Quantum Black Holes, Wall Crossing, and Mock Modular Forms

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 $Ad{S}_3/CF{T}_2$ 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.

151 pages, 1 figure. v2: typos corrected, table added, references added

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Cited by in corpus (106)

• Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS${}_5$ black holes
• Tensionless Strings and the Weak Gravity Conjecture
• Nonperturbative black hole entropy and Kloosterman sums
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• 3d Modularity
• Refined BPS invariants of 6d SCFTs from anomalies and modularity
• Eisenstein series and automorphic representations
• Resurgence in complex Chern-Simons theory
• Charged structure constants from modularity
• E + E $\to$ H
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• Trialities of $\mathcal{W}$-algebras
• Multiple D3-instantons and mock modular forms I
• On a modular property of N=2 superconformal theories in four dimensions
• Indefinite theta series and generalized error functions
• A 2d/1d Holographic Duality
• Mock Modular Mathieu Moonshine Modules
• Fricke S-duality in CHL models
• Dyson's Ranks and Appell-Lerch Sums
• Eichler-Selberg Type Identities for Mixed Mock Modular Forms
• S-duality and refined BPS indices
• Quantum black hole entropy and the holomorphic prepotential of N=2 supergravity
• Lessons on Black Holes from the Elliptic Genus
• Stringy corrections to the entropy of electrically charged supersymmetric black holes with ${\mathrm{AdS}}_5\times S^5$ asymptotics
• Umbral Moonshine and the Niemeier Lattices
• Higher depth quantum modular forms and plumbed $3$-manifolds
• Generalised Mathieu Moonshine
• Vafa-Witten theory and iterated integrals of modular forms
• Wall-crossing made smooth
• Single-centered black hole microstate degeneracies from instantons in supergravity
• A proof of the Thompson Moonshine Conjecture
• Weight One Jacobi Forms and Umbral Moonshine
• A Serre derivative for even weight Jacobi Forms
• Elliptic genera and real Jacobi forms
• Entanglement, Replicas, and Thetas
• On two arithmetic theta lifts
• Mixed Rademacher and BPS Black Holes
• Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine
• K3 String Theory, Lattices and Moonshine
• Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds
• Mathieu Moonshine and the Geometry of K3 Surfaces
• Generalised Umbral Moonshine
• Representations of affine superalgebras and mock theta functions III
• Modular Forms as Classification Invariants of 4D N=2 Heterotic--IIA Dual Vacua
• Proof of the Umbral Moonshine Conjecture
• Snapshots of Conformal Field Theory
• Umbral Moonshine and K3 Surfaces
• Quantum entropy of BMPV black holes and the topological M-theory conjecture
• N=2 Moonshine
• An Elliptic Triptych
• Renormalization and quantum modular forms, part II: Mock theta functions
• Symmetric Orbifold Theories from Little String Residues
• Squashed toric sigma models and mock modular forms
• Eulerian series, zeta functions and the arithmetic of partitions
• Rademacher Sums and Rademacher Series
• Givental J-functions, Quantum integrable systems, AGT relation with surface operator
• Polar harmonic Maaß forms and holomorphic projection
• Attractive Strings and Five-Branes, Skew-Holomorphic Jacobi Forms and Moonshine
• The McKay-Thompson series of Mathieu Moonshine modulo two
• Quantum Black Hole Entropy, Localization and the Stringy Exclusion Principle
• Mock Modular Index of M2-M5 Brane System
• Indefinite theta functions and black hole partition functions
• Two moonshines for $L_2(11)$ but none for $M_{12}$
• Horizon states and the sign of their index in $\mathcal{N}=4$ dyons
• On the Complete Evaluation of Jacobi Theta Functions
• Arithmetic of decay walls through continued fractions: a new exact dyon counting solution in $\mathcal{N}=4$ CHL models
• L-functions for Meromorphic Modular Forms and Sum Rules in Conformal Field Theory
• Counting spinning dyons in maximal supergravity: The Hodge-elliptic genus for tori
• An approach to BPS black hole microstate counting in an N=2 STU model
• Duality in N=4 Liouville Theory and Moonshine Phenomena
• Irreducibility of a holomorphic eta quotient is determinable
• Modular forms in the spectral action of Bianchi IX gravitational instantons
• H-Harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series
• Generalized Kloosterman Sums from M2-branes
• Temperature-reflection II: Modular Invariance and T-reflection
• Schur's partition theorem and mixed mock modular forms
• On the Fourier coefficients of negative index meromorphic Jacobi forms
• Modular Transformations of Ramanujan's Tenth Order Mock Theta Functions
• The Bloch-Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms
• Regularized Integrals on Riemann Surfaces and Modular Forms
• Umbral Moonshine and String Duality
• BPS Spectra, Barcodes and Walls
• Quasimodular forms and sl(m|m)^ characters
• Universal mock theta functions and two-variable Hecke-Rogers identities
• Locally harmonic Maass forms and the kernel of the Shintani lift
• Vector valued formal Fourier-Jacobi series
• Non-Compact SCFT and Mock Modular Forms
• U-duality Invariant Quantum Entropy from Sums of Kloosterman Sums
• State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture
• Modularity of generating series of winding numbers
• Topological String on elliptic CY 3-folds and the ring of Jacobi forms
• TASI Lectures on Moonshine
• Exact holography and black hole entropy in N=8 and N=4 string theory
• Mock Modular Forms and Class Number Relations
• Ramanujan's Thoughts from God
• Regularized inner products and weakly holomorphic Hecke eigenforms
• Periods of Jacobi forms and Hecke operator
• Radial limits of mock theta functions
• Harmonic Maass forms associated to real quadratic fields
• Negative index Jacobi forms and quantum modular forms
• On some vertex algebras related to $V_{-1}(\mathfrak{sl}(n))$ and their characters
• Properties of Lerch Sums and Ramanujan's Mock Theta Functions
• Zagier's weight $3/2$ mock modular form
• Algebraicity of higher Green functions at a CM point
• On the exact entropy of $\mathcal{N}=2$ black holes
• Infinite Type D-modules and Higher Depth Mock Modular Forms I