Loop Groups and Diffeomorphism Groups of the Circle as Colimits

In this paper, we show that loop groups and the universal cover of Diff+(S1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm Diff}_+(S^1)}$$\end{document} can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of S1. Analogous results hold for based loop groups and for the based diffeomorphism group of S1. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group.


Introduction and Statement of Results
In the category of groups, the concept of colimit is a simultaneous generalisation of the notions of direct limit, and amalgamted free product. Given a diagram of groups {G i } i∈I indexed by some poset I (i.e., a functor from I viewed as a category into the category of groups) the colimit colim I G i is the quotient colim I G i = * i∈I G i /N of the free product of the G i by the normal subgroup N generated by the elements g −1 f i j (g) for g ∈ G i , where f i j : G i → G j are the homomorphisms in the diagram. If the diagram takes values in the category of topological groups (i.e., if the G i are topological groups and the f i j are continuous), then we may take the colimit in the category of topological groups. The underlying group remains the same, but it is now endowed with the colimit topology: the finest group topology such that all the maps G i → G are continuous.
In the present paper, we show that loop groups and the universal cover of Diff + (S 1 ) can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals 1 of S 1 .
Let now G be a compact Lie group, and let LG := Map(S 1 , G) be the group of smooth maps of S 1 := {z ∈ C : |z| = 1} into G. This is the so-called loop group of G. For every interval I ⊂ S 1 , let L I G ⊂ LG denote the subgroup of loops whose support is contained in I . If G is simple and simply connected, then LG admits a well-known central extension by U (1) [GW84,Mic89,PS86]: Letting L I G be the restrictions of that central extension to the subgroups L I G, our main result about loop groups is that LG = colim I ⊂S 1 L I G and LG = colim where the colimit is taken in the category of topological groups. Let G be the subgroup of LG consisting of loops that map the base point 1 ∈ S 1 to e ∈ G, and all of whose derivatives vanish at that point. We call G the based loop group of G (this is the version of the based loop group that was used in [Hen15]). Letting G be the central extension of G induced by (1), we prove that whereI denotes the interior of an interval I , and colim H G i denotes the maximal Hausdorff quotient of colim G i , equivalently, the colimit in the category of Hausdorff topological groups. Let Diff + (S 1 ) be the group of orientation preserving diffeomorphisms of S 1 , and letDiff + (S 1 ) be its universal cover (with center Z). Given a subinterval I ⊂ S 1 of the circle, we write Diff 0 (I ) for the subgroup of diffeomorphisms that fix the complement of I pointwise. The groups Diff 0 (I ) are contractible and may therefore be treated as subgroups ofDiff + (S 1 ). The group Diff + (S 1 ) admits a well-known central extension by the reals, called the Virasoro-Bott group [Bot77, KW09,TL99]. We write Diff R + (S 1 ) for the Virasoro-Bott group and Diff R×Z + (S) for its universal cover. We then have the following system of central extensions: Let Diff * (S 1 ) be the subgroup of Diff(S 1 ) consisting of diffeomorphisms that fix the point 1 ∈ S 1 , and are tangent up to infinite order to the identity map at that point. We call it the based diffeomorphism group of S 1 . Let Diff R * (S 1 ) be the restriction of the central extension by R to Diff * (S 1 ). We also prove that Remark. All our results are formulated for the C ∞ topology (uniform convergence of all derivates), but they hold equally well for groups of C r loops S 1 → G, r ≥ 1, and for groups of C r diffeomorphisms of S 1 , r ≥ 2 (with the exception of Sect. 3.3.1, which seems to requires r ≥ 4 [CDVIT18]).
In the third section of this paper, we apply the above results about (central extensions of) LG and Diff(S 1 ) to the representations of loop group conformal nets. For each compact simple Lie group G and integer k ≥ 1, there is a conformal net called the loop group conformal net. It associates to an interval I ⊂ S 1 a von Neumann algebra version of the twisted group algebra of L I G.
We construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group. This last result is needed in order to fill a small gap between the statement of [Hen17, Thm. 1.1] and the results announced in [Hen15,§4 and §5].

