Title: 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
$\textbf{Tube}(\mathcal{C}/\psi) \cong \mathcal{C} \times (\mathcal{C}/\psi)$.
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.