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This article is based on a talk by S.S. at the Nambu Memorial Symposium atthe University of Chicago. We review ideas on the nature of the metallic statesof the hole-doped cuprate high temperature superconductors, with an emphasis onthe connections between the Luttinger theorem for the size of the Fermisurface, topological quantum field theories (TQFTs), and critical theoriesinvolving changes in the size of the Fermi surface. We begin with thederivation of the Luttinger theorem for a Fermi liquid, using momentum balanceduring a process of flux-insertion in a lattice electronic model with toroidalboundary conditions. We then review the TQFT of the Z2 spin liquid, anddemonstrate its compatibility with the toroidal momentum balance argument. Thisdiscussion leads naturally to a simple construction of `topological' Fermiliquid states: the fractionalized Fermi liquid (FL*) and the algebraic chargeliquid (ACL). We present arguments for a description of the pseudogap metal ofthe cuprates using Z2-FL* or Z2-ACL states with Ising-nematic order. Thesepseudogap metal states are also described as Higgs phases of a SU(2) gaugetheory. The Higgs field represents local antiferromagnetism, but theHiggs-condensed phase does not have long-range antiferromagnetic order: themagnitude of the Higgs field determines the pseudogap, the reconstruction ofthe Fermi surface, and the Ising-nematic order. Finally, we discuss the routeto the large Fermi surface Fermi liquid via the critical point where the Higgscondensate and Ising nematic order vanish, and the application of Higgscriticality to the strange metal.

The novel metallic states of the cuprates: Fermi liquids with topological order, and strange metals