Scattering amplitudes of stable curves

Scattering amplitudes of stable curvesJENIA TEVELEV Hypertree divisors on the moduli space of stable rational curves were introduced by Castravet and Tevelev (2013).Their equations appear as numerators of scattering amplitude forms for n particles in N D 4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka (2016).Rather than being a coincidence, this is just the tip of the iceberg of an exciting relation between algebraic geometry and high energy physics.We interpret leading singularities of scattering amplitudes of massless particles as probabilistic Brill-Noether theory: the study of statistics of images of n marked points under a random meromorphic function uniformly distributed with respect to the translation-invariant volume form of the Jacobian.We focus on the maximum helicity violating case, which leads to a beautiful physics-inspired geometry for various classes of complex algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.