Two-dimensional topological field theories as taffy

TL;DR

This paper introduces taffy-like operations on string worldsheets using defects, demonstrating invariance of physical properties through mathematical identities, with applications to topological string theories and string topology.

Contribution

It develops mathematical identities to verify invariance of spectra under taffy-folding in topological field theories, extending to string topology and other models.

Findings

Massless spectra are invariant under taffy operations.

Mathematical identities confirm topological invariance.

Applications to elliptic genera and string topology.

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.