Colimits and Central Extensions
The category of topological groups admits all colimits. If {G i } i∈I is a diagram of topological groups indexed by some small category I, then the colimit G = colim I G i can be computed as follows. As an abstract group, it is given by the colimit of the G i in the category of groups. The topology on G is the finest group topology that makes all the maps G i → G continuous.
Let us call a small category I connected if any two objects i, j ∈ I are related by a zig-zag i ← i 1 → i 2 ← i 3 → i 4 . . . i n ← j of morphisms. Similarly, let us call a diagram connected ('diagram' is just an other name for 'functor') if the indexing category I is connected.
We recall the notion of central extension of topological groups: Definition 1. Let G be a topological group, and let Z be an abelian topological group. A central extension of G by Z is a short exact sequence such that Z sits centrally in G, the map ι is an embedding (Z is equipped with the subspace topology) and there exist a continuous local section s : Note that for U and s as above, the map s · ι : U × Z →G is always an open embedding.
Proposition 2. Let {G i } i∈I be a connected diagram of topological groups, let G := colim I G i , and let ϕ i : G i → G be the canonical homomorphisms. Assume that there exists a neighbourhood of the identity U ⊂ G, and finitely many continuous maps f i : (the f i are indexed over an ordered finite subset of the objects of I).
Let 0 → Z → G → G → 0 be a central extension of G, and let G i be the induced central extensions of G i (the pullback ofG along the map ϕ i : G i → G). Then the canonical map is an isomorphism of topological groups. 2 2 For i → j a morphism in I, the mapG i →G j is the unique group homomorphism which makes the commute (by the universal property of the pullback definingG j ).
Proof. The diagram is connected, so all the central Z 's in the various G i get identified in colim I G i . Moreover, the canonical map ι Z : Z → colim I G i is an embedding, because the triangle commutes. The quotient is easily computed: so the sequence 0 → Z → colim I G i → G → 0 is a short exact sequence of groups. To show that it is also a short exact sequence of topological groups, we need to argue that the projection map colim I G i → G admits local sections. Pick neighbourhoods U i ⊂ G i of the neutral elements and local sections s i : is the desired local section. We have two exact sequences of topological groups, and a map between them: By the five lemma, the middle vertical map is an isomorphism of groups. It is an isomorphism of topological groups because bothG and colim I G i are locally homeomorphic to a product U × Z .

Loop groups.
We write S for a manifold diffeomorphic to the standard circle S 1 := {z ∈ C : |z| = 1}, and call such a manifold a circle. We write I for a manifold diffeomorphic to [0, 1], and call such a manifold an interval. All circles and intervals are oriented.

The free loop group
Let G be a compact, simple, simply connected Lie group. Throughout this section, we fix a circle S, and write LG := Map(S, G) for the group of smooth maps from S to G. Given an interval I , we denote by L I G be the group of maps I → G that send the boundary of I to the neutral element e ∈ G, and all of whose derivatives vanish at those points. If I is a subinterval of S, we identify L I G with the subgroup of LG of loops with support in I . The group LG = Map(S, G) admits a well-known central extension LG, constructed as follows. Let g be the Lie algebra of G. The Lie algebra C ∞ (S, g) of LG admits a wellknown 2-cocycle given by the formula ( f, g) → 1 2πi S f, dg . (Here, , : g×g → R is the basic inner product -the smallest G-invariant inner product whose restriction to any su(2) ⊂ g is a positive integer multiple of the pairing (X, Y ) → −tr(XY ).) This cocycle can be used to construct a central extension of C ∞ (S, g) by the abelian Lie algebra iR. The latter can be then integrated to a simply connected infinite dimensional Lie group LG with center U (1) [GW84,Mic89,PS86,TL99].
Let LG k be the quotient of LG by the central subgroup μ k ⊂ U (1) of k-th roots of unity.
is an isomorphism of topological groups.
Lemma 5. (i) Let S be a circle and let I = {I i } be a collection of subintervals whose interiors cover S. Then the subgroups L I i G generate LG.
(ii) Let I be an interval and let I = {I i } be a finite collection of subintervals whose interiors cover that of I . Then the subgroups L I i G generate L I G.
