Polyakov relation for the sphere and higher genus surfaces

TL;DR

This paper extends the Polyakov relation, originally for the sphere, to higher genus surfaces, linking Liouville action variations to source positions and surface moduli, with proofs for genus 1 and hyperelliptic cases.

Contribution

It formulates and proves the Polyakov relation for genus 1 and hyperelliptic surfaces, generalizing the known sphere case to more complex topologies.

Findings

Polyakov relation established for genus 1 surfaces.

Polyakov relation extended to hyperelliptic surfaces.

Provides mathematical proof for these cases.

Abstract

The Polyakov relation, which in the sphere topology gives the changes of the Liouville action under the variation of the position of the sources, in the case of higher genus is related also to the dependence of the action on the moduli of the surface. We write and prove such a relation for genus 1 and for all hyperelliptic surfaces.