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 τ which is known from the A-elliptic Knizhnik-Zamolodchikov-Bernard associator. The expressions for their τ-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 α′. In fact, we are led to conjectural matrix representations of certain derivations dual to Eisenstein series. Like this, also the α′-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 τ → i∞ 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 α′-expansions of n-point integrals over boundaries of cylinder- or Möbius-strip worldsheets, where any desired order is accessible from elementary operations.
Highlights
Recent studies of scattering amplitudes in string theories have extended our computational reach into several directions and led to a variety of structural insights
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
The selection rules on whether a given combination of γ(k1, k2, . . . , kr|τ ) descends from elliptic multiple zeta values (eMZVs) are encoded in a family of derivations 2m, m ≥ 0 firstly studied by Tsunogai [39]
Summary
Recent studies of scattering amplitudes in string theories have extended our computational reach into several directions and led to a variety of structural insights. We describe a new method to integrate over open-string punctures in generating functions of genus-one integrals in one-loop amplitudes of bosonic strings and superstrings. One-loop open-string amplitudes in turn were shown [20, 21] to yield elliptic multiple zeta values (eMZVs) defined by Enriquez [22] upon integration over punctures on a cylinder or Mobius-strip worldsheet. Both of these worldsheet topologies are captured by more general integrals over A-cycles of a torus by different specializations of its complex modular parameter τ.
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