Positive energy representations of Sobolev diffeomorphism groups of the circle

We show that any positive energy projective representation of Diff(S^1) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms D^s(S^1) with 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 D^s(S^1), s>3. Moreover every direct sum of irreducible representations of a conformal net is also D^s(S^1)-covariant.


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. [Mil84]. 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. [KL04].
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 ) [GW85,TL99]. It follows from [Car04,Theorem A.2], see also [CKLW18,Section 3.2], that all irreducible positive energy unitary projective representations of Diff + (S 1 ) arise in this way. Accordingly they are completely classified by the central charge c and the lowest conformal energy h [KR87]. Related results including reducible representations have been recently obtained in [NS15,Zel17].
It has been found that these representations of Vect(S 1 ) extend to certain non-smooth vector fields as linear maps [CW05]. Apart from that this fact had many applications (e.g. the uniqueness of conformal covariance in conformal nets [CW05], positivity of energy in DHR sectors [Wei06], split property in conformal nets [MTW18] and covariance of soliton representations [Hen17,DIT18]), it leads naturally to the question whether the group representations extend to non-smooth diffeomorphisms. In contrast to the wide range of results and applications concerning the algebraic, analytic and topological properties of the group Diff k + (M ) of C k diffeomorphisms and D s (M ) of Sobolev class diffeomorphisms (see e.g. [EM70,Mis97,Ban97,KW09,Fig10]) and on and some results on (true) representations [KL02,AM06,Kuz07,Mal08], there appears to be only few results in the literature on positive energy representations of these groups. Indeed, D s (M ) is an infinite-dimensional manifold modelled on the space H s (M ) of H s -vector fields, which is not a Lie algebra with the usual Lie bracket for Vect ∞ (M ). This makes the study of representations of D s (M ) rather subtle.
In this paper, we show that any positive energy (projective) representations of the diffeomorphism group extends to D s (S 1 ) for s > 3, by considering its action on vector fields, and therefore, by exploiting the representation theory of the Virasoro algebra. We also show that these representations can be locally made into multiplier representation by fixing the phase. This allows us to take the direct sum and it turns out that conformal nets are covariant with respect to this extended action.
For some special representations appearing in Fock space, another extension has been done to C 3 -diffeomorphisms [Vro13]. The arguments depend on realizing these representation in some specific conformal field theory, and it is open whether the results are valid for general central charge c. In contrast, by our argument representations extend to D s (S 1 ) for s > 3. There is no inclusion relations between these groups: one can only say Diff 3 + (S 1 ) ⊂ D 3 (S 1 ), while D s (S 1 ) ⊂ Diff 3 + (S 1 ) if s > 3 + 1 2 = 7 2 . Our proof follows in part the strategy in [GW85] for the integrability of the representations of the Virasoro algebra. The extension to non-smooth diffeomorphisms then follows from the above mentioned extension to non-smooth vector fields of the corresponding projective representation of Vect(S 1 ) given in [CW05]. Actually, our argument can be used to give a simpler proof of the results in [GW85], see Remark 3.8. This paper is organized as follows. In Section 2, we recall the relevant groups and algebras, their topologies and representations. In Section 3, we first extend the irreducible projective representations of Diff + (S 1 ) to D s (S 1 ) with s > 3. Then we lift them locally to multiplier representations, and show that the direct sum can make sense as projective representations. Section 4 demonstrates that two-dimensional chiral conformal field theories described by conformal nets of von Neumann algebras have this extended symmetry of D s (S 1 ). We summarize possible further continuation of this work in Section 5.

Diff + (S 1 ) and the Virasoro algebra
The diffeomorphism group. Let us denote by Diff + (S 1 ) the group of orientation preserving, smooth diffeomorphisms of the circle S 1 := {z ∈ C : |z| = 1} and Vect(S 1 ) denote the set of smooth vector fields on S 1 . Diff + (S 1 ) is an infinite dimensional Lie group whose Lie algebra is identified with the real topological vector space Vect(S 1 ) of smooth vector fields on S 1 with C ∞ topology [Mil84]. In the following we identify Vect(S 1 ) with C ∞ (S 1 , R) and for f ∈ C ∞ (S 1 , R) we denote by f ′ the derivative of f with respect to the angle θ, We consider a diffeomorphism γ ∈ Diff + (S 1 ) as a map from S 1 in S 1 ⊂ C. With this convention, its action on f ∈ Vect(S 1 ) is We denote by Diff k + (S 1 ) the group of C k -diffeomorphisms of S 1 . Note that this is not a Lie group, and indeed, the corresponding linear space Vect k (S 1 ) of C k -vector fields is not closed under the natural Lie bracket (see below).
If γ ∈ Diff + (S 1 ), its image under the covering map is in the following denoted byγ ∈ Diff + (S 1 ), whereγ(e iθ ) = e iγ(θ) . Conversely, if γ ∈ Diff + (S 1 ), there is an elementγ ∈ Diff + (S 1 ) whose image under the covering map is γ. Such aγ is unique up to 2π and called a lift of γ. The group Diff + (S 1 ) admits the Bott-Virasoro cocycle B : Diff + (S 1 ) × Diff + (S 1 ) → R (see e.g. [FH05]). The Bott-Virasoro group is then defined as the group with elements (γ, t) ∈ Diff + (S 1 ) × R and with multiplication Note that, given a true (not projective) unitary irreducible representation V of the universal covering of the Bott-Virasoro group, one can obtain a unitary multiplier representation 2 V (γ) := V (γ, 0) of Diff + (S 1 ) (with respect to the Bott-Virasoro cocycle B). Then the map where c ∈ R by irreducibility.
