  The loop representation of quantum gravity has many formal resemblances to a
background-free string theory. In fact, its origins lie in attempts to treat
the string theory of hadrons as an approximation to QCD, in which the strings
represent flux tubes of the gauge field. A heuristic path-integral approach
indicates a duality between background-free string theories and generally
covariant gauge theories, with the loop transform relating the two. We review
progress towards making this duality rigorous in three examples: 2d Yang-Mills
theory (which, while not generally covariant, has symmetry under all
area-preserving transformations), 3d quantum gravity, and 4d quantum gravity.
$SU(N)$ Yang-Mills theory in 2 dimensions has been given a string-theoretic
interpretation in the large-$N$ limit by Gross, Taylor, Minahan and
Polychronakos, but here we provide an exact string-theoretic interpretation of
the theory on $\R\times S^1$ for finite $N$. The string-theoretic
interpretation of quantum gravity in 3 dimensions gives rise to conjectures
about integrals on the moduli space of flat connections, while in 4 dimensions
there may be connections to the theory of 2-tangles.
