Abstract
Inspired by the definition of color-dressed amplitudes in string theory, we define analogous color-dressed permutations replacing the color-ordered string amplitudes by their corresponding permutations. Decomposing the color traces into symmetrized traces and structure constants, the color-dressed permutations define BRST-invariant permutations, which we show are elements of the inverse Solomon descent algebra and we find a closed formula for them. We then present evidence that these permutations encode KK-like relations among the different α′ corrections to the disk amplitudes refined by their MZV content. In particular, the number of linearly independent amplitudes at a given α′ order and MZV content is given by (sums of) Stirling cycle numbers. In addition, we show how the superfield expansion of BRST invariants of the pure spinor formalism corresponding to α′2ζ2 corrections is encoded in the descent algebra.
Highlights
It is well-known that superstring n-point color-ordered disk amplitudes satisfy monodromy relations which imply that the number of linearly independent amplitudes is (n − 3)!, for all α corrections [1, 2]
We present evidence that these permutations encode KK-like relations among the different α corrections to the disk amplitudes refined by their MZV content
In this paper we investigate a weaker set of relations, called KK-like relations [4], of higher α corrections to disk amplitudes refined by their MZV content [5–7]
Summary
It is well-known that superstring n-point color-ordered disk amplitudes satisfy monodromy relations which imply that the number of linearly independent amplitudes is (n − 3)!, for all α corrections [1, 2]. It is not yet known in general the form of the KK-like relations and the corresponding basis dimensions for the ζ2mζM components of (1.1) with m ≥ 2. We will see that the general KK-like relations are closely related to the mathematical framework of the Solomon descent algebra [10–16]. To see this we will define the colordressed permutation. In the appendices we review the descent algebra and collect various proofs and explicit expansions omitted from the main text
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