A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory

Abstract

The string corrections of tree-level open-string amplitudes can be described by Selberg integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. This allows for a recursion of the ${\alpha }^{\prime }$-expansion of tree-level string corrections in the number of external states using the Drinfeld associator. While the feasibility of this recursion is well-known, we provide a mathematical description in terms of twisted de Rham theory and intersection numbers of twisted forms. In particular, this leads to purely combinatorial expressions for the matrix representation of the Lie algebra generators appearing in the KZ equation in terms of directed graphs. This, in turn, admits efficient algorithms for symbolic and numerical computations using adjacency matrices of directed graphs and is a crucial step towards analogous recursions and algorithms at higher genera.

arXiv comment34 pages, 3 appendices, corrected typos, added some formulae, updated references, J. Phys. A: Math. Theor (2020)

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• Coaction and double-copy properties of configuration-space integrals at genus zero
• Modular graph forms from equivariant iterated Eisenstein integrals
• Higher-genus Fay-like identities from meromorphic generating functions
• Two dialects for KZB equations: generating one-loop open-string integrals
• BCJ, worldsheet quantum algebra and KZ equations
• Basis decompositions of genus-one string integrals
• Open-string integrals with multiple unintegrated punctures at genus one
• Advanced tools for basis decompositions of genus-one string integrals
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands
• Co-homology of Differential Forms and Feynman diagrams