correlators Let us consider the p → ∞ limit of the four-point functions of (1 , 2) operators withcharges given in case II in (7.38). While the case p (cid:48) = 2 is of most interest, we canstudy arbitrary fixed p (cid:48) ≥ p is alwaysco-prime with p (cid:48) ). Direct expansion of the result in (7.42) gives G II ( x, x ) = − π p (cid:48) p (cid:0) | x | + | − x | (cid:1) (7.59)+ π p (cid:48) p (cid:16) | x | log | x | + | − x | log | − x | − | x | + | − x | ) (cid:17) + O (1 / p ) . One can immediately see that assigning transcendentality weights T ( p ) = 1 and T ( p (cid:48) ) = T ( x ) = 0, the result is not uniformly transcendental. This fact can be fixedwith a corrected basis of conformal blocks.– 72 –o this end, we first recall the differential forms from section 2, which serve asbuilding blocks for the minimal basis. Specializing the Mandelstam variables s , s , s at ( n, p ) = (5 , 1) according to (7.38) we have: ω (5 , = − p (cid:48) p d log z, ω (5 , = p (cid:48) p d log(1 − z ) , (7.60) ν (5 , = d log zz − x , ν (5 , = d log z − xz − . (7.61)In order to project G II ( x, x ) onto the basis of sv F (5 , ab we only need to compute fourintersection numbers of the above forms with d z , giving (cid:104) ν (5 , | d z (cid:105) = p x p − p (cid:48) , (cid:104) ν (5 , | d z (cid:105) = p (1 − x ) p − p (cid:48) , (7.62) (cid:104) ω (5 , | d z (cid:105) = − p (cid:48) (1 + x )3( p − p (cid:48) ) , (cid:104) ω (5 , | d z (cid:105) = p (cid:48) ( x − p − p (cid:48) ) . (7.63)Steps needed to reproduce these results were spelled out in [27, Section 4B] in a verysimilar case. Using the basis expansion formula (7.29) we therefore find G II ( x, x ) = π pp (cid:48) p − p (cid:48) ) (cid:16) − (1+ x ) (cid:0) x sv F (5 , + (1 − x ) sv F (5 , (cid:1) +( x − (cid:0) x sv F (5 , + (1 − x ) sv F (5 , (cid:1)(cid:17) . (7.64)For completeness let us also give an expression for the basis of conformal blocks interms of F (5 , ab : F II a +1 ( x ) = (cid:88) b =1 (cid:104) ν (5 , b | d z (cid:105) F (5 , ab = pp − p (cid:48) (cid:16) x F (5 , a + (1 − x ) F (5 , a (cid:17) (7.65)for a = 1 , 2. ( G II can be expressed in terms of F II2 and F II3 using the same formulaas in (7.54) with p ↔ p (cid:48) .) One can compute their 1 / p -expansion using the formulaeexplained in section 3.2 with s = s = s = s = − p (cid:48) / p , z = x . (7.66)More precisely, with the α (cid:48) -expansion of F (5 , ab in (3.30) (also see appendix A.2 for theorders of α (cid:48)≤ ), the kinematic point (7.66) gives rise to the following leading orders oftheir single-valued imagessv F (5 , = 1 + 2 sG sv (0; x ) + 4 s G sv (0 , x ) + s G sv (0 , x ) + 8 s G sv (0 , , x )– 73 – 2 s G sv (0 , , x ) + 2 s G sv (0 , , x ) + 2 s G sv (0 , , x ) + 4 s ζ + O ( s ) , sv F (5 , = − sG sv (1; x ) − s G sv (1 , x ) − s G sv (1 , x ) − s G sv (1 , , x ) − s G sv (1 , , x ) − s G sv (1 , , x ) − s G sv (1 , , x ) + O ( s ) , (7.67)sv F (5 , = − sG sv (0; x ) − s G sv (0 , x ) − s G sv (0 , x ) − s G sv (0 , , x ) − s G sv (0 , , x ) − s G sv (0 , , x ) − s G sv (0 , , x ) + 4 s ζ + O ( s ) , sv F (5 , = 1 + 2 sG sv (1; x ) + s G sv (1 , x ) + 4 s G sv (1 , x ) + 2 s G sv (1 , , x )+ 2 s G sv (1 , , x ) + 2 s G sv (1 , , x ) + 8 s G sv (1 , , x ) + 12 s ζ + O ( s ) . To the weights shown, the single-valued polylogarithms G sv ( (cid:126)a ; z ) = sv G ( (cid:126)a ; z ) fromBrown’s construction [52] with (cid:126)a ∈ { , } × are given by G sv ( a ; x ) = G ( a ; x ) + G ( a ; x ) ,G sv ( a , a ; x ) = G ( a , a ; x ) + G ( a ; x ) G ( a ; x ) + G ( a , a ; x ) ,G sv ( a , a , a ; x ) = G ( a , a , a ; x ) + G ( a , a ; x ) G ( a ; x ) (7.68)+ G ( a ; x ) G ( a , a ; x ) + G ( a , a , a ; x ) , where the explicit expressions for single-valued polylogarithms are given in equations(6.8) and (6.9), also see (6.10) for a weight-four example involving a zeta value. Uponinsertion into the correlation function (7.64) with s = − p (cid:48) p , we arrive at the followinglarge-charge expansion G II ( x, x ) = − π pp (cid:48) p − p (cid:48) ) (cid:26) | x | + | − x | − p (cid:48) p (cid:20) | x | G sv (0; x ) + | − x | G sv (1; x ) (cid:21) (7.69)+ 3 p (cid:48) p (cid:20) | x | G sv (0 , x ) + x ( x − G sv (0 , x )+ x ( x − G sv (1 , x ) + 2 | − x | G sv (1 , x ) (cid:21) + O (cid:18) p (cid:19)(cid:27) . The decomposition (7.64) into uniformly transcendental sphere integrals F (5 , ab exem-plifies a key observation of this section: In a suitable normalization, certain four-pointcorrelation functions in minimal models furnish another family of physical quantitiesbesides amplitudes [22–24, 26, 124–127] and form factors [128, 129] that feature uni-form transcendentality. The natural normalization for (7.64) is to peel off the prefactor − π pp (cid:48) p − p (cid:48) ) , and uniform transcendentality then interlocks the transcendental weight ofthe polylogarithms and MZVs in (7.69) with the order in 1 / p in the large-charge expan-sion. It would be interesting to investigate if more general four- and n -point correlationfunctions exhibit similar transcendentality properties.– 74 – Summary and outlook In this work we have investigated configuration-space integrals over punctured Riemannspheres with an arbitrary number of integrated and unintegrated punctures z j . Sim-ilar to the Koba–Nielsen factor in string tree-level amplitudes, the integrands featureproducts of | z i − z j | s ij , whose non-integer exponents lead to twisted homologies and co-homologies. The exponents s ij may be either identified with dimensionless Mandelstaminvariants 2 α (cid:48) k i · k j containing the inverse string tension α (cid:48) , or with multiples of thedimensional-regularization parameter of Feynman integrals in spacetime dimensions ∈ N − (cid:15) .In this setting, we have given explicit bases of twisted cycles γ ( n,p ) a and cocycles ω ( n,p ) b , such that the coaction of the period matrix (cid:104) γ ( n,p ) a | ω ( n,p ) b (cid:105) lines up with the masterformula (1.1) with coefficients taken from the identity matrix. The coaction applies tothe MZVs and multiple polylogarithms in the Taylor expansion of the period-matrixentries with respect to s ij , and we have advanced their structural understanding by • introducing a systematic method for obtaining an explicit form of the s ij -expansions, • decomposing (cid:104) γ ( n,p ) a | ω ( n,p ) b (cid:105) into a matrix product which organizes MZVs and poly-logarithms at different arguments into separate factors, • pinpointing refined coaction formulae for the individual factors, i.e. for generatingseries of polylogarithms in different numbers of variables, • spelling out the analytic continuations between different orderings of the unintegratedpunctures on the real axis.The integrals (cid:104) γ ( n,p ) a | ω ( n,p ) b (cid:105) over paths in the configuration space C ( n,p ) are re-lated to complex integrals of ω ( n,p ) a ω ( n,p ) b over all of C ( n,p ) . Specifically, these complex C ( n,p ) -integrals are expressed both as single-valued versions or as complex bilinears in (cid:104) γ ( n,p ) a | ω ( n,p ) b (cid:105) . In this way, we generalize the KLT formula and the single-valued mapbetween open- and closed-string tree amplitudes beyond p = n − 3, i.e. to more generalintegrals with an arbitrary number of unintegrated punctures. Moreover, our resultsfor the complex C ( n,p ) -integrals yield a new