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We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin$_pm$ case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin$_-$ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the $mathbb{Z}_8$ classification of (1+1)d topological superconductors. We also compute the indicator formula of $mathbb{Z}_{16}$ valued time-reversal anomaly for (2+1)d pin$_+$ TQFT based on our construction.
We derive a canonical form for 2-group gauge theory in 3+1D which shows they are either equivalent to Dijkgraaf-Witten theory or to the so-called EF1 topological order of Lan-Wen. According to that classification, recently argued from a different poi
These notes offer an introduction to the functorial and algebraic description of 2-dimensional topological quantum field theories `with defects, assuming only superficial familiarity with closed TQFTs in terms of commutative Frobenius algebras. The g
Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-F
We propose a mechanism to pin skyrmions in chiral magnets by introducing local maximum of magnetic exchange strength, which can be realized in chiral magnetic thin films by engineering the local density of itinerate electrons. Thus we find a way to a
We address the problem of computing temperature correlation functions of the XXZ chain, within the approach developed in our previous works. In this paper we calculate the expected values of a fermionic basis of quasi-local operators, in the infinite