Positive energy representations of Sobolev diffeomorphism groups of the circle
- PDF / 628,318 Bytes
- 36 Pages / 439.37 x 666.142 pts Page_size
- 55 Downloads / 185 Views
Positive energy representations of Sobolev diffeomorphism groups of the circle Sebastiano Carpi1 · Simone Del Vecchio1 · Stefano Iovieno2 · Yoh Tanimoto1 Received: 2 July 2019 / Revised: 23 October 2020 / Accepted: 29 October 2020 © The Author(s) 2020
Abstract We show that any positive energy projective unitary representation of Diff + (S 1 ) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms Ds (S 1 ) for any real s > 3, and in particular to C k diffeomorphisms Diff k+ (S 1 ) with k ≥ 4. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on S 1 is covariant with respect to Ds (S 1 ), s > 3. Moreover every direct sum of irreducible representations of a conformal net is also Ds (S 1 )-covariant.
Sebastiano Carpi: Supported in part by ERC advanced grant 669240 QUEST “Quantum Algebraic Structures and Models” and GNAMPA-INDAM. Simone Del Vecchio: Supported by ERC advanced grant 669240 QUEST “Quantum Algebraic Structures and Models”. Yoh Tanimoto: Supported until February 2020 by Programma per giovani ricercatori, anno 2014 “Rita Levi Montalcini” of the Italian Ministry of Education, University and Research.
B
Yoh Tanimoto [email protected] Sebastiano Carpi [email protected] Simone Del Vecchio [email protected] Stefano Iovieno [email protected]
1
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
2
Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale Aldo Moro 5, 00185 Roma, Italy 0123456789().: V,-vol
12
Page 2 of 36
S. Carpi et al.
1 Introduction The group of (smooth) diffeomorphisms of a manifold has been extensively studied and there have been many interesting results concerning its algebraic and topological properties, see e.g. [31]. Among them, the group Diff + (S 1 ) of orientation preserving diffeomorphisms of the circle S 1 is of particular interest in connection with conformal field theory. In (1 + 1)-dimensional conformal field theory, the symmetry group of the chiral components is Diff + (R) and often this can be extended to Diff + (S 1 ). As this group contains spacetime translations, the relevant representations must be positive energy representations and they act on the space of local observables. The representation theory of positive energy representations has been exploited for construction and classification of a certain subclass of conformal field theories, see e.g. [24]. Non-trivial positive energy representations of Diff + (S 1 ) are necessarily projective. Any irreducible unitary positive energy representation of the Virasoro algebra extends to a projective representation of the Lie algebra Vect(S 1 ), the Lie algebra of vector fields on S 1 , and it integrates to a positive energy projective unitary representation of Diff + (S 1 ) [19,36,45]. It follows from [5, Theorem A.2],
Data Loading...
