# How do I understand different realizations of symmetry in the absence of fractionalization?

To use a simple example to ask my question, consider the two dimensional toric code with a $Z_2$ global symmetry acting in two ways:

1. The most boring trivial way possible.
2. By permuting the charge and flux excitations.

Are the phases corresponding to the two situations (neither of which involves any `symmetry fractionalization') distinct?

A more detailed version of my question- Consider a 2+1 dimensional topologically ordered system with a global symmetry $G$. If $\mathcal{C}$ is the Modular Tensor Category that describes the topological order and $\mathcal{A}$ is the group of Abelian anyons, the symmetry action is described by a map $\rho$ $$\rho: G \rightarrow \text{Aut}(\mathcal{C})$$ Given a specific $\rho$, $G$ and $\mathcal{C}$, it was shown in the paper by Barkeshli et al that (provided an obstruction class valued in $H^3_{[\rho]}(G,\mathcal{A})$ vanishes) different symmetry fractionalization classes correspond to the elements of $H^2_{[\rho]}(G,\mathcal{A})$ and these also label different SET phases. My question is about how to understand the trivial element of $H^2_{[\rho]}(G,\mathcal{A})$ for different $[\rho]$- do they correspond to different phases of matter?

• Well the answer is obviously yes, they are different phases, since there's no way to interpolate between two different actions on the anyon types. Maybe you should expand on why you are doubtful that they are different phases? – Dominic Else May 11 '18 at 22:08
• You are right. I feel less confused by this now. I guess I was expecting anything labeled by the 'trivial element' ($H^2_{[\rho]}(G,\mathcal{A})$ in this case) to correspond to the same thing which is trivial in some sense. This is clearly wrong for multiple reasons. Maybe I should list the reasons for anyone else who might stumble on this post with similar confusions as mine. – sawd May 15 '18 at 18:08

The reason I was confused about this question was because I was comparing the classification of Symmetry Enriched Topological (SET) phases with the classification Symmetry Protected Topological (SPT) phases. Consider, for example the classification of bosonic SPT phases with a unitary on-site action of a symmetry $G$. It is known that there exist SPT phases classified by elements of $H^{d+1}(G,U(1))$ meaning we can associate an element of $H^{d+1}(G,U(1))$ to each SPT phase. In particular, the trivial element of $H^{d+1}(G,U(1))$ corresponds to the trivial phase i.e with a ground state that can be adiabatically connected to the product state without breaking symmetry. This gave me the impression that in the case of SET phases, I should expect the trivial element of $H^2_{[\rho]}(G,\mathcal{A})$ to correspond to the same 'trivial fractionalization pattern' irrespective of what the $\rho$ is. This is wrong for two reasons.
2) Secondly, with a fixed action $\rho$ on the anyons, the different SET phases corresponding to different fractionalization classes are not labeled by $H^2_{[\rho]}(G,\mathcal{A})$ rather they form a $H^2_{[\rho]}(G,\mathcal{A})$ torsor meaning (unlike the similar situation with the classification of SPT phases) there is no canonical way to associate an element of $H^2_{[\rho]}(G,\mathcal{A})$ to a fractionalization class rather different fractionalization classes are related to each other by an element in $H^2_{[\rho]}(G,\mathcal{A})$. So my intuitive understanding of what the trivial element of $H^2_{[\rho]}(G,\mathcal{A})$ corresponds to was wrong.