The Lie algebra. The space Vect(S 1 ) is endowed with the Lie algebra structure with the Lie bracket given by As a Lie algebra, Vect(S 1 ) admits the Gelfand-Fuchs two-cocycle The Virasoro algebra Vir is the central extension of the complexification of the algebra generated by the trigonometric polynomials in Vect(S 1 ) defined by the two-cocycle ω. It can be explicitly described as the complex Lie algebra generated by L n , n ∈ Z, and the central element Consider a representation π : Vir → End(V ) of Vir on a complex vector space V endowed with a scalar product ·, · . We call π a unitary positive energy representation if the following hold 1. Unitarity: v, π(L n )w = π(L −n )v, w for every v, w ∈ V and n ∈ Z; 2. Positivity of the energy: , m = 3, 4, · · · , p = 1, 2, · · · , m − 1, q = 1, 2, · · · , p, (discrete series representation) [KR87] [DMS97]. In this case the representation space V is denoted by H fin (c, h). We denote by H(c, h) the Hilbert space completion of the vector space H fin (c, h) associated with the unique irreducible unitary positive energy representation of Vir with central charge c and lowest weight h.
In these representations, the conformal Hamiltonian π(L 0 ) is diagonalized, and on the linear span of its eigenvectors H fin (c, h) (the space of finite energy vectors), the Virasoro algebra acts algebraically as unbounded operators.
The stress-energy tensor. Let H(c, h) as above and, with abuse of notation, we denote by L n the elements of Vir represented in H(c, h). For a smooth complex-valued function f on S 1 with finitely many non-zero Fourier coefficients, the (chiral) stress-energy tensor associated with f is the operator by the linear energy bounds, yielding a self-adjoint unbounded operator T (f ). Moreover it can be extended to a particular class of non-smooth functions [CW05], retaining its self-adjointness. This fact will be used in this article and will be thus resumed in some detail in Section 2.2. It is a crucial fact that the irreducible representations H(c, h) of Vir integrate to irreducible unitary strongly continuous representations of the universal covering of the Bott-Virasoro group [FH05]. In other words, denoting by q the quotient map q : U (H(c, h)) → U (H(c, h))/C (we denote by U (K) the group of unitary operators on K), there is an irreducible, unitary, strongly continuous multiplier representation U of Diff + (S 1 ), the universal covering of Diff + (S 1 ), such that q(U (Exp(f ))) = q(e iT (f ) ) for all f ∈ Vect(S 1 ), where Exp is the Lie-theoretic exponential map of Diff + (S 1 ) (see [Mil84]). For the stress-energy tensor T , we have the following covariance [FH05, Proposition 5.1, Proposition 3.1].
Proposition 2.1. The stress-energy tensor T on H(c, h) transforms according to hold for arbitrary f, g ∈ C ∞ (S 1 ), on vectors ψ ∈ H fin (c, h). Here and ω(·, ·) are related by  (2) T (f ) * is an extension of the operator T (f ) + := n∈Z L nfn (this is again understood as an operator on the domain H fin (c, h)).
(3) T (f ) is closable and T (f ) = (T (f ) + ) * , where T (f ) and T (f ) + are considered as operators on the domain H fin (c, h). In particular, iff n =f −n for all n ∈ Z (i.e. if f is a real-valued function), then T (f ) is essentially self-adjoint on H fin (c, h).
(4) If f is real, then for every ξ ∈ D(L 0 ) we have the following energy bounds where r is a positive constant. Consequently, D(L 0 ) ⊂ D(T (f )).
(5) If {f n } (n ∈ N) is a sequence 3 of continuous real functions on S 1 of finite · 3 2 norm and f − f n 3 2 converges to 0 as n tends to ∞, then in the strong resolvent sense.
It has been also shown that the class S Proposition 2.3. If a real-valued function f on the circle is piecewise smooth and once continuously differentiable on the whole S 1 , then f ∈ S 3 2 (S 1 , R).

Groups of diffeomorphisms of Sobolev class H q
We introduce (see [EK14, Section 2] and [EK14, Definition 2.2], respectively) whereγ is a lift of γ to R. Actually, in literature there are various definitions of these Sobolev spaces. Although it is well-known that they coincide, for the convenience of the reader we recall them and show their equivalence in Appendix. From these definitions, it is immediate that Diff k + (S 1 ) is continuously embedded in D k (S 1 ). Conversely, by the Sobolev-Morrey embedding [IKT13, Proposition 2.2], it holds that D s ֒→ Diff k + (S 1 ) if s > k + 1 2 . The first statement of the following is a straightforward adaptation of [IKT13, Lemma 2.3]. One can also find various different elementary proofs, for example [tim, Smy]. The second statement is an adaptation of [IKT13, Lemma B.4].
The following is a special case of [IKT13, Theorem B.2] and an analogue of [IKT13, Proposition B.7], see also Appendix. According to [Kol13, P.12], Lemma 2.5(a) for integer s has been first established in [Ebi68].
By applying these results, we obtain the following Lemma 2.6. We have the following.
Proof. (a) is obtained from the following inequality for any ǫ > 0.

Projective and multiplier representations
A strongly continuous unitary projective representation of a topological group G is a pair (U, H) where H is a Hilbert space and U is a continuous group homomorphism from G to U (H)/T, where U (H) is equipped with the strong operator topology and U (H)/T with the quotient topology by the quotient map q. Namely, the subbasis elements which contain q(u) are {U q(u),ξ,ε } ξ∈H,ε>0 , where Therefore, it is clear that a net {q(u λ )} has limit q(u) if and only if for each ξ ∈ H there is z ξ,λ ,ẑ ξ,λ ∈ T such that z ξ,λ u λ ξ −ẑ ξ,λ uξ → 0 if and only if there is z ξ,λ ∈ T such that 4 z ξ,λ u λ ξ → uξ. Actually, z ξ,λ does not depend on ξ (because, if z ξ,λ u λ η were not convergent for η ⊥ ξ, z ξ,λ u λ (ξ + η) would not be convergent in H/T, hence convergence holds for any η), hence q(u λ ) is convergent if and only if there is a net z λ ∈ T such that z λ u λ is convergent in the strong operator topology.
