  Supernova remnants (SNRs) are believed to be the source of Galactic cosmic
rays (CRs). SNR shocks accelerate CR protons and electrons which reveal key
insights into the non-thermal physics by means of their synchrotron and
$\gamma$-ray emission. The remnant SN 1006 is an ideal particle acceleration
laboratory because it is observed across all electromagnetic wavelengths from
radio to $\gamma$-rays. We perform three-dimensional (3D) magnetohydrodynamics
(MHD) simulations where we include CR protons and follow the CR electron
spectrum. By matching the observed morphology and non-thermal spectrum of SN
1006 in radio, X-rays and $\gamma$-rays, we gain new insight into CR electron
acceleration and magnetic field amplification. 1. We show that a mixed
leptonic-hadronic model is responsible for the $\gamma$-ray radiation: while
leptonic inverse-Compton emission and hadronic pion-decay emission contribute
equally at GeV energies observed by Fermi, TeV energies observed by imaging air
Cherenkov telescopes are hadronically dominated. 2. We show that quasi-parallel
acceleration (i.e., when the shock propagates at a narrow angle to the upstream
magnetic field) is preferred for CR electrons and that the electron
acceleration efficiency of radio-emitting GeV electrons at quasi-perpendicular
shocks is suppressed at least by a factor ten. This precludes extrapolation of
current one-dimensional plasma particle-in-cell simulations of shock
acceleration to realistic SNR conditions. 3. To match the radial emission
profiles and the $\gamma$-ray spectrum, we require a volume-filling,
turbulently amplified magnetic field and that the Bell-amplified magnetic field
is damped in the immediate post-shock region. Our work connects micro-scale
plasma physics simulations to the scale of SNRs.
