BCJ, worldsheet quantum algebra and KZ equations

We exploit the correspondence between twisted homology and quantum group to construct an algebra explanation of the open string kinematic numerator. In this setting the representation depends on string modes, and therefore the cohomology content of the numerator, as well as the location of the punctures. We show that quantum group root system thus identified helps determine the Casimir appears in the Knizhnik-Zamolodchikov connection, which can be used to relate representations associated with different puncture locations.


Introduction
Ever since its discovery by Bern, Carrasco and Johansson (BCJ) in [1,2], the notion of colour-kinematics duality has been playing a prominent role in improving practical calculation efficiency as well as providing insights into understanding gravity and gauge theory amplitudes, in the sense that the trivalent graph setting of duality drastically reduces the number of independent factors taken into account at loop level , and that its symmetry structure has proved advantageous when analysing noval formulations [26][27][28][29] and various theories [12,17,[30][31][32][33][34][35][36][37][38][39][40][41]. For more details we refer the readers to a recent review [42] and references therein. The BCJ duality itself has an interesting understanding from string monodromy relations [43][44][45][46], which incidentally was known as the Plahte identities [47] in earlier literature. In recent years we have witnessed a series of exciting development unfolded from string perspectives [15,[48][49][50][51][52][53][54][55][56], in particular it was shown by Casali, Mizera and Tourkine [55] that the monodromy relations can be naturally described in terms of twisted homology ( , ( ), Φ ) defined on a uni-valued branch of the Koba-Nielsen factor Φ . From this viewpoint the number of independent integrands can be determined from the topology of the worldsheet, and the monodromy relations generalises to generic particle insertions for arbitrary genus.
Incidentally, twisted homology groups ( , ( ), Φ ) are known to the quantum groups community to be isomorphic to tensor representations of the quantum universal enveloping algebras (QUEA) ( ) [57]. Indeed, suppose if we denote a 1-chain on the -punctured plane as ∈ ℂ ⋅ ( , 0 ) (Fig 1), properly normalised so that it is mapped by differential to the 0-form 0 as 1 ∶ 0 , the homology requirement for a 1-cycle yields exactly the same coefficients as the highest weight vector expanded by the tensors 1 ⋯⊗ 0 ⊗⋯⊗ . Namely, the requirement that In the correspondence described above, adding a 1-chain is related to the action of the algebra generator 0 on the -th site. In particular if we permute the location of any pair of the punctures, while keeping the 1-chains continuously deformed in the process, the result is equivalent to the action of a universal -matrix. The above identification is known to generalise to higher homology group [58]. In light of the above relation between twisted homology and quantum groups we feel that perhaps it would be interesting to explore an alternative narrative. In this paper we start with a more algebraic setting for the monodromy story. We show that both ( − 2)! and ( − 3)! basis BCJ numerators are expressible respectively as bilinear form and quantum Clebsch-Gordan coefficient raised or lowered by the action of screening vertex operators, which are contour representations of the QUEA ( ), with its root system determined from external leg momenta. In this setting standard Hopf algebra operations can be understood as contour manipulations on the screening operators. When taking the infinite string tension limit ′ → 0 the algebra reduces to the undeformed Lie algebra (or more generically a Kac-Moody algebra) defined by the same set of roots. Both the bilinear form and the quantum Clebsch-Gordan interpretation of the numerators will depend on the representation modules, whose roles are played by the vertex operators fixed at specific locations by (2, ℝ) invariance. Modules located at different 's will be related by the Knizhnik-Zamolodchikov (KZ) connection. We show that the root system inherited from the screening operator picture helps determine the Casimir of the KZ equation [59]. From KZ solutions associated with various tensor modules one can reconstruct -amplitudes and open string numerators, and we feel this part of the story serves as an algebra-oriented supplement to the recent development on -theory and KZ [60].
We organise this paper as the following. In section 2 we briefly review two slightly different versions of the screening operators relevant to the discussion. In section 3 we explain BCJ numerators using the newly introduced settings. Section 4 begins with a very quick review of the KZ equation with emphasis on its algebra contents followed by a discussion on -theory amplitudes and KZ coefficients in section 4.2. We conclude our paper in section 5 .

