Tree-level open string amplitudes are integrals over the moduli space of n points on the projective line, and their field theory limit is the Cachazo-He-Yuan formula for tr(\Phi^3) amplitudes. Joint work with Freddy Cachazo, Alfredo Guevara and Sebastián Mizera (CEGM) replaces the line by P^(k-1), giving amplitude-like objects whose Feynman diagrams are higher-dimensional polyhedral complexes rather than trees. After reviewing this framework, I will discuss recent progress on new factorization patterns which appear and linear zero-preserving deformations, which produce nonlinear sigma model amplitudes at k=2 by a result of Arkani-Hamed, Cao, Dong, Figueiredo and He, and new factorization behavior at k greater than or equal to 3.
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