  We discuss two scenarios of emergent gravity. In one of them the quantum
vacuum is considered as superplastic crystal, and the effective gravity
describes the dynamical elastic deformations of this crystal. In the other one
the gravitational tetrads emerge as the bilinear form of the fermionic fields.
In spite of the essentially different mechanisms of emergent gravity, these two
scenarios have one important common property: the metric field has dimension of
the inverse square of length $[g_{\mu\nu}]=1/[l]^2$, as distinct from the
conventional dimensionless metric, $[g_{\mu\nu}]=1$, in general relativity. As
a result the physical quantities, which obey diffeomorphism invariance, become
dimensionless. This takes place for such quantities as Newton constant, the
scalar curvature, the cosmological constant, particle masses, fermionic and
scalar bosonic fields, etc. This may suggest that the dimensionless physics can
be the natural consequence of the diffeomorphism invariance, and thus can be
the general property of any gravity, which emerges in the quantum vacuum. One
of the nontrivial consequences of the shift of dimensions is related to
topology. Due to the shift of dimensions some operators become topological, and
contain the integer or fractional prefactors in the action. This in particular
concerns the intrinsic 3+1 quantum Hall effect and the Nieh-Yan quantum anomaly
in terms of torsion.
