Cone Vertex Algebras, Mock Theta Functions, and Umbral Moonshine Modules

Area of Science:

• Number Theory
• Algebraic Geometry
• Representation Theory

Background:

• Indefinite theta functions are generalizations of theta functions with applications in various mathematical fields.
• Vertex algebras provide a framework for studying quantum field theory and conformal field theory.
• Umbral moonshine connects number theory and string theory through modular forms and vertex algebras.

Purpose of the Study:

• To introduce a new family of indefinite theta functions of signature (1,1).
• To establish connections between these theta functions, mock theta functions, and Appell-Lerch sums.
• To utilize these connections to interpret McKay-Thompson series in umbral moonshine.

Main Methods:

• Expressing indefinite theta functions using trace functions of vertex algebras.
• Analyzing the relationship between indefinite theta functions and mock theta functions.
• Applying these relationships to specific cases of umbral moonshine (lambency ℓ = 8, 12, 16).

Main Results:

• A family of indefinite theta functions is characterized by trace functions of vertex algebras.
• Connections to mock theta functions and Appell-Lerch sums are demonstrated.
• McKay-Thompson series for umbral moonshine at ℓ = 8, 12, 16 are expressed via vertex algebra trace functions.

Conclusions:

• The study provides explicit vertex algebra modules for specific instances of umbral moonshine.
• This work deepens the understanding of the interplay between indefinite theta functions, vertex algebras, and moonshine phenomena.
• The findings offer new perspectives on the structure and representation theory underlying umbral moonshine.