  One of the virtues of the Ashtekar variables is the simplification of the
initial value constraints for gravity. In the case of self-dual variables this
entails a complexification of the phase space which comes at the expense of
having to implement reality conditions in the Lorentzian signature case. A
reformulation of the theory in terms of real variables eliminates this
difficulty, albeit at the expense of having to deal with a more complicated
Hamiltonian constraint. The set of available gravitational theories classically
equivalent to Einstein's is parametrized by a parameter $\beta$, known as the
Immirzi parameter. We rephrase the Hamiltonian constraint into the form of an
affine Lie algebra for arbitrary $\beta$, and perform a quantization.
