Some Thoughts About the Kapustin-Kitaev Cobordism Conjecture

Explore advanced mathematical physics in this 44-minute conference talk examining the Kapustin–Kitaev cobordism conjecture and its implications for understanding gapped invertible phases in quantum lattice models. Learn about Kitaev's 2013 framework organizing bosonic lattice model phases into an Ω-spectrum and Kapustin's subsequent conjecture connecting this spectrum to oriented bordism MSO for bosonic systems and spin bordism for fermionic systems. Discover the Freed-Hopkins proof for continuous unitary quantum field theories and examine ongoing research investigating the conjecture through deeper category theory approaches. Understand the construction of universal target categories for phases with finite semisimplicity hypotheses and explore how any spectrum of invertible finite-semisimple phases relates to Thom spectra for topological groups acting on sphere spectra. Investigate the connection between bosonic phases condensable from vacuum and the piecewise linear group, leading to the bordism spectrum MSPL for oriented piecewise smooth manifolds. Examine the proposed conjecture that MSPL, rather than MSO, classifies invertible gapped phases of bosonic lattice models, supported by a surgery exact sequence for topological phases that mirrors the MSPL surgery sequence. Analyze which invertible phases admit gapped boundary conditions, including the specific case of Arf–Kervaire invariants admitting finite-semisimple gapped boundary conditions.