Fermion condensation and super pivotal categories

Abstract

We study fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain a fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call "m-type" and "q-type" particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if $\mathcal{C}$ is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies $\text{Tube}(\mathcal{C}/\psi )\cong \mathcal{C}\times (\mathcal{C}/\psi )$. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum. A large portion of this work is devoted to three detailed examples of performing fermion condensation to produce fermionic topological phases: we condense fermions in the Ising theory, the $SO(3{)}_6$ theory, and the $\frac{1}{2}{\text{E}}_6$ theory, and compute the quasiparticle excitation spectrum in each of these examples.

161 pages; v2: corrected typos (including 18 instances of "the the") and added some references

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• Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d

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