Folds in 2D string theories

We study maps from a 2D world sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area-preserving diffeomorphisms of the target space. The contribution to the partition function of maps associated with a given fold configuration is computed. We derive a description of folds in terms of Feynman diagrams. A scheme to sum up the contributions of folds to the partition function in a special case is suggested and is shown to be related to the Baxter-Wu lattice model. An interpretation of folds as trajectories of particles in the adjoint representation of SU(N) gauge group in the large-N limit which in eract in an unusual way with the gauge fields is discussed.