# 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 gene rating 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 $\cal D$ , at the large volume attractor point. For $\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\cal D$ and is therefore known to be modular. Instead, when $\cal D$ is the sum of $n$ irreducible divisors ${\cal D}_i$, 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 $n−1$.
Document type :
Preprints, Working Papers, ...
Domain :
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01852413
Contributor : L2c Aigle <>
Submitted on : Tuesday, August 28, 2018 - 3:21:56 PM
Last modification on : Wednesday, April 10, 2019 - 5:34:04 PM
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1808.08479.pdf
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### Identifiers

• HAL Id : hal-01852413, version 2
• ARXIV : 1808.08479

### Citation

Sergei Alexandrov, Boris Pioline. Black holes and higher depth mock modular forms. 2018. ⟨hal-01852413v2⟩

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