  We investigate the low-energy effective theory of a Fermi surface coupled to
an Ising-nematic quantum critical point in (2+1) spacetime dimensions with
translation symmetry. We formulate the system using the large $N$ Yukawa-SYK
model, whose saddle point is described by the Migdal-Eliashberg equations. The
low-energy physics can be revealed by studying the Gaussian fluctuation
spectrum around the saddle point, which is generated by the Bethe-Salpeter
kernel $K_\text{BS}$. Based on the Ward identities, we propose an inner product
on the space of two point functions, which reveals a large number of soft modes
of $K_\text{BS}$. These soft modes parameterize deformation of the Fermi
surface, and their fluctuation eigenvalues describe their decay rates. We
analytically compute these eigenvalues for a circular Fermi surface, and we
discover the odd-parity modes to be parametrically longer-lived than the
even-parity modes, due to the kinematic constraint of fermions scattering on a
convex FS. The sign of the eigenvalues signals an instability of the
Ising-nematic quantum critical point at zero temperature for a convex Fermi
surface. At finite temperature, the system can be stabilized by thermal
fluctuations of the critical boson. We derive an effective action that
describes the soft-mode dynamics, and it leads to a linearized Boltzmann
equation, where the real part of the soft-mode eigenvalues can be interpreted
as the collision rates. The structure of the effective action is similar to the
theory of linear bosonization of a Fermi surface. As an application, we
investigate the hydrodynamic transport of non-Fermi liquid. Analyzing the
Boltzmann equation, we obtain a conventional hydrodynamic transport regime and
a tomographic transport regime. In both regimes, the conductance of the system
in finite geometry can be a sharp indicator for the soft-mode dynamics and
non-Fermi liquid physics.
