Gauge theory of Gravity based on the correspondence between the $1^{st}$ and the $2^{nd}$ order formalisms

Abstract: This is a shortened version of an invited talk at the XIII International Workshop "Lie Theory and its Applications in Physics", June 17-23, Varna, Bulgaria. A covariant canonical gauge theory of gravity free from torsion is studied. Using a metric conjugate momentum and a connection conjugate momentum, which takes the form of the Riemann tensor, a gauge theory of gravity is formulated, with form-invariant Hamiltonian. By the metric conjugate momenta, a correspondence between the Affine-Palatini formalism and the metric formalism is established. For, when the dynamical gravitational Hamiltonian $\tilde{H}_{Dyn}$ does not depend on the metric conjugate momenta, a metric compatibility is obtained from the equation of motions, and the equations of motion correspond to the solution is the metric formalism.