perspective on double-copy structures ofcorrelation functions in minimal models, generalizing earlier p = n − Acknowledgments We are grateful to Samuel Abreu, Claude Duhr, Lorenz Eberhardt, Einan Gardi, Mar-tijn Hidding, Daniel Kapec, Nils Matthes and Bram Verbeek for combinations of in-spiring discussions and collaboration on related topics. S.M. thanks Uppsala Universityfor hospitality during parts of this project. O.S. is grateful to Trinity College Dublinfor hospitality during early stages of this project. This research was supported by theMunich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excel-lence “Origin and Structure of the Universe.” S.M. gratefully acknowledges the fundingprovided by Frank and Peggy Taplin as well as the grant DE-SC0009988 from the U.S.– 76 –e( z )Im( z ) • z =0 • z • z =1 • + ∞•−∞ γ (5 , γ (5 , γ (5 , γ (5 , Figure 6 : The closed contour C relevant to the monodromy relations (A.1) at ( n, p ) =(5 , 1) consists of subsets γ (5 , , γ (5 , , γ (5 , , γ (5 , of the real line drawn in red while thedashed semicircle simply indicates that z → ±∞ are identified on the Riemann sphereand does not contribute to the integral (A.1). Department of Energy. R.B. is supported by the European Research Council undergrant ERC-CoG-647356 (CutLoops). C.R. and O.S. are supported by the EuropeanResearch Council under ERC-STG-804286 UNISCAMP. A Further details on the α (cid:48) -expansion This appendix complements the discussion of section 3 on the α (cid:48) -expansion of theintegrals F ( n,p ) ab . A.1 Monodromy relations for F (5 , In this appendix, we infer the formal initial value at z = 0 for the five-point integrals F (5 , a with cocycles in (3.20) and (3.21) from monodromy relations [83, 84]. On theintegration contour γ (5 , = γ (5 , = { z < z < } for z under consideration, the limit z → α (cid:48) -expansion. Hence, the goal of this appendix isto infer the latter from integrals over γ (5 , = γ (5 , = { < z < z } and γ (5 , = { 6. At n = 6, for instance, the monodromy relation0 = sin( πs ) F (6 , a + sin (cid:0) π ( s + s ) (cid:1) F (6 , a + sin (cid:0) π ( s + s + s ) (cid:1) F (6 , a + sin (cid:0) π ( s + s + s + s ) (cid:1) (cid:90) γ (6 , KN (6 , ω (6 , a (A.6)involving one contour γ (6 , = { < z < ∞} outside the basis (3.33) can be used toinfer the z → F (6 , a which does not commute with α (cid:48) -expansion. The z → γ (6 , can be identified with the kinematic limits of F (5 , ab seen in(3.43) and (3.42). In this way, we arrive at the initial values (3.45) and (3.46) as wellas their n -point generalizations to be given in appendix A.3. A.2 α (cid:48) -expansion of F (5 , The method of section 3.2 to obtain the α (cid:48) -expansion of F (5 , via (3.30) gives rise tothe leading orders F (5 , = 1 + s G (0; z ) + s G (0; z ) + ( s + s ) G (0 , z )+ s s G (0 , z ) − s s ζ + O ( α (cid:48) ) F (5 , = − s G (1; z ) − s s G (1 , z ) − s G (1 , z ) − ( s + s ) s G (1 , z ) + O ( α (cid:48) ) (A.7) F (5 , = − s G (0; z ) − s G (0 , z ) − s s G (0 , z ) − s s G (0 , z ) − s s G (0 , z ) − s s ζ − s s ζ + O ( α (cid:48) ) F (5 , = 1 + ( s + s ) G (1; z ) + ( s + s ) G (1 , z )+ s s G (1 , z ) − ( s + s ) s ζ + O ( α (cid:48) ) . A.3 The explicit form of P ( n, and M ( n, The derivation of the z → F (5 , ab and F (6 , ab in (3.28) and (3.42) frommonodromy relations generalizes to F (7 , ab ( z → , z , z ) = | z | s | z | s F (4 , s ˆ F (6 , s + s − K (7 , ˆ F (6 , ˆ F (6 , ˆ F (6 , s ˆ F (6 , s + s ˆ F (6 , ˆ F (6 , ˆ F (6 , s ˆ F (6 , s + s ˆ F (6 , ˆ F (6 , ˆ F (6 , (A.8)– 79 –nd more generally F ( n, ab ( z → , z , . . . , z n − ) = (cid:16) (cid:81) n − k =4 | z k | s k (cid:17) F (4 , . . . s ˆ F ( n − , s + s − K ( n, ˆ F ( n − , ˆ F ( n − , . . . ˆ F ( n − , ,n − s ˆ F ( n − , s + s ˆ F ( n − , ˆ F ( n − , . . . ˆ F ( n − , ,n − ... ... ... ... s ˆ F ( n − , n − , s + s ˆ F ( n − , n − , ˆ F ( n − , n − , . . . ˆ F ( n − , n − ,n − , (A.9)where the ( a, b ) = (2 , 1) entries involve K (7 , = sin( πs )sin( π ( s + s )) | z | s | z | s F (4 , (A.10)= | z | s | z | s s s + s Γ(1+ s )Γ(1 − s − s )Γ(1 − s ) K ( n, = sin( πs )sin( π ( s + s )) (cid:16) n − (cid:89) k =4 | z k | s k (cid:17) F (4 , (A.11)= (cid:16) n − (cid:89) k =4 | z k | s k (cid:17) s s + s Γ(1+ s )Γ(1 − s − s )Γ(1 − s ) . The hat notation instructs to change the arguments of F ( n − , ab to (cf. (3.43)) s → s + s , s ,j → s ,j +1 , z k → z k +1 (A.12)for j = 3 , , . . . , n − k = 3 , , . . . , n − 3. The four-point integral ( d (4 , = 1) yieldsthe standard Euler beta function (3.31). The formal z j → W ( s , s , s ) = F (4 , ( s + s , s ) − F (4 , ( s , − s − s ) s + s = − ( s + s ) ζ + (cid:2) ( s + s + s ) s − s s (cid:3) ζ + O ( α (cid:48) ) W ( s , s , s , s ) = F (4 , ( s + s + s , s ) − F (4 , ( s , − s − s − s ) s + s + s = − ( s + s ) ζ + ( s + s + s + s ) s ζ (A.13) − ( s + s ) s ζ + O ( α (cid:48) ) W ( s , s , . . . , s ) = F (4 , ( s + s + s + s , s ) − F (4 , ( s , − s − s − s − s ) s + s + s + s – 80 – − ( s + s ) ζ + ( s + s + s + s + s ) s ζ − ( s + s + s ) s ζ + O ( α (cid:48) ) , and more generally W j ( s , s , . . . , s ,j +1 ) = F (4 , ( s + (cid:80) ji =3 s i , s ,j +1 ) − F (4 , ( s j , − s − (cid:80) ji =3 s i ) s + s + . . . + s j . (A.14)This notation yields the compact representations:lim z → F (5 , ( z ) = (cid:18) F (4 , ( s , s ) 0 s W F (4 , ( s + s , s ) (cid:19) , (A.15)lim z ,z → F (6 , ( z , z ) = F (4 , ( s , s ) 0 0 s W F (4 , ( s + s , s ) 0 s W ( s + s ) W F (4 , ( s + s + s , s ) , (A.16)lim z ,z ,z → F (7 , ( z , z , z ) = (A.17) F (4 , ( s , s ) 0 0 0 s W F (4 , ( s + s , s ) 0 0 s W ( s + s ) W F (4 , ( s + s + s , s ) 0 s W ( s + s ) W ( s + s + s ) W F (4 , ( s + s + s + s , s ) . These expressions follow from the initial values (A.8) and generalize as follows to highermultiplicity: lim z k → F ( n, ab = b > aF (4 , (cid:16) s + (cid:80) b +1 m =3 s ,m , s ,b +2 (cid:17) : b = a (cid:16) s + (cid:80) b +1 m =3 s ,m (cid:17) W a +1 : b < a (A.18) A.4 The explicit form of P (6 , and M (6 , We shall finally give the key steps towards the α (cid:48) -expansion of the integrals F (6 , ab overbasis formsˆ ω (6 , = s z (cid:18) s z + s z (cid:19) , ˆ ω (6 , = s z (cid:18) s z + s z + s z (cid:19) ˆ ω (6 , = (cid:18) s z + s z (cid:19)(cid:18) s z + s z + s z (cid:19) , ˆ ω (6 , k = ˆ ω (6 , k − (cid:12)(cid:12)(cid:12) ↔ , k = 1 , , ↔ z ij and s ij . Again,the basis of integration contours γ (6 , = { < z < z < z } , γ (6 , = { < z < z < z < } γ (6 , = { z < z < z < } , γ (6 , k = γ (6 , k − (cid:12)(cid:12)(cid:12) ↔ , k = 1 , , γ (6 , j at j = 3 , , , α (cid:48) -expansion does not commute withthe z → n, p ) = (5 , 1) case in appendix A.1,we use monodromy relations to relate these problematic contours to auxiliary ones α (6 , = { ( z , z ) ∈ R | < z < z and 1 < z < ∞} α (6 , = { ( z , z ) ∈ R | < z < z < ∞} (A.21) α (6 , = α (6 , (cid:12)(cid:12)(cid:12) ↔ , α (6 , = α (6 , (cid:12)(cid:12)(cid:12) ↔ depicted in figure 7. These α (6 , j are engineered to have commutative limits z → α (cid:48) → P (6 , and M (6 , in (3.7).We will make use of the monodromy relations γ (6 , = − α (6 , sin ( π ( s + s + s + s )) + γ (6 , sin ( π ( s + s )) + γ (6 , sin ( πs )sin ( π ( s + s + s )) ,γ (6 , = − sin ( πs ) sin ( πs ) γ (6 , + sin ( πs ) sin ( π ( s + s + s + s )) γ (6 , sin ( π ( s + s + s )) sin ( π ( s + s + s + s + s )) (A.22)+ sin ( πs ) sin ( π ( s + s + s + s )) α (6 , sin ( π ( s + s + s )) sin ( π ( s + s ))+ sin ( π ( s + s + s + s )) sin ( πs ) α (6 , sin ( π ( s + s + s + s + s )) sin ( π ( s + s ))+ sin ( π ( s + s + s + s )) sin ( π ( s + s + s + s + s + s )) α (6 , sin ( π ( s + s + s + s + s )) sin ( π ( s + s ))and two similar relations for γ (6 , and γ (6 , that are obtained from relabelling theMandelstam invariants via 2 ↔ γ (6 , ↔ γ (6 , as well as α (6 , ↔ α (6 , and α (6 , ↔ α (6 , on the right-hand sides of (A.22). A.4.1 z → limits on the α (6 , i contours The integrals F (6 , α i ,b = (cid:104) α (6 , i | ω (6 , b (cid:105) with i = 3 , , , z → z → F (6 , α ,b = − s + s + s s + s + s + s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) Γ(1 − s − s − s − s )Γ(1+ s )Γ(1 − s − s − s )– 82 –e( z )Re( z ) z z γ (6 , γ (6 , × × α (6 , α (6 , × α (6 , × α (6 , × × Figure 7 : We will determine the initial conditions for F (6 , ab from the depicted six-dimensional basis of contours γ (6 , , γ (6 , , α (6 , , α (6 , , α (6 , , α (6 , . For these contours,the z → α (cid:48) -expansion which is not the case for the contoursmarked with × such as γ (6 , j with j = 3 , , , × (cid:18) s + s s + s + s , s s + s + s , , , , (cid:19) b (A.23)lim z → F (6 , α ,b = − s + s + s s + s + s + s Γ(1+ s )Γ(1+ s )Γ(1+ s + s ) Γ(1 − s − s − s − s )Γ(1+ s )Γ(1 − s − s − s ) × (cid:18) s s + s + s , s + s s + s + s , , , , (cid:19) b (A.24)as well aslim z → F (6 , α , ,b = (cid:18) u ˆ F (5 , α , , + u ˆ F (5 , α , , , u ˆ F (5 , α , , + u ˆ F (5 , α , , ,s s + s ˆ F (5 , α , , , s s + s ˆ F (5 , α , , , ˆ F (5 , α , , , ˆ F (5 , α , , (cid:19) b . (A.25)The gamma functions in (A.24) stem from the unique component of F (4 , in (3.31),and the coefficients u ij in (A.25) are given by u ij = (cid:32) s ( s + s )( s + s )( s + s + s ) − s s ( s + s )( s + s + s ) − s s ( s + s )( s + s + s ) s ( s + s )( s + s )( s + s + s ) (cid:33) ij . (A.26)– 83 –urthermore, the integration contours of the ˆ F (5 , α , ,i on the right-hand side of (A.25) canbe reduced to a basis of γ (5 , , via monodromy relationsˆ F (5 , α ,i = sin( π ( s + s )) sin( π ( s + s + s + s )) ˆ F (5 , ,i − sin( πs ) sin( π ( s + s )) ˆ F (5 , ,i sin( π ( s + s + s + s )) sin( π ( s + s + s + s + s ))ˆ F (5 , α ,i = sin( π ( s + s )) sin( π ( s + s + s + s )) ˆ F (5 , ,i − sin( πs ) sin( π ( s + s )) ˆ F (5 , ,i sin( π ( s + s + s + s )) sin( π ( s + s + s + s + s )) . (A.27)Finally, the hat denotes the following replacement of the arguments of F (5 , ,ˆ F (5 , ab ( s , s , s , s , s ) = F (5 , ab ( s + s , s + s , s , s , s ) , (A.28)which can be traced back to the z → z → KN (6 , = | z | s + s | z | s + s | z | s | − z | s | − z | s . (A.29) A.4.2 Assembling the initial value By combining the monodromy relations (A.22) with the z → F (6 , α i ,b ( z ), wearrive at the following initial values of F (6 , ab ( z ) in the basis of (A.20): P (6 , M (6 , = lim z → F (6 , = (A.30) F (5 , F (5 , F (5 , F (5 , H (6 , H (6 , F (4 , ( s , s ) F (4 , ( s , s + s + s ) 0 0 0 H (6 , H (6 , F (4 , ( s , s ) F (4 , ( s , s + s + s ) 0 0 J (6 , J (6 , K (6 , s s + s ˆ F (5 , ˆ F (5 , ˆ F (5 , J (6 , J (6 , s s + s ˆ F (5 , K (6 , ˆ F (5 , ˆ F (5 , . The entries H (6 , j are given by H (6 , = s + s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s ) (A.31) − sin( π ( s + s ))sin( π ( s + s + s )) F (5 , − sin( πs )sin( π ( s + s + s )) F (5 , H (6 , = s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s ) (A.32) − sin( π ( s + s ))sin( π ( s + s + s )) F (5 , − sin( πs )sin( π ( s + s + s )) F (5 , , – 84 –hile the entries H (6 , j can be obtained from H (6 , j by relabeling 2 ↔ H (6 , = s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s ) (A.33) − sin( π ( s + s ))sin( π ( s + s + s )) F (5 , − sin( πs )sin( π ( s + s + s )) F (5 , H (6 , = s + s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s ) (A.34) − sin( π ( s + s ))sin( π ( s + s + s )) F (5 , − sin( πs )sin( π ( s + s + s )) F (5 , . The entries J (6 , j are given by J (6 , = u ˆ F (5 , + u ˆ F (5 , (A.35) − sin( πs )sin( π ( s + s )) s + s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s )+ (cid:104) − sin( πs ) sin( πs ) F (5 , + sin( πs ) sin( π ( s + s )) F (5 , (cid:105) sin( π ( s + s + s )) sin( π ( s + s + s )) J (6 , = u ˆ F (5 , + u ˆ F (5 , (A.36) − sin( πs )sin( π ( s + s )) s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s )+ (cid:104) − sin( πs ) sin( πs ) F (5 , + sin( πs ) sin( π ( s + s )) F (5 , (cid:105) sin( π ( s + s + s )) sin( π ( s + s + s )) , where the u ij are defined in (A.26). The entries J (6 , j can again be obtained from J (6 , j by relabeling 2 ↔ J (6 , = J (6 , (cid:12)(cid:12) (2 ↔ = u ˆ F (5 , + u ˆ F (5 , (A.37) − sin( πs )sin( π ( s + s )) s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s )+ (cid:104) − sin( πs ) sin( πs ) F (5 , + sin( πs ) sin( π ( s + s )) F (5 , (cid:105) sin( π ( s + s + s )) sin( π ( s + s + s )) J (6 , = J (6 , (cid:12)(cid:12) (2 ↔ = u ˆ F (5 , + u ˆ F (5 , (A.38) − sin( πs )sin( π ( s + s )) s + s s + s + s F (4 , ( s , s ) F (4 , ( s , s + s + s )+ (cid:104) − sin( πs ) sin( πs ) F (5 , + sin( πs ) sin( π ( s + s )) F (5 , (cid:105) sin( π ( s + s + s )) sin( π ( s + s + s )) . – 85 –astly, the entries K (6 , ii related by 2 ↔ K (6 , = s ˆ F (5 , s + s − sin( πs )sin( π ( s + s )) F (4 , ( s , s ) F (4 , ( s , s + s + s ) K (6 , = s ˆ F (5 , s + s − sin( πs )sin( π ( s + s )) F (4 , ( s , s ) F (4 , ( s , s + s + s ) . (A.39)With the known α (cid:48) -expansions of the four- and five-point integrals F (4 , and F (5 , ab in open-string tree amplitudes, one can expand (A.30) to any desired order. Whilethe expansion of F (4 , is given by (3.31), all-order results for F (5 , can for instance beobtained from the methods in [21, 23, 73], and certain orders are available for downloadfrom the website [81]. A.4.3 Further comments Several entries of the initial value (A.30) feature spurious poles such as ( s + s ) − and ( s + s + s ) − within the individual terms of (A.35). It is a strong consistencycheck of both the assembly of the initial value and the α (cid:48) -expansion of the F (5 , ab thateach order in α (cid:48) conspires to polynomials in s ij . The coefficient of ζ , for instance, hasthe following entries in the first, third and fifth line,( P (6 , ) a = (cid:16) − s s − s s − s s − s s − s s , s s , , , , (cid:17) a , ( P (6 , ) a = (cid:16) s s − s s − s s − s s − s s , − s ( s + s + s ) , − s s − s s − s s − s s , , , (cid:17) a , (A.40)( P (6 , ) a = (cid:16) − s ( s + s + s + s + s ) , s ( s + s + s + s + s ) , − s ( s + s + s + s ) , s s , − s s − s s − s s − s s − s s − s s − s s , s ( s + s ) (cid:17) a , while the remaining entries can be reconstructed from relabelling 2 ↔ 3. The explicitform of the matrices P (6 , w , M (6 , w up to and including w = 9 and the braid matricesin (A.42) below can be found in an ancillary file within the arXiv submission of thiswork.In contrast to the initial values (A.18) of the F ( n, ab which boil down to Riemann zetavalues ζ k , the F ( n,p ) ab with p ≥ ≥ ζ , . The MZVs in the α (cid:48) -expansion of F (5 , known from string amplitudes [7, 145]propagate to the initial value of F (6 , as spelt out above. We have verified up toand including α (cid:48) that the initial values (A.30) obey the coaction principle (3.19), e.g.