In this final part of a series of three papers, we will assemble supersymmetric expressions for one-loop correlators in pure-spinor superspace that are BRST invariant, local, and single valued. A key driving force in this construction is the generalization of a so far unnoticed property at tree-level; the correlators have the symmetry structure akin to Lie polynomials. One-loop correlators up to seven points are presented in a variety of representations manifesting different subsets of their

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... ining properties. These expressions are related via identities obeyed by the kinematic superfields and worldsheet functions spelled out in the first two parts of this series and reflecting a duality between the two kinds of ingredients. Interestingly, the expression for the eight-point correlator following from our method seems to capture correctly all the dependence on the worldsheet punctures but leaves undetermined the coefficient of the holomorphic Eisenstein series G 4 . By virtue of chiral splitting, closed-string correlators follow from the double copy of the open-string results. Note that V 123 Z tree 123 + V 132 Z tree 132 in (2.10) is symmetric in 1, 2, 3 even though only two out of 3! permutations are spelled out. This is a consequence of the Lie-polynomial structure of the correlator; 2 the right-hand side of (2.8) is permutation symmetric in 1, a 1 , a 2 , . . . , a |A| even though only the weaker symmetry in a 1 , a 2 , . . . , a |A| is manifest. The Lie-polynomial structure of the building blocks in the tree-level correlator (2.9) motivates us to search for a similar organization of the one-loop correlators. Assembling one-loop correlators Let us summarize what we have seen in part I and II in order to better understand the motivation behind the general form of the one-loop correlators K n (ℓ) to be proposed shortly. • In section I.3.3, we reviewed the definition of local superfields that satisfy generalized Jacobi identities and, in section I.4.4, we showed how they can be assembled in several classes of local building blocks. • In section II.5.5, we constructed functions composed of the expansion coefficients of the Kronecker-Eisenstein series that obey shuffle symmetries when the vertex insertion points are permuted. Let us thread the above points together in view of the tree-level structure discussed above. Firstly, since the short-distance singularities within the correlator are independent on the global properties of the Riemann surface, the shuffle symmetries of the worldsheet functions should also be a property of the worldsheet functions at one loop. And secondly, the shuffle symmetry obeyed by the functions are the driving force in the Lie-polynomial organization of the tree-level correlators with local kinematic building blocks. When taken together these points suggest that the superfields and worldsheet functions of one-loop correlators have the same symmetry structure of Lie polynomials. This realization will lead to a beautiful organization of superstring one-loop correlators. The Lie-polynomial structure of one-loop correlators The additional zero modes at genus one, in particular the availability of loop momenta, allow for a significantly richer system of kinematic building blocks as compared to the treelevel kinematics V 1A V n−1,B V n in (2.9). Also their accompanying worldsheet functions must accommodate the different OPE singularities and powers of loop momentum characteristic to each zero-mode saturation pattern, see e.g. (I.3.23) and (I.3.24). The corresponding Lie polynomials will therefore differ with respect to these features but will preserve their mathematical characterization as sums over products of shuffle-and Lie-symmetric objects. 2 This follows from the identity A 1 |A| ZAVA = B ZiBViB [8].