Siegel Modular Forms and Black Hole Entropy

Abstract

We discuss the application of Siegel Modular Forms to Black Hole entropy counting. The role of the Igusa cusp form ${\chi }_{10}$ in the D1D5P system is well-known, and its transformation properties are what allows precision microstate counting in this case. We apply a similar method to extract the Fourier coefficients of other Siegel modular and paramodular forms, and we show that they could serve as candidates for other types of black holes. We investigate the growth of their coefficients, identifying the dominant contributions and the leading logarithmic corrections in various regimes. We also discuss similarities and differences to the behavior of ${\chi }_{10}$, and possible physical interpretations of such forms both from a microscopic and gravitational point of view.

41 pages + appendices. V2: typos fixed and references added

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• No More Walls! A Tale of Modularity, Symmetry, and Wall Crossing for 1/4 BPS Dyons
• Logarithmic correction to black hole entropy in universal low-energy string theory models
• The Holographic Landscape of Symmetric Product Orbifolds
• Non-topological logarithmic corrections in minimal gauged supergravity
• Siegel Paramodular Forms and Sparseness in AdS${}_3$/CFT${}_2$
• Dyon degeneracies from Mathieu moonshine
• Logarithmic correction to the entropy of a Kerr-Newman family of black holes in $U(1{)}^2$-charged STU supergravity models
• Quantum Black Hole Entropy, Localization and the Stringy Exclusion Principle
• Central charges, elliptic genera, and Bekenstein-Hawking entropy in $\mathcal{N}=(2,2)$ AdS${}_3$/CFT${}_2$