S-duality and refined BPS indices

Abstract

Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $\mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $\mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(\mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on ${\mathbb{P}}^2$, and make them explicit for $U(4)$.

34+24 pages, 6 figures; misprints fixed

References in corpus (12)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Mock Theta Functions
• R-Twisting and 4d/2d Correspondences
• Wall-Crossing from Boltzmann Black Hole Halos
• A Farey Tail for Attractor Black Holes
• D-instantons and twistors
• The M5-Brane Elliptic Genus: Modularity and BPS States
• $R^3$ index for four-dimensional $N=2$ field theories
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• Examples of M5-Brane Elliptic Genera
• Ample D4-D2-D0 Decay
• Twistorial Topological Strings and a tt* Geometry for N=2 Theories in 4d

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• 4d N=2 supergravity observables from Nekrasov-like partition functions
• Duality and Mock Modularity
• Vafa-Witten invariants from modular anomaly
• Heavenly metrics, BPS indices and twistors
• Curve counting and S-duality
• Scaling Black Holes and Modularity
• BPS Dendroscopy on Local $P^2$
• $\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
• Mock modularity at work, or black holes in a forest
• Topological correlators of $SU(2)$, $\mathcal{N}=2^{\ast }$ SYM on four-manifolds
• Refinement and modularity of immortal dyons
• Attractor invariants, brane tilings and crystals
• On the existence of scaling multi-centered black holes
• Type IIB parabolic ($p,q$)-strings from M2-branes with fluxes
• Modular anomaly of BPS black holes
• Quantum TBA for refined BPS indices