The Hydrotope in the Water-Wave Action

Abstract

The Hydrotope gives a geometric representation of tree-level water-wave amplitudes with two negative spatial momenta as the volume of a hyperplane slice of a box. We trace the origin of this geometry directly to the water-wave action. Writing the $n$-point contact interaction as $\mathcal{V}_n=\sum_{i<j}w_iw_jh_{ij}^{(n)}$, we show that when the two marked momenta have the same sign and every spectator has the opposite sign, the corresponding coefficient is $h_{ij}^{(n)}=2H_n$, where $H_n/(n{-}3)!$ is the Hydrotope volume. More generally, every fixed-pair coefficient admits a denominator-free ordered-flag representation as an oriented sum of $(n{-}3)$-dimensional box volumes. We then sum all two-minus trees by cutting each at the unique vertex joining its two minus branches. The coefficients multiplying minus-minus, minus-plus, and plus-plus frequency bilinears reduce, respectively, to $(2^{n{-}1}{-}2)H_n$, $0$, and $2H_n$, and immediately reproduce the known amplitude $2^{n{-}1}w_1w_2H_n$. Thus, the Hydrotope-and a broader class of related box-slice geometries-is already encoded locally in the water-wave action. This points to further hidden simplicity and geometric structure in general water-wave amplitudes.