At the end of chapter 6 on p. 210 in David's Griffiths' book Introduction to Elementary Particle Physics he says that 't Hooft proved that all gauge theories are renormalizable. I have also read somewhere that General Relativity is a gauge theory. Why then is Quantum Gravity not renormalizable?
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As QMechanic said, when Griffiths says gauge theory he really means Yang-Mills theory, of which gravity is not an example. Also, regarding it's renormalizability, it's worth mentioning that in three spacetime dimensions, quantum gravity actually is renormalizable. That's something that Witten showed by rewriting the action in new variables, which makes it look a lot like Chern-Simons theory, a topological field theory which has a pretty well defined quantum mechanical behavior. This equivalence is just an equivalence of the actions of the two theories.
But a QFT is more than just an action: to really be the same, the entire partition functions must be equal. Clearly, the measures of the two theories (3d gravity and Chern-Simons) must be different. To see this, note that the ground state of 3d gravity will always be full rank (and hence, invertible as a matrix) for any reasonable boundary conditions for the metric, but the ground state of Chern-Simons is $A_\mu = 0$ which turns out to correspond the zero metric $g_{\mu\nu} = 0$.$^1$ As pointed out in a comment below, see Virasoro TQFT for more details on a better candidate dual of 3d gravity.
$^1$ Really, 3d gravity is "equivalent" to $SL(2,\mathbb{R})_L\times SL(2,\mathbb{R})_R $ Chern-Simons, so the more correct statement is that the pair of connections$A^L_\mu = A^R_\mu = 0$.
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