[Paper Review] Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds

Victor V. Batyrev, Lev Borisov|ArXiv.org|Feb 3, 1994

Geometry and complex manifoldsMathematics11 references106 citations

TL;DR

This paper introduces reflexive Gorenstein cones as a combinatorial framework to generalize mirror symmetry for Calabi-Yau manifolds, extending it to generalized Calabi-Yau varieties of dimension $d + 2(r-1)$ via toric geometry. The duality of these cones corresponds to mirror symmetry, unifying known constructions and providing a mathematical explanation for mirrors of rigid Calabi-Yau manifolds.

ABSTRACT

We introduce a special class of convex rational polyhedral cones which allows to construct generalized Calabi-Yau varieties of dimension $(d + 2(r-1))$, where $r$ is a positive integer and d is the dimension of critical string vacua with central chatge $c = 3d$. It is conjectured that the natural combinatorial duality satisfies by these cones corresponds to the mirror involution. Using the theory of toric varieties, we show that our conjecture includes as special cases all already known examples of mirror pairs proposed by physicists and agrees with previous conjectures of the authors concerning explicit constructions of mirror manifolds. In particular we obtain a mathematical framework which explains the construction of mirrors of rigid Calabi-Yau manifolds.

Motivation & Objective

• To develop a mathematical framework that explains mirror symmetry for generalized Calabi-Yau manifolds beyond the standard Calabi-Yau case.
• To extend the combinatorial duality of reflexive polyhedra to higher-dimensional generalized Calabi-Yau varieties using Gorenstein cones.
• To reconcile the mirror symmetry of rigid Calabi-Yau manifolds with the duality of reflexive Gorenstein cones.
• To unify known mirror constructions—especially those of Schimmrigk and others—within a single toric-geometric formalism.

Proposed method

• Defining reflexive Gorenstein cones as rational polyhedral cones in a lattice with a canonical dual, ensuring the associated toric variety has Gorenstein singularities and a sheaf whose r-th tensor power is canonical.
• Using the duality of these cones to define a mirror involution analogous to that in $N=2$ superconformal field theories.
• Constructing generalized Calabi-Yau manifolds as zero loci of global sections of the sheaf $\mathcal{O}_{{\bf P}_\sigma}(1)$ on the toric Fano variety ${{\bf P}_\sigma}$.
• Reducing complete intersections in Gorenstein toric Fano varieties to hypersurfaces in higher-dimensional toric varieties via combinatorial reduction techniques.
• Establishing a correspondence between nef-partitions defining Calabi-Yau complete intersections and reflexive Gorenstein cones via duality.
• Computing Hodge numbers and cohomological invariants using the duality of cones and the geometry of quotient singularities, particularly for $({\bf P}_\Delta)^d/G$.

Experimental results

Research questions

• RQ1 Can the duality of reflexive Gorenstein cones provide a unified framework for mirror symmetry in generalized Calabi-Yau manifolds?
• RQ2 How does the duality of these cones relate to the mirror involution in $N=2$ superconformal field theories?
• RQ3 Can the construction of mirrors for rigid Calabi-Yau manifolds be explained via this cone duality?
• RQ4 Does the duality of reflexive Gorenstein cones reproduce known mirror pairs, including those of Schimmrigk?

Key findings

• The duality of reflexive Gorenstein cones corresponds to the mirror involution in $N=2$ superconformal field theories, providing a combinatorial realization of mirror symmetry.
• The construction yields generalized Calabi-Yau manifolds of dimension $d + 2(r-1)$ as zero loci of sections of $\mathcal{O}_{{\bf P}_\sigma}(1)$, with $r$-th tensor power isomorphic to the anticanonical sheaf.
• For $d=3$, the construction yields a rigid Calabi-Yau 3-fold $Z'$ as a quotient of $E_0 \times E_0 \times E_0$ by a ${\bf Z}/3{\bf Z}$ action, with its mirror as a quotient of a 7-dimensional cubic hypersurface.
• The Hodge $h^{1,1}$-number of the resolution $\hat{Z}$ of the quotient $({\bf P}_\Delta)^d/G$ is computed as $\frac{d(3d-1)(3d-2)}{2}$, confirming cohomological consistency.
• The duality between reflexive Gorenstein cones agrees with the duality of nef-partitions used in constructing Calabi-Yau complete intersections in toric varieties.