Quantum gravity at the corner

Understand

We investigate the quantum geometry of $2d$ surface $S$ bounding the Cauchy slices of 4d gravitational system.

• We investigate in detail and for the first time the symplectic current that naturally arises boundary term in the first order formulation of general relativity in terms of the Ashtekar-Barbero connection.
• This current is proportional to the simplest quadratic form constructed out of the triad field, pulled back on $S$.
• We show that the would-be-gauge degrees of freedom---arising from $SU(2)$ gauge transformations plus diffeomorphisms tangent to the boundary, are entirely described by the boundary $2$-dimensional symplectic form and give rise to a representation at each point of $S$ of $SL(2,\mathbb{R}) \times SU(2)$.