# A Duality in Two-Dimensional Gravity

## TL;DR

This paper establishes a duality between two integrable systems in two-dimensional gravity, enabling systematic computation of amplitudes in non-compact topological gravity and revealing an equivalence between different matter central charge phases.

## Contribution

It introduces a duality in integrable flows that connects different phases of topological gravity with varying matter central charges.

## Key findings

- Demonstrates equivalence of integrable flows in 2D gravity
- Proposes amplitudes for non-compact topological gravity
- Shows phase equivalence for different matter central charges

## Abstract

We demonstrate an equivalence between two integrable flows defined in a polynomial ring quotiented by an ideal generated by a polynomial. This duality of integrable systems allows us to systematically exploit the Korteweg-de Vries hierarchy and its tau-function to propose amplitudes for non-compact topological gravity on Riemann surfaces of arbitrary genus. We thus quantise topological gravity coupled to non-compact topological matter and demonstrate that this phase of topological gravity at N=2 matter central charge larger than three is equivalent to the phase with matter of central charge smaller than three.