Local equivalence of representations of Diff${}^{+}(S^1)$ corresponding to different highest weights

Abstract

Let $c,h$ and $c,\tilde{h}$ be two admissible pairs of central charge and highest weight for ${\mathrm{Diff}}^{+}(S^1)$. It is shown here that the positive energy irreducible projective unitary representations $U_{c,h}$ and $U_{c,\tilde{h}}$ of the group ${\mathrm{Diff}}^{+}(S^1)$ are locally equivalent. This means that for any $I⋐S^1$ open proper interval, there exists a unitary operator $W_I$ such that $W_I{U}_{c,h}(\gamma )W_I^{\ast }=U_{c,\tilde{h}}(\gamma )$ for all $\gamma \in {\mathrm{Diff}}^{+}(S^1)$ 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.

15 pages, no figures

References in corpus (2)

• Nuclearity and Thermal States in Conformal Field Theory
• Restricting positive energy representations of ${\mathrm{Diff}}^{+}(S^1)$ to the stabilizer of $n$ points

Cited by in corpus (6)

• Representation theory in chiral conformal field theory: from fields to observables
• Local energy bounds and strong locality in chiral CFT
• Unitary representations of the ${\mathcal{W}}_3$-algebra with $c\ge 2$
• Positive energy representations of Sobolev diffeomorphism groups of the circle
• Rotational KMS states and type I conformal nets
• Conformal nets from minimal W-algebras