Moduli spaces of (co)closed $\mathrm{G}_2$-structures on nilmanifolds

TL;DR

This paper investigates the moduli spaces of left-invariant closed and coclosed G2-structures on 7-dimensional nilmanifolds, revealing their dimensions and properties, and contrasting automorphism groups between the two types.

Contribution

It provides explicit computations of moduli space dimensions and demonstrates that automorphism groups of coclosed G2-structures can be non-abelian, unlike the closed case.

Findings

Moduli space dimensions are computed and shown to be unrelated to the third Betti number.

Automorphism groups of coclosed G2-structures can be non-abelian.

Contrasts the properties of closed and coclosed G2-structures on nilmanifolds.

Abstract

We compute the dimensions of some moduli spaces of left-invariant closed and coclosed $\mathrm{G}_2$-structures on 7-dimensional nilmanifolds, showing that they are not related to the third Betti number. We also prove that, in contrast to the case of closed $\mathrm{G}_2$-structures, the group of automorphisms of a coclosed $\mathrm{G}_2$-structure is not necessarily abelian.