The above continuity is equivalent to the following, see [Bar54]: whenever g λ → g, it holds for any x ∈ B(H) that Ad U (g λ )(x) → Ad U (g)(x) (note that Ad U (g) is well-defined for U (g) ∈ U (H)/T). We show it here for the convenience of the reader. It is straightforward that if U (g λ ) → U (g) in U (H)/T, then one can fix phases of U (g λ ) and U (g) so that U (g λ ) → U (g) in the strong operator topology (with a slight abuse of notation) by the previous paragraph, hence for any x ∈ B(H), and fix the phases of U (g λ ), U (g). For any one-dimensional projection p ξ to a unit vector ξ, we have Ad U (g λ )(p ξ ) → Ad U (g)(p ξ ), and the latter is again a one-dimensional projection to, say, Cξ g , and we may assume Then ξ g , U ′ (g λ )ξ is positive and tends to 1, implying that U ′ (g λ )ξ → ξ g . Since this holds for an arbitrary ξ, it yields the convergence in U (H)/T again by the previous paragraph.
We can consider U (g) as an operator acting on H determined up to a phase factor. Two projective unitary representations (U 1 , H 1 ) and (U 2 , H 2 ) are said to be equivalent if exists an unitary W : H 1 → H 2 such that W U 1 (g) = U 2 (g)W for every g ∈ G up to a phase factor.
A unitary multiplier representation of G is a pair (U, H) were U : G → U (H) is a map such that U (g 1 )U (g 2 ) = ω(g 1 , g 2 )U (g 1 g 2 ) and ω : G × G → T is a map which satisfies the equality 3 Extension of the Diff + (S 1 ) representations to Sobolev diffeomorphisms

Irreducible case
Our purpose of this section is to extend the (positive energy projective) representation U on H(c, h) of Diff + (S 1 ) to D s (S 1 ) with s > 3. In the following s > 3 will be always assumed. An element γ ∈ D s (S 1 ) acts on f ∈ Vect(S 1 ) via (2.1). If T is the energy-momentum operator associated with a positive energy unitary representation of the Virasoro algebra Vir with central charge c and lowest weight h, we define a new class of operators where f ∈ Vect(S 1 ) and β(γ, f ) = c 24π S 1 {γ, z}izf (z)dz, which makes sense for γ ∈ D s (S 1 ) by Lemma 2.6 and Proposition 2.2(1). The fact that γ * f is in S 3 2 (S 1 , R) ensures that T (γ * f ) is an essentially self-adjoint operator on H fin (c, h) and so is T γ (f ) by Proposition 2.2(3). We denote its closure by the same symbol T γ (f ), so long as no confusion arises.
Note that, if γ ∈ Diff + (S 1 ), then we have ) and by Proposition 2.1, (3.1) holds on D(L 0 ), and the both operators are essentially self-adjoint there, hence they must coincide. As they are unitarily implemented, the energy bound holds as well: where L γ 0 := T γ (1). We define for γ 1 , γ 2 ∈ D s (S 1 ) Proof. Using the properties of the Schwarzian derivative [OT05] where we used the change of variables e iϕ = γ 2 (e iθ ), hence e iθ dθ dϕ dγ 2 where L γ 0 := T γ (1) and here we denote by 1 the constant function with the value 1.
Proof. By Lemma A.4 we can take a sequence {γ n } in Diff + (S 1 ) convergent to γ in the topology of D s (S 1 ). We observe that 1 = lim n γ n * (γ −1 * (1)) in the topology of S 3 2 (S 1 ). For ξ ∈ D(L 0 ) we know from Proposition 2.2(5) and (3.2) that Recall that we know that D(L 0 ) ⊂ D(L γ 0 ) from Proposition 2.2(4) and L γ 0 is essentially selfadjoint on D(L 0 ). From the above inequality, we infer that any sequence ξ n ∈ D(L 0 ) converging to ξ ∈ D(L γ 0 ) in the graph norm of L γ 0 is also convergent in the graph norm of L 0 , and therefore, we have D(L γ 0 ) = D(L 0 ).
Proposition 3.3 (energy bounds for T γ ). Let γ ∈ D s (S 1 ). Then Proof. Let {γ n } a sequence of elements in Diff + (S 1 ) converging to γ ∈ D s (S 1 ) as in Lemma A.4. By Proposition 2.2(5) and (3.2), which is the desired inequality. Proof. We are going to prove the Virasoro relations on C ∞ (L γ 0 ). For this purpose, we have to take under control the action of various exponentiated operators.
