Partition function for quantum gravity in 4 dimensions as a $1/\mathcal{N}$ expansion

Quantum gravity is the solution ascribed to rendering the geometric description of classical gravity, in any dimensions, completely consistent with principles of quantum theory. The serendipitous theoretical discovery that black holes are thermodynamical objects that must participate in the second law has led to these purely gravitational objects to be dubbed, `the hydrogen atom for quantum gravity', analogous to the atomic spectrum of hydrogen, effectively used by Neils Bohr and his contemporaries to successfully formulate quantum mechanics in the early 20th century. Here, we employ the temperature and entropy formulae describing Schwarzschild black holes to consider the emergence of Einstein Field Equations from a complex-Hermitian structure, [Ricci tensor $\pm \sqrt{-1}$ Yang-Mills field strength], where the gravitational degrees of freedom are the SU($N$) colors with $N \in \mathbb{Z} \geq 0$ and a condensate comprising color pairs, $k = N/2$, appropriately coupled to the Yang-Mills gauge field. The foundations of our approach reveal the complex-Hermitian structure is analogous to Cayley-Dickson algebras, which aids in formulating the appropriate action principle for our formalism. The SU($N$) gauge group is broken into an effective SU($4$) $\rightarrow$ SO($4$) $\leftrightarrow$ SO($1,3$) field theory on the tangent space with two terms: the Einstein-Hilbert action and a Gauss-Bonnet topological term. Moreover, since topologically classifying all $n = 4$ (dimensional) Riemannian manifolds is not a clear-cut endeavor, we only consider the Euclidean path integral as the sum over manifolds with distinct topologies, $h \in \mathbb{Z} \geq 0$ homeomorphic to connected sums of an arbitrary number of $n = 4$-spheres and $h$ number of $n = 4$-tori. Consequently, the partition function adopts a reminiscent form of the sum of the vacuum Feynman diagrams for a large $\mathcal{N} = \exp(\beta M/2)$ theory, provided $\mathcal{S} = \beta M/2 = \pi N$ is the Schwarzschild black hole entropy, $\beta = 8\pi GM$ is the inverse temperature, $G$ is gravitational constant, $M$ is the black hole mass and horizon area, $\mathcal{A} = 2G\beta M = 4\pi GN$ is pixelated in units of $4\pi G$. This leads us to conclude that the partition function for quantum gravity is equivalent to the vacuum Feynman diagrams of a yet unidentified large $\mathcal{N}$ theory in $n = 4$ dimensions. Our approach also sheds new light on the asymptotic behavior of dark matter-dominated galaxy rotation curves (the empirical baryonic Tully-Fisher relation) and emergent gravity in condensed matter systems with defects such as layered materials with cationic vacancies as topological defects.