Diagrammatic state sums for 2D pin-minus TQFTs

Abstract

The two dimensional state sum models of Barrett and Tavares are extended to unoriented spacetimes. The input to the construction is an algebraic structure dubbed half twist algebras, a class of examples of which is real separable superalgebras with a continuous parameter. The construction generates pin-minus TQFTs, including the root invertible theory with partition function the Arf-Brown-Kervaire invariant. Decomposability, the stacking law, and Morita invariance of the construction are discussed.

Topological superconductors on superstring worldsheets

“…For recent work on this invertible phase, see e.g. [47][48][49]. The ABK invariant can be thought of as the effective action of the Kitaev chain protected by time-reversal.…”

Pin TQFT and Grassmann integral

“…In this section, we construct the 2d pin − invertible TQFT [23] for the Arf-Brown-Kervaire (ABK) invariant via the Grassmann integral on lattice, whose state sum definition was initially given in [15]. In condensed matter literature, this invertible theory describes (1+1)d topological superconductors in class BDI [16].…”

“…(1. 2) In contrast, it is sometimes useful to consider a fermionic topological phase on an unoriented manifold [11][12][13][14][15], when the system has a symmetry that reverses the orientation of spacetime. In such a situation, the corresponding theory requires a pin structure, which encodes the orientation reversing symmetry.…”

“…We further show that pin ± Gu-Wen SPT phases always admit a gapped boundary, by explicitly constructing the Grassmann integral for the coupled bulk and boundary system on an unoriented manifold. In addition, we propose a lattice definition of 2d pin − TQFT whose partition function is the Arf-Brown-Kervaire (ABK) invariant [15,22,23], which generates the Z 8 classification of (1+1)d topological superconductors. Finally, we discuss a way to compute the Z 16 -valued (2+1)d pin + anomaly from the data of (2+1)d anomalous theory.…”

Classification of String Theories via Topological Phases

“…Then, we have the choice of invertible phases on the world sheet, given by ℧ 2 Pin − ðptÞ ¼ Z 8 . These invertible phases have been studied previously in [8,28,29], and in the condensed matter literature in [10,30,31]. The partition functions are given by…”