Quantum Gravity: Noncommutative Geometry, The Gauge/Gravity Correspondence And Matrix Models

 Noncommutative geometry (and its matrix models) presents a distinct solution to the problem of quantum gravity whereas the gauge/gravity correspondence is currently the most successful proposal for quantum gravity. The two approaches intersect within the quantum mechanics of the BFSS (or M-(atrix)) theory and also within the IKKT matrix model which should be viewed as providing the starting unifying framework.
The BFSS-type Yang-Mills quantum mechanics and the IKKT-type Yang-Mills matrix models provide a non-perturbative formulation of superstring theory and its underlying eleven-dimensional M-theory. But they also provide a quantum gravitational formulation (gravitational Feynman path integral) of Connes' noncommutative geometry (classical phase space). In other words, we should think of noncommutative geometry as a "first quantization of geometry" (classical gravity) and think of the corresponding matrix models as a "second quantization" of geometry " (quantum gravity).
Poisson manifolds play therefore the fundamental role of "curved spacetime", the Darboux theorem plays the role of the "equaivalence principle" while Moyal-Weyl spaces are what defines our "flat spacetime".
The nature of quantum geometry can also be probed by means of multitrace matrix models where both the renormalization group equation, the large N saddle point analysis and the Monte Carlo method come together in a symphony of mathematical and computational methods applied to the same theoretical problem (which is quite rare). The multitrace matrix models is in fact an alternative to Yang-Mills matrix models which allow for emergent geometry (quantum geometry), emergent gravity (quantum gravity) and emergent time (quantum cosmology).
Another important gauge/gravity duality (besides the BFSS-type Yang-Mills quantum mechanics) is the AdS/CFT correspondence. The case of two dimensions is the most mysterious and is the most important for quantum black holes as well as it is the case most closely related to noncommutative geometry which is very intriguing indeed.

Question 1: Towards "computational physics of string theory"!
Answer 1: Preliminary results are reported in https://arxiv.org/abs/2007.04488.

Question 2: What is the relation between multitrace matrix models and quantum geometry?
Answer 2: The discussion of the fixed points of a cubic multitrace matrix model (which is important to a very important case of emergent noncommutative geometry in two dimensions, i.e. the fuzzy sphere, the noncommutative torus and the Moyal-Weyl plane) is discussed in https://arxiv.org/abs/2008.09564.

Question 3: Can we reformulate a noncommutative theory of the AdS/CFT correspondence and black hole evaporation problem?
Answer 3 (Sep 2021): Update number 9 (The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry). See also https://badisydri.blogspot.com/2021/09/the-ads2cft1-correspondence-and.html