Mirror symmetry and T-duality in the complement of an anticanonical divisor

Abstract: We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.

• The paper demonstrates how mirror symmetry and T-duality extend beyond Calabi-Yau settings by dualizing special Lagrangian fibrations in the complement of an anticanonical divisor.
• It employs rigorous analyses of moduli spaces, Lagrangian Floer homology, and disc counting to connect classical symplectic geometry with quantum-corrected Landau-Ginzburg models.
• The study reveals significant implications for managing singular fibers and wall-crossing phenomena, thereby refining mirror constructions in toric and non-toric Fano manifolds.

Mirror Symmetry and T-Duality in the Complement of an Anticanonical Divisor

The paper authored by Denis Auroux explores intricate connections between mirror symmetry and T-duality within the framework of specific geometric constructs, focusing on Lagrangian submanifolds situated in the complement of an anticanonical divisor within a compact Kähler manifold. The analysis is rooted in examining the Strominger-Yau-Zaslow conjecture, which posits that mirrors of Calabi-Yau manifolds can be constructed by dualizing fibrations of special Lagrangian tori, extended in this work to a broader context encompassing Fano manifolds via Landau-Ginzburg models with superpotentials.

The central thesis hinges on the attempt to reconcile and elucidate the roles played by T-duality and mirror symmetry beyond the conventional Calabi-Yau setting, with substantial attention paid to toric and non-toric Fano manifolds. Noteworthy examples discussed include Del Pezzo surfaces, highlighting the nuanced geometric symmetries transcending the classical Calabi-Yau confines. The proposed model examines how moduli spaces of special Lagrangians exhibit duality and adapts it to Fano varieties through quantum-corrective measures.

One of the study's pivotal claims challenges the naivety of Conjecture 1.1 by recognizing the intrinsic geometric and topological corrections required in constructing the mirror manifold M. Such corrections are necessitated by singular fibers in Lagrangian torus fibrations and wall-crossing phenomena, demanding a nuanced formulation in terms of quantum corrections in absence of a neat definition for the superpotential's fibers.

The theoretical framework is substantiated with a detailed exploration of the moduli space of special Lagrangians, laying out foundational propositions regarding the ψ-harmonic structures on L, consequently forming the basis for the conjectured quantum corrections. The analysis further explores the toric case, revealing insights consistent with existing constructs like those derived by Cho and Oh. In particular, for toric manifolds, the superpotential aligns with known Laurent polynomials, reinforcing its compatibility with T-duality.

The discussion extensively tackles the implications of Maslov index computations and disc counting phenomena, which crucially delineate boundary conditions and anticipate wall-crossing transitions between moduli spaces. Such wall-crossing events necessitate a reevaluation in defining the mirror symmetry, where β and α are notably considered in the practical scenarios of CP^2 and CP^1 × CP^1, offering significant analytical and numerical clarity.

By anchoring the study in thorough mappings between Lagrangian Floer homology and quantum cohomology, with meaningful reflection on eigenvalues of quantum cup-product actions, the work raises prominent theoretical and practical implications. The conjectures poised regarding the mirror construction aligning with a renormalization limit in a toric scenario, suggest a new dimension to mirror geometry through symplectic inflations along anticanonical divisors.

Contemplating an extension to non-Fano contexts and addressing the quantum correction adjustments, this study ventures prospectively into synthesizing the constructs of mirror symmetry with contemporary geometric formalisms and their prospective confluence in theoretical physics. Through this rigorous investigation, Auroux lays a compelling foundation for advancing studies in symplectic geometry, seeking to deepen the comprehensive understanding of the interconnected role played by T-duality and mirror symmetry within broader geometrical landscapes.