Black holes and higher depth mock modular forms
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866911049459957760 |
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| author | Alexandrov, Sergei Pioline, Boris |
| author_facet | Alexandrov, Sergei Pioline, Boris |
| contents | 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 $n-1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1808_08479 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Black holes and higher depth mock modular forms Alexandrov, Sergei Pioline, Boris High Energy Physics - Theory Mathematical Physics Algebraic Geometry Number Theory 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 $n-1$. |
| title | Black holes and higher depth mock modular forms |
| topic | High Energy Physics - Theory Mathematical Physics Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/1808.08479 |