  The evolution operator for states of gauge theories in the graph
representation (closely related to the loop representation of Gambini and
Trias, and Rovelli and Smolin) is formulated as a weighted sum over worldsheets
interpolating between initial and final graphs. As examples, lattice $SU(2)$ BF
and Yang-Mills theories are expressed as worldsheet theories, and (formal)
worldsheet forms of several continuum $U(1)$ theories are given. It is argued
that the world sheet framework should be ideal for representing GR, at least
euclidean GR, in 4 dimensions, because it is adapted to both the
4-diffeomorphism invariance of GR, and the discreteness of 3-geometry found in
the loop representation quantization of the theory. However, the weighting of
worldsheets in GR has not yet been found.