(i) C-shaped contour screening operators
Suppose if we denote the open string Hilbert space as , the vertex operators are algebra valued distributions  ∈ ()[[ , −1 ]] satisfying the state-field correspondence principle. For the purpose of discussion let us for the moment focus on bosonic open strings, so that a typical vertex operator takes the explicit form A screening operator ∈ () associated with any vertex operator ( ) = ⋅ ( ) of interest is defined as the following integral over the C-shaped contour along both sides of the branch cut [61,62].
In this paper we follow the same conventions as in [63] and [64]. Variables appear on the left are assumed to start with a larger value on the real line than those on the right, correspondingly shifted by a larger on the complex plane, and then analytically continued to their designated values. Note however that in terms of figures, conventionally the real line points to the right instead of left, so that everything illustrated in figures will be the mirror image to what appears in the equation. For example the action of is represented by Fig. 2(a). The analytic continuation we use here leads to the following braiding relations for 1 > 2 .
In the presence of successive actions, the contour associated with operator that comes later is defined so as to surround the pre-existing contours ( Fig. 2(b)). The action of an operator is defined to annihilate the contour integral created by using the conformal property of vertex operator that reduces an integral to its boundaries, Explicitly this is defined to carry a normalisation factor so that The screenings thus defined together with the fixed point vertex operator ( ), which serves as the highest weight Verma module, provides a representation of the QUEA ( ) [65][66][67][68], where ∶= ′ 2 and is the operator that measures momentum, or charge in the original settings [61,62,69], so that 's and 's were supposed to lower or raise the background charge produced by the module, hence the name screenings.
A natural representation for tensor of modules ⊗ can be obtained by simply taking the product of vertex operator at distinct fixed points ( 1 ) ( 2 ). The coproduct of a screening Δ( ) is then defined by the corresponding action on this product followed by integration over a contour that surrounds both vertices (Fig. 3), which in turn can be translated into the actions on individual modules by breaking the original contour into two smaller ones surrounding each modules and then swap the ordering using braiding relation (2.4).
. The above result is the same as the following tensors of screenings as was expected for a quantum group. The action of antipode and counit are represented by reversing and removing the contour of a screening respectively.

The -matrix
In the screening representation of quantum groups the universal -matrix is defined as the composition of a plain permutation that swaps modules along with their screenings, together with the application of braiding relations (2.3) and (2.4) that eventually restores modules back to their original order.
The above effect is the same as the action that successively removes screenings from one of the modules within the tensor product and then reapply them onto the other 1 .
Generically the complete formula for can be derived term by term following similar reasoning [70,71]. Quasitriangular condition Δ( ) = Δ ′ ( ) can be seen from the fact that has no effect on a screening as long as it encompasses both modules (Fig. 5). In particular that Yang-Baxter equation 12 13 23 = 23 13 12 is indeed satisfied can be seen from the fact that derives from braiding. 1 For simplicity we have neglected here the braiding factor produced by swapping modules, which will result in an overall ⊗ 2 in the -matrix

(ii) Line interval screenings
An alternative version of the screening operator [72] that turns out to be also relevant to our BCJ problem is defined as the line integral over a fixed interval along the real line, for example over [0, 1], whereas the charge operator is defined by the same closed integral as before (2.10). In the case of line interval screenings it is sometimes convenient to restrict our considerations to only the positive Borel subalgebra + ( ) = ⊕ ⊕ of the full quantum universal enveloping algebra generated by Cartan subalgebra and positive root vectors but without 's, because the boundary of a line interval is less symmetric than the C-shaped contour, making it less natural to define the action of using conformal generator as in (2.5), even though one can simply define it as manually removing one line screening. As we will see in section 3 that the positive part of the full algebra will be enough as far as numerators and amplitudes are concerned. In the settings of line screenings tensors of modules ⊗ are represented by products of vertex operators ( 1 ) ( 2 ) as before. Action of an on individual module is defined similar to (2.16) but with the integral carried out only to a manually fixed point between 1 and 2 , whereas in the coproduct it is carried out over the full segment [0, 1]. The antipode and counit are represented by reversing and removing the contour respectively as before. More details can be found for example in [72].
For the purpose of discussions it is useful to consider right action of a line screening ( ) , defined the same as (2.16) but with the ordering of the two vertex operators swapped. According to our convention this corresponds to a line integral starting with a point 2 on the real line that is larger than , and then analytically continued, so that the contours associated with left and right actions corresponds to the blue and lilac lines illustrated in Fig. 6 respectively. From this perspective a C-shaped screening can be identified as the -deformed adjoint action of line screening, where and represent taking the left and right actions respectively and is the antipode, ( ) = − −1 .
Explicitly we have which is the same as (2.2) up to an overall factor that we will discard through redefinition.

