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Summary credit to author Ryan Thorngren
Thought I'd give this a try, especially since Xiao-Gang did as well. :)
I was originally interested in the question of describing boundary topological order for bosonic phases protected by time reversal, though I believe the applications of this sort of investigation are far-reaching in physics. Even though these systems are bosonic, it is known that ``fermionic" things happen in the topological order, eg. Kramers degeneracy and fermionic spin-statistics. This is because the unoriented cobordism invariants, the Stiefel-Whitney classes, are the obstruction classes for things like spin structures. As we go to higher dimensions, what does the topological order--eg. in 3+1d--look like? It turns out to have some sort of fermionic string.
The approach is to think of the unoriented bordism class of spacetime as the configuration of a gauge field. Since this is specified by the Stiefel-Whitney classes, we can consider this as a collection of discrete Z/2 gauge fields in each form degree. Then I define physically the electric and magnetic operators corresponding to such a thing and discuss how they can end.
Some elementary algebraic topology is used to figure out pictorially what it means for a string to be fermionic.
At the end of the paper I show that QED with fermionic electron and fermionic magnetic monopole has a global gravitational anomaly.