Vertex operator algebras, the Verlinde conjecture, and modular tensor categories

Area of Science:

• Algebraic Quantum Field Theory
• Representation Theory
• Conformal Field Theory

Background:

• Vertex operator algebras (VOAs) are algebraic structures crucial in 2D conformal field theory and string theory.
• The Verlinde conjecture relates fusion rules of VOA modules to modular transformations, a key problem in the field.

Purpose of the Study:

• To prove the Verlinde conjecture for a specific class of simple vertex operator algebras.
• To explore consequences of the conjecture, including the Verlinde formula and properties of modular tensor categories.

Main Methods:

• Focuses on simple vertex operator algebras satisfying specific conditions (N-gradability, C(2)-cofiniteness).
• Utilizes properties of contragredient modules and the action of modular transformations on characters of irreducible modules.

Main Results:

• Announces a proof of the Verlinde conjecture, demonstrating diagonalization of fusion rule matrices by modular transformations.
• Establishes rigidity and nondegeneracy for the braided tensor category of VOA modules, leading to a modular tensor category structure.

Conclusions:

• The Verlinde conjecture holds for the studied class of vertex operator algebras.
• The category of VOA modules possesses a natural modular tensor category structure, with significant implications for theoretical physics and mathematics.