Quantum Liouville Theory On The Riemann Sphere With $n>3$ Punctures

TL;DR

This paper develops a quantum Liouville theory framework on the Riemann sphere with more than three punctures, deriving exchange relations for monodromy parameters to establish a consistent quantum description.

Contribution

It extends quantum Liouville theory to Riemann spheres with n>3 punctures, providing new exchange relations for monodromy parameters and exploring symmetries.

Findings

Quantum theory near a puncture via canonical quantization.

Exchange relations for monodromy parameters with multiple punctures.

Identification of symmetries in the quantum Liouville framework.

Abstract

We have studied the quantum Liouville theory on the Riemann sphere with n>3 punctures. While considering the theory on the Riemann surfaces with n=4 punctures, the quantum theory near an arbitrary but fixed puncture can be obtained via canonical quantization and an extra symmetry is explored. While considering more than four distinguished punctures, we have found the exchange relations of the monodromy parameters from which we can get a reasonable quantum theory.