[Paper Review] The Geometry Underlying Mirror Symmetry

David R. Morrison|ArXiv.org|Aug 5, 1996

Algebraic Geometry and Number Theory39 references41 citations

TL;DR

This paper proposes a geometric characterization of mirror symmetry in Calabi–Yau manifolds by identifying one manifold as the compactified, complexified moduli space of special Lagrangian tori on its mirror partner. The key contribution is a mathematically rigorous definition of geometric mirror pairs in arbitrary dimensions, linking quantum mirror symmetry to T-duality and establishing a correspondence between homology subspaces and Hodge-theoretic structures via Mukai vectors and Fourier–Mukai transforms.

ABSTRACT

The recent result of Strominger, Yau and Zaslow relating mirror symmetry to the quantum field theory notion of T-duality is reinterpreted as providing a way of geometrically characterizing which Calabi-Yau manifolds have mirror partners. The geometric description---that one Calabi-Yau manifold should serve as a compactified, complexified moduli space for special Lagrangian tori on the other Calabi-Yau manifold---is rather surprising. We formulate some precise mathematical conjectures concerning how these moduli spaces are to be compactified and complexified, as well as a definition of geometric mirror pairs (in arbitrary dimension) which is independent of those conjectures. We investigate how this new geometric description ought to be related to the mathematical statements which have previously been extracted from mirror symmetry. In particular, we discuss how the moduli spaces of the `mirror' Calabi-Yau manifolds should be related to one another, and how appropriate subspaces of the homology groups of those manifolds could be related. We treat the case of K3 surfaces in some detail.

Motivation & Objective

• To formulate a geometric criterion for mirror symmetry in Calabi–Yau manifolds independent of physical conjectures.
• To define geometric mirror pairs in arbitrary dimensions using moduli spaces of special Lagrangian submanifolds.
• To relate the new geometric characterization to established topological and Hodge-theoretic invariants of mirror symmetry.
• To investigate the relationship between homology subspaces of mirror Calabi–Yau manifolds and their Hodge structures.
• To verify the geometric mirror correspondence in the case of K3 surfaces using Mukai's theory of moduli spaces of sheaves.

Proposed method

• Define geometric mirror pairs via the compactified and complexified moduli space of special Lagrangian tori on a Calabi–Yau manifold.
• Use the Strominger–Yau–Zaslow conjecture as a foundation, interpreting mirror symmetry as T-duality on special Lagrangian fibrations.
• Apply the notion of quantum cohomology and correlation functions to define the quantum moduli space of conformal field theories.
• Employ Mukai vectors and the Riemann–Roch formula to compute dimensions of moduli spaces of sheaves on K3 surfaces.
• Utilize Fourier–Mukai transforms to relate sheaf categories on mirror K3 surfaces and verify mirror symmetry at the level of homology and Hodge structures.
• Use algebraic geometry techniques, including Simpson’s results, to compactify moduli spaces of semistable sheaves on algebraic K3 surfaces.

Experimental results

Research questions

• RQ1 How can mirror symmetry be characterized geometrically in terms of special Lagrangian torus fibrations?
• RQ2 What is the precise mathematical relationship between the homology subspaces of mirror Calabi–Yau manifolds?
• RQ3 How do the Hodge structures on the mirror pair relate under the proposed geometric mirror correspondence?
• RQ4 Can the moduli space of special Lagrangian 2-cycles on a K3 surface be compactified and complexified to yield a mirror partner?
• RQ5 To what extent does the Fourier–Mukai transform realize the geometric mirror map between sheaf categories on mirror K3 surfaces?

Key findings

• The moduli space of simple sheaves with Mukai vector $v = (0, u, 0)$ on a K3 surface has dimension two and is compact when the surface is algebraic, as guaranteed by Simpson’s theorem.
• For algebraic K3 surfaces, the moduli space of semistable sheaves with $v = (0, u, 0)$ forms a projective variety, providing a natural compactification of the special Lagrangian torus moduli space.
• The mirror map in physics sends the Mukai vector $v = (0, u, 0)$ to $(0, 0, 1)$, corresponding to the structure sheaf of a point, matching the expected mirror correspondence.
• The Hodge structure on the mirror space is realized as $v^ot / v$ in $H^2$, which matches the Hodge structure found by Mukai for moduli spaces of sheaves on K3 surfaces.
• A Fourier–Mukai transform exists that maps the class of a special Lagrangian section (homology class $(1,0,1)$) to the fundamental cycle on the mirror, confirming the geometric mirror map in this case.
• The geometric mirror correspondence for K3 surfaces unifies the physical mirror map with Mukai’s duality, showing that special Lagrangian 2-cycles and zero-cycles are exchanged under mirror symmetry.