Introduction to Combinatorial Mirror Symmetry

Combinatorial mirror symmetry suggests a purely combinatorial approach to mirror symmetry which is based on polar duality in the class of special lattice polytopes introduced by the lecturer and called reflexive. The Platonic solids provide the most famous examples of polar dual pairs of polyhedra: cube-octahedron, icosahedron-dodecahedron. In combinatorial mirror symmetry, an essential circumstance is that the vertices of reflexive polyhedra Δ are elements of a lattice M and the vertices of dual reflexive polyhedra Δ* belong to the dual lattice N. The lattice M can be identified with the lattice of characters of some algebraic torus T, and the dual lattice N becomes the lattice of one-parameter subgroups in T. For this reason, the main tool of the combinatorial approach is the theory of toric varieties. Combinatorial mirror symmetry allows us to interpret the mirror symmetry discovered by physicists for 3-dimensional Calabi-Yau manifolds by passing from M to N and from Δ to Δ*.

The purpose of the lectures is to explain in the most accessible way possible the connection between reflexive polyhedra and Calabi-Yau manifolds and to inform students about further results obtained using this combinatorial duality.

Meetings

Prerequisites

Algeba, commutative algebra and basics of algebraic geometry.

Content

• Quintic 3-folds in projective 4-space and their mirrors.
• Toric varieties associated with fans of rational polyhedral cones. Toric varieties associated with lattice polyhedra. Smoothness.
• Resolution of singularities. Cohomology rings of smooth projective toric varieties.
• Construction of Calabi-Yau varieties as hypersurfaces in toric varieties associated with reflexive polyhedra.
• A combinatorial formula for Hodge numbers of Calabi-Yau 3-folds. Monomial-divisor correspondence.
• Combinatorial mirror construction for Calabi-Yau complete intersections. Mirrors of rigid Calabi-Yau varieties.
• Computation of periods of Calabi-Yau hypersurfaces using generalized hypergeometric functions.
• Stringy Hodge numbers of singular Calabi-Yau varieties.
• Moduli spaces. Boundary points in moduli spaces of Calabi-Yau hypersurfaces and secondary polytopes.
• Computation of Gromov-Witten invariantes of Calabi-Yau complete intersections.
• A combinatorial approach to Berglund-Hübsch mirror symmetry.

References

• V. V. Batyrev, Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties, J. Alg. Geom. 3 (1994), no. 3, 493–535.
• V. V. Batyrev, D. van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys., 168:3 (1995), 493–533.
• V. V. Batyrev and L. A. Borisov, Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, Mirror Symmetry II, AMS/IP Stud. Adv. Math. 1, Amer. Math. Soc., Providence, RI (1997), 71–86.
• D. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, Vol. 68, AMS, (1999).
• I. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Springer-Birkhäuser, (1994).
• M. Jinzenji, Classical Mirror Symmetry, SpringerBriefs in Mathematical Physics, volume 29, (2018).