  Using ideas from Jones, lattice gauge theory and loop quantum gravity, we
construct 1+1-dimensional gauge theories on a spacetime cylinder. Given a
separable compact group $G$, we construct localized time-zero fields on the
spatial torus as a net of C*-algebras together with an action of the gauge
group that is an infinite product of $G$ over the dyadic rationals and, using a
recent machinery of Jones, an action of Thompson's group $T$ as a replacement
of the spatial diffeomorphism group. Adding a family of probability measures on
the unitary dual of $G$ we construct a state and obtain a net of von Neumann
algebras carrying a state-preserving gauge group action. For abelian $G$, we
provide a very explicit description of our algebras. For a single measure on
the dual of $G$, we have a state-preserving action of Thompson's group and
semi-finite von Neumann algebras. For $G=\mathbf{S}$ the circle group together
with a certain family of heat-kernel states providing the measures, we obtain
hyperfinite type III factors with a normal faithful state providing a
nontrivial time evolution via Tomita-Takesaki theory (KMS condition). In the
latter case, we additionally have a non-singular action of the group of
rotations with dyadic angles, as a subgroup of Thompson's group $T$, for
geometrically motivated choices of families of heat-kernel states.
