Berezin Quantization, Conformal Welding and the Bott-Virasoro Group

Area of Science:

• Mathematical Physics
• Representation Theory
• Complex Analysis

Background:

• The study builds upon the Nag-Sullivan framework for representing the group of circle diffeomorphisms.
• It explores the Hilbert space of holomorphic functions and its connection to symplectic transformations.
• Conformal welding is a key concept providing triangular decompositions.

Purpose of the Study:

• To investigate the representation of the group of diffeomorphisms of the circle on a Hilbert space.
• To quantize the concept of conformal welding using operator formalism.
• To find a new representative for the Bott-Virasoro cocycle class and explore energy functionals.

Main Methods:

• Utilizing the Berezin formalism to lift decompositions to operators acting on the Fock space.
• Applying operator theory on Hilbert spaces of holomorphic functions.
• Analyzing symplectic transformations derived from conformal welding.

Main Results:

• A successful quantization of conformal welding is achieved.
• A novel representative of the Bott-Virasoro cocycle class is identified.
• A surprising identity for the Takhtajan-Teo energy functional on the circle is derived.

Conclusions:

• The operator-theoretic approach provides a powerful tool for studying conformal field theory and related areas.
• The findings offer new insights into the structure of the Bott-Virasoro cocycle and energy functionals.
• This work bridges concepts from representation theory, complex analysis, and mathematical physics.