topological conformal algebra for any $d$.[^4][By comparing the conformal weight and the statistics of the each component fields, we can see that our algebra in  and  are not the same as the twisted $N=4$ superconformal algebras analyzed by Nojiri \[\].]{} Our observation thus suggests the topological nature of the $N=2$ super-Liouville theory (or the $N=2$ fermionic string theory) as the quantum theory.

We note that the first relation in  implies , and the second relation in  implies the BRST invariance of the BRST supercurrent, therefore the BRST charge is nilpotent.

The anomaly-free property of the operator algebra in  and  might be understood from the absence of the ghost number anomaly in $N=2$ theory: In our construction, which is analogous to the one in \[,\], the correction of the ghost number current due to the Liouville mode is determined from the ghost number anomaly. In $N=2$ case, since we have no ghost number anomaly in , the form of the effective ghost number supercurrent in  has no correction due to the Liouville mode. From experiences on the $N=0$ and $N=1$ (super-)Liouville theories \[,,\], we know the following relation for the [ *effective*]{} currents: $$j_B=-\left\{Q_B,j_{\rm gh}\right\},
\eqn\fourtwelve$$ and we can see that 