  We study the coupling between mechanical motion and Dirac electrons in a
dynamical sheet of graphene. We show that this coupling can be understood in
terms of an effective gauge field acting on the electrons, which has two
contributions: quasistatic and purely dynamic of the Berry-phase origin. As is
well known, the static gauge potential is odd in the K and K' valley index,
while we find the dynamic coupling to be even. In particular, the mechanical
fluctuations can thus mediate an indirect coupling between charge and valley
degrees of freedom.
