(\[IONS\]), (\[IOR\]) we can find the expressions for the asymptotic vacuum states. As they have infinite norm we should think of them, as we usually do for boundary states, as vectors in the dual space ${\cal H}_{0}^{*}$.

The details are as follows. The normalisation factors ${\cal N}(\nu)$, $\tilde {\cal N}(\nu)$ each goes to zero due to the Casimir energy factors \[Casimir\] in (\[N1\]), (\[N2\]). This reflects the fact that the conformal boundary states to which the massive vacuum approaches have infinite norm. Stripping off this vanishing factor we obtain the following asymptotic representations of the vacuum sectors of ${\cal H}_{\infty}$ in ${\cal H}_{0}$ |0\_[NS]{}\^[+]{}\_[m+]{} D\_[NS]{}|0\_[NS]{} = \_[n=0]{}\^(ia\^\_[n+1/2]{}|a\^\_[n+1/2]{})|0 , |0\_[R]{}\^[+]{}\_[m+]{} D\_[R]{}|0\_[R]{} = 2\^[1/4]{} \_[n=1]{}\^(ia\^\_[n]{}|a\^\_[n]{})| . These states can be decomposed into conformal Ishibashi states[^4] [@Cardy] using the identification of the energy primary state \[eps\_state\] |= ia\^\_[1/2]{}|a\_[1/2]{}\^|0 . We obtain |0\_[NS]{}\^[+]{} = |0+ | , |0\_[R]{}\^[+]{} = 2\^[1/4]{} | . We observe that these states already have correct relative normalisations to be combined into conformal boundary states. By multiplying them by $1/\sqrt{2}$ and choosing the appropriate relative phases between the NS- and R- sector components we combine them into two Cardy states which represent local boundary conditions corresponding to fixed boundary spin: 