Topological field theories and symmetry protected topological phases with fusion category symmetries

TL;DR

The paper advances the classification of 1+1d bosonic SPT phases with fusion category symmetries by formulating 2d unoriented TQFTs and showing that TR-invariant cases are encoded by orientation-reversing data together with fiber functors. Without time-reversal symmetry, SPT phases are in one-to-one correspondence with isomorphism classes of fiber functors of the fusion category. With TR symmetry, phases are classified by equivalence classes of quintuples (Z, M, i, s, φ), which extend the oriented case via orientation-reversing data and cross-cap amplitudes. The authors illustrate the framework with explicit examples including finite group symmetries and Tambara–Yamagami duality categories, deriving detailed phase counts and highlighting potential mixed anomalies; they also outline future directions toward non-split symmetries and fermionic generalizations.

Abstract

Fusion category symmetries are finite symmetries in 1+1 dimensions described by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic symmetry protected topological (SPT) phases with fusion category symmetry by using topological field theories. We first formulate two-dimensional unoriented topological field theories whose symmetry splits into time-reversal symmetry and fusion category symmetry. We then solve them to show that SPT phases are classified by equivalence classes of quintuples $(Z, M, i, s, φ)$ where $(Z, M, i)$ is a fiber functor, $s$ is a sign, and $φ$ is the action of orientation-reversing symmetry that is compatible with the fiber functor $(Z, M, i)$. We apply this classification to SPT phases with Kramers-Wannier-like self-duality.

Figures (10)

• Figure 1: The left and right (co)evaluation morphisms describe the operation of folding topological defect lines to the left and right respectively. In the folded diagrams, topological point operators corresponding to (co)evaluation morphisms are often implicit.
• Figure 2: The vector space on a circle with multiple topological defects is the same as the vector space on a circle with a single topological defect labeled by the tensor product of the topological defects. To determine the order of the tensor product, we need to specify the base point on a circle, which is represented by the cross mark in the above figure.
• Figure 3: There are two types of transition amplitudes for a cylinder. The left figure represents the transition amplitude $Z(f): V_x \rightarrow V_{x^{\prime}}$ corresponding to a topological point operator $f \in \mathop{\mathrm{Hom}}(x, x^{\prime})$. The right figure represents the transition amplitude $X_{xy}: V_{x \otimes y} \rightarrow V_{y \otimes x}$ corresponding to a change of the base point.
• Figure 4: The transition amplitude $Z(g \circ f)$ corresponding to the composition of topological point operators agrees with the composition $Z(g) \circ Z(f)$ of the transition amplitudes corresponding to each topological point operator. This is because putting a topological point operator $g \circ f$ can be regarded as putting two topological point operators $f$ and $g$ successively.
• Figure 5: The functor $Z: \mathcal{C} \rightarrow \mathrm{Vec}$ is $\mathbb{C}$-linear in morphisms.