# The Complex Structure of 2D Surfaces

## TL;DR

This paper investigates the complex structures of 2D surfaces generated by dynamical triangulation, revealing stability for matter fields with central charge ≤1 and instability beyond, supported by numerical evidence.

## Contribution

It provides a numerical analysis linking the complex structure stability of 2D surfaces to matter central charges and supports the conjecture on resistor networks behaving as continuous sheets.

## Key findings

- Surfaces with matter central charge ≤1 have well-defined complex structures.
- Surfaces become unstable when the matter central charge exceeds 1.
- Resistor network behavior aligns with continuous resistivity sheets for c<1.

## Abstract

The complex structure of a surface generated by the two-dimensional dynamical triangulation(DT) is determined by measuring the resistivity of the surface. It is found that surfaces coupled to matter fields have well-defined complex structures for cases when the matter central charges are less than or equal to one, while they become unstable beyond c=1. A natural conjecture that fine planar random network of resistors behave as a continuous sheet of constant resistivity is justified numerically for c<1.