Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

Abstract

We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.

The Swampland Distance Conjecture for Kähler moduli

“…In general, this is an extremely hard question, since it requires information about the global properties of the moduli space and the D-brane states existent at various other points. For elliptic fibrations, however, there is much literature [39][40][41][42][43][44] on how to leave the large volume point using the map v 0 → 1/v 0 , where we recall that v 0 is the volume of the elliptic fiber. In the following, we will use these results to present a charge orbit for the limit…”

“…which is the specialization of (3.50) to elliptic fibrations. The form of the matrix S can be explicitly calculated following [39,46,47] as we show in detail in appendix B. The resulting…”

“…It is non-trivial to show (3.66), since it equates central charges at different points in moduli space. However, it was argued in [39][40][41][42][43][44] that the transformation S effectively maps…”

Modular fluxes, elliptic genera, and weak gravity conjectures in four dimensions

“…[87] and extended in ref. [88]. The upshot there was that four-point correlators are modular forms of weight −2.…”

“…This is what happens for the example discussed in ref. [88], where it was shown how a modular eigenbasis of H 2,2 vert,0 (Y 4 ) can be chosen such as to realize the off-diagonal part of the intersection form (2.5).…”