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.
Highlights
That the field theories encode the universal, long-distance effective behavior — the “phase” — of gapped quantum systems, which means characterizing their responses to topological probes and reproducing the ground state expectation values of nonlocal order parameters [2, 3]
The case of spin theories in two spacetime dimensions was recently studied by Barrett and Tavares [7]. They exploit the relation between spin structures on a surface M and immersions of M into R3 to construct, for each spin surface, a ribbon diagram, the twists and crossings of which keep track of the spin structure
We will not give one here; instead our focus will be on the pin TQFTs that arise from the diagrammatic state sum construction introduced below
Summary
2.1 Pin structures, immersions, and quadratic enhancements The goal of this section is to review the following equivalences: pin− structures / isom. ↔ quadratic enhancements ↔ immersions / reg. homot. The obstruction class vanishes in two dimensions, so each surface supports exactly |H1(M ; Z/2)| pin structures, up to isomorphism. Another characterization of pin structures on a surface M can be given in terms of immersions of M into R3. Pin structures on surfaces have a third characterization: their isomorphism classes are in bijective correspondence with quadratic enhancements of the intersection form [14]; that is, functions q : H1(M ; Z/2) → Z/4. Every quadratic enhancement arises from both a pin structure and an immersion, and the constructions are isomorphism and regular homotopy invariant, respectively. As all linear automorphisms α that preserve the intersection form are induced by diffeomorphisms of M [15, 18], all equivalences of quadratic enhancements arise from equivalences of immersions. Quadratic enhancements form a torsor for H1(M ; Z/2) by the action q → q + 2 · A, with respect to which the correspondence with pin structures is equivariant [14]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
