Black holes and higher depth mock modular forms

Abstract

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\mathcal{D}$, at the large volume attractor point. For $\mathcal{D}$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\mathcal{D}$ and is therefore known to be modular. Instead, when $\mathcal{D}$ is the sum of $n$ irreducible divisors ${\mathcal{D}}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.

43+35 pages, 11 figures; v2: various cosmetic changes, index of notations added to aid the reader; v3: minor corrections and clarifications, added refs and new prop. 9 on the shadow of the completion

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Cited by in corpus (14)

• Vafa-Witten invariants from exceptional collections
• S-duality and refined BPS indices
• Duality and Mock Modularity
• Heavenly metrics, BPS indices and twistors
• Scaling Black Holes and Modularity
• Supersymmetric black holes and $T\bar{T}$ deformation
• Mock modularity at work, or black holes in a forest
• Refinement and modularity of immortal dyons
• On the existence of scaling multi-centered black holes
• Modular anomaly of BPS black holes
• Higher depth mock theta functions and $q$-hypergeometric series
• Quantum TBA for refined BPS indices
• S-duality and the universal isometries of q-map spaces
• Special Joyce structures and hyperkähler metrics