Abstract
We give an infinite family of embeddings of $\mathbb{R}P^2$ into $S^4$ such that they are mutually topologically isotopic but are not smoothly isotopic to each other. Moreover, they are topologically isotopic to the standard $P^2$-knot. To prove that these $P^2$-knots are not smoothly isotopic to each other, we construct a gauge theoretic invariant of embedded surfaces in $4$-manifolds using a variant of the Seiberg–Witten theory, which is called the Real Seiberg–Witten theory.
