Pin TQFT and Grassmann integral

TL;DR

This work develops a lattice form of pin$_{\pm}$ TQFTs by extending the Gu-Wen Grassmann integral to unoriented manifolds with orientation-reversing walls. It provides concrete constructions: (i) pin$_{\pm}$ Gu-Wen SPT phases with gapped boundaries, (ii) a lattice realization of the (1+1)d pin$_{-}$ ABK invertible theory realizing the $\mathbb{Z}_8$ classification, and (iii) an indicator formula for the $\mathbb{Z}_{16}$ time-reversal anomaly of (2+1)d pin$_{+}$ TQFT via a Walker-Wang shadow. The framework includes detailed retriangulation invariants, wall data, and bulk-boundary couplings that ensure gauge-invariant, non-orientable lattice theories. By connecting Grassmann-refined state sums with bordism data, the paper clarifies anomaly inflow on unoriented spacetimes and provides practical lattice tools for studying fermionic topological phases with time-reversal symmetry.

Abstract

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.

Figures (5)

• Figure 1: Assignment of Grassmann variables on 1-simplices in the case of $d=2$. $\theta$ (resp. $\overline{\theta}$) is represented as a black (resp. white) dot.
• Figure 2: (a): The signs of $d$-simplices near the orientation reversing wall, which is represented as a red line. (b): Assignment of Grassmann variables on the wall specifies a deformation of the wall that intersects the wall transversally at $(d-2)$-simplices.
• Figure 3: When $\lambda(v)=1$ on a single $(d-2)$ simplex $v$, Grassmann variables on $(d-1)$-simplices surrounding $v$ are counted in the integral. In the expression of the integral, we encounter $\pm d\vartheta_{2i}d\vartheta_{2i+1}$ measure factors from $(d-1)$-simplices, and $\pm \vartheta_{2i+1}\vartheta_{2i+2}$ integrand factors from $d$-simplices. The sign $\pm$ from the measure (resp. integrand) is expressed by the orange (resp. green) arrow. For instance, the arrow is directed from $\vartheta_{2i}$ to $\vartheta_{2i+1}$ if we have a $+$ sign on the measure, otherwise directed in the opposite direction. (a): If $v$ is away from the orientation reversing wall, we can see that all the signs from the measure share the same sign. We can also check that signs from the integrand have the same sign. In such a situation, we have $\sigma(\delta\lambda)=-1$. (b): If $v$ is placed on the orientation reversing wall (red thick line), we have to flip the direction of all arrows on one side of the wall. The total number of flipped arrows is odd; odd number of orange arrows and even number of green arrows. Thus, the value of the integral in (b) has the opposite sign from that of (a). Hence, we have $\sigma(\delta\lambda)=+1$, when the two Grassmann variables attached on one side of the wall have different colors. (b'): On the other hand, we have $\sigma(\delta\lambda)=-1$, when the two Grassmann variables attached on one side of the wall have the same color.
• Figure 4: (a): An example of $K$ such that $\partial K=(N\times\{0\})\sqcup (M\times[0,1])\sqcup (N\times\{1\})$. (b): Away from the orientation reversing wall, the colors of Grassmann variables on $M$ match those of $\overline{M}$. (b'): On the orientation reversing wall (red plane), the Grassmann variable is assigned such that (1) the color of the Grassmann variable of $e'$ on one side of the wall is the same as that of $f$, and (2) the color of the Grassmann variable of $e$ on one side of the wall is different from that of $e'$.
• Figure 5: Illustration of the gluing relation.