Some thoughts about Kapustin’s cobordism conjecture

In 2013, Kitaev explained that, under some reasonable locality hypotheses, gapped invertible phases of bosonic lattice models in different dimensions are naturally organized into an $\Omega$-spectrum. The following year, Kapustin conjectured that this spectrum is dual to a Thom spectrum, specifically (smooth) oriented bordism MSO, and that for fermionic lattice models one sees instead the dual to spin bordism. In 2016, Freed and Hopkins proved Kapustin’s conjecture for invertible phases of continuous unitary QFTs valued in an at-the-time conjectural universal target category. Freed and Hopkins put bordism categories into the statement of the problem, by working from the beginning with continuous QFTs. Kapustin’s conjecture for lattice models remains open.
David Reutter and I, in ongoing work in progress, have investigating Kapustin’s conjecture from the perspective of deeper category theory. We have built the universal target category for phases satisfying a finite semisimplicity hypothesis, and we are working on relaxing finite semisimplicity. We can show that any spectrum of invertible finite-semisimple phases will indeed be dual to a Thom spectrum for some topological group G acting on the spectrum of spheres. For example, if one looks just at those bosonic phases which can be topologically condensed from the vacuum, G is almost the (oriented) piecewise linear group, whose Thom spectrum is the bordism spectrum MSPL is the (oriented) *piecewise* smooth manifolds; the difference between MSPL and MSO is only visible in dimensions 7 and above. I say almost because in fact our G is what you would get if you tried to build MSPL, but could only make finitary measurements, which surely is explained by our restriction to condensable semisimple TQFTs. We conjecture that MSPL, rather than MSO, classifies invertible gapped phases of bosonic lattice models.
The general relation between MSPL and topological phases is explained by a certain “surgery exact sequence” for topological phases that mirrors the surgery sequence for MSPL. By studying this sequence, we can also answer the question of which invertible phases admit a gapped boundary condition. In particular that only (the trivial phase and) the Arf–Kervaire invariants admit finite-semisimple gapped boundary conditions.