An algebraic approach to minimal models in CFTs

TL;DR

This paper explores algebraic methods to construct minimal models in conformal field theories, emphasizing their modular properties and solutions via algebraic geometry and differential equations.

Contribution

It introduces an algebraic framework for minimal models in CFTs, connecting their modular invariance to algebraic geometry and hypergeometric differential equations.

Findings

Minimal models exhibit covariance under the mapping class group.

Partition functions relate to algebraic solutions of hypergeometric differential equations.

Algebraic geometry provides tools for solving rational CFTs.

Abstract

CFTs are naturally defined on Riemann surfaces. The rational ones can be solved using methods from algebraic geometry. One particular feature is the covariance of the partition function under the mapping class group. In genus $g=1$, this yields modular forms, which can be linked to ordinary differential equations of hypergeometric type with algebraic solutions.