– 86 –hat the coefficient of ζ , in P (6 , M (6 , is given by [ M (6 , , M (6 , ]. Moreover, the α (cid:48) -expansion at finite z (cf. (4.13)), F (6 , ( z ) = P (6 , M (6 , G (6 , { , } ( z ) , (A.41)involves the series in polylogarithms G (6 , { , } ( z ) in (3.49) that depends on the transposes E (6 , ,z = ( e (6 , ) t and E (6 , ,z = ( e (6 , ) t of the braid matrices e (6 , = s + s + s − s − s − s − s s s + s + s − s − s − s s − s s + s − s 00 0 0 s + s − s ,e (6 , = − s s + s − s s + s − s s − s − s − s s + s + s s − s − s − s − s s + s + s . (A.42)We have checked that the combination M (6 , G (6 , { , } ( z ) obeys the formulation (4.26)of the coaction principle up to and including α (cid:48) . Moreover, since the F (6 , ab arethe simplest instance where MZVs beyond depth one and polylogarithms coexist inthe α (cid:48) -expansion, we have made the following crosscheck at the α (cid:48) -order: The co-efficients of G (1; z ) ⊗ ζ , and G (1; z ) ⊗ ζ ζ in ∆ G (6 , { , } ( z ) are indeed given by[[( e (6 , ) t , M (6 , ] , M (6 , ] and [[ M (6 , , M (6 , ] , ( e (6 , ) t ], respectively, in agreement with(4.24).Finally, the F (5 , in five-point string amplitudes exhibit a first dropout among theMZVs at weight 18, which is due to the vanishing of [[ M (5 , , M (5 , ] , [ M (5 , , M (5 , ]][7, 61]. By their assembly from ( n, p ) = (4 , , (5 , 2) integrals in (A.30), the F (6 , must share this dropout, and we have cross-checked its consistency with the coactionprinciple by verifying [[ M (6 , , M (6 , ] , [ M (6 , , M (6 , ]] = 0.Note that the soft limit s , s → e (6 , , e (6 , in (A.42) reproduces the five-pointinstances of the arguments of the 6 × α (cid:48) -expansionof F (5 , [23, 70]. – 87 – Braid group, monodromies and analytic continuation B.1 Obtaining X ( n,p ) ( g ) for any g ∈ S n − p In section 5 we have determined the analytic continuation of the F ( n,p ) -integrals from z i < z i +1 to z i +1 < z i for unintegrated punctures i = p +2 , . . . , n − 2. The group actionof such braid operations σ i,i +1 of neighboring punctures was explicitly given by matrices X ( σ i,i +1 ) in (5.31). In this appendix we will discuss the composition of group operations X ( g g ) for g , g ∈ B N to reduce more general analytic continuations to the X ( σ i,i +1 ),and mainly refer to [92] for facts about the braid group. This will also be relevant toshow that X is indeed compatible with the group structure and that we can recovermonodromies by doing the same braiding operation twice.It is convenient to remember that there exists a canonical projection,proj : B N → S N (B.1)given by forgetting the details of how the punctures braid around each other. Letus call g pr ∈ S N the image of an element g ∈ B N under this projection. Then, we canrewrite the content of (5.30) as follows:˜ G ( n,p ) (cid:0) σ pr i,i +1 ( z p +2 , . . . , z n − ) (cid:1) = X ( n,p ) ( σ i,i +1 ) G ( n,p ) ( z p +2 , . . . , z n − ) , (B.2)where the permutation σ pr i,i +1 acts on the indices of the punctures z i . In (B.2) we candescribe the braiding due to the element σ − i,i +1 by changing the sign in the exponentialin X ( n,p ) ( σ i,i +1 ). For g ∈ B n − p , we can generalize (B.2) to˜ G ( n,p ) ( g pr ( z p +2 , . . . , z n − )) = X ( n,p ) ( g ) G ( n,p ) ( z p +2 , . . . , z n − ) , (B.3)where we can define X ( n,p ) ( g ) recursively for the following formula for composing twogroup elements, g , g ∈ B n − p : ˜ G ( n,p ) (cid:0) ( g pr1 g pr2 ) ( z p +2 , . . . , z n − ) (cid:1) = ˜ G ( n,p ) (cid:0) g pr1 ( g pr2 ( z p +2 , . . . , z n − )) (cid:1) (B.4)= g pr1 (cid:0) X ( n,p ) ( g ) (cid:1) X ( n,p ) ( g ) G ( n,p ) ( z p +2 , . . . , z n − ) . The permutation g pr1 acts on the indices of the braid matrices in X ( n,p ) ( g ), and not inany way on the signs of the exponentials in this expression. From equation (B.4) wecan read off a formula for X ( n,p ) ( g g ): X ( n,p ) ( g g ) = g pr1 (cid:0) X ( n,p ) ( g ) (cid:1) X ( n,p ) ( g ) . (B.5) One can also conveniently perform this map by replacing the generators of the braid group, σ i,i +1 by transpositions ( i, i +1) ∈ S N . We are using a convention of composition of braidings and permutations consistent with σ pr34 σ pr45 =(34)(45) = (345). – 88 –ecause we can decompose any g ∈ B n − p into generators σ i,i +1 of the braid group, andwe know the form of X ( n,p ) ( σ i,i +1 ), we have obtained a prescription to compute any X ( n,p ) ( g ).As a sanity check, we should verify that X ( n,p ) satisfies the equations of the pre-sentation of the braid group, (5.27). The first of these equations, exemplified in, X ( n,p ) ( σ , σ , ) = X ( n,p ) ( σ , σ , ) follows easily from the algebra of braid matrices. Wefound the second of these equations, exemplified by X ( n,p ) ( σ , σ , σ , ) = X ( n,p ) ( σ , σ , σ , ),harder to prove in general, but checked explicitly that it holds for ( n, p ) = (7 , 1) up toweight α (cid:48) . B.2 Example: Monodromies from braiding twice in ( n, p ) = (5 , G ( n,p ) can be identified by the kernel of proj, which is a normalsubgroup of B n − p , called the pure braid group, P B n − p . From our description of thegenerators of B n − p , the simplest elements to describe in P B n − p are the squares of thegenerators, σ i,i +1 . Using X (5 , ( σ , ) = exp( iπE (5 , , ), the effect of braiding twice isdescribed in a way consistent with (5.2): X (5 , ( σ , ) = M ,z . (B.6) B.3 Example: Analytic continuation from two braidings For concreteness, we shall discuss an example with ( n, p ) = (7 , 1) and suppress thissuperscript. We will analytically continue from the integration domain 0 < z < z PolyLogTools [90] when the respective arguments obey arg( z ) > arg( z ) > arg( z ). We have per-formed such checks for several terms up to and including α (cid:48) , i.e. for MPLs up to weight5. One can generate equations valid in other regions of { z , z , z } by just changing thesigns of the exponentials in (B.7), or equivalently, by using the inverses of the braidoperations, σ − , or σ − , . For instance, the analogue of (B.7) for the braiding ( σ − , , σ , )is consistent in the region where arg( z ) > arg( z ) > arg( z ). Note that in the ( n, p ) = (5 , 1) case, there is no puncture z after integration, so punctures z = 0and z are neighbors. Furthermore, notice that we are braiding a puncture SL(2 , C )-fixed to 0, whichcould cause some problems due to regularization of terms G ( (cid:126)u ; 0) with (cid:126)u ∈ { , z } × . This is not a bigproblem if the end result lies in P B n − p . – 89 – .4 Initial values in an alternative fibration basis of polylogarithms We shall here spell out examples of the modified initial conditions of the F (6 , in section5.3 that arise from a change of fibration basis for the polylogarithms in (5.21). Morespecifically, the simplest instances of the matrices ˜ P (6 , w and ˜ M (6 , k +1 in (5.24) read˜ P (6 , = s − ( s + s ) 00 s = − ( e (6 , ) t , ˜ P (6 , = − s ( s +2 s ) − (2 s − s − s ) s s ( s + s ) − s + s s − s s − s − s ( s + s , ) 3 s − s s +2 s s − s , s − s , s , (B.8)˜ P (6 , = s s ( s + s ) ( s − s − s ) s − s ( s + s ) s , ( s − s s +3 s s + s ) 0 s s ( s + s ) − s ( s s , − s s ) 0 , ˜ M (6 , = s ( s , s + s ) s ( s s − s , s ) 0 s ( s s − s , s ) s ( s + s , s ) 0 s ( s + s , s − s s ) s , s s , − ( s + s ) s s , s s , , where we use the shorthand s ij...k, = s i + s j + . . . + s k . 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