On symmetrization of 6j-symbols and Levin-Wen Hamiltonian

Abstract

It is known that every ribbon category with unimodality allows symmetrized $6j$-symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category $\mathcal{E}$. We define the mirror conjugate symmetry of $6j$-symbols instead and show that $6j$-symbols of any unitary spherical category can be normalized to have this property. As an application, we discuss an exactly soluble model on a honeycomb lattice. We prove that the Levin-Wen Hamiltonian is exactly soluble and hermitian on a unitary spherical category.

15 pages,many figures

References in corpus (1)

• Some non-braided fusion categories of rank 3

Cited by in corpus (1)

• Counterexamples in Levin-Wen string-net models, group categories, and Turaev unimodality