Let $c,h$ and $c,\tilde{ h}$ be two admissible pairs of central charge and highest weight for ${ \rm Diff}^+(S^1)$. It is shown here that the positive energy irreducible projective unitary representations $U_{ c,h}$ and $U_{ c,\tilde{ h}}$ of the group ${ \rm Diff}^+(S^1)$ are locally equivalent. This means that for any $I\Subset S^1$ open proper interval, there exists a unitary operator $W_I$ such that $W_I U_{ c,h}(\gamma)W_I^* = U_{ c,\tilde{ h}}(\gamma)$ for all $\gamma \in { \rm Diff}^+(S^1)$ which act identically on $I^c\equiv S^1\setminus I$ (i.e. which can "displace" or "move" points only in $I$). This result extends and completes earlier ones that dealt with only certain regions of the "$c,h$-plane", and closes the gap in the full classification of superselection sectors of Virasoro nets.
Categories: math-ph math.MP math.OA math.RT