  Magnetic fields are believed to drive accretion and relativistic jets in
black hole accretion systems, but the magnetic-field structure that controls
these phenomena remains uncertain. We perform general relativistic (GR)
polarized radiative transfer of time-dependent three-dimensional GR
magnetohydrodynamical (MHD) simulations to model thermal synchrotron emission
from the Galactic Center source Sagittarius A$^\ast$ (Sgr A*). We compare our
results to new polarimetry measurements by the Event Horizon Telescope (EHT)
and show how polarization in the visibility (Fourier) domain distinguishes and
constrains accretion flow models with different magnetic field structures.
These include models with small-scale fields in disks driven by the
magnetorotational instability (MRI) as well as models with large-scale ordered
fields in magnetically-arrested disks (MAD). We also consider different
electron temperature and jet mass-loading prescriptions that control the
brightness of the disk, funnel-wall jet, and Blandford-Znajek-driven funnel
jet. Our comparisons between the simulations and observations favor models with
ordered magnetic fields near the black hole event horizon in Sgr A*, although
both disk- and jet-dominated emission can satisfactorily explain most of the
current EHT data. We show that stronger model constraints should be possible
with upcoming circular polarization and higher frequency ($349~{\rm GHz}$)
measurements.
