  We construct the complete set of boundary states of two-dimensional fermionic
CFTs using that of the bosonic counterpart. We see that there are two groups of
boundary conditions, which contributes to the open-string partition function by
characters with integer coefficients, or with $\sqrt{2}$ times integer
coefficients. We argue that, using the argument of [JHEP 09 (2020) 018], this
$\sqrt{2}$ indicates a single unpaired Majorana zero mode, and that these two
groups of boundary conditions are mutually incompatible. We end the paper by
mentioning a possible interpretation of the result in terms of the entanglement
entropy.
