Folds in 2D String Theories

TL;DR

This paper investigates the geometry and contributions of folds in 2D string theories, introducing a fold metric, analyzing their impact on partition functions, and relating them to Feynman diagrams and lattice models.

Contribution

It provides a new geometric framework for folds in 2D string theories, computes their partition function contributions, and connects them to lattice models and gauge theory interpretations.

Findings

Fold contributions to the partition function are explicitly computed.

A Feynman diagram description of folds is developed.

A summation scheme for fold contributions relates to the Baxter-Wu model.

Abstract

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 interact in an unusual way with the gauge fields is discussed.