  We obtain constraints on the variation of the fundamental constants from the
full shape of the redshift-space correlation function of a sample of luminous
galaxies drawn from the Data Release 9 of the Baryonic Oscillations
Spectroscopic Survey. We combine this information with data from recent CMB,
BAO and H_0 measurements. We focus on possible variations of the fine structure
constant \alpha and the electron mass m_e in the early universe, and study the
degeneracies between these constants and other cosmological parameters, such as
the dark energy equation of state parameter w_DE, the massive neutrinos
fraction f_\nu, the effective number of relativistic species N_eff, and the
primordial helium abundance Y_He. When only one of the fundamental constants is
varied, our final bounds are \alpha / \alpha_0 = 0.9957_{-0.0042}^{+0.0041} and
m_e /(m_e)_0 = 1.006_{-0.013}^{+0.014}. For their joint variation, our results
are \alpha / \alpha_0 = 0.9901_{-0.0054}^{+0.0055} and m_e /(m_e)_0 = 1.028 +/-
0.019. Although when m_e is allowed to vary our constraints on w_DE are
consistent with a cosmological constant, when \alpha is treated as a free
parameter we find w_DE = -1.20 +/- 0.13; more than 1 \sigma away from its
standard value. When f_\nu and \alpha are allowed to vary simultaneously, we
find f_\nu < 0.043 (95% CL), implying a limit of \sum m_\nu < 0.46 eV (95% CL),
while for m_e variation, we obtain f_nu < 0.086 (95% CL), which implies \sum
m_\nu < 1.1 eV (95% CL). When N_eff or Y_He are considered as free parameters,
their simultaneous variation with \alpha provides constraints close to their
standard values (when the H_0 prior is not included in the analysis), while
when m_e is allowed to vary, their preferred values are significantly higher.
In all cases, our results are consistent with no variations of \alpha or m_e at
the 1 or 2 \sigma level.
