Abstract
We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers and additional rational sections or just N -sections, in order to study string dualities in four and five dimensions as well as rigid limits in which gravity decouples. The generating functions are Jacobi-forms of Γ1(N) with the complexified fiber volume as modular parameter. The string coupling λ, or the ϵ± parameters in the rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gauge bosons are elliptic parameters. To understand this structure, we show that specific auto-equivalences act on the category of topological B-branes on these geometries and generate an action of Γ1(N) on the stringy Kähler moduli space. We argue that these actions can always be expressed in terms of the generic Seidel-Thomas twist with respect to the 6-brane together with shifts of the B-field and are thus monodromies. This implies the elliptic transformation law that is satisfied by the generating functions. We use Higgs transitions in F-theory to extend the ansatz for the modular bootstrap to genus one fibrations with N -sections and boundary conditions fix the all genus generating functions for small base degrees completely. This allows us to study in depth a wide range of new, non-perturbative theories, which are Type II theory duals to the CHL ℤN orbifolds of the heterotic string on K3 × T2. In particular, we compare the BPS degeneracies in the large base limit to the perturbative heterotic one-loop amplitude with {R}_{+}^2{F}_{+}^{2g-2} insertions for many new Type II geometries. In the rigid limit we can refine the ansatz and obtain the elliptic genus of superconformal theories in 5d.
Highlights
In this paper we solve the all-genus topological string partition function Ztop. on compact genus one fibered Calabi-Yau 3 folds M in a large base expansion, extending the approach of [1, 2] to elliptic fibrations with reducible fibers and in particular to geometries that do not exhibit a section but only N -sections.An elliptic curve is a genus one curve with a marked point which is the zero O in the additive group law on the elliptic curve [3]
The modular bootstrap for multi-section geometries that we developed in the previous sections allows us to compare the topological string partition function against the heterotic one-loop computations that have been performed in [51]
We considered the case that M had multiple N -sections as well as fibral divisors, which respectively lead to Abelian and non-Abelian gauge symmetry enhancements in the Type II — and F-theory vacua
Summary
In this paper we solve the all-genus topological string partition function Ztop. on compact genus one fibered Calabi-Yau 3 folds M in a large base expansion, extending the approach of [1, 2] to elliptic fibrations with reducible fibers and in particular to geometries that do not exhibit a section but only N -sections. We obtain the correct Ansatz for general Zβ on general genus one fibered Calabi-Yau threefolds with N -section from the corresponding Ansatz for elliptic fibrations by considering Higgs transitions in F-theory, see subsection 4.5. Using a maximal supersymmetric dual pair in 6d between a Type II Z2 orbifold on K3 and a heterotic CHL Z2 orbifold on T 4 [45], as well as an adiabatic argument on the Type II side, the Z2 action on a dual elliptically fibered Calabi-Yau 3-fold was identified in [39] This leads to a 2-section geometry with a compatible K3 fibration. In appendix B we prove an identity between charateristic classes that is crucial to obtain the auto-equivalences and monodoromies in subsection 3.3 as well as in the derivation of the modular anomaly equation in subsection 4.4
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