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We present a geometric derivation of all-multiplicity (single-trace)tree-level Einstein Yang-Mills (EYM) amplitudes ${\cal A}(n;r)$ involving $n$gluons and $r$ gravitons by a bilinear of two twisted differential forms on themoduli space of Riemann spheres with $m\!:=\!n\!+\!r$ punctures. Thedifferential forms are gained by studying the underlying superstring diskamplitude and proposing an embedding of the disk onto the sphere. This map canbe interpreted as a geometrical map from the open superstring to a heterotic orambitwistor string structure. Then, the twisted intersection number of the two$m$-forms, which is obtained by integrating over the moduli space of Riemannsphere with $m$ punctures, reproduces in the infinite inverse string tensionlimit $\alpha'\!\rightarrow\! \infty$ the corresponding CHY formula of the EYMamplitude. To bolster our findings we study the disk amplitude of open andclosed strings using the Grassmann description of the underlying superstringamplitude, map it to a closed string amplitude and consider the$\alpha'\!\rightarrow\! \infty$ limit. Finally, we present an all-multiplicitydecomposition formula of any EYM amplitude ${\cal A}(n;r)$ as linearcombination over $(m\!-\!3)!$ pure $m$ gluon subamplitudes.

Einstein Yang-Mills Amplitudes from Intersections of Twisted Forms