Holes in Calabi-Yau Effective Cones

Abstract

Motivated by their role in non-perturbative potentials in string theory, we study divisors in effective cones of Calabi-Yau threefolds. We give examples of geometries for which some divisor classes in the effective cone are not themselves effective: i.e., they have no global sections. We call these non-holomorphic divisor classes "holes," and characterize their behavior in an ensemble of toric hypersurface Calabi-Yau threefolds. We prove some necessary and sufficient conditions for the existence of holes, show consequences of holes that follow from the minimal model program, and demonstrate that a class of holes come in semigroups (with this class conjectured to constitute all holes). Furthermore, we provide moduli-dependent bounds on the volumes of four-cycles representing holes.

Figures (9)

• Figure 1: 3D cartoon of the effective cone $\mathcal{E}_X$ of a Calabi--Yau hypersurface $X$ in a toric variety $V$. In blue is the extended Kähler cone $\mathcal{K}_X$, containing the movable divisors on $X$; its complement, the non-movable divisors, is in red. Hashed is the subcone $\mathcal{E}_V$ of the $\mathcal{E}_X$ inherited from effective divisors of the ambient variety $V$. The extended Kähler cone itself decomposes as a union of Kähler/nef cones (see, e.g., Fig. 1 in Gendler:2022ztv). The (open) cone of big divisors on $X$ is the interior of $\mathcal{E}_X$, and the (open) cone of divisors inherited from big divisors on $V$ is the interior of $\mathcal{E}_V$. The boundaries of these open cones are the divisors that are effective but not big on $X$ and $V$, respectively.
• Figure 2: A scan through the Kreuzer--Skarke database for Calabi--Yau threefolds with non-trivial Hilbert basis elements up to $h^{1,1}=200$. Here, for each given $h^{1,1}$, we randomly sample $\sim 10^4$ polytopes within the Kreuzer--Skarke database using CYTools Demirtas:2022hqf. In the right figure, the box indicates 50% of the number of non-trivial Hilbert basis elements, the whiskers indicate the minimum and maximum value of the number of non-trivial Hilbert basis elements within $3/2$ of the interquartile range, the orange horizontal line indicates the median, and the green triangle indicates the mean.
• Figure 3: The toric effective cone $\mathcal{E}_{V_{2,106}}$ for the Calabi--Yau hypersurface $X_{2,106}$ (red) with non-effective divisor classes in the effective cone --- holes --- in blue. The variables $p_1, p_2$ denote the components of divisor classes when expressed in the basis of \ref{['eq:X2106 GLSM']}.
• Figure 4: The toric effective cone $\mathcal{E}_{V_{3,51}}$ for the Calabi--Yau hypersurface $X_{3,51}$ (red) with holes in blue. The variables $p_1, p_2, p_3$ denote the components of divisor classes when expressed in the basis of \ref{['eq:351_charge_matrix']}.
• Figure 5: The toric effective cone $\mathcal{E}_{V_{3,165}}$ for the Calabi--Yau hypersurface $X_{3,165}$ (red) with holes on the boundary and on the interior of $\mathcal{E}_{V_{3,165}}$ in blue and green, respectively. The variables $p_1, p_2, p_3$ denote the components of divisor classes when expressed in the basis of \ref{['eq:GLSM_3_165']}.

Theorems & Definitions (40)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Definition 8
• Proposition 9
• Theorem 10