Computations on D(L 0 ). We start by noting that e iT γ (g) D(L 0 ) ⊂ D(L 0 ). Indeed, using [FH05, Proposition 3.1] we have, for ξ ∈ D(L 0 ) and γ n ∈ Diff + (S 1 ) as in Lemma A.4, and the right-hand side converges as n → ∞ by Proposition 2.2(5). Therefore, since both e iT γn (g) ξ and L 0 e iT γn (g) ξ are convergent, it follows that e iT γ (g) ξ ∈ D(L 0 ) and For vectors ξ ∈ D(L 0 ) and γ n ∈ Diff + (S 1 ), by Proposition 2.1 we have the operator equality and we saw above that for ξ ∈ D(L 0 ) and γ n ∈ Diff + (S 1 ), it holds that e −iT γn (g) ξ ∈ D(L 0 ) ⊂ D(T γn (f )), therefore, we have We apply to the operator equality the function where χ is the characteristic function of the interval (−k, k) ⊂ R. By bounded functional calculus, we obtain for any ξ ∈ D(L 0 ) and the right-hand side tends to e iT γ (g) h k (T γ (f ))e −iT γ (g) ξ as n → ∞, because we have convergence of T γn (f ) to T γ (f ) and T γn (g) to T γ (g) in the strong resolvent sense, and their bounded functional calculus e iT γn (g) , h k (T γn (f )) converge to e iT γ (g) , h k (T γn (f )), respectively. On the other hand, the left-hand side of (3.3) can be rewritten as and this converges to as n → ∞, again by the convergence of {T γn (Exp(g) * (f ))} in the strong resolvent sense and bounded functional calculus with h k . Altogether, we know that the following equality holds: By taking the limit for k → ∞, we get for every ξ ∈ D(L 0 ) Computations on C ∞ (L γ 0 ). The right-hand side of (3.5) is differentiable with respect to t when ξ ∈ D(L 0 ) since for the right hand side we get The first term converges to −iT γ (f )L 0 ξ. Indeed, by Proposition 3.3, Since ξ ∈ C ∞ (L γ 0 ), by Stone's theorem [RS80, Theorem VIII.7(c)] the above converges to 0 as t → 0. Thus the limit exists also for the second term of (3.6), and by applying Stone's theorem The Virasoro relations. Finally we show that the stress-energy tensor T γ indeed yields a representation of Vect(S 1 ).
Let us see the right-hand side of (3.9) term by term. As for the first term, we have (3.10) The first term of (3.10) goes to 0 by Stone's theorem [RS80, Theorem VIII.7(c)]. The second term can be treated by (3.4) and (3.7) as follows: each term can be seen to converge to 0: the first term is done by noting that L γ 0 = T γ (1), continuity of T γ (Proposition 3.3), [g, 1] = g ′ and unitarity of e −itT γ (g) . The second term vanishes by using Stone's theorem. The last term also converges to zero by (2.2) and using the fact that ω(g, 1) = 0. To summarize, the first term of the right-hand side of (3.9) tends to −iT γ (f )T γ (g).
The second term of (3.9) is equal to iT γ (g)T γ (f ). Indeed, since C ∞ (L γ 0 ) is invariant under the action of T γ (f ), this follows by Stone's theorem.
Altogether, we obtained the equality , which is the Virasoro commutation relation.
Note that until here we have only used that T is a positive energy representation of the Virasoro algebra with the central charge c with diagonalizable L 0 , but not irreducibility. Therefore, one can iterate our construction for another element in D s (S 1 ). In particular, by taking γ −1 , we obtain by Proposition 3.1 We claim that the new representation T γ is irreducible and has the same lowest weight h. Indeed, by (3.11), one can approximate T (f ) by T γ (γ −1 n * f ) + β(γ, (γ −1 n ) * (f )) in the strong resolvent sense, where {γ n } ⊂ Diff + (S 1 ) and γ n → γ in the topology of D s (S 1 ). As {e iT (f ) : f ∈ Vect(S 1 )} generates B (H(c, h)), so does {e iT γ (f ) : f ∈ Vect(S 1 )}, and this shows that T γ is a irreducible representation of the Virasoro algebra. Furthermore, the new conformal Hamiltonian L γ 0 = T γ (1) has spectrum which is a subset of the spectrum of the old conformal Hamiltonian L 0 since it is obtained as a limit in the strong resolvent sense of {Ad U (γ n )(L 0 )} with the same spectrum [RS80, Theorem VIII.24(a)]. Again by iteration, we have therefore, all these sets must coincide. In particular, h is the lowest eigenvalue of L γ 0 .
As T and T γ are equivalent as irreducible representations of Vect(S 1 ) and thus of the Virasoro algebra, there is an intertwiner U (γ), defined up to a scalar: U (γ)T (f ) = T γ (f )U (γ).
Proof. We know that for γ 1 , γ 2 ∈ D s (S 1 ) hold for every f ∈ Vect(S 1 ). So Consequently by the computations of Proposition 3.1 therefore, U (γ 1 • γ 2 ) = U (γ 1 )U (γ 2 ) up to a phase because we are dealing with irreducible representations of the Virasoro algebra. Proof. The only thing that remains to be proven is continuity, namely that the action α : D s (S 1 ) → Aut(B (H(c, h))), γ → Ad U (γ) is pointwise continuous in the strong operator topology of B (H(c, h)).
Let {γ n } ⊂ Diff + (S 1 ), γ ∈ D s (S 1 ) with γ n → γ in the topology of D s (S 1 ). Then where the limit is meant in the strong topology. By taking f = 1, we obtain the convergence of L γn 0 to L γ 0 in the strong resolvent sense. As they are in the (c, h)-representation of the Virasoro algebra, the lowest eigenprojections E 0 , E γ 0 are one-dimensional, and it holds that Let Ω, Ω γ be the lowest eigenvectors. By fixing the scalars, we may assume that Ω γn := U (γ n )Ω → Ω γ , see the arguments of Section 2.4.
With this U (γ n ) with fixed phase, the sequence U (γ n )e iT (f 1 ) · · · e iT (f k ) Ω = e iT γn (f 1 ) · · · e iT γn (f k ) Ω γn is convergent to e iT γ (f 1 ) · · · e iT γ (f k ) Ω γ , because all the operators e iT γn (f 1 ) , · · · , e iT γn (f k ) are uniformly bounded and convergent in the strong operator topology. Since vectors of the form e iT (f 1 ) · · · e iT (f k ) Ω span a dense subspace of the whole Hilbert space H(c, h), together with the uniform boundedness of U (γ n ), we obtain the convergence of U (γ n ) to U (γ) in the strong operator topology. The claimed continuity is follows from this, because for any x ∈ B(H), Ad U (γ n )(x) is convergent in the strong operator topology, again because U (γ n ) is uniformly bounded.
