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Black holes and higher depth mock modular forms 
Auteur(s): Alexandrov S., Pioline Boris
(Article) Publié:
Communications In Mathematical Physics, vol. 374 p.549–625 (2019)
Texte intégral en Openaccess : 
Ref HAL: hal-01852413_v2
Ref Arxiv: 1808.08479
DOI: 10.1007/s00220-019-03609-y
WoS: 000527910200005
Ref. & Cit.: NASA ADS
Exporter : BibTex | endNote
Résumé: 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$\cal D$ , 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 $n−1$.
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