A family of simple non-weight modules over the twisted $N=2$ superconformal algebra

We construct a class of non-weight modules over the twisted $N=2$ superconformal algebra $T$. Let $mathfrak{h}=C L_0oplusC G_0$ be the Cartan subalgebra of $T$, and let $mathfrak{t}=C L_0$ be the Cartan subalgebra of even part $T_{bar 0}$. These modules over $T$ when restricted to the $mathfrak{h}$ are free of rank $1$ or when restricted to the $mathfrak{t}$ are free of rank $2$. We provide the sufficient and necessary conditions for those modules being simple, as well as giving the sufficient and necessary conditions for two $T$-modules being isomorphic. We also compute the action of an automorphism on them. Moreover, based on the weighting functor introduced in cite{N2}, a class of intermediate series modules $A_sigma$ are obtained. As a byproduct, we give a sufficient condition for two $T$-modules are not isomorphic.