Jacobi forms of higher index and paramodular groups in $ \mathcal{N} $ = 2, D = 4 compactifications of string theory

Abstract

We associate a Jacobi form over a rank s lattice to N = 2, D = 4 heterotic string compactifications which have s Wilson lines at a generic point in the vector multiplet moduli space. Jacobi forms of index m = 1 and m = 2 have appeared earlier in the context of threshold corrections to heterotic string couplings. We emphasize that higher index Jacobi forms as well as Jacobi forms of several variables over more generic even lattices also appear and construct models in which they arise. In particular, we construct an orbifold model which can be connected to models that give index m = 3, 4 or 5 Jacobi forms through the Higgsing process. Constraints from being a Jacobi form are then employed to get threshold corrections using only partial information on the spectrum. We apply this procedure for index m = 3, 4 or 5 Jacobi form examples and also for Jacobi forms over A 2 and A 3 root lattices. Examples with a single Wilson line are examined in detail and we display the relation of Siegel forms over a paramodular group Γ m to these models, where Γ m is associated with the T-duality group of the models we study. Finally, results on the heterotic string side are used to clarify the linear mapping of vector multiplet moduli to Type IIA duals without using the one-loop cubic part of the prepotential on the Type II side, and also to give predictions for the geometry of the dual Calabi-Yau manifolds.

Heterotic strings on (K3 × T2)/ℤ3 and their dual Calabi-Yau threefolds

“…We take the necessary branching rules from [60]. For more on higgsing see for example [19,[61][62][63].…”

Weight one Jacobi forms and umbral moonshine

“…It is interesting to note that A t also appears as a moduli space describing massless degrees of freedom in certain compactifications of heterotic string theory, and associated paramodular forms have been shown [38][39][40] to govern one-loop corrections of their interaction terms. This may be a good setting in which to understand the physical significance of Q 1,1 .…”

Weight One Jacobi Forms and Umbral Moonshine

“…It is interesting to note that A t also appears as a moduli space describing massless degrees of freedom in certain compactifications of heterotic string theory, and associated paramodular forms have been shown [38][39][40] to govern one-loop corrections to their interaction terms. This may be a good setting in which to understand the physical significance of Q 1,1 .…”