String BCJ numerators
In a previous paper [73] we showed that the on-shell limit of the multiple C-shaped contour integrals derived originally from KLT in [64] serves as a natural string theory generalisation of the ( − 2)! basis BCJ numerator.
Let us also recall that, in addition, the ( − 3)! basis numerator was identified in the context of CHY in [75] as the product of momentum kernel with partial amplitudes, which in terms of vertex operators can be expressed also as multiple C-shaped contour integrals [64], but with three of the vertices ( 1 , 2 , ) fixed instead of two 2 , Suppose if we focus on the vertex operators in (3.1) and (3.3), ignoring for the moment a common final leg ( ) that is frequently pushed to infinity, the multiple C-shaped contour integrals appear in ( − 2)! and ( − 3)! basis numerators can be identified as screenings acting on single and tensor modules respectively. Explicitly, for the The above settings naturally defines a representation of quantum group ( ) with the 3 , 4 , … , −1 identified as the simple root vectors. Comparing with the definition of a screening (2.2) and equations (2.6) to (2.8) we see that the corresponding simple roots are identified with the momenta 3 , 4 , … , −1 carried by external legs. In the infinite string tension limit ′ → 0, therefore → 1 and the QUEA ( ) reduces to the classical Lie (or Kac-Moody) algebra with the (symmetrised) Cartan matrix defined by the same roots, so that in the field theory limit the BCJ kinematic algebra should be isomorphic to the algebra determined by external leg momenta. Starting with 3 , 4 , … , −1 as building blocks the QUEA thus defined contains non-simple root vectors generated by -commutators [ ] up to a noramlisation that depend on its root . The result of the -commutator can be seen from Fig. 7 to be by itself a screening operator, but with its vertex operator calculated from the following operator product.
where we have chosen 1 , 2 to be tachyons as an example. This process continues generating new root vectors indefinitely until it is interrupted by quantum Serre relations where 's are ordered operator product line integrals, and likewise for the other ordering. When the root system is identical to that of the (3), suppose if we choose 1 = , 2 = 3 = , we see that (3.9) becomes which indeed vanishes because for (3) the roots satisfy ⋅ ( + ) + ⋅ = 0, and therefore ′ (( + ) ⋅ ) = − ′ ( ⋅ ). For generic momentum configuration the algebra is infinite dimensional. To make the algebra finite one can chose to work with compactified sapcetime such that all momenta live on a rational lattice [76], in this case all ′ ⋅ eventually become integers after being superposed large enough number of times and the overall sinusoidal factor appears in (3.8) vanishes. Note that the BCJ amplitude relation can be regarded as a special type of the Serre relations (3.9) even though it eliminates only the root vector that carries zero root constrained by momentum conservation 1 +⋯+ = 0, especially that roots containing multiple copies of the same momentum 1 1 + 2 2 +… remain and the algebra is infinite unless rest of the constraints just described were imposed.
In the settings of string theory asymptotic states provides a natural definition for a bilinear form that can be used to normalise modules and root vectors. Note in particular from this perspective the ( − 3)! basis numerator can be regarded as a quantum Clebsch-Gordan coefficient. Different choices of vertex operator screenings in this setting correspond to representations associated with different modes. When the C-shaped contour extends to infinity the norm of a lowered state, ( , ) for example, can be calculated through flipping the contour surrounding as shown in Fig. 8, The above result can be expressed as linear combination of ordered line integrals, which in this example are beta functions. Similar calculation applies to multiply lowered state by 's and to different choices of vertex operator representations as well. Generically the bilinear form can be similarly defined for arbitrary values of 1 and , different choices of bilinear form are related by KZ equations. We leave this part of the discussion to section 4.1.

Jacobi-like identities
In light of the original idea of BCJ duality it is perhaps more or less expected that the numerator is expressable as successive adjoint actions that mimics the colour dependence of the amplitude, and indeed it was realised in [29, 73, 74] that such structure is accounted for in string theory by deformed brackets. In addition we note that because the C-shaped screening can be regarded as the -deformed adjoint action of line screenings, the ( − 2)! basis numerator (3.1) can be recast into the following BCJ manifest form.
where the 's above are line screenings, and the numerator is therefore the successive adjoint actions of QUEA determined by external leg momenta. If our only purpose is to explain the BCJ duality originally observed in field theory amplitudes, it is not strictly necessary to come up with a -deformed analogue of the Jacobi-like identity satisfied by string numerators. However we do actually have an identity that explains the expected relation at quantum level,

