  This article is based on a talk by S.S. at the Nambu Memorial Symposium at
the University of Chicago. We review ideas on the nature of the metallic states
of the hole-doped cuprate high temperature superconductors, with an emphasis on
the connections between the Luttinger theorem for the size of the Fermi
surface, topological quantum field theories (TQFTs), and critical theories
involving changes in the size of the Fermi surface. We begin with the
derivation of the Luttinger theorem for a Fermi liquid, using momentum balance
during a process of flux-insertion in a lattice electronic model with toroidal
boundary conditions. We then review the TQFT of the Z2 spin liquid, and
demonstrate its compatibility with the toroidal momentum balance argument. This
discussion leads naturally to a simple construction of `topological' Fermi
liquid states: the fractionalized Fermi liquid (FL*) and the algebraic charge
liquid (ACL). We present arguments for a description of the pseudogap metal of
the cuprates using Z2-FL* or Z2-ACL states with Ising-nematic order. These
pseudogap metal states are also described as Higgs phases of a SU(2) gauge
theory. The Higgs field represents local antiferromagnetism, but the
Higgs-condensed phase does not have long-range antiferromagnetic order: the
magnitude of the Higgs field determines the pseudogap, the reconstruction of
the Fermi surface, and the Ising-nematic order. Finally, we discuss the route
to the large Fermi surface Fermi liquid via the critical point where the Higgs
condensate and Ising nematic order vanish, and the application of Higgs
criticality to the strange metal.
