Hamiltonian theory: Generalizations to higher dimensions, supersymmetry, and modified gravity

Loop quantum gravity in its Hamiltonian form relies on a connection formulation of the gravitational phase space with three key properties: (1.) a compact gauge group, (2.) real variables, and (3.) canonical Poisson brackets. In conjunction, these properties allow to construct a well-defined kinematical quantization of the holonomy-flux algebra, on top of which the remaining constraints can be implemented. While this idea has traditionally been mainly used for Einstein gravity, any gravitational theory with the above properties can be accommodated. In this chapter, we are going to review three strands of work building on this observation, namely, the study of higher-dimensional loop quantum gravity, supersymmetric extensions of loop quantum gravity, as well as the quantization of modified gravitational theories.