Modular functors with infinite dimensional Hilbert spaces

TL;DR

This paper introduces a generalized concept of modular functors with infinite-dimensional Hilbert spaces, linking conformal field theory, wave functions, and quantum Teichmüller theory, and provides a specific example from Liouville theory.

Contribution

It defines generalized modular functors with infinite-dimensional spaces and demonstrates their connection to conformal blocks and quantum Teichmüller theory, expanding the theoretical framework.

Findings

Wave functions as flat sections of generalized modular functors

Conformal blocks identified with eigenfunctions in quantum Teichmüller theory

Example derived from Liouville conformal field theory

Abstract

We introduce a notion of generalized modular functors with Hilbert spaces of infinite dimension in general, and show that a generalized modular functor with data of conformal dimensions determines uniquely wave functions as its flat sections. Furthermore, we study an example of generalized modular functors derived from the Liouville conformal field theory. In this example, wave functions are seen to be gluing conformal blocks which are identified with eigenfunctions in the quantum Teichm\"uller theory.