Two-dimensional topological field theories as taffy

Abstract

In this paper, we use trivial defects to define global taffy-like operations on string worldsheets, which preserve the field theory. We fold open and closed strings on a space $X$ into open strings on products of multiple copies of $X$, and perform checks that the “taffy-folded” worldsheets have the same massless spectra and other properties as the original worldsheets. Such folding tricks are a standard method in the defects community; the novelty of this paper lies in deriving mathematical identities to check that e.g., massless spectra are invariant in topological field theories. We discuss the case of the B model extensively, and also derive the same identities for string topology, where they become statements of homotopy invariance. We outline analogous results in the A model, B-twisted Landau–Ginzburg models, and physical strings. We also discuss the understanding of the closed string states as the Hochschild homology of the open string algebra, and outline possible applications to elliptic genera.