On the automorphism group of a symplectic half-flat 6-manifold
Fabio Podestà, Alberto Raffero
TL;DR
The paper analyzes automorphism groups of compact and noncompact 6-manifolds endowed with symplectic half-flat SU(3)-structures $(\omega,\psi)$. It proves that the identity component of the automorphism group is abelian with $\dim\mathfrak{g}\le \min\{5,b_1(M)\}$, and that the infinitesimal action embeds into the space of harmonic 1-forms via $X\mapsto i_X\omega$, constraining isotropy and implying that a cohomogeneity-one action can occur only on the torus $\mathbb{T}^6$ in the compact strict setting. The authors also show there are no compact homogeneous examples with invariant strict structures and construct new complete noncompact cohomogeneity-one examples on $T\mathbb{S}^3$ with an ${\mathrm{SO}}(4)$-action, including a Calabi–Yau (Stenzel) case and numerous non-homogeneous models. These results highlight a dichotomy between compact and noncompact cases and provide explicit, highly symmetric geometries with potential applications in string theory.
Abstract
We prove that the automorphism group of a compact 6-manifold $M$ endowed with a symplectic half-flat SU(3)-structure has abelian Lie algebra with dimension bounded by min$\{5,b_1(M)\}$. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new complete examples on $T\mathbb{S}^3$ which are invariant under a cohomogeneity one action of SO(4).