Let $K$ be an imaginary quadratic field, and let $A$ be an abelian surface over $\mathbb{C}$ with complex multiplication by $K\times K$, meaning that $K\times K\subseteq End(A)\otimes\mathbb{Q}$. Then $A$ is isogeneous to $\mathbb{C}^2/\frak{a}$ where $\frak{a}=\frak{a}_1\times\frak{a}_2$ is the product of two fractional ideals of $K$ (see §3.4 of https://www.math.u-bordeaux.fr/~mbilu/Complexmultiplication.pdf). Moreover, the torsion points of $A$ are in bijection with $K^2/\frak{a}$.
The usual theory of complex multiplication tells us what is $A^\sigma$ and how the action of a Galois automorphism $\sigma\in Gal(\bar{\mathbb{Q}}/K)$ on the torsion points of $A$ can be interpreted as an analythic action on $K^2/\frak{a}$.
Is there a way to interpret also the action of the complex conjugation on $A$ as an analythic action on $K^2/\frak{a}$?