On constrained analysis and diffeomorphism invariance of generalised Proca theories

We consider a whole class of theories, each of which describes dynamics of a vector field coupled to any external background field. Furthermore, each of these theories is diffeomorphism invariance, has time–time and time-space components of Hessian of the vector sector equal zero but with space–space part being non-degenerate, and the vector sector is free from Ostrogradski instability. By using the Faddeev–Jackiw constrained analysis, we derive the conditions that theories in this class have to satisfy in order for the vector sector to have d-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d-1$$\end{document} propagating degrees of freedom in d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-dimensional spacetime. Most of these conditions are trivialised due to diffeomorphism invariance requirements. This leaves only a condition that a complicated combination of terms should not be trivially zero. Being an inequality, this condition is naturally easy to be fulfilled. We verify our result by using the Dirac constrained analysis to obtain the same result. Furthermore, we have also made a small remark on how the absence of diffeomorphism invariance affects the form of Faddeev–Jackiw brackets.