Three-manifold quantum invariants and mock theta functions

Area of Science:

• Number Theory
• Topology
• Mathematical Physics

Background:

• Mock modular forms, introduced by Ramanujan, have broad applications in mathematics.
• Three-manifold invariants are crucial in understanding the topology of 3D spaces.
• Quantum invariants offer a powerful tool for classifying and distinguishing manifolds.

Purpose of the Study:

• To explore the emerging role of mock modular forms in the study of three-manifold invariants.
• To propose a conjecture regarding the mock modular properties of a new quantum invariant for Seifert three-manifolds.
• To provide computational evidence for the conjecture using a specific example.

Main Methods:

• Investigating the relationship between mock modular forms and quantum invariants.
• Formulating a conjecture based on theoretical considerations.
• Performing explicit calculations for the Brieskorn sphere Σ(2, 3, 7).

Main Results:

• A conjecture is presented linking mock modular forms to a quantum invariant of Seifert three-manifolds.
• Concrete computations for the Brieskorn sphere Σ(2, 3, 7) support the proposed conjecture.
• This work highlights a novel connection between number theory and low-dimensional topology.

Conclusions:

• Mock modular forms are a relevant tool for understanding three-manifold invariants.
• The proposed conjecture opens new avenues for research in quantum topology.
• Further investigation into these connections is warranted for a deeper understanding of both fields.