Fermionic Modular Categories and the 16-fold Way

TL;DR

This work develops a systematic framework for fermionic topological orders via spin and super-modular categories, formulating the 16-fold way conjecture for minimal modular extensions and establishing that exactly 16 extensions (up to ribbon/Witt equivalence) arise when a minimal extension exists. Central to the approach is zesting, a flexible construction that modifies tensor products and associators along a $\mathbb{Z}_2$-grading to generate new modular closures from a given one, notably producing eight distinct variants per base extension and, with central-charge data, yielding the full 16. The authors provide explicit constructions and data for PSU$(2)_{4m+2}$-based families, deriving modular data via quantum-group methods and demonstrating how to realize and distinguish closures through graphical calculus for zesting. The results advance the classification of fermionic topological orders, offering concrete recipes to build and compare spin modular categories and their super-modular truncations, with implications for fermionic quantum Hall states and related topological phases. Overall, the paper integrates categorical gauging of fermion parity, obstruction theory, and a robust zesting toolkit to map the landscape of fermionic modular extensions and their Witt classes.

Abstract

We study spin and super-modular categories systematically as inspired by fermionic topological phases of matter, which are always fermion parity enriched and modelled by spin TQFTs at low energy. We formulate a $16$-fold way conjecture for the minimal modular extensions of super-modular categories to spin modular categories, which is a categorical formulation of gauging the fermion parity. We investigate general properties of super-modular categories such as fermions in twisted Drinfeld doubles, Verlinde formulas for naive quotients, and explicit extensions of $PSU(2)_{4m+2}$ with an eye towards a classification of the low-rank cases.

Figures (8)

• Figure 1: The birth and death on odd $x$ in $\mathcal{C}^\boxtimes$.
• Figure 2: Figures for the rigidity equations.
• Figure 3: An incomplete picture morphism. The gluing strands terminating at the black dots need to be connected to other strands in order to define a composition. Different compositions may result, but any two give the same morphism up to crossing factors.
• Figure 4: One way to connect the gluing objects. The constant factor is $r^{-2} = \theta_e$ since a gluing object crosses an odd object in each of $\operatorname{ev}^{\boxtimes}_{\operatorname{coev}^\boxtimes}$ and in $\operatorname{ev}^{\boxtimes}_{a^\boxtimes}$. If you don't like the presence of births, deaths, and pivotal isomorphisms on the gluing objects, connect the gluing objects for the domains of $f^{**}$ and $f^{\boxast\boxast}$ along a straight line path, and verify that after accounting for constant factors the same morphism results.
• Figure 5: The pivotal structure for odd $x$ in $\mathcal{C}^\boxtimes$.

Theorems & Definitions (61)

• Definition 2.1
• Remark 2.1
• Remark 2.2
• Proposition 2.1
• proof
• Remark 2.3
• Proposition 2.2
• proof
• Example 2.1
• Example 2.2