  We present a renormalization group theory for the onset of Ising-nematic
order in a Fermi liquid in two spatial dimensions. This is a quantum phase
transition, driven by electron interactions, which spontaneously reduces the
point-group symmetry from square to rectangular. The critical point is
described by an infinite set of 2+1 dimensional local field theories, labeled
by points on the Fermi surface. Each field theory contains a real scalar field
representing the Ising order parameter, and fermionic fields representing a
time-reversed pair of patches on the Fermi surface. We demonstrate that the
field theories obey compatibility constraints required by our redundant
representation of the underlying degrees of freedom. Scaling forms for the
response functions are proposed, and supported by computations by up to three
loops. Extensions of our results to other transitions of two-dimensional Fermi
liquids with broken point-group and/or time-reversal symmetry are noted. Our
results extend also to the problem of a Fermi surface coupled to a U(1) gauge
field.
