  We perform the dual transformation of the Yang-Mills theory in d=3 dimensions
using the Wilson action on the cubic lattice. The dual lattice is made of
tetrahedra triangulating a 3-dimensional curved manifold but embedded into a
flat 6-dimensional space (for the SU(2) gauge group). In the continuum limit
the theory can be reformulated in terms of 6-component gauge-invariant scalar
fields having the meaning of the external coordinates of the dual lattice
sites. These 6-component fields induce a metric and a curvature of the
3-dimensional dual colour space. The Yang-Mills theory can be identically
rewritten as a quantum gravity theory with the Einstein-Hilbert action but
purely imaginary Newton constant, plus a homogeneous `matter' term.
Interestingly, the theory can be formulated in a gauge-invariant and local form
without explicit colour degrees of freedom.
