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.Comment: 61 pages, LaTeX; v2: typos fixe
