IAS Amplitudes Group Meeting

Abstract: We consider the scattering problem of weakly nonlinear gravity waves on the surface of a two-dimensional incompressible fluid of infinite depth. The leading nonlinear dynamics is governed by four-wave interactions; however, a surprising result by Dyachenko & Zakharov (1994) showed that the four-wave interaction coefficient vanishes identically on the non-trivial resonant manifold. Craig & Worfolk (1995) demonstrated that non-vanishing resonant terms appear at fifth order in the Birkhoff normal form. Lvov (1997) then provided a complete classification of five-wave interaction coefficients for all orientations of wavevectors, obtaining expressions that, while remarkably simple, required case-by-case treatment across multiple kinematic regimes. We argue that the structure of these coefficients is governed by a selection rule bearing a striking analogy to the classification of gluon scattering amplitudes in Yang–Mills theory. Parke & Taylor (1986) showed that tree-level n-gluon amplitudes vanish when fewer than two gluons carry negative helicity, with the first non-vanishing case - the maximally helicity violating (MHV) amplitude - taking a strikingly compact closed form. An analogous result for free surface water waves exists: the single-minus tree-level n-wave scattering amplitude vanishes identically for all n, while the two-minus amplitude, the first non-vanishing case, admits a geometric interpretation - it is the volume of a hyperplane slice of a hypercube. Its combinatorial structure automatically unifies all of Lvov's sub-cases into a single closed-form expression. If time permits, we will also discuss leading-order soft theorems.