Abstract
We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on K3 fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi forms generate the dimensions of the refined cohomology of the Hilbert schemes of symmetric products of the fiber and allow us to refine the BPS indices in the fiber and therefore to obtain, conjecturally, actual state counts. Using the properties of the vector-valued lattice Jacobi forms we also provide a mathematical proof of the non-Abelian weak gravity conjecture for F-/M-theory compactified on this general class of fibered Calabi-Yau 3-folds.
Highlights
Analyzing supergravity theories [1, 2] and superconformal quantum field theories (SCFT) without gravity [3] in six dimensions geometrically, using F-theory on elliptic — or more general genus one fibered Calabi-Yau 3-folds E → M → B, has been very successful
There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings
While supergravity theories are defined on compact Calabi-Yau 3-folds M, the SCFTs are geometrically engineered on non-compact local models, that are often realizable as limits of compact M, consisting of elliptic fibered surfaces S over curves Cβ ∈ B, β ∈ H2comp(B, Z) with negative self intersection β2 < 0
Summary
Analyzing supergravity theories [1, 2] and superconformal quantum field theories (SCFT) without gravity [3] in six dimensions geometrically, using F-theory on elliptic — or more general genus one fibered Calabi-Yau 3-folds E → M → B, has been very successful. The infinite tower of states at points of maximal unipotent monodromy (MUM), that exists generically in Calabi-Yau 3-folds compactification at infinite distance in their complex moduli space Mcs has been explained as being mirror dual to the BPS bound states of D2-D0 branes as counted by the topological string on the mirror by the unrefined BPS indices [45] in the large volume limit. As in 6d F-theory the volume of the base sets the Planck scale vol ∼ m4Pl and the volume of a base curve Cβ2 supporting a 7-brane Yang-Mills theory sets the Yang-Mills coupling to vol(CβYM) ∼ ge−l2 ∼ vol, one can attempt to challenge the weak gravity conjecture by holding vol(B) finite, while sending vol(CβYM) → ∞ It was meticulously shown in [53] that in this scenario β2 = 0 and the K3 in the limit leads to a light Heterotic string spectrum with small string tension, i.e. scenario 2). We discuss the validity for genus one fibrations in 7
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
