Explicit gauge-invariant variables in multifield inflation beyond linear order and Hamiltonian dynamics

General relativity coupled to multiple scalar fields is a diffeomorphism-invariant constrained system. Consequently, a naive counting of the perturbative degrees of freedom unavoidably overestimates the true number of physical modes propagating in the theory, as gauge redundancies and constraint equations remove non-dynamical ones. While this problem has been solved for linear fluctuations, this work presents the first explicit calculation of all large-scale gauge-invariant phase-space variables in multifield inflation and at second order in perturbation theory, in a Hamiltonian language. Building upon the well-known Sasaki-Mukhanov variables, we show how to construct a finite-dimensional basis of quadratic corrections which are invariant under gauge transformations. Although our procedure is generic to any number of fields and at any scale, we restrict to super-Hubble scales for their explicit solution, which we deliver. Henceforth, we prove that it is possible to recover the usual flat-gauge and comoving-gauge fluctuations as large-scale gauge-invariant combinations, making for a robust consistency check of the gauge-fixed procedure to connect theoretical predictions above the horizon to observations. We derive the quadratic and cubic Hamiltonian of multifield inflation in a gauge independent manner, then we gauge fix our theory by going into the flat gauge, and we show perfect agreement with the literature on this topic, usually based on a Lagrangian approach. After these concrete steps, we propose a more formal proof of the existence of gauge-invariant variables at quadratic order, and we provide a sketch of the procedure that should allow to go to higher orders in perturbation theory.