Moduli spaces of spacefilling branes in symplectic 4-manifolds

TL;DR

This paper studies the moduli space of spacefilling branes on symplectic 4-manifolds, showing smoothness and dimension for certain compact cases using Torelli theorems.

Contribution

It establishes the smoothness and computes the dimension of the moduli space of spacefilling branes on holomorphic symplectic compact Kähler 4-manifolds.

Findings

Moduli space is smooth for holomorphic symplectic compact Kähler 4-manifolds.

The dimension of the moduli space is explicitly determined.

The proof uses the local Torelli theorem for K3 surfaces and tori.

Abstract

On a symplectic manifold $(M, \omega)$, a spacefilling brane structure is a closed 2-form $F$ which determines a complex structure, with respect to which $F +i\omega$ is holomorphic symplectic. For holomorphic symplectic compact K\"ahler 4-manifolds, we show that the moduli space of spacefilling branes is smooth, and determine its dimension. The proof relies on the local Torelli theorem for K3 surfaces and tori.