Refined BPS invariants of 6d SCFTs from anomalies and modularity

Abstract

F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function $Z_{top}$ of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing $Z_{top}$ in terms of contributions $Z_{\beta }$ at base degree $\beta$ of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kaehler class of the elliptic fiber and elliptic parameters the couplings and mass parameters. The indices with regard to the multiple elliptic parameters are fixed by the refined holomorphic anomaly equations, which we show to be completely determined from knowledge of the chiral anomaly of the corresponding SCFT. We express $Z_{\beta }$ as a quotient of weak Jacobi forms, with a universal denominator inspired by its pole structure as suggested by the form of $Z_{top}$ in terms of 5d BPS numbers. The numerator is determined by modularity up to a finite number of coefficients, which we prove to be fixed uniquely by imposing vanishing conditions on 5d BPS numbers as boundary conditions. We demonstrate the feasibility of our approach with many examples, in particular solving the E-string and M-string theories including mass deformations, as well as theories constructed as chains of these. We make contact with previous work by showing that spurious singularities are cancelled when the partition function is written in the form advocated here. Finally, we use the BPS invariants of the E-string thus obtained to test a generalization of the Goettsche-Nakajima-Yoshioka $K$-theoretic blowup equation, as inspired by the Grassi-Hatsuda-Marino conjecture, to generic local Calabi-Yau threefolds.

64 pages; v2: typos corrected

References in corpus (9)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories
• The Refined Topological Vertex
• 6d Conformal Matter
• Remodeling the B-model
• Extended Holomorphic Anomaly in Gauge Theory
• Anomaly of strings of 6d $\mathcal{N}=(1,0)$ theories
• E + E $\to$ H
• Operators and higher genus mirror curves
• On the Elliptic Genus of Three E-strings and Heterotic Strings

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• Revisiting the classifications of 6d SCFTs and LSTs
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• Blowup Equations for 6d SCFTs. I
• Bootstrapping BPS spectra of 5d/6d field theories
• Blowup Equations for Refined Topological Strings
• Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains
• The Asymptotic Weak Gravity Conjecture in M-theory
• On elliptic genera of 6d string theories
• Instantons from Blow-up
• Singular Geometry and Higgs Bundles in String Theory
• Black Hole Entropy from Quantum Mechanics
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• Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions
• Computing the elliptic genus of higher rank E-strings from genus 0 GW invariants
• Vafa-Witten invariants from modular anomaly
• Twisted 6d $(2,0)$ SCFTs on a Circle
• Refined large N duality for knots
• Refined topological vertex with ON-planes
• Rank $N$ Vafa-Witten invariants, modularity and blow-up
• Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane
• Cardy Limits of 6d Superconformal Theories
• The Fate of Discrete 1-Form Symmetries in 6d
• Blowup Equations for Little Strings
• Towards Refining the Topological Strings on Compact Calabi-Yau 3-folds
• New results in $\mathcal{N}=2$ theories from non-perturbative string
• Weyl invariant Jacobi forms along Higgsing trees
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• Bootstrapping ADE M-strings
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• State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture
• Recent Developments in Topological String Theory
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