Smooth moduli spaces of associative submanifolds

TL;DR

This paper studies the deformation spaces of associative submanifolds in G_2-manifolds, showing how perturbations and boundary conditions affect their smoothness and providing explicit examples.

Contribution

It introduces methods to achieve smooth moduli spaces via perturbations and boundary adjustments, extending McLean's and others' results on associative submanifolds.

Findings

Perturbing the G_2-structure ensures smoothness of the moduli space near Y.

A generic boundary perturbation makes the moduli space with boundary smooth.

A vanishing theorem using Bochner technique indicates conditions for smoothness.

Abstract

Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $\phi$, and $Y^3$ be an closed compact $\phi$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space $\bm_{Y,\phi}$ of the $\phi$-associative deformations of $Y$ has vanishing virtual dimension. In this paper, we perturb $\phi$ into a $G_2$-structure $\psi$ in order to ensure the smoothness of $\bm_{Y,\psi}$ near $Y$. If $Y$ is allowed to have a boundary moving in a fixed coassociative submanifold $X$, it was proved in \cite{GaWi} that the moduli space $\bm_{Y,X}$ of the associative deformations of $Y$ with boundary in $X$ has finite virtual dimension. We show here that a generic perturbation of the boundary condition $X$ into $X'$ gives the smoothness of $\bm_{Y,X'}$. In another direction, we use the Bochner technique to prove a vanishing theorem that forces $\bm_Y$ or $\bm_{Y,X}$ to be smooth near $Y$. For every case, some explicit families of examples will be given.