Good Introductory Text for Quantizing Geometry? In this video she makes an interesting claim. In loop quantum gravity on can think of the gravity as:
$$ G^{\mu \nu} = \kappa T^{\mu \nu} $$
Now in loop quantum gravity one can quantize geometry as well along with the stress-energy tensor:
$$ \hat G^{\mu \nu} = \kappa \hat  T^{\mu \nu} $$
(Author: Prof. Dr. Kristina Giesel; Youtube Title: The Big Bounce, Signs in the CMB? A Loop Quantum Gravity update; Institute: Institute of Theoretical Physics)
Question
Is there a good introductory text for methods in which they quantize geometry? (including other approaches than Loop Quantum Geometry)
 A: Believe it or not, this is quite a broad topic, and depends upon who you ask on what you mean by, ''quantizing geometry.'' Since this term can sometimes appear broad, I will also outline the different quantum gravity schemes and attempt to differentiate them from quantum geometry schemes:

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*String theory: this is not a theory of quantized geometry. String theory is essentially taking the worldline of a particle, extending it in one dimension, and anlyzing the worldline (turns into a worldsheet) physics (also adding more fields and symmetries to eventually create the 9+1 superstrings we know and love). If you wanted to study strings, I would recommend the textbook String Theory in a Nutshell by Kirtsis and use Polchinski's text to fill in some of the gaps (like derivations). For a quick overview of the field then these notes look promising, ''String theory: a perspective over the last 25 years'' by Mukhi, but they are missing the latest developments of string theory's Swampland program which Palti has the best introduction and review thereof, ''Swampland.''

*Loop Quantum Gravity (LQG): LQG is a theory that quantizes the geometry directly and uses the tetrad formalism ($g_{\mu\nu} = \eta_{ab}e^a_\mu e^b_\nu$) to quantize the geometry of the metric. Rovelli has a textbook out himself (the newer one) which is here: Covariant Loop Quantum Graivty. But, a review by Ashtekar is on arxiv, ''A Short Review of Loop Quantum Gravity.''

*Symplectic geometry: there is currently a field of study that is looking into what quantization actually means which involves a field of mathematics called symplectic geometry. The best notes I have found and read/used came from an introduction by John Baez on his website here: From Classical to Quantum and Back.

*Canonical methods: the original methodology to quantize gravity was to use the normal methods at the time (around 1940's I believe) by stating the commutators the fields should obey. Or, a little more in-depth, finding the Hamiltonian formulation of General Relativity, and then act on a state/wave-function. This way led to the famous Wheeler-de Witt equation, which turned out to be unsolvable (this is where LQG came into play).
-Miscellaneous theories: the 4-dimensional version of quantizing general relativity has a lot of issues. One way to simplify this is by a dimension reduction to form 2+1 gravity. Steven Carlip is one of the people on the forefront of this research and has a text on it here: Quantum Gravity in 2+1 Dimensions. For a good overview of all the theories listed above, Claus Kiefer has a text here: Quantum Gravity.

*Others: I know there are a few other methods such as emergent gravity, gauge field formulations, non-commutative geometries, and twistor field theory, which all deal with quantizing geometry itself, but my knowledge is quite limited. Your google search would be as good as mine in finding a resource.

I hope this helps you decide which direction you want to look at!
