Abstract
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 defining 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 G4. By virtue of chiral splitting, closed-string correlators follow from the double copy of the open-string results.
Highlights
This is the third part of a series of papers [1,2,3] towards the derivation of one-loop correlators of massless open- and closed-superstring states using techniques from the pure-spinor formalism [4, 5]
The reason why Kn(0)(l) in (2.16) cannot be the full expression for the one-loop correlator for n ≥ 7 is related to BRST invariance; it is not difficult to show that the seven-point instance is not BRST invariant using the worldsheet functions discussed in part II
The correlator (3.19) takes the manifestly single-valued form: K5(l) = V1T2m,3,4,5E1m|2,3,4,5 + V1T23,4,5E1|23,4,5 + (2, 3|2, 3, 4, 5). This representation reverses the roles of worldsheet functions and kinematic factors in comparison to (3.17):6 Manifest BRST invariance is traded for manifest monodromy invariance
Summary
This is the third part of a series of papers [1,2,3] towards the derivation of one-loop correlators of massless open- and closed-superstring states using techniques from the pure-spinor formalism [4, 5]. The main result of this paper is the assembly of local one-loop correlators in pure-spinor superspace [6] up to eight points. This will be done by combining two main ingredients:. We will relate a double-copy structure of the open-string correlators [13] to the low-energy limit of the closed-string amplitudes This incarnation of the duality between kinematics and worldsheet functions is checked in detail up to multiplicity seven, and we describe the problems and perspectives in the quest for an n-point generalization at the end of section 4
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