Corollary 3.7. Let U (c,h) be the irreducible unitary projective representation of Diff + (S 1 ) with central charge c and lowest weight h. U (c,h) extends to a strongly continuous irreducible unitary projective representation of Diff k + (S 1 ) with k ≥ 4. Proof. This is an immediate corollary of the continuous embedding Diff k + (S 1 ) ֒→ D s (S 1 ), s ≤ k.
Remark 3.8. Our argument for the construction of projective representations of D s (S 1 ) can be used to simplify the proof the integrability of the irreducible unitary positive energy representations of the Virasoro algebra to strongly continuous projective unitary representations of Diff + (S 1 ). Such a proof was first given in by realizing them in the oscillator algebra [GW85, Section 3, Theorem 4.2]. One can do it now only within the Virasoro algebra as follows.
Besides the energy-bounds (a priori estimates) in [GW85, Section 2], see also [BSM90], which are used in [CW05] and are crucial to our proof, we also used (3.1) coming from [GW85]. More precisely, we used the fact that for every γ ∈ Diff + (S 1 ) there is an unitary operator U (γ) such that U (γ)T (f )U (γ) * = T γ (f ) for all f ∈ Vect(S 1 ) and U (γ)D(L 0 ) = D(L 0 ). This can be proved directly following the strategy in pages 1100-1101 of [CKL08], see also the proof of [CKLW18, Proposition 6.4]. One only needs some of the direct consequences of the energy bounds proved in [TL99, Section 2]. We outline the arguments here: • Since Diff + (S 1 ) is simple [Mil84, Remark 1.7], it is generated by exponentials, because the subgroup generated by exponentials is a normal subgroup.
• By the proof of Corollary 3.5, the set of γ such that an unitary U (γ) with the required properties exists forms a subgroup of Diff + (S 1 ). Hence, it is enough to consider the special case where γ = Exp(g) for g ∈ Vect(S 1 ).
• It follows from the linear energy-bounds by [TL99, Proposition 2.1] that e itT (g) D(L k 0 ) = D(L k 0 ) for all positive integers k and all t ∈ R. As a consequence e itT (g) C ∞ (L 0 ) = C ∞ (L 0 ) for all t ∈ R.
• Now, let ξ ∈ C ∞ (L 0 ) and let ξ(t) = T Exp(tg) (f )e itT (g) ξ. By [TL99, Corollary 2.2] we have d dt e itT (g) ξ = ie itT (g) T (g)ξ in the graph topology of D(L k 0 ) for all positive integers k. It then follows from the energy bounds that d dt ξ(t) = iT (g)ξ(t). Hence, ξ(t) = e itT (g) T (f )ξ for all ξ ∈ C ∞ (L 0 ) so that T Exp(tg) (f ) = e itT (g) T (f )e −itT (g) which is the required relation. Continuity of U follows as in Corollary 3.6.

Direct sum of irreducible representations
Here we prove that every positive energy projective unitary representation of Diff + (S 1 ) extends to a unitary projective representation of D s (S 1 ) for s > 3. A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. This is not an immediate consequence of Corollary 3.6, because, in general, the direct sum of projective representations does not make sense: U (H j )/C is not a linear space. On the other hand, if we have multiplier representations of a group G with the same cocycle, U j (g 1 )U j (g 2 ) = ω(g 1 , g 2 )U j (g 1 g 2 ) where ω(g 1 , g 2 ) is a 2-cocycle H 2 (G, C) of G, then the direct sum j U j (g) is again a multiplier representation with the same cocycle ω. If we are interested in a projective representation of a certain quotient G/H by a normal subgroup H we have to make sure that the direct sum U j (h) reduces to a scalar when h ∈ H.
Continuous fragmentation of D s (S 1 ). Let I be a proper open interval of S 1 and I ′ = (S 1 \I) • the interior of its complement. We denote by I the closure of I. Diff + (I) (resp. D s (I)) denotes the subgroup of diffeomorphisms Diff + (S 1 ) (resp. D s (S 1 )) such that γ(x) = x for x ∈ I ′ . We also say that γ ∈ Diff + (I) (resp. γ ∈ D s (I)) is supported in I. Let {I j } j=1,2,3 be a cover of the unit circle as Fig. 1. Let us name the end points of the intervals: I k = (a k , b k ). We also take a slightly smaller intervalÎ k = (â k ,b k ) ⊂ I k which still consist a cover of S 1 pointsȃ 1 ,b 1 , c.f. [DFK04]. Furthermore, we takeb 2 ,b 2 such that a 1 <b 2 <b 2 < b 2 . Any given diffeomorphism γ can be written as a product of elements supported in I k . This is known as fragmentation (see [Man15] and references therein). We need a slightly refined version of it, namely, if γ is in a small neighborhood V of the unit element ι, then we can take the fragments γ k also in a small, but larger neighborhoodV. The precise statement is the following.
We choose ε such that γ ′ 1 is positive for γ ∈ V ε . Now the assignment V ε → D s (S 1 ), γ → γγ −1 1 is continuous by Lemma 2.5. We take V ⊂ V ε to be the neighborhood of the identity of D s (S 1 ) such that for γ ∈ V we have γγ −1 1 ∈ V ε 1 where ε 1 is small enough that we obtain γ 2 ∈ D s (S 1 ) (in particular γ ′ 2 is positive) if we do an analogous construction on I 2 for γγ −1 1 . For γ ∈ V we set χ 1 (γ) = γ 1 . The continuity of the map χ 1 in the topology of D s (S 1 ) is clear from (3.16) and (3.12)(3.14).