Relation to the KZ equations
Recall that the KZ equations [59] is a set of differential equations for Lie algebra (or more generally Kac-Moody algebra)-module with coordinate dependence, For a Kac-Moody algebra with simple roots and Cartan subalgebra ℎ , satisfying the following (classical) commutation relations, Its Verma modulẽ Λ is generated by the highest weight modulẽ Λ , and root vectors , The module that appears in the KZ equation is a map from the space = {( 1 , 2 , … , ) ∈ ℂ | ≠ } to the space of tensor modules̃ Λ 1 ⊗̃ Λ 2 ⊗ ⋯ ⊗̃ Λ . For example when = 2, On the other hand the operator Ω in the KZ is the Casimir, (4.6) and Ω is understood to act only on the -th and -th site of the tensor. By construction Ω commutes with all coproducts in the algebra, in particular [Ω, Δ( )] = 0, so that it only mixes tensors with the same overall weights. In light of this the solutions can be assorted into top modules, 1-level lowered modules and so on, as was shown by equations (4.4) and (4.5). Explicitly the coefficient functions are given by where Φ = − 1 An depends only on the (twisted) homology once we have chosen a particular -form, so that when 's vary along a path in the base space the contour deforms continuously, and the action of the KZ provides a Gauss-Manin connection on the bundle with base and twisted homology ( , ( ), Φ ) as its fibre. In particular when the end point { 1 , 2 , … , } is a permutation of the starting point ( 0 1 , 0 2 , … , 0 ), the action of KZ braids (Fig. 9) and defines an -matrix on ( , ( ), Φ ). The twisted homology group is known to be isomorphic to the quantum group ( ) that corresponds to the -deformation of (4.2) [58].

Correlators, bilinear forms and KZ solutions
In this section we temporarily remove all integrals present in an amplitude or BCJ numerators (3.1), (3.3) and focus exclusively on their correlator . When all vertices are tachyons apparently the correlator is the 0-form coefficient 0 = ∏ , ( − ) ′ ⋅ of the top highest weight module 0 = 0 ⊗ ⋯ ⊗ 2 ⊗ 1 and the KZ equations (4.1) in this case translate to the differential equations of 0 , whereas the action of Casimir Ω can be read off directly from the module, giving (Assuming that we identify ′ = 1∕ .) Starting with 0 ( 0 1 , 0 2 , … , 0 ) at a specific set of 's the KZ connection uniquely determines the value ( ′ 1 , ′ 2 , … , ′ ) through parallel transport, and therefore 0 ( ′ 1 , ′ 2 , … , ′ ) at any set of 's. Especially when the number of punctures is restricted to 2 we see the bilinear form ⟩ for all ( 1 , 2 ) including the asymptotics (0, ∞) are related to each other in the same manner.
In the case where one gluon is present, suppose instead of direct substitution with a gluon vertex operator gluon 1 ( 1 ) = ⋅̇ 1 ⋅ ( 1 ) we choose to represent gluon by the Del Giudice-Di Vecchia-Fubini (DDF) constructed vertex [81], where 0 ( ) = ⋅̇ 0 ⋅ ( ) . The polarisation is taken to be in the orthogonal direction ⋅ 0 = 0, and 0 + 0 = 1 is the original gluon momentum. Note the settings of DDF demands ′ 0 ⋅ 0 = −1 so that equation (4.9) can be regarded as a special case of the screening acting on a module 0 ( 1 ), where we can safely close the contour on the same sheet to form a closed loop 1 around 1 (the same contour 1 as was shown earlier in Fig. 1). From this perspective the one gluon correlator is the pairing of the cycle 1 with a 1-form derived from OPE, which in turn can be easily identified term by term with the 1-forms generated by KZ coefficients (0,…,1,…,0) = ∫ In light of the 1-level lowered KZ solution is given by the following sum of tensor modules, the one gluon correlator can be expressed as 1 projected onto a dual vector 0 (1,0,…,0) + ⋅ 2 (0,1,…,0) + ⋯ + ⋅ (0,…,0,1) . The action of Casimir Ω can be again read off directly from the module, yielding a slightly more complicated set of differential equations that relates correlators at different 's. As a quick consistency check of the relations just described, recall that in the zero string tension limit ′ → ∞, the correlator should only have support on the Gross-Mende saddle points [83]. Suppose if we fix the values of ( 0 , 2 , … , ) while maintaining the condition ′ 0 ⋅ 0 = −1, so that 0 ∼ 1∕ ′ → 0 and the action of root vector 0 on modules becomes negligible, the KZ equations of 1 implies therefore we see that scattering equations indeed must be satisfied if the 's are to localise. It is straightforward to generalise the above reasoning to incorporate more higher modes in the string spectrum, for example an -gluon correlator is given by the pairing of an -cycle with the -form derived from the OPE, whereas the -cycle, when visualised on the punctured plane, is given by the independent closed loops 1 , … , surrounding each puncture . The corresponding -form, on the other hand, can be spanned by the -forms generated by KZ coefficients through straightforward term by term identifications.

