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تسجيل مستخدم جديدFermion condensation and super pivotal categories
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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.
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