Gluing theory of Riemann surfaces and Liouville conformal field theory

TL;DR

This paper develops a formal algebraic geometric framework for gluing Riemann surfaces, deriving relations between parameters and constructing sheaves of conformal blocks with key properties relevant to Liouville CFT.

Contribution

It introduces a novel algebraic geometric approach to gluing Riemann surfaces and constructs sheaves of conformal blocks with factorization and symmetry properties.

Findings

Derived computable relations for gluing parameters.

Constructed sheaves of tempered conformal blocks.

Established the factorization property and Teichmüller groupoid action.

Abstract

We study the gluing theory of Riemann surfaces using formal algebraic geometry, and give computable relations between the associated parameters for different gluing processes. As its application to the Liouville conformal field theory, we construct the sheaf of tempered conformal blocks on the moduli space of pointed Riemann surfaces which satisfies the factorization property and has a natural action of the Teichm\"{u}ller groupoid.