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We give an infinite family of embeddings of $\mathbb{R} P^2$ to $S^4$ suchthat they are mutually topologically isotopic however are not smoothly isotopicto 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, weconstruct a gauge theoretic invariant of embedded surfaces in $4$-manifoldsusing a variant of the Seiberg--Witten theory, which is called the RealSeiberg--Witten theory.

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