I'd like to bring together the following two notions of "non-chiral":

On the abstract algebraic side, a modular fusion category describing the anyon content of some physical system is said to be non-chiral if it is the Drinfeld center of some fusion category, and thus has a vanishing chiral central charge. This is clearly fulfilled by the Levin-Wen string-net models aka TVBW state-sum.

On the other hand, in condensed matter physics, "non-chiral" refers to the presence of a "chiral symmetry", so a (anti-)unitary representation of the group $Z_2$ on the many-body Hilbert space that commutes with the Hamiltonian. Clearly, for any Hamiltonian one can find thousands of such representations, so in order to make "presence of a chiral symmetry" something meaningful, there have to be some conditions on the representation that make it a valid "chiral" symmetry.

Now my question: What kinds of representations of $Z_2$ are called a valid "chiral symmetry" and how is such a chiral symmetry implemented in each Levin-Wen model?

[Edit: I know that "chiral symmetry" is well-defined in the context of free-fermion systems. However in the Levin-Wen models there is not even a Fock space, we just have local (super-)Hilbert spaces distributed over a 2D lattice. I have no idea how to generalize the free fermion definition to this setting. This is the key point of the question I think.]

Edit: Meanwhile it came to my mind that "chiral" is a very generic word and the two notions of chiral might simply be different. My current guess of what a chiral symmetry could be on the many-body level is "complex conjugation in some local basis", and it seems like there are Turaev-Viro models that do not have such a symmetry.

anticommuteswith the Hamiltonian; then it cannot be removed by block-diagonalization; also note that the presence of both time-reversal and particle-hole symmetry implies a chiral symmetry, but not the other way around. $\endgroup$ – Carlo Beenakker May 21 '18 at 18:342more comments