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Moduli of special Lagrangians with boundary, II: Lagrangian Flux and Affine Structures

Vasanth Pidaparthy

TL;DR

The paper advances the theory of moduli spaces of special Lagrangians with boundary by developing Hitchin-type affine structures and a Hessian $L^2$ metric on the SLb moduli. It introduces two flux functionals, the relative flux $RF$ and the special Lagrangian flux $SF$, and proves their well-definedness and end-point homotopy invariance, enabling canonical coordinate charts. By combining these fluxes, it constructs two affine coordinate systems whose transitions preserve volume, and proves an isometric Lagrangian embedding of the moduli into a product of cohomology spaces, yielding a Hessian metric interpretation. The framework generalizes to almost Calabi--Yau manifolds via a conformal metric, preserving the core structure of the fluxes, affine structures, and Hessian geometry. This work deepens understanding of the SYZ picture and the global geometry of SLb moduli, with potential implications for mirror symmetry and Lagrangian boundary problems.

Abstract

This article continues the study of moduli spaces of special Lagrangians with boundary in a Calabi--Yau manifold. The moduli space was shown to be a smooth finite-dimensional manifold in the prequel arXiv:2503.6321918. This article investigates geometric structures on the moduli space of special Lagrangians with boundary and constructs a pair of special affine structures and a Hessian metric on this moduli space.

Moduli of special Lagrangians with boundary, II: Lagrangian Flux and Affine Structures

TL;DR

The paper advances the theory of moduli spaces of special Lagrangians with boundary by developing Hitchin-type affine structures and a Hessian metric on the SLb moduli. It introduces two flux functionals, the relative flux and the special Lagrangian flux , and proves their well-definedness and end-point homotopy invariance, enabling canonical coordinate charts. By combining these fluxes, it constructs two affine coordinate systems whose transitions preserve volume, and proves an isometric Lagrangian embedding of the moduli into a product of cohomology spaces, yielding a Hessian metric interpretation. The framework generalizes to almost Calabi--Yau manifolds via a conformal metric, preserving the core structure of the fluxes, affine structures, and Hessian geometry. This work deepens understanding of the SYZ picture and the global geometry of SLb moduli, with potential implications for mirror symmetry and Lagrangian boundary problems.

Abstract

This article continues the study of moduli spaces of special Lagrangians with boundary in a Calabi--Yau manifold. The moduli space was shown to be a smooth finite-dimensional manifold in the prequel arXiv:2503.6321918. This article investigates geometric structures on the moduli space of special Lagrangians with boundary and constructs a pair of special affine structures and a Hessian metric on this moduli space.

Paper Structure

This paper contains 16 sections, 25 theorems, 130 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Special Lagrangians and Lagrangian boundary conditions
  5. Smooth paths and tangents in the moduli space of Lagrangians with boundary
  6. L2̂ metric on the moduli space
  7. Lagrangian flux
  8. Relative Lagrangian flux
  9. Proof of Theorem \ref{['thm:HWB-rel-flux-well-defined']}
  10. Special Lagrangian flux
  11. Proof of Theorem \ref{['thm:HWB-dual-flux-well-defined']}
  12. Canonical coordinates and affine structures
  13. Canonical coordinates
  14. Proof of Theorem \ref{['thm:HWB-affine-structure-on-moduli-space']}
  15. Isometric embedding and Hessian metric
  16. ...and 1 more sections

Key Result

Theorem 1.1

vasanth-hitchin-1 Under the assumptions of Proposition prop:HWB-tangentspacetospeciallagrangianswithboundary-1, the moduli space $\mathcal{S} \mathcal{L}(X,L; \Lambda_1,\dots, \Lambda_d)$ is a finite-dimensional manifold whose tangent space at an immersed special Lagrangian is isomorphic to the spac

Figures (4)

  • Figure 1: Left: The boundary Lagrangians $\Lambda_i,\Lambda_j$ (vertical planes), the immersed Lagrangian end points $\mathcal{Z}_0, \mathcal{Z}_1$ (horizontal planes), and the images of the squares $l_0,l_1$ depicted as rectangles parallel to the page. Right: The integral of $\omega$ on the surface of the box formed by $\Lambda_0, \Lambda_1, \mathcal{Z}_0, \mathcal{Z}_1, l_0([0,1]^2)$ and $l_1([0,1]^2)$ is zero since $\omega$ is closed. The top, bottom, left and right side of the cube are Lagrangian, and so the integral of $\omega$ over them vanishes. Thus the integral of $\omega$ over the front and back sides has opposite sign when oriented with the outward normal.
  • Figure 2: The boundary Lagrangians $\Lambda_i, \Lambda_j$ (vertical planes), the immersed special Lagrangian end points $\mathcal{Z}_0, \mathcal{Z}_1$ (horizontal planes), and the image of $[0,1]\times B$ under $\beta_u$ represented as a cylinder with bounadry on $\mathcal{Z}_0$ and $\mathcal{Z}_1$. The arrows depict $\frac{d\beta_{u}}{du}$ on $\{0\}\times B$ and $\{1\} \times B$, which are respectively tangent to $\mathcal{Z}_0$ and $\mathcal{Z}_1$.
  • Figure 3: Change of base point from $(\mathcal{Z}_0,f_0)$ to $(\overline{\mathcal{Z}}_0 , \overline{f}_0)$.
  • Figure 4: Change of base point from $(\mathcal{Z}_0,f_0)$ to $(\mathcal{Z}_2,f_2)$ on $\mathcal{U}_0$, and from $(\mathcal{Z}_1,f_1)$ to $(\mathcal{Z}_2,f_2)$ on $\mathcal{U}_1$.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 65 more