One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points

Abstract

We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless $n$-point one-loop amplitudes of open superstrings and open bosonic strings. These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t. the modular parameter $\tau$ which is known from the A-elliptic Knizhnik--Zamolodchikov--Bernard associator. The expressions for their $\tau$-derivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes. These differential equations manifest the uniformly transcendental appearance of iterated integrals over holomorphic Eisenstein series in the low-energy expansion w.r.t. the inverse string tension ${\alpha }^{\prime }$. In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the ${\alpha }^{\prime }$-expansion of non-planar integrals is manifestly expressible in terms of iterated Eisenstein integrals without referring to twisted elliptic multiple zeta values. The degeneration of the moduli-space integrals at $\tau \to i\infty$ is expressed in terms of their genus-zero analogues -- $(n+2)$-point Parke--Taylor integrals over disk boundaries. Our results yield a compact formula for ${\alpha }^{\prime }$-expansions of $n$-point integrals over boundaries of cylinder- or Moebius-strip worldsheets, where any desired order is accessible from elementary operations.

73 pages, v2: assorted small corrections

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• Local BCJ numerators for ten-dimensional SYM at one loop
• Elliptic modular graph forms I: Identities and generating series
• A double copy from twisted (co)homology at genus one
• Two dialects for KZB equations: generating one-loop open-string integrals
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• Basis decompositions of genus-one string integrals
• Symmetric Orbifold Theories from Little String Residues
• Open-string integrals with multiple unintegrated punctures at genus one
• Advanced tools for basis decompositions of genus-one string integrals
• The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes
• Integral of depth zero to three basis of Modular Graph Functions
• A construction of single-valued elliptic polylogarithms