Proof. The two statements are entirely analogous. We only prove the first one. First of all, since S is compact, we may assume without loss of generality that n := |I| < ∞. Let exp : g → G be the exponential map, and let u ⊂ g be a convex neighbourhood of 0 such that exp| u : u → G is a diffeomorphism onto its image U := exp(u). Given a partition of unity φ i : I i → [0, 1], every loop γ ∈ Map(S, U ) ⊂ LG can be factored as . (Note that the γ i commute.) The subgroup generated by the L I i G therefore contains Map(S, U ). The latter is open and hence generates LG.
Let I be a collection of subintervals of S whose interiors form a cover and that is closed under taking subintervals: (I 1 ∈ I and I 2 ⊂ I 1 ) ⇒ I 2 ∈ I. If I 1 , I 2 ∈ I are such that I 1 ∩ I 2 is an interval, then the diagram (3) clearly commutes. When I 1 , I 2 ∈ I have disconnected intersection and γ has support in I 1 ∩ I 2 , it is not clear, a priori, that ι 1 (γ ) = ι 2 (γ ). Letting J 1 , J 2 be the connected components of I 1 ∩ I 2 , we can rewrite γ as a product γ = γ 1 γ 2 , with γ i ∈ L J i G. We then have So the diagram (3) always commutes, even when I 1 ∩ I 2 is disconnected. Given γ ∈ L I G for some I ∈ I, we also write γ for its image in colim I L I G. This element is welldefined by the commutativity of (3).
The following result is a strengthening of Theorem 3: Theorem 6. Let I be a collection of subintervals of S whose interiors form a cover, and that is closed under taking subintervals. Let N colim I L I G be the normal subgroup generated by commutators of loops with disjoint supports: Then the natural map is an isomorphism of topological groups.
Before embarking in the proof, let us show how Theorems 3 and 4 follow from the above result.
Proof of Theorem 3. Let I be the poset of all subintervals of S. By Theorem 6, the map (colim I ⊂S L I G)/N → LG is an isomorphism. So it suffices to show that N is trivial. Given two loops γ 1 , γ 2 ∈ LG with disjoint support, let I ⊂ S be an interval that contains the union of their supports. The commutator of γ 1 and γ 2 is trivial in L I G. It is therefore also trivial in the colimit.
Proof of Theorem 4. It is enough to show that the colimit which appears in Theorem 3 satisfies the assumption of Proposition 2. The maps γ → γ i used in Eq. (2) provide the required factorization. So Theorem 3 implies Theorem 4.
Proof of Theorem 6. Given γ ∈ L I G for some I ∈ I, we write [γ ] for its image in (colim I L I G)/N .
Let {J j } j = 1...n be a cover of S such that each J j ∩ J j+1 is an interval (in particular J j ∩ J j+1 is non-empty) and the other intersections are empty (cyclic numbering). The J j may be chosen small enough so that each union J j−1 ∪ J j ∪ J j+1 is in I (cyclic numbering). Let U ⊂ G be as in the proof of Lemma 5. By (2), any loop γ ∈ Map(S, U ) ⊂ LG can be factored as γ = γ 1 . . . γ n , with γ j ∈ L J j G. Moreover, that factorisation may be chosen to depend continuously on γ . This provides a local section of the map in (4): LG The map (colim I L I G)/N → LG is surjective by Lemma 5. Since there exists a continuous local section, all that remains to do in order to show that it is an isomorphism is to prove injectivity.
Let g be an element in the kernel. By Lemma 5, we may rewrite g as a product [γ 1 ][γ 2 ] · · · [γ N ], with γ i ∈ L J i G for J i ∈ I. Since g is in the kernel of the map to LG, the relation holds in LG. Any loop γ ∈ Map(S, U ) can be factored as γ = γ 1 · · · γ n with γ j ∈ L J j G. The set U := L J 1 G · · · · · L J n G = {γ i . . . γ n | γ j ∈ L J j G} is therefore a neighbourhood of e ∈ LG. Moreover, it is visibly path connected. Since π 2 (G) = 0 [PS86, §8.6], we have So, by Lemma 7, Eq. (5) is a formal consequence of relations of length 3 between the elements of U.
Lemma 7. Let G be a simply connected topological group, let U ⊂ G be a pathconnected neighbourhood of e ∈ G, and let F be the free group on U. Then the kernel of the map F → G is generated as a normal subgroup by words of length 3.
