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

2019-08-28

Authors: Carlos R. Mafra and Oliver Schlotterer

Preprint number: UUITP--36/19

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'. In fact, we are led to matrix representations of certain derivations dual to Eisenstein series. Like this, also the alpha'-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 -> i infinity 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'-expansions of n-point integrals over boundaries of cylinder- or Möbius-strip worldsheets, where any desired order is accessible from elementary operations.