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Mock Modularity at Work, or Black Holes in a Forest.
1Laboratoire Charles Coulomb (L2C), Université de Montpellier, CNRS, F-34095 Montpellier, France.
Entropy (Basel, Switzerland)
|July 29, 2025
Summary
Mock modular forms generalize modular forms and are key to understanding BPS black hole microstates in string theory. This review explores their applications in computing topological invariants and constructing non-commutative structures.
Area of Science:
- Number Theory
- String Theory
- Mathematical Physics
Background:
- Mock modular forms, introduced by Ramanujan, are generalizations of modular forms.
- They have recently been identified as crucial for describing BPS black hole microstates in string theory.
Purpose of the Study:
- To review the recent findings on mock modular forms in string theory.
- To describe their applications in computing generating functions and exploring mathematical structures.
Main Methods:
- The review synthesizes results connecting mock modular forms to BPS black hole microstates.
- It covers the computation of generating functions for compact and non-compact manifolds.
- It also discusses the construction of non-commutative structures.
Main Results:
- Mock modular forms successfully capture the generating functions for BPS black hole microstates in string theory compactifications with 8 and 16 supercharges.
- These generating functions correspond to Donaldson-Thomas and Vafa-Witten topological invariants.
- New non-commutative structures on Calabi-Yau threefold moduli spaces have been constructed.
Conclusions:
- Mock modular forms offer a powerful framework for understanding black hole microstates and topological invariants in string theory.
- Their study leads to advancements in both theoretical physics and pure mathematics, including non-commutative geometry.
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