Quantum Black Holes, Wall Crossing, and Mock Modular Forms (1208.4074v2)

Abstract: We show that the meromorphic Jacobi form that counts the quarter-BPS states in N=4 string theories can be canonically decomposed as a sum of a mock Jacobi form and an Appell-Lerch sum. The quantum degeneracies of single-centered black holes are Fourier coefficients of this mock Jacobi form, while the Appell-Lerch sum captures the degeneracies of multi-centered black holes which decay upon wall-crossing. The completion of the mock Jacobi form restores the modular symmetries expected from $AdS_3/CFT_2$ holography but has a holomorphic anomaly reflecting the non-compactness of the microscopic CFT. For every positive integral value m of the magnetic charge invariant of the black hole, our analysis leads to a special mock Jacobi form of weight two and index m, which we characterize uniquely up to a Jacobi cusp form. This family of special forms and another closely related family of weight-one forms contain almost all the known mock modular forms including the mock theta functions of Ramanujan, the generating function of Hurwitz-Kronecker class numbers, the mock modular forms appearing in the Mathieu and Umbral moonshine, as well as an infinite number of new examples.

• The paper demonstrates how mock modular forms elucidate the microstate structure of quarter-BPS black holes affected by wall crossing.
• It employs meromorphic Jacobi forms, Hecke-like operators, and Atkin-Lehner involutions to analyze state degeneracies in a rigorous framework.
• Numerical results reveal the asymptotic growth of Fourier coefficients, linking quantum black hole entropy with classical number theory.

Quantum Black Holes, Wall Crossing, and Mock Modular Forms

The paper "Quantum Black Holes, Wall Crossing, and Mock Modular Forms" by Atish Dabholkar, Sameer Murthy, and Don Zagier presents a detailed discussion on quantum degeneracies related to black holes through the lens of mock modular forms. The primary focus of this research is the analytic exploration of the degeneracies of quarter-BPS states in $\mathcal{N}=4$ supersymmetric string theories and their relation to mock modular forms, a relatively new class of modular objects that defy conventional modularity properties.

Summary and Contributions

• Meromorphic Jacobi Forms and Quantum Black Holes: The authors explore the connection between meromorphic Jacobi forms, mock modular forms, and the counting of black hole microstates in string theory. They demonstrate that the partition functions counting quarter-BPS states—which should be holomorphic due to their modular nature—fail this property due to wall crossing effects. Here, the partition functions are expressed in terms of a mock Jacobi form complemented by an Appell-Lerch sum.
• Wall Crossing and Degeneracies: A central problem addressed is the wall crossing phenomenon, where the degeneracies jump due to changes in the stability of multi-centered black hole configurations. The distinction between single-centered "immortal" black holes, which remain stable across walls, and multi-centered configurations is analyzed using mock modular forms as a mathematical tool.
• Mock Modular Forms: The remarkable structure of mock modular forms allows them to account for the loss of modularity observed in the partition functions. The paper details how these forms can be constructed from their shadows (holomorphic modular forms of a complementary weight), thus restoring modular symmetry at least in a completed form. This is crucial for understanding the quantum properties of black holes.
• The Role of Atkin-Lehner Involutions and Hecke-like Operators: The paper employs advanced techniques, such as Hecke-like operators and Atkin-Lehner involutions, to decompose and analyze the forms associated with various configurations of black hole charge distributions. This is used to construct detailed expressions for the degeneracies of states.
• Strong Numerical Results: The authors provide detailed numerical results and insights into the asymptotic growth of the Fourier coefficients of mock Jacobi forms. These are crucial for understanding the entropy and degeneracy of the associated black holes, tying the growth rates to classical number-theoretical functions like class numbers.

Implications and Future Directions

The implications of this paper are profound in both the theoretical realms of mathematics and physics:

• Theoretical Physics and Holography: This work underlines the role of modular forms in analyzing the holographic nature of quantum black holes, advancing our understanding of AdS/CFT correspondence and string theory.
• Mathematics of Modular Forms: The research expands the toolbox available for tackling problems related to modularity, providing deeper insights into the structure and utility of mock modular forms.
• Applications to Quantum Gravity: The results suggest new directions for exploring quantum gravity, potentially offering insights into the microscopic structure of spacetime through the modular symmetries.

Future research might explore the new types of mock modular forms introduced in this work and explore their broader applications within mathematical physics. Considerations could include the relation of these forms to other aspects of dualities in string theory, and extensions to other classes of black holes, particularly those relevant in different dimensions or theories of quantum gravity.