Examples of real stable bundles on K3 surfaces

Abstract: Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-S\'a Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ with Picard rank $2$ using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples.