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.

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    $\begingroup$ I don't know anything about the Levin-Wen model nor anything about fusion categories but in condensed matter physics, when people discuss chiral symmetry (as well as particle-hole and time-reversal symmetries) of a Hamiltonian, the symmetry is assumed to be "local", so that if you write your Hamiltonian in a position representation, the matrix elements of the symmetry operator that relate components of the wavefunction at one point in space to those someplace else are required to vanish. $\endgroup$ – j.c. May 21 '18 at 18:27
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    $\begingroup$ I'm having a hard time finding places where this is spelled out explicitly but see e.g. the start of section 3.3 of Freed's "Short-range entanglement and invertible field theories" arxiv.org/abs/1406.7278 . $\endgroup$ – j.c. May 21 '18 at 18:28
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    $\begingroup$ I am a bit confused by the reference to a chiral symmetry as "commuting" with the Hamiltonian; that would allow one to block-diagonalize the Hamiltonian and the symmetry within each block would become trivial; instead, a chiral symmety is defined as a unitary operator that anticommutes with 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 18:34
  • $\begingroup$ Symmetries may anti-commute with a "Hamiltonian" of a free-fermion system on the single-particle level, but they always commute with the Hamiltonian on the many-body level. In the Levin-Wen models however, there is not even a notion of a single-particle Hilbert space. $\endgroup$ – Andi Bauer May 21 '18 at 18:57
  • $\begingroup$ in my understanding the following statements are equivalent: 1) non-chiral topological phase; 2) chiral central charge = 0; 3) thermal quantum Hall conductance = 0; 4) presence of time-reversal symmetry ––– the string-net models satisfy all 4 criteria; a fifth statement "presence of chiral symmetry" means something else (a system may very well have chiral central charge $\neq 0$ in the presence of chiral symmetry, for example, graphene in a magnetic field); hence I do not subscribe to the statement "in condensed matter physics, non-chiral refers to the presence of a chiral symmetry". $\endgroup$ – Carlo Beenakker Jul 28 '18 at 21:00

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