# Ising anyon topological order and its edge $c=1/2$ CFT

We know that conformal field theories are closely related to two-dimensional topological orders via edge-boundary correspondence. An Ising topological order can be obtained by gauging the fermion parity from a $p+ip$ superconductor. The anyon fusion rule $\sigma\times \sigma = 1+ \psi$ (where $\sigma$ is the vortex excitation that binds a Majorana zero mode, from which the above fusion rule is easily identified) indicates its relation with the free (Majorana) fermion CFT with $c=1/2$.

Indeed, on its edge, there is a chiral mode with $c=1/2$. Besides the fermion mode, such a CFT has a twist operator $\sigma$ with $h=1/16$. My question is what is this operator in the context of $p+ip$ superconductor? What is its relation with the Majorana-binding vortex in the bulk? How do I understand its fusion rule $\sigma\times \sigma = 1+ \psi$ in the edge CFT sense?

The $SU(2)_2$ QH state $\chi_1(z_i)\chi_2^2(z_i)$ and the Paffian QH state have Ising topological order. They have $\sigma$ non-abelian particle as intrinsic bulk excitations. (Here $\chi_n$ is the many-fermion wave function with $n$ filled Landau levels.)
• Yes. Just gauge $Z_2^f$ fermion-parity symmetry. Aug 3, 2017 at 0:35