The naive $2$-dimensional Fermi sea in $k$-space (with a convex structure and positive Gaussian curvature, some nice properties, etc) in $2+1$-dimensional spacetime may be viewed as an infinite collection of the $1$-dimensional in k-space chiral Conformal Field Theory perturbed around from the Fermi surface. For example, based on the picture of this Ref.
More generally, the naive $d$-dimensional Fermi surface in $k$-space (with convex structure and positive Gaussian curvature, some nice properties, etc) in $d+1$-dimensional spacetime may be viewed as an infinite collection of the $(d-1)$-dimensional chiral Conformal Field Theory perturbed around from the Fermi surface.
I suppose that this scheme works for non-interacting fermions or Landau Fermi liquid theory with good quasi-particle description (where quasi-particle is dressed by the renormalized effect on effective mass, spectral weight, from interactions). However, we know there are phenomena beyond Landau Fermi liquid theory, such as (1) 1+1 d Luttinger liquid theory, or the (2) higher dimensional (2+1d above) where Landau Fermi liquid theory breaks down: e.g. critical Fermi surface and non-Fermi liquid theory.
Does the infinite many Conformal Field Theory on the Fermi surface picture still hold its validity for those exotic cases?
