Ricci-Flat Mirror Hypersurfaces in Spaces of General Type

TL;DR

The paper expands Calabi–Yau mirror symmetry to Calabi–Yau hypersurfaces embedded in spaces of general type with maximal toric symmetry, using Laurent anticanonical sections to smooth degenerations. It introduces a broad computational framework and demonstrates it through Calabi–Yau hypersurfaces in Hirzebruch scrolls, revealing an infinite yet algebro-geometrically tractable family with transposition mirrors. A key development is smoothing unsmoothable Tyurin-degenerate models via Laurent deformations, including intrinsic-limit constructions and extended GLSMs that accommodate non-regular defining equations. The work also develops a panoramic surgical view with VEX multitopes and flip-folded multifans, arguing for an extensive network of multiple mirrors and conjecturing that many toric varieties (and their CY hypersurfaces) may be realized as generalized complete intersections in products of projective spaces. Taken together, these results point to a rich, computationally accessible landscape of Ricci-flat geometries in non-Fano ambient spaces with potential implications for string compactifications and beyond.

Abstract

Complex Ricci-flat (i.e., Calabi-Yau) hypersurfaces in spaces admitting a maximal (toric) $U(1)^n$ gauge symmetry of general type (encoded by certain non-convex and multi-layered multitopes) may degenerate, but can be smoothed by rational (Laurent) anticanonical sections. Nevertheless, the phases of the Gauged Linear Sigma Model and an increasing number of their classical and quantum data are just as computable as for their siblings encoded by reflexive polytopes, and they all have transposition mirror models. Showcasing such hypersurfaces in so-called Hirzebruch scrolls shows this class of constructions to be infinitely vast, yet amenable to standard and well-founded algebro-geometric methods of analysis.

Figures (5)

• Figure 1: Left-most, $g{=}0$: the 0-dimensional reduction, $\mathbb{P}^4[5]\mkern2mu{\to}\mkern2mu \cdots\mkern2mu{\to}\mkern2mu \mathbb{P}^1[2]$, Calabi-Yau 0-fold (two red points), the anticanonical hypersurface in $S^2{=}\mathbb{P}^1$. This Calabi--Yau 0-fold easily embeds in increasingly higher-genus Riemann surfaces; those with $g\geqslant2$ are called "of general type."
• Figure 2: The $m=1,2,3$ sequence of anticanonical monomials, $\Gamma(\mathcal{K}^*_{{F^{(2)}_{m}}})$, where "$X^{abcd}$" stands for $X_1\!^aX_2\!^bX_3\!^cX_4\!^d$
• Figure 3: Change in the system of anticanonical sections used to define the Calabi--Yau hypersurface; the $F^{(2)}_{3}$-system on the left is over-complete, but factorizes without the rational monomials.
• Figure 4: The $m=1,2,3$ sequence of (trans)polar pairs $({{\mit\Delta}^{\mkern-3mu\raisebox{1pt}{$$}} _{\mkern-1mu{F^{(2)}_{m}}}},{{\mit\Delta}^{\mkern-3mu\raisebox{1pt}{$\star$}} _{\mkern-1mu{F^{(2)}_{m}}}})$; note that $\sphericalangle\Theta_3=\sphericalangle\Theta_2\cap\sphericalangle\Theta_4$ accounts for the overlap --- this corresponds to the non-convexity of ${{\mit\Delta}^{\mkern-3mu\raisebox{1pt}{$\star$}} _{\mkern-1mu{F^{(2)}_{3}}}}$ at the concave vertex $\theta_3$
• Figure 5: The flip-folded Newton multitope ${{\mit\Delta}^{\mkern-3mu\raisebox{1pt}{$$}} _{\mkern-1mu{F^{(2)}_{3}}}}$ being "repaired" by splicing in the Newton polytope of ${}^\triangledown\!\mathbb{P}^2\mkern2mu{=}\mkern2mu \mathrm{Bl}^\uparrow[\mathbb{P}^2/\mathbb{Z}_3]$ along the two $\mathbb{P}^1$s (dashed arrows), to produce the 2-sheeted (doubly winding) Newton multitope ${{\mit\Delta}^{\mkern-3mu\raisebox{1pt}{$$}} _{\mkern-1mu{\widehat{F}^{(2)}_3}}}$

Theorems & Definitions (8)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Conjecture 4
• Conjecture 5
• Definition 1
• Conjecture 6
• Conjecture 7