Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

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 Γ 1 (N) with the complexified fiber volume as modular parameter. The string coupling λ, or the ± 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 Γ 1 (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 Z N orbifolds of the heterotic string on K3 × T 2. 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.

Instanton corrections and Emergent Strings

“…As already alluded to earlier, in the present case, the Fourier-Mukai transform corresponds to a monodromy action M on the Kähler moduli space [38]. We can therefore use that Donaldson-Thomas invariants do not change under monodromies, i.e.…”

“…However, for non-trivial fibrations we have to account for possible I 1 -singularities and curvature of the base. This simple double T-duality should then be replaced by a certain Fourier-Mukai transform 1 [37], which has been shown to corresponds to a monodromy action on the Kähler moduli space [38].…”

“…One may apply the same reasoning to see how the tension of other strings scale in the quantum-corrected regime. For instance, the tension of a D3-string wrapped on some curve γ a ⊂ X behaves as: 38) reflecting the fact that the tension of the D3-string in string units scales like the action of the D1-instanton wrapped on the same curve. As a consequence the quantum corrections increase the speed at which the D3-string becomes tensionless to χ −1 ∼ e −3σ/2 , but not as much as they do for the D1-string.…”

Modular curves and Mordell-Weil torsion in F-theory

“…In a related F-theory context, fibrations with restricted monodromies have also been considered in[23,24] and recently[25].…”

“…Alternatives might be quotients of type (K3 × T k )/Z n[25,37] as well as quotients, that do not require a finite Mordell-Weil group.…”