Multiple D3-instantons and mock modular forms I

Abstract

We study D3-instanton corrections to the hypermultiplet moduli space in type IIB string theory compactified on a Calabi-Yau threefold. In a previous work, consistency of D3-instantons with S-duality was established at first order in the instanton expansion, using the modular properties of the M5-brane elliptic genus. We extend this analysis to the two-instanton level, where wall-crossing phenomena start playing a role. We focus on the contact potential, an analogue of the Kahler potential which must transform as a modular form under S-duality. We show that it can be expressed in terms of a suitable modification of the partition function of D4-D2-D0 BPS black holes, constructed out of the generating function of MSW invariants (the latter coincide with Donaldson-Thomas invariants in a particular chamber). Modular invariance of the contact potential then requires that, in case where the D3-brane wraps a reducible divisor, the generating function of MSW invariants must transform as a vector-valued mock modular form, with a specific modular completion built from the MSW invariants of the constituents. Physically, this gives a powerful constraint on the degeneracies of BPS black holes. Mathematically, our result gives a universal prediction for the modular properties of Donaldson-Thomas invariants of pure two-dimensional sheaves.

arXiv comment33 pages; v2: minor corrections and clarifications

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• Attractor flow trees, BPS indices and quivers
• Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I
• The Asymptotic Weak Gravity Conjecture in M-theory
• S-duality and refined BPS indices
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• Vafa-Witten theory and iterated integrals of modular forms
• Rigid limit for hypermultiplets and five-dimensional gauge theories
• Donaldson-Witten theory and indefinite theta functions
• Black holes and higher depth mock modular forms
• Vafa-Witten invariants from modular anomaly
• Heavenly metrics, BPS indices and twistors
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• Scaling Black Holes and Modularity
• Supersymmetric black holes and $T\bar{T}$ deformation
• Mock modularity from black hole scattering states
• Donaldson-Witten theory, surface operators and mock modular forms
• Squashed Toric Manifolds and Higher Depth Mock Modular Forms
• Three Avatars of Mock Modularity
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Refinement and modularity of immortal dyons
• Mock modularity at work, or black holes in a forest
• Modular anomaly of BPS black holes
• Quantum Modular $\hat{Z}{}^G$-Invariants
• S-duality and the universal isometries of instanton corrected q-map spaces
• Fourier expansions of vector-valued automorphic functions with non-unitary twists
• An exact formula for $\mathrm{U}(3)$ Vafa-Witten invariants on ${\mathbb{P}}^2$
• Quantum Curve for strip geometries, Topological Recursion and open GW/DT invariants
• S-duality and the universal isometries of q-map spaces
• On the exact entropy of $\mathcal{N}=2$ black holes