  Any variation of the fundamental physical constants, and more particularly of
the fine structure constant, $\alpha$, or of the mass of the electron, $m_e$,
would affect the recombination history of the Universe and cause an imprint on
the cosmic microwave background angular power spectra. We show that the Planck
data allow one to improve the constraint on the time variation of the fine
structure constant at redshift $z\sim 10^3$ by about a factor of 5 compared to
WMAP data, as well as to break the degeneracy with the Hubble constant, $H_0$.
In addition to $\alpha$, we can set a constraint on the variation of the mass
of the electron, $m_{\rm e}$, and on the simultaneous variation of the two
constants. We examine in detail the degeneracies between fundamental constants
and the cosmological parameters, in order to compare the limits obtained from
Planck and WMAP and to determine the constraining power gained by including
other cosmological probes. We conclude that independent time variations of the
fine structure constant and of the mass of the electron are constrained by
Planck to ${\Delta\alpha}/{\alpha}= (3.6\pm 3.7)\times10^{-3}$ and ${\Delta
m_{\rm e}}/{m_{\rm e}}= (4 \pm 11)\times10^{-3}$ at the 68% confidence level.
We also investigate the possibility of a spatial variation of the fine
structure constant. The relative amplitude of a dipolar spatial variation of
$\alpha$ (corresponding to a gradient across our Hubble volume) is constrained
to be $\delta\alpha/\alpha=(-2.4\pm 3.7)\times 10^{-2}$.
