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On counting special Lagrangian homology 3-spheres

Dominic Joyce

TL;DR

The paper proposes an invariant $I(\delta)$ for almost Calabi–Yau 3-folds by counting special Lagrangian (SL) homology 3-spheres in each class, weighted by a topology-dependent function $w(N)$, with $I(\delta)=\sum_{N\in S(\delta)} w(N)$. It analyzes the SL singularity landscape via explicit local models $L_0$, $L^a_t$, and $K_{\boldsymbol{\phi},A}$ to derive how $I$ should behave under deformations, proposing $w(N)=|H_1(N,\mathbb{Z})|$ to satisfy key invariance and transition identities; a partial transformation rule across hypersurfaces in $H^3(M,\mathbb{C})$ is formulated, and the framework is extended to include multiple covers and stable singularities. The work connects these counts to physics by interpreting SL 3-folds as isolated 3-branes and discusses potential roles in Mirror Symmetry, outlining conjectures that would render $I$ a robust, deformation-sensitive, yet computable invariant. The results suggest a rich interplay between calibrated geometry, topology, and string-theoretic dualities, with concrete local models guiding global deformation behavior. Overall, the paper lays a conjectural foundation for a Gromov–Witten–style invariant in the SL setting and highlights explicit topological mechanisms governing invariance and transformation of counts under deformations.

Abstract

We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class. In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau structure pass through some topologically determined hypersurfaces in the cohomology of M. As we deform the underlying almost Calabi-Yau 3-fold, the collection of special Lagrangian homology 3-spheres only change when they become singular. Thus, to determine the stability of the invariant under deformations we need know about the singular behaviour of special Lagrangian 3-folds, which is not well understood. We describe two kinds of singular behaviour of special Lagrangian 3-folds, and derive identities on the weight function w(N) for I to be unchanged or transform well under them. The weight function w(N)=|H_1(N,Z)| satisfies these identities. We conjecture that an invariant I defined with this weight is independent of the Kahler class, and changes in certain ways as the holomorphic 3-form passes through some real hypersurfaces in H^3(M,C). Finally we consider connections with String Theory. We argue that our invariant I counts isolated 3-branes, and that it should play a part in the Mirror Symmetry story for Calabi-Yau 3-folds.

On counting special Lagrangian homology 3-spheres

TL;DR

The paper proposes an invariant for almost Calabi–Yau 3-folds by counting special Lagrangian (SL) homology 3-spheres in each class, weighted by a topology-dependent function , with . It analyzes the SL singularity landscape via explicit local models , , and to derive how should behave under deformations, proposing to satisfy key invariance and transition identities; a partial transformation rule across hypersurfaces in is formulated, and the framework is extended to include multiple covers and stable singularities. The work connects these counts to physics by interpreting SL 3-folds as isolated 3-branes and discusses potential roles in Mirror Symmetry, outlining conjectures that would render a robust, deformation-sensitive, yet computable invariant. The results suggest a rich interplay between calibrated geometry, topology, and string-theoretic dualities, with concrete local models guiding global deformation behavior. Overall, the paper lays a conjectural foundation for a Gromov–Witten–style invariant in the SL setting and highlights explicit topological mechanisms governing invariance and transformation of counts under deformations.

Abstract

We attempt to define a new invariant I of (almost) Calabi-Yau 3-folds M, by counting special Lagrangian rational homology 3-spheres N in M in each 3-homology class, with a certain weight w(N) depending on the topology of N. This is motivated by the Gromov-Witten invariants of a symplectic manifold, which count the J-holomorphic curves in each 2-homology class. In order for this invariant to be interesting, it should either be unchanged by deformations of the underlying (almost) Calabi-Yau structure, or else transform according to some rigid set of rules as the periods of the almost Calabi-Yau structure pass through some topologically determined hypersurfaces in the cohomology of M. As we deform the underlying almost Calabi-Yau 3-fold, the collection of special Lagrangian homology 3-spheres only change when they become singular. Thus, to determine the stability of the invariant under deformations we need know about the singular behaviour of special Lagrangian 3-folds, which is not well understood. We describe two kinds of singular behaviour of special Lagrangian 3-folds, and derive identities on the weight function w(N) for I to be unchanged or transform well under them. The weight function w(N)=|H_1(N,Z)| satisfies these identities. We conjecture that an invariant I defined with this weight is independent of the Kahler class, and changes in certain ways as the holomorphic 3-form passes through some real hypersurfaces in H^3(M,C). Finally we consider connections with String Theory. We argue that our invariant I counts isolated 3-branes, and that it should play a part in the Mirror Symmetry story for Calabi-Yau 3-folds.

Paper Structure

This paper contains 23 sections, 15 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Special Lagrangian geometry
  3. Special Lagrangian submanifolds in $\mathbb{C}^m$
  4. Almost Calabi--Yau $m$-folds and SL $m$-folds
  5. Singularities of SL $m$-folds
  6. A model degeneration of SL 3-folds
  7. Three families of SL 3-folds in $\C^3$
  8. Using $L_0$ and the $L^a_t$ as local models
  9. Interpretation in terms of holomorphic discs
  10. Topological behaviour of these singularities
  11. SL 3-folds with $T^2$-cone singularities
  12. The case $\chi\ne 0$: desingularizing $N_0$
  13. The case $\chi\ne 0$: homology 3-spheres
  14. The case $\chi=0$: stable SL singularities
  15. Another model degeneration of SL 3-folds
  16. ...and 8 more sections

Key Result

Proposition 2.3

Let $L$ be a real $m$-dimensional submanifold of $\C^m$. Then $L$ admits an orientation making it into an SL submanifold of $\C^m$ if and only if $\om\vert_L\equiv 0$ and $\Im\Om\vert_L\equiv 0$.

Theorems & Definitions (21)

  • Proposition 2.3
  • Lemma 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Conjecture 2.12
  • Conjecture 2.13
  • Lemma 3.2
  • Corollary 3.3
  • Corollary 3.4
  • ...and 11 more