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We describe the phase diagram of a 2+1 dimensional SU(2) gauge theory offluctuating incommensurate spin density waves for the hole-doped cuprates. Ourprimary assumption is that all low energy fermionic excitations are gaugeneutral and electron-like, while the spin density wave order is fractionalizedinto Higgs fields transforming as adjoints of the gauge SU(2). The confiningphase of the gauge theory is a conventional Fermi liquid with a large Fermisurface (and its associated $d$-wave superconductor). There is a quantum phasetransition to a Higgs phase describing the `pseudogap' at lower doping.Depending upon the quartic terms in the Higgs potential, the Higgs phaseexhibits one or more of charge density wave, Ising-nematic, time-reversal oddscalar spin chirality, and $\mathbb{Z}_2$ topological orders. It is notablethat the emergent broken symmetries in our theory of fluctuating spin densitywaves co-incide with those observed in diverse experiments. For theelectron-doped cuprates, the spin density wave fluctuations are at wavevector$(\pi,\pi)$, and then the corresponding SU(2) gauge theory only has a crossoverbetween the confining and Higgs regimes, with an exponentially largeconfinement scale deep in the Higgs regime. On the Higgs side, for both theelectron- and hole-doped cases, and at scales shorter than the confinementscale (which can be infinite when $\mathbb{Z}_2$ topological order is present),the electron spectral function has a `fractionalized Fermi liquid (FL*)' formwith small Fermi surfaces. We also describe the deconfined quantum criticalityof the Higgs transition in the limit of a large number of Higgs flavors, andperturbatively discuss its coupling to fermionic excitations.

Gauge theory for the cuprates near optimal doping