Involutions of the Moduli Space of Principal E₆-Bundles over a Compact Riemann Surface

In this paper, the fixed points of involutions on the moduli space of principal $E_6$-bundles over a compact Riemann surface X are investigated. In particular, it is proved that the combined action of a representative $\sigma$ of the outer involution of $E_6$ with the pull-back action of a surface involution $\tau$ admits fixed points if and only if a specific topological obstruction in $H^2\left(X/\tau ,{\pi }_0\left(E_6^{\sigma }\right)\right)$ vanishes. For an involution $\tau$ with $2k$ fixed points, it is proved that the fixed point set is isomorphic to the moduli space of principal H-bundles over the quotient curve $X/\tau$, where H is either $F_4$ or $PSp(8,\mathbb{C})$ and it consists of $2^{g-k+1}$ components. The complex dimensions of these components are computed, and their singular loci are determined as corresponding to H-bundles admitting non-trivial automorphisms. Furthermore, it is checked that the stability of fixed $E_6$-bundles implies the stability of their corresponding H-bundles over $X/\tau$, and the behavior of characteristic classes is discussed under this correspondence. Finally, as an application of the above results, it is proved that the fixed points correspond to octonionic structures on $X/\tau$, and an explicit construction of these octonionic structures is provided.