Stability of Fermi Surfaces and $K$ Theory

Nonrelativistic Fermi liquids in $d+1$ dimensions exhibit generalized Fermi surfaces: $(d-p)$-dimensional submanifolds in the $(\mathbf{k},\omega )$-space supporting gapless excitations. We show that the universality classes of stable Fermi surfaces are classified by $K$ theory, with the pattern of stability determined by Bott periodicity. The Atiyah-Bott-Shapiro construction implies that the low-energy modes near a Fermi surface exhibit relativistic invariance in the transverse $p+1$ dimensions. This suggests an intriguing parallel between nonrelativistic Fermi liquids and D-branes of string theory.