F-theory with hyperelliptic fibrations
E. Ballico, E. Gasparim, M. P. García del Moral, C. las Heras
TL;DR
This work develops a mathematical framework for incorporating hyperelliptic fibrations into F-theory by embedding a genus‑2 fibered surface $Y$ into a noncompact Calabi–Yau threefold $X=\mathrm{Tot}(\omega_Y)$ and into a CY$_4$, enabling a two‑fold perspective (upstairs and downstairs). The authors propose identifying the Type IIB axio-dilaton with the determinant of the genus‑2 period matrix, $\lambda=\det\Omega$, and derive conditions under which this determinant transforms under $SL(2,\mathbb{Z})$, yielding new types of monodromies and singularities not present in elliptic fibrations. They establish that for suitable $Y$ (including those with split Jacobians and $D_4$ symmetry) the determinant‑based axio-dilaton can be made to transform correctly, and they show that every even value of $c_2(X)$ is realizable via $X=\mathrm{Tot}(\omega_Y)$, with blowups allowing all even integers. In the downstairs picture, placing the hyperelliptic fibration inside the CY$_3$ base of an elliptically fibered CY$_4$ enables large Euler characteristics to aid tadpole cancellation through brane fluxes, while keeping D7/D3 charge accounting consistent without orientifold planes in certain regimes. Overall, the paper provides a concrete geometric route to extend F-theory beyond elliptic fibrations, offering new tools for brane engineering and flux compactifications while highlighting novel singularity structures and monodromies arising from genus‑two fibrations.
Abstract
We discuss the role of hyperelliptic fibrations in F-theory. For each even integer $n$ we give a noncompact Calabi--Yau threefold $X$ containing a hyperelliptically fibered surface $Y$, such that $X$ and $Y$ are homotopy equivalent and $c_2(X) = n$. We investigate two distinct cases depending on the position of the hyperelliptic fibration. First, we propose to extend F-theory considering hyperelliptic fibrations, giving an identification between the determinant of the period matrix and the axio-dilaton. Such an identification requires that the curve satisfies an appropriate criterium which we describe. Our explicit examples have split Jacobian, preserve the same number of degrees of freedom of usual F-theory, while allowing for the appearance of a greater variety of singularities. Second, when the hyperelliptic fibration is contained in the base of a Calabi--Yau fourfold, we show that tadpole cancellation conditions are satisfied for arbitrarily large values of $c_2(X)$.