A gauge theoretic invariant of embedded surfaces in $4$-manifolds and exotic $P^2$-knots

Abstract: We give an infinite family of embeddings of $\mathbb{R} P<sup>2$ to $S<sup>4$ such that they are mutually topologically isotopic however are not smoothly isotopic to each other. Moreover, they are topologically isotopic to the standard $P<sup>2$-knot. To prove that these $P<sup>2$-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.