If γ is already localized, we can have the following improvement.
Lemma 3.10. Let k ∈ {1, 2, 3} mod 3 andĨ k = I k ∪ I k+1 . There is a neighborhood V of the unit element ι of D s (S 1 ) and continuous localizing maps Proof. Without loss of generality, we may assume k = 2. This is done by applying the steps of constructing χ 2 and χ 3 in the proof of Lemma 3.9 to slightly enlarged I 2 andÎ 2 , so that χ The last assertion follows from [Wei17, Lemma 2.1].
We are going to show that we can take the direct sum of irreducible projective representations of D s (S 1 ), {U (c,h j ) }, with the same central charge c but possibly different lowest weights {h j } where differences h j − h j ′ are integers. We split the proof into two steps. First, we make U (c,h j ) into continuous multiplier representations with the same cocycle in some neighborhood V of the identity diffeomorphism ι ∈ D s (S 1 ). Then it is straightforward to take the direct sum. Next, we show that the direct sum representation reduced to a projective representation of D s (S 1 ) if the differences h j − h j ′ are integers.
Let G and G ′ be two topological groups. Given a neighborhood V of the identity in G, a continuous map µ : V → G ′ is a local homomorphism if µ(g 1 )µ(g 2 ) = µ(g 1 g 2 ) for all g 1 , g 2 ∈ V and g 1 g 2 ∈ V.
We say that a map U is a local unitary multiplier representation of a topological group G on a neighborhood V of the identity if U is a map from V to the unitary group U (H) of a Hilbert space H which satisfies the equality U (g 1 )U (g 2 ) = ω(g 1 , g 2 )U (g 1 g 2 ), where ω : V × V → T and ω(g 1 , g 2 )ω(g 1 g 2 , g 3 ) = ω(g 1 , g 2 g 3 )ω(g 2 , g 3 ) whenever g 1 , g 2 , g 3 , g 1 g 2 and g 2 g 3 are in V. The following is obtained by reversing the idea of [Tan18]. Proof. Let us take h 1 . By [Bar54][Mor17, Proposition 12.44], in a neighborhoodV of the identity ι ∈ Diff 4 + (S 1 ), U (c,h 1 ) lifts to a continuous multiplier representation, with some continuous cocycle c(·, ·), which we will denote by U 1 .
We fix a covering {I k } of S 1 as in Lemma 3.9. For γ ∈ p(V), we define U j as follows: By Lemma 3.11, there are unitary intertwiners {V j,k } between U (c,h 1 ) and U (c,h j ) restricted to D s (I k ). We set U j (χ k (γ)) = Ad V j,k (U 1 (γ k )), which makes sense because p(V) ⊂V. Note that U j (χ k (γ)) does not depend on the choice of unitary intertwiner V j,k , since, if V j,k andV j,k are both unitary intertwiners, then by Lemma 3.11 for γ smooth, and by continuity of U 1 for χ k (γ) ∈ D s (I k ) ∩V. Let us denote γ k = χ k (γ) for simplicity. Now, since γ = γ 1 γ 2 γ 3 with γ k ∈ D s (I k ) ∩V, we can define U j (γ) by and note that the corresponding equation holds for U 1 .
Well-definedness. We used a particular set of maps χ k to define U j , but actually they do not depend on the choice of such map χ k if γ satisfies certain properties and is sufficiently close to ι. Namely, we take two decompositions γ = γ 1 γ 2 γ Furthermore, as U 1 is a multiplier representation inV, we have By putting all factors in one side, we obtain Note that U j is unitarily equivalent to U 1 on any proper interval, therefore, has support in I 2 ∪ I 3 . Then we can again use the unitary equivalence between U j and U 1 on I 2 ∪ I 3 to obtain which is, by (3.18), equivalent to the equality In other words, U j is well-defined on p 6 (V).
have to compute where we used local equivalence between U j and U 1 in and 4th equalities, and the well-definedness (independence of the partition of a group element into D s (I k )∩p 5 (V)) in the 5th equality. Namely, U j has the cocycle c on V = p 11 (V).
Direct sum of multiplier representations. Since all the projective representations U j can be made into the local multiplier representations with the same cocycle c, the direct sum U := j U j is again a local multiplier representation of D s (S 1 ) on V. By forgetting the phase, we can interpret that U is a local projective representation of V ⊂ D s (S 1 ), or in other words, a continuous local group homomorphism from V into U (  Proof. LetŨ (c,h j ) the irreducible global multiplier representation of Diff + (S 1 ) with central charge c and lowest weight h j associated to the Bott-Virasoro cocycle. As a projective representation, we have U Diff + (S 1 ) = jŨ(c,hj) : this is because, by definition of U , they agree on a neighborhood of the identity of Diff + (S 1 ), and since Diff + (S 1 ) is simply connected they agree globally. Since The lift to a true representation of PSL(2, R) is unique, since if V 1 and V 2 are true representations which give rise to the same projective representation, we have that V 1 (g) = χ(g)V 2 (g) for all g ∈ PSL(2, R), where χ is a character. Since PSL(2, R) is a perfect group, χ(g) = 1 for all g. By the uniqueness of the lift of U PSL(2,R) to a true representation V , we have that where V (c,h j ) is the lift ofŨ (c,h j ) PSL(2,R) to a true representation. As we assume that h j − h j ′ are integers, V (ρ(2π)) ∈ C.
