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Moduli of special Lagrangians with boundary, I: Unobstructed Deformations

Vasanth Pidaparthy

TL;DR

This work extends McLean's deformation theory to special Lagrangian submanifolds with boundary (SLb) constrained along prescribed Lagrangian boundary components in a Calabi–Yau manifold. The authors encode infinitesimal SLb deformations as 1-forms on the boundary Lagrangian that vanish on the boundary and use a boundary-compatible vector-bundle framework to formulate an elliptic special Lagrangian operator $F$, whose linearization $d_0F$ is surjective with kernel given by harmonic Dirichlet fields. Through an implicit function theorem, they show the SLb moduli space is a finite-dimensional smooth manifold, with tangent space isomorphic to the space of closed, co-closed 1-forms on $L$ vanishing on $\partial L$, hence of dimension $\dim H^1(L,\partial L;\mathbb{R})$. The results unify and generalize prior cylinder and boundary-boundary boundary-condition cases, and extend to almost Calabi–Yau settings, providing a robust framework for boundary Lagrangian deformation theory with potential applications in mirror symmetry and geodesic boundary problems.

Abstract

This article studies the deformation problem for compact special Lagrangians with boundary in a Calabi--Yau manifold, with each boundary component constrained along a given Lagrangian submanifold. The tangent vectors generating such deformations are identified with harmonic 1-forms vanishing on the boundary of the special Lagrangian, and the deformation generated by any such tangent vector is unobstructed. Consequently, the moduli space of special Lagrangians with boundary is a smooth manifold whose dimension equals the dimension of the first relative cohomology of the special Lagrangian.

Moduli of special Lagrangians with boundary, I: Unobstructed Deformations

TL;DR

This work extends McLean's deformation theory to special Lagrangian submanifolds with boundary (SLb) constrained along prescribed Lagrangian boundary components in a Calabi–Yau manifold. The authors encode infinitesimal SLb deformations as 1-forms on the boundary Lagrangian that vanish on the boundary and use a boundary-compatible vector-bundle framework to formulate an elliptic special Lagrangian operator , whose linearization is surjective with kernel given by harmonic Dirichlet fields. Through an implicit function theorem, they show the SLb moduli space is a finite-dimensional smooth manifold, with tangent space isomorphic to the space of closed, co-closed 1-forms on vanishing on , hence of dimension . The results unify and generalize prior cylinder and boundary-boundary boundary-condition cases, and extend to almost Calabi–Yau settings, providing a robust framework for boundary Lagrangian deformation theory with potential applications in mirror symmetry and geodesic boundary problems.

Abstract

This article studies the deformation problem for compact special Lagrangians with boundary in a Calabi--Yau manifold, with each boundary component constrained along a given Lagrangian submanifold. The tangent vectors generating such deformations are identified with harmonic 1-forms vanishing on the boundary of the special Lagrangian, and the deformation generated by any such tangent vector is unobstructed. Consequently, the moduli space of special Lagrangians with boundary is a smooth manifold whose dimension equals the dimension of the first relative cohomology of the special Lagrangian.

Paper Structure

This paper contains 14 sections, 19 theorems, 58 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Special Lagrangians and Lagrangian boundary conditions
  5. Deformations with Lagrangian boundary conditions
  6. Smooth paths and tangents in the moduli space of Lagrangians with boundary
  7. Hodge decomposition for manifolds with boundary
  8. Moduli space of special Lagrangians
  9. Hodge dual of tangent 1-form
  10. Proof of Proposition \ref{['prop:HWB-tangentspacetospeciallagrangianswithboundary-1']}
  11. Special Lagrangian operator
  12. Proof of Theorem \ref{['thm:HWB-tangentspacetospeciallagrangianswithboundary-2']}
  13. Generalizations
  14. Proof of Corollary \ref{['cor:HWB-boundary-lagrangian-deformation-space']}(ii)

Key Result

Theorem 1.4

Mclean1998 Let $(X, \omega, J, \Omega)$ be a Calabi-Yau manifold and $L \subset X$ a compact special Lagrangian submanifold without boundary. Then the set of special Lagrangian submanifolds sufficiently close to $L$ is a finite-dimensional manifold parametrized by the set of harmonic 1-forms on $L$.

Figures (2)

  • Figure 1: Consider the case $d=2$ boundary components $\Lambda_0$ and $\Lambda_1$. The flow of the normal vector field on the left does not keep the boundary of $L$ constrained along $\Lambda_0$ and $\Lambda_1$, while the vector field on the right preserves the boundary constrains $\Lambda_0, \Lambda_1$ (Definition \ref{['def:HWB-boundary-conditions-on-immersions-and-lagrangians']}).
  • Figure 2: $f_1:L \rightarrow X$ is an immersion in the neighborhood $\mathcal{I} \subset C^\infty(L,X; \Lambda_1,\dots, \Lambda_d)$ of $f$, and $\widetilde{f_1}$ is a lifting of this immersion.

Theorems & Definitions (61)

  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Lemma 3.4
  • proof
  • Definition 3.5
  • ...and 51 more