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Canonical description of Exotic General Massive Gravity

Exotic General Massive Gravity is the next-to-simplest gravitational theory fulfilling the so-called third-way consistency, the simplest being Minimal Massive Gravity. We investigate the canonical structure of the first-order formulation of Exotic General Massive Gravity. By using the Dirac Hamiltonian formalism, we systematically discover the complete set of physical constraints, including primary, secondary, and tertiary ones, and explicitly compute the Poisson bracket algebra between them. In particular, we demonstrate that the consistency condition for the tertiary constraints provides explicit expressions which can be solved algebraically for the auxiliary fields $f$ and $h$ in terms of the dreibein $e$. In this configuration, to confirm that the theory is ghost-free, the whole set of constraints is classified into first and second-class ones showing the existence of only two physical degrees of freedom corresponding to one massive graviton. Furthermore, we identify the transformation laws for all of the dynamical variables corresponding essentially to gauge symmetries, generated by the first-class constraints. Finally, by taking into account all the second-class constraints, we explicitly compute the Dirac matrix together with the Dirac's brackets.