The low-energy TQFT of the generalized double semion model

Abstract: The generalized double semion (GDS) model, introduced by Freedman and Hastings, is a lattice system similar to the toric code, with a gapped Hamiltonian whose definition depends on a triangulation of the ambient manifold $M$, but whose space of ground states does not depend on the triangulation, but only on the underlying manifold. In this paper, we use topological quantum field theory (TQFT) to investigate the low-energy limit of the GDS model. We define and study a functorial TQFT $Z_{\mathrm{GDS}}$ in every dimension $n$ such that for every closed $(n - 1)$-manifold $M$, $Z_{\mathrm{GDS}}(M)$ is isomorphic to the space of ground states of the GDS model on $M$; the isomorphism can be chosen to intertwine the actions of the mapping class group of $M$ that arise on both sides. Throughout this paper, we compare our constructions and results with their known analogues for the toric code.