Non-planar contact structures with Calabi-Yau Fillings and topological quantum computation

Abstract

This paper is a non-planar counterpart of the author's earlier work on planar contact structures and topological quantum computation. On a planar page every branch point is paired by a liftable arc, and the pairing matches the Ising representation exactly, in that fixing the channels of the pairs already fixes the state. Here the page has genus g and connected binding, presented as a simple degree-three cover of the disk. Every maximal system of disjoint liftable arcs then has exactly g members and leaves two branch points unused, and the half-twist interchanging the two points is not liftable, although its cube is and lifts to the identity. Ising fusion does not see that leftover pair, determining its channel from the other g and acting there only by scalars, whereas Fibonacci fusion can leave the channel undetermined. The powers of the cube then act by ten distinct relative phases on a single joint eigenspace, while the Calabi-Yau Stein filling stays fixed. An integral residue criterion supplies the Calabi-Yau test, a factorization localized on a contact connected sum is automatically Calabi-Yau, and, unlike in the planar case, a Calabi-Yau factorization need not be the shortest.