From the previous theorem, it follows that every positive energy projective unitary representation of Diff + (S 1 ) extends to a unitary projective representation of D s (S 1 ) using the following well-known fact that we here prove for completeness.
Proposition 3.14. Let U be a positive energy unitary projective representation of Diff + (S 1 ) on the Hilbert space H. Then U is unitarily equivalent to a direct sum of irreducible positive energy unitary projective representation of Diff + (S 1 ) and extends to D s (S 1 ), s > 3.
Proof. As in the proof of Theorem 3.13, we have that U PSL(2,R) can be lifted to a true representation of PSL(2, R). Thus we can take the generator of rotations L 0 and, since e i2πL 0 ∈ C½ from the fact that U is a projective representation of Diff + (S 1 ), it follows that L 0 is diagonalizable with spectrum Sp(L 0 ) ⊂ {h 1 + N} with h 1 ∈ R, h 1 ≥ 0. Let H fin be the dense subspace of H generated by the eigenvectors of L 0 . We can apply [CKLW18,Theorem 3.4] to conclude that there exists a positive energy unitary representation π U of Vir on H fin .
The representation of Vir on H fin is equivalent to an algebraic orthogonal direct sum of multiples of irreducible positive energy representations of Vir in the following sense. Let V 1 be the smallest π U -invariant subspace of H fin which contains ker(L 0 − h 1 ½ H fin ) where h 1 is the smallest eigenvalue of L 0 . By induction let V n be the smallest π U -invariant subspace of It is straightforward to see that H fin = n V n in the algebraic sense. Now choose an orthonormal basis {e n j } of W n := V n ∩ ker(L 0 − h n ½ H fin ). We define H n j to be the smallest π U -invariant subspace of W n which contains the vector e n j . By construction H n j has no proper π U -invariant subspaces, H n j and H n k are orthogonal subspaces for j = k and V n = j H n j . Let T be the stress-energy tensor associated to the representation π U of Vir. By construction T (f )| H n j is essentially self-adjoint on H n j .
To conclude the decomposition of U , we have to show that e iT (f ) H n j ⊂ H n j for all f ∈ Vect(S 1 ). We note that D ( Proof. This again follows from Proposition 3.14 and the continuous embedding Diff k We do not know whether our local multiplier representations can be extended to a global multiplier representation of D s (S 1 ). It is also open whether the global multiplier representation of Diff + (S 1 ) with the Bott-Virasoro cocycle [FH05, Proposition 5.1] extends to D s (S 1 ) by continuity. (1) Isotony: A(I 1 ) ⊂ A(I 2 ), if I 1 ⊂ I 2 , I 1 , I 2 ∈ I.
(4) Positivity of energy: the representation U has positive energy, i.e. the conformal Hamiltonian L 0 (the generator of rotations) has non-negative spectrum.
Additionally Ω is cyclic for the algebra I∈I A(I).
(7) Haag duality: for every I ∈ I, A(I ′ ) = A(I) ′ where A(I) ′ is the commutant of A(I).
(9) Semicontinuity: if I n ∈ I is a decreasing family of intervals and I = ( n I n ) • then A(I) = n A(I n ).
By a conformal net (or diffeomorphism covariant net) we shall mean a Möbius covariant net which satisfies the following: (10) The representation U extends to a projective unitary representation of Diff + (S 1 ) such that for all I ∈ I we have U (γ)A(I)U (γ) * = A(γI), γ ∈ Diff + (S 1 ), where Diff + (I ′ ) denotes the subgroup of diffeomorphisms γ such that γ(z) = z for all z ∈ I.
A positive energy representation U of Diff + (S 1 ) is equivalent to a direct sum of irreducible representations, see Proposition 3.14. Every irreducible component U j in decomposition has the same value of the central charge c and if h j is the lowest weight of U j , h j − h k ∈ Z for every j, k. This fact is crucial for our purpose, which is to extend the conformal symmetry of the net to the larger group D s (S 1 ), s > 3, in the sense that we want to show that the conditions in (10) are satisfied for arbitrary γ in D s (S 1 ) and D s (I ′ ) respectively.
Proof. Let {γ n } be a sequence of diffeomorphisms in Diff + (S 1 ) converging to γ ∈ D s (S 1 ) in the topology of D s (S 1 ) as in Lemma A.4. For all n ∈ N it holds that where we used isotony of the net A. For x ∈ A(I), it follows for m ≤ n that U (γ n )xU (γ n ) * ∈ A( n k=m γ k I) = ∞ k=m A(γ k I), by additivity. By Proposition 3.6 it follows that U (γ)xU (γ) * = lim n→∞ U (γ n )xU (γ n ) * (convergence in the strong operator topology) is in ∞ k=m A(γ k · I) for any m, hence we have by upper semicontinuity that The other inclusion follows by applying Ad U (γ −1 ). Now consider γ ∈ D s (I ′ ) and x ∈ A(I). We know from lemma A.4 that exists a sequence {γ n } ⊂ Diff + (I ′ n ) converging to γ in the topology of D s (S 1 ) and a decreasing sequence of intervals I ′ n ⊃ supp (γ n ) ⊃ I ′ such that n I ′ n = I ′ . For x ∈ A(I n ), U (γ m )xU (γ m ) * = x if m ≥ n, hence by Proposition 3.6 we obtain U (γ)xU (γ) * = x. As n is arbitrary, this holds for any x ∈ A( n I n ) = A(I) by additivity.

