Dynamical Triangulations and Topological Recursion Reformulate Schwinger-Dyson Equations in 2D Gravity

Understanding the fundamental nature of gravity remains one of the most challenging problems in physics, and recent work by Hiroyuki Fuji from Kobe University, Masahide Manabe from Tottori University, and Yoshiyuki Watabiki from the Institute of Science Tokyo, offers new insights into this elusive force. The researchers demonstrate a surprising connection between two seemingly disparate areas of theoretical physics: dynamical triangulations, a method for defining gravity using building blocks of space, and topological recursion, a powerful mathematical technique originating in random matrix theory. This achievement establishes a reformulation of equations describing two-dimensional gravity in terms of topological recursion, explicitly calculating key components of the theory, and potentially opening new avenues for exploring quantum gravity and the geometry of spacetime itself. The work provides a crucial link between discrete models and the continuous limit, offering a novel approach to understanding gravity at a fundamental level.

This approach allows for a non-perturbative definition of the theory by summing over all possible triangulations of a given topology, weighted by the Einstein-Hilbert action. The results demonstrate that the scaling limit of this sum reproduces the continuum limit of two-dimensional gravity, confirming the method’s consistency. Researchers also establish a connection between dynamical triangulations and topological recursion, a powerful technique used in random matrix theory and integrable systems. Analysis reveals that the scaling behaviour of the triangulations is governed by a phase transition, separating smooth spacetime geometry from a fractal structure, with critical exponents agreeing with predictions from conformal field theory.

String Field Theory via Surface Decomposition

Scientists developed a new methodology to reformulate Schwinger-Dyson equations within non-critical string field theory, successfully connecting them to the Chekhov-Eynard-Orantin recursion. This work involved explicitly calculating low-order amplitudes for two discrete models, the basic and strip types, as well as the continuum limit of dynamical triangulations. The team pioneered a “slicing decomposition” technique, iteratively removing boundaries from a two-dimensional surface and selecting new boundaries based on their appearance. Researchers also introduced a “peeling decomposition” method, analogous to peeling an apple, where triangles are sequentially removed from a surface, altering the boundary length.

Within the string field theory framework, scientists defined creation and annihilation operators for boundaries of varying lengths, establishing commutation relations and defining vacuum states. The study expressed the time evolution of these configurations using a Hamiltonian, recognising that in dynamical triangulations, time corresponds to geodesic distance. Crucially, the team enforced a “no big-bang condition” on the Hamiltonian, ensuring vacuum state stability, and extended the method to a multi-peeling decomposition for complex configurations.

Dynamical Triangulations and Topological Recursion Unified

This work establishes a profound connection between Schwinger-Dyson equations, derived within a Hamiltonian framework of non-critical string field theory, and the Chekhov-Eynard-Orantin recursion. Researchers successfully reformulated these equations for two discrete models, the basic and strip types, and for the continuum limit of dynamical triangulations, demonstrating a unified mathematical structure. The team rigorously defined and calculated amplitudes for pure dynamical triangulations, introducing a “slicing decomposition” to systematically count triangulated surfaces with specific boundary conditions. This decomposition allows for the iterative removal of triangles, updating boundary conditions, and facilitating amplitude calculation. Crucially, the researchers demonstrated that the Schwinger-Dyson equations directly lead to the topological recursion, confirming a deep equivalence between the Hamiltonian formulation of string field theory and this established topological approach.

Schwinger-Dyson and Topological Recursion are Equivalent

This research establishes a significant connection between two approaches to understanding two-dimensional gravity, specifically through the reformulation of Schwinger-Dyson equations in terms of the Chekhov-Eynard-Orantin recursion. The team successfully demonstrated this equivalence for both discrete models and the continuum limit of dynamical triangulations, providing a novel derivation of the topological recursion for the dynamical triangulation model at a continuous level. The work clarifies the relationship between the Schwinger-Dyson equations and the topological recursion, providing a deeper understanding of the underlying mathematical structure of two-dimensional gravity.