KZ solutions and Z-amplitudes
We return to amplitudes and numerators. In the previous section we directly identified the KZ coefficients needed to span a correlator. Generically a correspondence is known as the Drinfeld-Kohno theorem [79,80] which identifies given quntum algebra ( ) behaviour with the monodromy of 's, which lives in the representation space of classical Kac-Moody algebra with the same roots as ( ). In view of the discussions in section 3 we see that an (2, ℝ) fixed -point amplitude or numerator is described by the QUEA with simple roots 2 , 3 , ..., −2 read off from its external legs as in (3.4), it is therefore natural to look for KZ coefficients in the ∑ −2

=2
-weight lowered level subspace of the classical tensor modulẽ 1 ⊗̃ −1 . For example at four points, suppose if we fix ( 1 , 3 , 4 ) at two arbitrary points and infinity respectively, the associated algebra is then given by (4.2) with only one simple root 2 . The KZ equation is solved on the maps from ( 1 , 3 ) ∈ ℂ 2 | | | 1 ≠ 3 to vectors iñ 1 ⊗̃ 3 , which has the following form.
Here we denote the coefficients as ({ 2 },∅) to emphasise generically they should be labeled by an ordered set that clarifies in which order the root vector 's are applied to the corresponding tensor vector. In terms of this notation the solution to the KZ equations is given by the following. where and is any closed contour, for example the Pochammer encircling 1 and 3 . A -theory amplitude (1, 2, 3, 4) [28,29,77] for example is known to be expressible (up to proportionality factors produced when translating between ordered integrals and Pochammer) as the linear combination ({ 2 },∅) − (∅,{ 2 }) and therefore satisfies the KZ equations, in the sense that it can be expressed as scalar product of 4 and a independent dual module, assuming the orthonormal duals to the two basis tensor vectors in (4.5) are ({ 2 },∅) and (∅,{ 2 }) . The solution for the first three 's takes the following form.
with the Koba-Nielsen factor Φ given by The rest three coefficients can be obtained by permutations of 2 and 3 . Note that the solution 5 [53,84,85].) The KZ equation used in [60] is a nomalised version of KZ equation from Kac-Moody algebra with simple roots 2 , 3 , ..., −2 solving for on the weigh- lowered submodule a tri-tensor spacẽ matrix representation of (4.26) acting on -integrals can be used to build the Drinfeld associator. And the specific form of matrices used in [60] can be achieved by suitable linear transformations.

Conclusions
In this paper we showed, with the help of screening vertex operators, that the string generalisation of the BCJ numerators previously derived in [64,73] have a natural quantum group explanation. The associated algebra structure depends on the specific root system, which in turn is entirely defined by external leg momenta. The definition of a screening involves a string vertex operator followed by a contour integration. For this setting to be interpreted as a representation of the quantum group, a screening operator only needs to create a 1-chain on the punctured plane ℂ − { 1 , 2 , … , } while various choices for the vertex operator in the string spectrum leads to different cohomology contents and corresponds to different representations. Generically the representation depends on string modes as well as on the exact location of the modules. We showed that modules built from the same string modes (and therefore lead to the same cohomology structures) but located at different 's are related to each other by the flat KZ connection. From this perspective the action of a universal -matrix has a explicit graphical interpretation as the braiding of two punctures. In other words, quantum algebraic structure of string amplitude can be represented by sections of a local system over configuration spaces of s, with modules of the kinematic algebra as its fibre and KZ connection as its flat connection. This local system can be isomorphically mapped to the local system used in discussing twisted homology for string amplitude, with each element in the module mapped to a class of twisted forms and the KZ connection mapped to the Gauss-Manin connection. In fact this identification is known to the quantum groups community as part of the complex of isomorphic algebraic structures build from a discirminantal hyperplane arrangement including Orlik-Solomon algebra, flag complex, twisted homology and cohomology, and representation of Kac-Moody algebra. It would be interesting to see if such complex of isomorphisms can be respectively translated to the interplay between different formalisms of amplitude such as twistor formalism, cluster algebra, positive geometry and string scattering form, and the algebra of screening operators.