Interacting fermionic SPT phases in 2d with time-reversal symmetry Interacting fermionic SPT phases in 1d and 3d with $\mathbb{Z}_2^T$ symmetry are classified by $\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ respectively, as shown in the paper by Fidkowski and Kitaev http://arxiv.org/abs/1008.4138, and Wang and Senthil http://arxiv.org/abs/1401.1142. I'd like to know the classification for the 2d case. Can anyone suggest some reference on that?
 A: Here is the classification table of fermionic SPT phases, copied from A. Kapustin et. al., arXiv:1406.7329. In the table, the arrow denotes the reduction of classification by interaction: free fermion classification $\to$ interacting fermion classification.
$$\begin{array}{c|ccc}
d= & 1 & 2 & 3\\
\hline
\text{BDI} & \mathbb{Z}\to\mathbb{Z}_8 & 0\to 0 & 0\to 0 \\
\text{D} & \mathbb{Z}_2\to\mathbb{Z}_2 & \mathbb{Z}\to\mathbb{Z} & 0\to 0 \\
\text{DIII} & \mathbb{Z}_2\to\mathbb{Z}_2 & \mathbb{Z}_2\to\mathbb{Z}_2 & \mathbb{Z}\to\mathbb{Z}_{16} \\
\end{array}$$
BDI class: $\mathbb{Z}_2^T$ symmetry with $\mathcal{T}^2=+1$. 
DIII class: $\mathbb{Z}_2^T$ symmetry with $\mathcal{T}^2=-1$. 
D class: no symmetry (apart form fermion parity).
One must note that, the 1d $\mathbb{Z}_8$ and the 3d $\mathbb{Z}_{16}$ classified fermionic SPT phases mention in the question are actually protected by different types of time-reversal symmetries, and belongs to different symmetry classes. Depending on the signature of $\mathcal{T}^2$ being $+1$ or $-1$, the $\mathbb{Z}_2^T$ symmetry is actually ascribed to either BDI or DIII symmetry class. In 1d and 3d, there are $\mathbb{Z}$ classified free fermionic SPT phases (1d BDI or 3d DIII) to start with. The phenomenon of interaction reduced classification only happens if we start from a $\mathbb{Z}$ classified free fermionic SPT state. In 2d, there is no such free fermion SPT state to start with, so there is no analogous interaction reduced classification in 2d with $\mathbb{Z}_2^T$ symmetry protection.
A: The non-interacting classification was obtained in the seminal "periodic table" papers by Kitaev and Ryu/Snyder/Furusaki/Ludwig. The interacting classification of 2D fermionic SPT phases with time-reversal symmetry has been considered by several groups already, including:
Gu and Wen's group super-cohomology theory, http://arxiv.org/abs/1201.2648
Kapustin et. al. cobordism group, as already mentioned in the previous answer.
and a recent treatment based on tensor category http://arxiv.org/abs/1501.01313
All approaches agree that there are no 2D nontrivial fermionic SPT with $T^2=1$.
