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Abstract We propose a geometric relation between closed and open string amplitudes at one-loop. After imposing a homological splitting on the world-sheet torus, twisted intersection theory is used to establish a one-loop double copy relation. The latter expresses a closed string amplitude by a pair of open string amplitudes and twisted intersection numbers. These inner products on the vector space of twisted differential forms are related to the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral.

Abstract We derive a massive double copy construction within string theory. To this end, we use massive vectors of the open string spectrum that appear in compactifications to four dimensions and construct massive spin-2 tensors as closed string states, thereby mimicking the structure of the massless graviton. We then compute three-point amplitudes for the scattering of massless and massive spin-2 closed string states and reveal the double copy structure of the latter. With these results being finite in the string scale, we are further able to reproduce the cubic Lagrangian of ghost-free bimetric theory around flat spacetime for bulk massive spin-2 states originating in products of vectors of extended brane supersymmetry.

Abstract We present a geometric derivation of all-multiplicity (single-trace) tree-level Einstein Yang-Mills (EYM) amplitudes A $$ \mathcal{A} $$ (n; r) involving n gluons and r gravitons by a bilinear of two twisted differential forms on the moduli space of Riemann spheres with m := n + r punctures. The differential forms are gained by studying the underlying superstring disk amplitude and proposing an embedding of the disk onto the sphere. This map can be interpreted as a geometrical map from the open superstring to a heterotic or ambitwistor string structure. Then, the twisted intersection number of the two m-forms, which is obtained by integrating over the moduli space of Riemann sphere with m punctures, reproduces in the infinite inverse string tension limit α′ → ∞ the corresponding CHY formula of the EYM amplitude. To bolster our findings we study the disk amplitude of open and closed strings using the Graßmann description of the underlying superstring amplitude, map it to a closed string amplitude and consider the α′ → ∞ limit. Finally, we present an all-multiplicity decomposition formula of any EYM amplitude A $$ \mathcal{A} $$ (n; r) as linear combination over (m − 3)! pure m gluon subamplitudes.