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A spinorial approach to the construction of balanced Spin(7) manifolds Martín-Merchán, Lucía


In this talk we explain how to describe pure types of $\mathrm{Spin}(7)$ structures in terms of spinors and focus on the construction of balanced examples. An $8$-dimensional Riemannian manifold admitting a $\mathrm{Spin}(7)$ structure determined by a $4$-form $\Omega$ is spin and the structure can also be described in terms of a spinor $\eta$. Balanced $\mathrm{Spin}(7)$ structures are a pure class and are characterized by the equation $(\ast d\Omega)\wedge \Omega=0$ or, equivalently, by the condition that $\eta$ is harmonic, that is, $D \eta=0$ where $D$ is the Dirac operator. For our purposes, the description of balanced structures in terms of spinors turns out to be much simpler. Our examples are products $(N\times T,g+g_k)$, where $(N,g)$ is a $k$-dimensional nilmanifold endowed with a left-invariant metric, $(T,g_k)$ is an $(8-k)$-dimensional flat torus, and $k=5,6$. Under these assumptions, the presence of a left-invariant balanced $\mathrm{Spin}(7)$ structure on the product is equivalent to the fact that $(N,g)$ admits a left-invariant non-zero harmonic spinor. For this reason we search left-invariant metrics on $N$ that admit left-invariant harmonic spinors. The results of our investigation are a list of $5$ and $6$-dimensional nilmanifolds that verify this condition, and the description of the set of left-invariant metrics with left-invariant harmonic spinors in the particular case $k=5$.

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