Moduli spaces of (co)closed $\mathrm{G}_2$-structures on nilmanifolds
Giovanni Bazzoni, Alejandro Gil-García
TL;DR
The paper studies moduli spaces of left-invariant closed and coclosed G2-structures on 7 dimensional nilmanifolds by reducing questions to Lie algebra data. It defines finite dimensional moduli spaces M^c(g) and M^cc(g) via V^c(g) and V^cc(g) and shows their embedding into the full moduli on nilmanifolds, providing dimension counts dim V^c(g) = dim Z^3(g^*) and dim V^cc(g) = dim Z^4(g^*), together with a formula dim M^c(g) = dim V^c(g) - (dim Aut(g) - dim Aut(g)_phi) and its coclosed counterpart. The computations across a classified set of 7-dimensional NLAs show no universal relation between moduli dimensions and the third Betti number b3, and reveal distinct behavior with respect to nilpotency steps and decomposability. They also demonstrate that the automorphism group of a coclosed G2-structure need not be abelian, in contrast to the closed case, highlighting richer symmetry structures in the coclosed setting.
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.