  We present a polymer(loop) quantization of a two dimensional theory of
dilatonic gravity known as the CGHS model. We recast the theory as a
parametrized free field theory on a flat 2-dimensional spacetime and quantize
the resulting phase space using techniques of loop quantization. The resulting
(kinematical) Hilbert space admits a unitary representation of the spacetime
diffeomorphism group. We obtain the complete spectrum of the theory using a
technique known as group averaging and perform quantization of Dirac
observables on the resulting Hilbert space. We argue that the algebra of Dirac
observables gets deformed in the quantum theory. Combining the ideas from
parametrized field theory with certain relational observables, evolution is
defined in the quantum theory in the Heisenberg picture. Finally the dilaton
field is quantized on the physical Hilbert space which carries information
about quantum geometry.
