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Moment maps, monodromy and mirror manifolds

R. P. Thomas

TL;DR

The paper develops a moment-map perspective and complexified gauge framework to study special Lagrangian geometry on Calabi–Yau manifolds, pairing holomorphic Chern–Simons functionals with their complexified Lagrangian counterparts. It proposes a stability condition for graded Lagrangians, linking Lagrangian connect sums to extension data and wall-crossing phenomena paralleling Kontsevich’s mirror conjecture. A central contribution is a concrete conjecture that a special Lagrangian representative exists within a fixed Hamiltonian deformation class, with rigorous validation in the two-dimensional case $T^2$ and a detailed exploration of monodromy, Floer theory, and mirror symmetry. This work bridges symplectic reduction, Fukaya categories, and derived categories to provide a unified stability framework for Lagrangians and their mirrors, with a tangible testbed in dimension two supporting the broader conjecture.

Abstract

Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians, and can be proved for the simple case of $T^2$.

Moment maps, monodromy and mirror manifolds

TL;DR

The paper develops a moment-map perspective and complexified gauge framework to study special Lagrangian geometry on Calabi–Yau manifolds, pairing holomorphic Chern–Simons functionals with their complexified Lagrangian counterparts. It proposes a stability condition for graded Lagrangians, linking Lagrangian connect sums to extension data and wall-crossing phenomena paralleling Kontsevich’s mirror conjecture. A central contribution is a concrete conjecture that a special Lagrangian representative exists within a fixed Hamiltonian deformation class, with rigorous validation in the two-dimensional case and a detailed exploration of monodromy, Floer theory, and mirror symmetry. This work bridges symplectic reduction, Fukaya categories, and derived categories to provide a unified stability framework for Lagrangians and their mirrors, with a tangible testbed in dimension two supporting the broader conjecture.

Abstract

Via considerations of symplectic reduction, monodromy, mirror symmetry and Chern-Simons functionals, a conjecture is proposed on the existence of special Lagrangians in the hamiltonian deformation class of a given Lagrangian submanifold of a Calabi-Yau manifold. It involves a stability condition for graded Lagrangians, and can be proved for the simple case of .

Paper Structure

This paper contains 6 sections, 2 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Chern-Simons-type functionals and critical points
  3. Gauge equivalence and moment maps
  4. Relationship to Kontsevich's mirror conjecture
  5. Stability
  6. The 2-torus

Key Result

Lemma 3.4

For $t>0$ the SLags $L^t$ are all in the same hamiltonian deformation class. Similarly for $L_1^t,\ L^t_2$, and for $t<0$.

Figures (3)

  • Figure 1: $\left(\int_{L_1}\Omega\right)$-space, as $\Omega$ on $K3$ varies, with polar coordinates $(R,\,\phi(L_1))$
  • Figure 2: $\left(\int_{L_1}\Omega\right)$-space, as $\Omega$ on a 3-fold varies, with polar coordinates $(R,\,\phi(L_1))$
  • Figure 3: $L_1\# L_2$ and $L_2\#(L_1[1])$, equivalent SLags, and their mirror sheaves

Theorems & Definitions (4)

  • Lemma 3.4
  • Lemma 3.6
  • Definition 5.1
  • Conjecture 5.2