We investigate ($3+1$)d topological orders (TOs) in fermionic systems with anomalous ${\mathbb{Z}}_{2N}^F$ symmetry, where its ${\mathbb{Z}}_2^F$ subgroup is the fermion parity. Such an anomalous symmetry arises as a discrete subgroup of the chiral U(1) symmetry of $\nu$ copies of Weyl fermions of the same chirality. Inspired by the crystalline correspondence principle, we deform the anomalous ${\mathbb{Z}}_{2N}^F$ symmetry of a (3 + 1)d Weyl fermion to the anomalous $C_N\times {\mathbb{Z}}_2^F$ symmetry. Then we microscopically construct symmetry preserving gapped boundary states of the closely related ($4+1$)d $C_N\times {\mathbb{Z}}_2^F$ symmetry protected topological state (with $C_N$ being the $N$-fold rotation), whenever it is possible. For $\nu =N$, we show that the ($3+1$)d symmetric gapped state admits a topological ${\mathbb{Z}}_4$ gauge theory description at low energy and propose that a similar theory saturates the corresponding ${\mathbb{Z}}_{2N}^F$ anomaly. For $N\nmid \nu$, our construction admits no topological quantum field theory (TQFT) symmetric gapped state; while for $\nu =N/2$, we find a non-TQFT symmetric gapped state via stacking lower-dimensional ($2+1$)d non-discrete-gauge-theory TO inhomogeneously. For other values of $\nu$, no symmetric gapped state is possible within our construction, which is consistent with the no-go theorem by Cordova-Ohmori.

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• Quantum field theory
• Topological order
• Strongly correlated systems
• Discrete symmetries in condensed matter