[Paper Review] Moduli, Motives, Mirrors (Plenary talk at the 3rd ECM, Barcelona, July 10-14, 2000)

Yuri I. Manin|ArXiv.org|May 15, 2000

Nonlinear Waves and Solitons61 references3 citations

TL;DR

This paper explores Mirror Symmetry in Calabi–Yau manifolds through two dual frameworks: Frobenius manifold isomorphisms arising from quantum cohomology and Hodge theory, and Lagrangian/complex duality via real toric fibrations. The key contribution is a precise cohomological mirror isomorphism $\mu_{X,Y}: H^*(X,\mathbb{C}) \to H^*(Y,\wedge^*(\mathcal{T}_Y)) \otimes V_Y^{-2}$, linking Gromov–Witten invariants to motivic structures, and proposing a unified motivic language for mirror symmetry beyond transcendental maps.

ABSTRACT

This talk is dedicated to various aspects of Mirror Symmetry. It summarizes some of the mathematical developments that took place since M. Kontsevich's report at the Zürich ICM and provides an extensive, although not exhaustive, bibliography.

Motivation & Objective

• To clarify the mathematical structure of Mirror Symmetry beyond the original quintic identity, particularly through Frobenius manifold isomorphisms.
• To unify two dual perspectives: quantum cohomology (A-model) and extended Hodge structures (B-model) in mirror pairs.
• To investigate how Gromov–Witten invariants and motivic geometry relate to mirror symmetry via algebraic correspondences.
• To propose a motivic framework for mirror symmetry, treating mirror maps as motivated morphisms in a category of motives.
• To explore the role of derived categories and automorphism groups in realizing mirror symmetries at the level of motivic fundamental groups.

Proposed method

• Constructing Frobenius manifolds from quantum cohomology of Calabi–Yau manifolds and from variations of Hodge structure on mirror families.
• Using Givental's equivariant cohomology and torus action techniques to prove the original quintic mirror identity.
• Defining mirror partnership via isomorphism of Frobenius manifolds built from A- and B-model data.
• Introducing real toric fibrations with special Lagrangian fibers to realize the Lagrangian/complex duality in the sense of Kontsevich and Strominger–Yau–Zaslow.
• Tracing mirror isomorphisms between cohomology spaces via flat vector fields and identifying trace functionals and metrics.
• Extending the category of motives to include mirror isomorphisms, treating them as new motivated morphisms, and linking to derived category autoequivalences.

Experimental results

Research questions

• RQ1 How can the Mirror Identity for quintic threefolds be understood as a special case of a deeper duality between Frobenius manifolds?
• RQ2 What is the precise geometric and cohomological structure of the mirror isomorphism $\mu_{X,Y}$ between Calabi–Yau manifolds $X$ and $Y$?
• RQ3 How do Gromov–Witten invariants, which are motivic in nature, reflect in the mirror geometry of $Y$?
• RQ4 Can the highly transcendental mirror map be formalized within a motivic framework, and what new structures emerge?
• RQ5 What is the role of special Lagrangian fibrations and their monodromy in realizing the duality between symplectic and complex structures?

Key findings

• The mirror isomorphism $\mu_{X,Y}: H^*(X,\mathbb{C}) \to H^*(Y,\wedge^*(\mathcal{T}_Y)) \otimes V_Y^{-2}$ is established as a cohomological identification induced by Frobenius manifold isomorphisms.
• The isomorphism identifies $H^{p,q}(X)$ with $H^q(Y, \wedge^p(\mathcal{T}_Y))$, showing a precise Hodge-theoretic duality.
• Near a cusp in moduli space, the mirror map becomes a ring isomorphism after trivializing $V_Y = H^0(Y, \Omega_Y^{\text{max}})$ via a volume form.
• The trace functionals and flat metrics on both sides of the mirror correspondence are identified via the Frobenius structure.
• Gromov–Witten invariants on the A-side arise from algebraic correspondences between $X^n$ and $\overline{M}_{0,n}$, indicating a motivic origin.
• The paper proposes extending the category of motives by adding mirror isomorphisms as new motivated morphisms, suggesting a deeper categorical structure underlying mirror symmetry.