From Graph Laplacians to String Partition Functions: A Rigorous Pathway from Discrete Spectra to Emergent Geometry

Abstract

This work establishes rigorous mathematical foundations connecting spectral graph theory, algebraic geometry, and string theory. We construct a canonical mapping whereby any finite graph \(G\) defines a compact Riemann surface \(X_{G}\) (the spectral curve) whose period matrix \(Ω_{G}\) encodes the graph's coarse-grained spectral information. We demonstrate that in the continuum limit of graph sequences converging to Riemannian manifolds, these spectral curves converge in the Deligne-Mumford compactification sense to the classical stable curves associated with the manifold. We establish connections to the topological recursion framework of Eynard-Orantin, showing that under appropriate conditions the spectral curve satisfies the loop equations of multi-cut matrix models. The spectral memory field \(Φ_{G}(u)\) is introduced and shown to provide a discrete regularization of minimal string partition functions. We construct quantum scattering operators on spectral curves and prove that their unitarity is equivalent to a positivity condition on the spectral memory field. Furthermore, we apply this framework to resolve spacelike singularities in general relativity, proving that the Belinski-Khalatnikov-Lifshitz (BKL) chaotic regime is isospectral to a critical random graph ensemble. The classical singularity is replaced by an infinite nodal chain of rational curves, and the Bekenstein-Hawking entropy emerges from the automorphism group of the spectral curve. This work provides rigorous mathematical underpinnings for discrete approaches to quantum gravity and establishes new connections between graph theory, algebraic geometry, and theoretical physics.