State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

TL;DR

The work develops a comprehensive shadow-fermion framework for 2+1d spin-TFTs by relating spin theories to bosonic shadows through gauging fermionic parity and fermionic anyon condensation. It furnishes explicit state-sum (Turaev-Viro) and lattice (Levin-Wen/string-net) constructions for fermionic phases, including a complete lattice realization of fermionic SPT phases via Gu-Wen data and Ising pull-backs, uncovering a rich group structure with notable quaternionic and dihedral extensions. It further clarifies how one-form symmetries and their anomalies govern the shadow-spin correspondence, and shows how to extend these ideas to gapped boundaries, global symmetries, and higher-categorical formalisms, yielding concrete Hamiltonians and state sums for both shadows and spin theories. The results illuminate the practical synthesis of fermionic topological matter from bosonic categories, with significant implications for explicit models of fermionic SPTs and symmetry-enriched topological phases in 2+1 dimensions.

Abstract

It is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic "shadow" theories, which are obtained from the original theory by "gauging fermionic parity". The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by the product of fermion parity and internal symmetry G. The quaternion group makes a surprise appearance.

Figures (42)

• Figure 1: A graphical depiction of the map from ${\mathfrak T}_{{\mathbb Z}_2}$ to ${\mathfrak T}_{b}$. On the right we have the partition function of ${\mathfrak T}_{{\mathbb Z}_2}$ on a three-dimensional manifold, equipped with a ${\mathbb Z}_2$ flat connection. We represent the connection as a collection of domain walls implementing ${\mathbb Z}_2$ symmetry transformations $g$, $g'$, etc. On the left we have the partition function of ${\mathfrak T}_{b}$, obtained by summing the ${\mathfrak T}_{{\mathbb Z}_2}$ partition function over all possible choices of ${\mathbb Z}_2$ flat connection.
• Figure 2: Wilson lines in ${\mathbb Z}_2$ gauge theory have trivial statistics and can be freely recombined. We use a double-line notation for quasi-particles and line defects to indicate a choice of framing, but the Wilson loops have no framing dependence, i.e. represent bosonic quasi-particles. In general, these abstract properties characterize the quasi-particle generators $B$ of non-anomalous ${\mathbb Z}_2$ 1-form symmetries.
• Figure 3: A graphical depiction of the map from ${\mathfrak T}_{b}$ to ${\mathfrak T}_{{\mathbb Z}_2}$. On the right we have the partition function of ${\mathfrak T}_{b}$ on a three-dimensional manifold, possibly decorated with Wilson line operators $B$ along non-trivial cycles, dual to the domain walls of the previous picture. Abstractly, the choice of Wilson lines equips the manifold with a flat connection $[\beta_2]$ for a dual ${\mathbb Z}_2$ 1-form symmetry of ${\mathfrak T}_{b}$. Summing over all choices gives back the ${\mathfrak T}_{{\mathbb Z}_2}$ partition function.
• Figure 4: The $\Pi$ lines have fermionic statistics and thus extra signs may occur as the worldlines are recombined.
• Figure 5: A topological field theory with a gapped boundary condition. Boundary lines are labelled by objects $L_i$ in a spherical fusion category ${\mathcal{C}}$ which controls their topological fusion and junctions. Bulk lines are labelled by objects $Y_a$ in a modular tensor category which can be recovered as the Drinfeld center $Z[{\mathcal{C}}]$ of the boundary lines. Junctions of lines are labelled by choices of local operators, i.e. elements in certain morphism spaces. We use a double-line notation to indicate the dependence of bulk lines on a choice of framing. The partition function can be computed by a Turaev-Viro state sum.