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 $U(4)$.

arXiv comment34+24 pages, 6 figures; misprints fixed

References in corpus (54)

• Microscopic Origin of the Bekenstein-Hawking Entropy
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• A Strong Coupling Test of S-Duality
• Black Hole Entropy in M-Theory
• Four-dimensional wall-crossing via three-dimensional field theory
• Holomorphic Anomalies in Topological Field Theories
• Mock Theta Functions
• Supergravity flows and D-brane stability
• Split States, Entropy Enigmas, Holes and Halos
• Quantum Black Holes, Wall Crossing, and Mock Modular Forms
• R-Twisting and 4d/2d Correspondences
• Wall-Crossing from Boltzmann Black Hole Halos
• Refined stable pair invariants for E-, M- and [p,q]-strings
• A Farey Tail for Attractor Black Holes
• E-Strings and N=4 Topological Yang-Mills Theories
• The non-compact elliptic genus: mock or modular
• Black hole bound states in AdS_3 x S^2
• Deformation quantization and superconformal symmetry in three dimensions
• D-instantons and twistors
• Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces
• Line operators on S^1xR^3 and quantization of the Hitchin moduli space
• D-instantons and twistors: some exact results
• Stability and duality in N=2 supergravity
• D3-instantons, Mock Theta Series and Twistors
• The M5-Brane Elliptic Genus: Modularity and BPS States
• Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces
• D0-branes in Black Hole Attractors
• Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces
• Multiple D3-instantons and mock modular forms I
• Indefinite theta series and generalized error functions
• Equivariant K-theory and refined Vafa-Witten invariants
• No entropy enigmas for N=4 dyons
• Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces
• Attractor flow trees, BPS indices and quivers
• Virtual refinements of the Vafa-Witten formula
• $R^3$ index for four-dimensional $N=2$ field theories
• Aspects of $\Omega$-deformed M-theory
• N=4 Supersymmetric Yang-Mills Theory on a Kaehler Surface
• Monopole contributions to refined Vafa-Witten invariants
• The Betti numbers of the moduli space of stable sheaves of rank 3 on P2
• Multiple D3-instantons and mock modular forms II
• Invariants of moduli spaces of stable sheaves on ruled surfaces
• Examples of M5-Brane Elliptic Genera
• $r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth
• Vafa-Witten theory and iterated integrals of modular forms
• Rigid limit for hypermultiplets and five-dimensional gauge theories
• D4-D0 Branes on the Quintic
• TBA for non-perturbative moduli spaces
• Black holes and higher depth mock modular forms
• Ample D4-D2-D0 Decay
• Intersection cohomology of moduli spaces of sheaves on surfaces
• Twistorial Topological Strings and a tt* Geometry for N=2 Theories in 4d
• Higher depth quantum modular forms, multiple Eichler integrals, and ${\mathfrak{sl}}_3$ false theta functions
• Attractive Strings and Five-Branes, Skew-Holomorphic Jacobi Forms and Moonshine

Cited by in corpus (18)

• Vafa-Witten invariants from exceptional collections
• Exponential BPS graphs and D brane counting on toric Calabi-Yau threefolds: Part I
• 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