Abstract
F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function Ztop of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing Ztop in terms of contributions Zβ at base degree β of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kähler 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 Zβ as a quotient of weak Jacobi forms, with a universal denominator inspired by its pole structure as suggested by the form of Ztop 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 Göttsche-Nakajima-Yoshioka K-theoretic blowup equation, as inspired by the Grassi-Hatsuda-Mariño conjecture, to generic local Calabi-Yau threefolds.
Highlights
How well and how generally we can compute the topological string partition functionZtop serves as a benchmark for how well we understand topological string theory
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
The natural generalization of the blowup equation that we propose, and that is satisfied by the E-string, can be applied to any local Calabi-Yau geometry on which the refined topological string can be formulated
Summary
How well and how generally we can compute the topological string partition function. Ztop serves as a benchmark for how well we understand topological string theory. In this paper, following [10], we will use modular methods in conjunction with vanishing conditions on 5d BPS invariants Njκ−j+ to compute the refined topological string partition function on a class of non-toric geometries, consisting of elliptically fibered local Calabi-Yau manifolds M. We will compute the refined topological string partition function on elliptic Calabi-Yau spaces with at least one zero section recursively as an expansion in the base classes β. We obtain explicit expressions for Z(tb, τ, tm, ǫ1, ǫ2) which, aside from passing the stringent test of integrality for all 5d BPS invariants encompassed, match all results available in the literature computed by other means
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