In a Dirac fermion BCFT with SU(2) conformal boundary conditions parametrized by a three-sphere, the filled Fermi sea carries a higher Berry curvature whose integral is quantized, realizing Berry curvature flow and Chern number pumping in Fock space.
Friedan, The space of conformal boundary conditions for the c=1 gaussian model (1993), https://www.physics.rutgers.edu/~friedan/papers/ boundary_c=1_1999.pdf
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Space of conformal boundary conditions from the view of higher Berry phase: Flow of Berry curvature in parametrized BCFTs
In a Dirac fermion BCFT with SU(2) conformal boundary conditions parametrized by a three-sphere, the filled Fermi sea carries a higher Berry curvature whose integral is quantized, realizing Berry curvature flow and Chern number pumping in Fock space.