Representations of conformal nets
Let (A, U, Ω) a conformal net. A representation ρ of (A, U, Ω) is a family ρ = {ρ I }, I ∈ I, where ρ I are representations of A(I) on a common Hilbert space H ρ and such that ρ J | A(I) = ρ I if I ⊂ J. The representation ρ is said to be locally normal if ρ I is normal for every I ∈ I (this is always true if the representation space H ρ is separable [Tak02, Theorem 5.1]). We say that a representation ρ of a conformal net (A, U, Ω) is diffeomorphism covariant if there exists a positive energy representation U ρ of Diff + (S 1 ) such that whereġ is the image of g in Diff + (S 1 ) under the covering map. Now let ρ a locally normal representation of the conformal net A and assume that e i2πL ρ 0 has pure point spectrum (this is always the case if ρ is a direct sum of irreducibles). By using [Car04, Proposition 2.2] and arguing as in the proof of [Car04, Proposition 3.7] it is not hard to see that ρ is diffeomorphism covariant (this will be directly proved in [Tan18]) and that the corresponding positive energy projective unitary representation U ρ of Diff + (S 1 ) is a direct sum of irreducibles. By our previous results U ρ extends to D s (S 1 ), s > 3, and this extension makes ρ D s (S 1 )-covariant. We summarize this fact in the following proposition.
Proposition 4.2. Let ρ be a a locally normal representation of the conformal net A and assume that e i2πL ρ 0 has pure point spectrum. Then ρ is D s (S 1 )-covariant, s > 3.

Outlook
It has been shown that for some c, h the irreducible unitary representation U (c,h) can be extended to C 3 -diffeomorphisms [Vro13]. It would be interesting to further determine to what point the regularity of the diffeomorphisms can be weakened in such a way that the representations U (c,h) may be extended to such a class in a continuous way. It might turn out, as in the case of T , that the map U (c,h) extends to a certain set of diffeomorphisms by continuity, but the set is not closed under the group operation. Another interesting question is whether the global multiplier representations in [FH05] extend to D s (S 1 ). The question is whether these representations are continuous in the D s (S 1 )-topology. Instead, what we used in Proposition 3.12 is the continuity of our extensions as projective representations, and the existence of local multiplier representations follows. In particular, we do not know whether there is a multiplier representation of D s (S 1 ) with the Bott-Virasoro cocycle.
Assuming that the statement holds for s − 1, we can conclude induction by applying it to f ′ and using Lemma A.1.
The whole Appendix B of [IKT13] can be adapted to H s (S 1 ) and D s (S 1 ) using these norms and one obtains Lemma 2.5 for s / ∈ Z, corresponding to [IKT13, Lemmas B.5,B.6] (for s ∈ Z, see [IKT13, Section 2]). We believe this is the fastest way for the reader not familiar with Sobolev spaces.
By local Sobolev spaces. Alternatively, one may start with the Sobolev spaces on R, following [IKT13]: wheref denotes the Fourier transform for f ∈ L 2 (R) (with a slight abuse of notation:f depends on f ∈ L 2 (R) or f ∈ L 2 (S 1 )). For an open connected set U ⊂ R, we set (see [IKT13,Definition B.1], where the boundary ∂U is required to be of Lipschitz, but in R it is not necessary): For s > 1 2 , we may take another definition for H s (S 1 ) = H s (S 1 , R): {f ∈ C(S 1 , R) : for each θ ∈ S 1 there is U ∋ θ s.t. f | U ∈ H s (U , R)}. Now, from Lemma A.2, it is clear that being in H s (S 1 ) is a local property (note that S 1 is compact). If f ∈ H s (S 1 ) (in the sense of (A.1)), for any smooth function ψ with compact support, ψf ∈ H s (S 1 ) by Lemma A.2. To a chart U in the atlas, take a smooth function ψ which is 1 on U and supported in a non-dense interval. Then ψf can be considered as an element of H s (R, R) by [IKT13, Lemma B.2], hence the definition (A.1) is stronger. Conversely, if for each θ ∈ S 1 there is U ∋ θ such that f | U ∈ H s (U , R), by compactness of S 1 one can take a finite cover {U k } of S 1 and a smooth partition of unity {ψ k } subordinate to it, and it follows that f = k ψ k f ∈ H s (S 1 ) in the sense of (A.1), therefore, two definitions are equivalent. It is also clear that the following definition [Tay11, Section 4.3] H s (U , R) := {f ∈ L 2 (U , R) : ψf ∈ H s (R, R) for any ψ ∈ C ∞ (U , R), supp ψ ⊂ U } is equivalent to the definition through Fourier coefficients. Now, we define H s (S 1 , S 1 ) to be the maps f : S 1 → S 1 such that there are two atlases {U k }, {V κ } of S 1 such that f | U k ∈ H s (U k , R) where we identified f (U k ) ⊂ V k as a subset of R by the chart (see [IKT13, Section 3.1]) and D s (S 1 ) = {γ ∈ Diff 1 + (S 1 ) : γ ∈ H s (S 1 , S 1 )}. Recall that the definition (A.2) is local, and id is a smooth function, hence it is equivalent to the definition above which is manifestly local. Now that we have the equivalence of definitions, we can use [IKT13, Theorem B.2], which we cited and specialized as Lemma 2.5.
Local approximation of diffeomorphisms. We also need that elements in D s (S 1 ) with compact support can be approximated by elements D s (S 1 ) with slightly larger support.
Lemma A.3. For a fixed f ∈ H s (S 1 ), the rotation R ∋ t → f t = f (e i(·−t) ) ∈ H s (S 1 ) is continuous.
This means f − f t H s → 0.
Lemma A.4. For every γ ∈ D s (S 1 ), there exists a sequence {γ n } converging to γ in the topology of D s (S 1 ). Furthermore, if γ is supported in I, we can take γ n such that supp γ n ⊃ γ n+1 and n supp γ n = I.
To obtain the claim, it is enough to show that ψ − ψ * g n H s → 0 as n → 0. This follows from ψ − ψ * g n H s ≤