K3-fibered Calabi-Yau Threefolds Research Articles

Hecke Operators on Vector-Valued Modular Forms

We study Hecke operators on vector-valued modular forms for the Weil representation $\rho_L$ of a lattice $L$. We first construct Hecke operators $\mathcal{T}_r$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of type $\rho_{L(r)}$, where $L(r)$ is the lattice $L$ with rescaled bilinear form $(\cdot, \cdot)_r = r (\cdot, \cdot)$, by lifting standard Hecke operators for scalar-valued modular forms using Siegel theta functions. The components of the vector-valued Hecke operators $\mathcal{T}_r$ have appeared in [Comm. Math. Phys. 350 (2017), 1069-1121] as generating functions for D4-D2-D0 bound states on K3-fibered Calabi-Yau threefolds. We study algebraic relations satisfied by the Hecke operators $\mathcal{T}_r$. In the particular case when $r=n^2$ for some positive integer $n$, we compose $\mathcal{T}_{n^2}$ with a projection operator to construct new Hecke operators $\mathcal{H}_{n^2}$ that map vector-valued modular forms of type $\rho_L$ into vector-valued modular forms of the same type. We study algebraic relations satisfied by the operators $\mathcal{H}_{n^2}$, and compare our operators with the alternative construction of Bruinier-Stein [Math. Z. 264 (2010), 249-270] and Stein [Funct. Approx. Comment. Math. 52 (2015), 229-252].

Sections of Calabi-Yau threefolds with K3 fibration

We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and that the group of algebraic 1 1 -cycles generated by these sections modulo algebraic equivalence is not finitely generated. We also give examples of K 3 3 surfaces over the function field F F of a complex curve with Zariski dense F F -rational points, whose geometric model is Calabi-Yau.

ZETA-FUNCTIONS OF CERTAIN K3-FIBERED CALABI–YAU THREEFOLDS

We consider certain K3-fibered Calabi–Yau threefolds. One class of such Calabi–Yau threefolds is constructed by Hunt and Schimmrigk using twist maps. They are realized in weighted projective spaces as orbifolds of hypersurfaces. Our main goal of this paper is to investigate arithmetic properties of these K3-fibered Calabi–Yau threefolds. In particular, we give detailed discussions on the construction of these Calabi–Yau varieties, singularities and their resolutions. We then determine the zeta-functions of these Calabi–Yau varieties. Next we consider deformations of our K3-fibered Calabi–Yau threefolds, and we study the variation of the zeta-functions using p-adic rigid cohomology theory.

STABLE SHEAVES OVER K3 FIBRATIONS

We construct stable sheaves over K3 fibrations using a relative Fourier-Mukai transform which describes the sheaves in terms of spectral data similar to the construction for elliptic fibrations. On K3 fibered Calabi–Yau threefolds we show that the Fourier-Mukai transform induces an embedding of the relative Jacobian of spectral line bundles on spectral covers into the moduli space of sheaves of given invariants. This makes the moduli space of spectral sheaves a generic torus fibration over the moduli space of curves of the given arithmetic genus on the Calabi–Yau manifold.

String junctions and non-simply connected gauge groups

Relations between the global structure of the gauge group in elliptic F-theory compactifications, fractional null string junctions, and the Mordell-Weil lattice of rational sections are discussed. We extend results in the literature, which pertain primarily to rational elliptic surfaces and obtain pi^1(G) where G is the semi-simple part of the gauge group. We show how to obtain the full global structure of the gauge group, including all U(1) factors. Our methods are not restricted to rational elliptic surfaces. We also consider elliptic K3's and K3-fibered Calabi-Yau three-folds.

String duality and modular forms

Tests of duality between heterotic strings on K3 × T 2 (restricted on certain Narain moduli subspaces) and type IIA strings on K3-fibered Calabi-Yau threefolds are attempted in the weak coupling regime on the heterotic side by identifying pertinent modular forms related to the computations of string threshold corrections. Concretely we discuss in parallel the three cases associated with Calabi-Yau manifolds ( A) : X(6, 2, 2, 1, 1) 2 −252, ( B) : X(12, 8, 2, 1, 1) 3 −480 and ( C) : X(10, 3, 3, 2, 2) 4 −132 on the type IIA side.