(In other words, any relation between elements of U is a formal consequence of relations of length 3 between elements of U.) , be a word in the kernel of the map F → G. We want to show that the relation is a formal consequence of relations of length 3 between elements of U.
For every i, pick a path γ i : [0, 1] → U from e to g i . Since G is simply connected, there exists a disk D 2 → G that bounds the loop Triangulate D 2 finely enough and orient all the edges so that, for each oriented edge x − − y, the ratio x −1 y is in U. The orientations along the boundary are chosen compatibly with the ε i 's in (6). Now forget the map D 2 → G and only remember the triangulation of D, along with the labelling of its vertices by elements of G. Before subdividing D, the word that one could read along the boundary of D was Each little triangle x − y − z − x of the triangulation corresponds to a 3-term relation among elements of U. Depending on the orientation of the edges, this 3-term relation could be any one of the following eight possibilities: The whole disc is a van Kampen diagram exhibiting the relation as a formal consequence of the above 3-term relations (see [Ol'91, Chapt 4] for generalities about van Kampen diagrams). The relation (6) is a formal consequence of the relations (9) and (8). Therefore, in order to finish the lemma, it remains to show that (8) is a formal consequence of relations of length 3 between elements of U. By construction, is a 3-term relation between elements of U. One checks easily that (8) is a formal consequence of the above 3-term relations.
We have shown that Eq. (5) is a formal consequence of relations of length 3 between elements of U = L J 1 G · · · · · L J n G. It is therefore a formal consequence of certain relations of length 3n between elements of the subgroup The implication (10) ⇒ (5) is formal: any group generated by subgroups isomorphic to the L J j G's in which the relations (10) hold also satisfies the relation (5).
In order to prove that the equation when 2n + 1 ≤ i ≤ 3n. At this point, it is useful to note that, for any γ ∈ L J j G and δ ∈ L J k G, (J j , J k ∈ I), the following equation holds: Let σ ∈ S 3n be a permutation such that α σ (3k−2) , α σ (3k−1) , α σ (3k) ∈ L J k G for every k ∈ {1, . . . , n}. By Lemma 8, there exist words w i ∈ (colim where, in the right hand side, we have identified w i with its image in LG. Let β i := α σ (i) . Recall that our goal is to show that Eq. (12) holds. So far, we have shown that Letting Lemma 8. Let F n = a 1 , . . . , a n be the free group on n letters. Then for any permutation σ ∈ S n , there exist words w i ∈ F n so that Moreover, the w i may be chosen so that each a i appears at most once in each w i . (1) . . . a n . Now use induction on n to rewrite a 1 . . . a σ (1) . . . a n as n i=2 w i a σ (i) w −1 i .

The based loop group
Fix a base point p ∈ S, and let G ⊂ LG be the subgroup consisting of loops that map p to the neutralelement of G, and all of whose derivatives vanish at that point. We call G the based loop group of G. Let G be the central extension of G induced by the basic central extension (1) of LG. The arguments of the previous section can be adapted without difficulty to prove the following variants: (The proofs are identical to those in the previous section: replace every occurrence of LG by G, and every occurrence of L I G by L I G ∩ G.) It is also possible to express G and G as colimits over the poset of subintervals whose interior does not contain p, provided one works in the category of Hausdorff topological groups as opposed to the category of topological groups.
Definition 9. Given a diagram {G i } i∈I of Hausdorff topological groups, let us write colim H I G i for the colimit in Hausdorff topological groups. Equivalently, this is the maximal Hausdorff quotient of colim I G i .
Given an interval I , we writeI for its interior. The next result does not seem to hold when the colimit is taken in the category of topological groups: Proposition 10. The natural maps are isomorphisms of topological groups.
Proof. Let I be the poset of subintervals of S whose interior does not contain p, and let N colim I L I G be the normal subgroup generated by commutators of loops whose supports have disjoint interiors. The proof of Theorem 6 applies verbatim (using a cover {J j } j=1...n for which the J j ∩ J j+1 are intervals for 0 < j < n, J 0 ∩ J n = {p}, and all other intersections empty) and shows that the map (colim I L I G) N → G is an isomorphism of topological groups. Since G is Hausdorff, the natural map is therefore an isomorphism, where N H denotes the image of N in colim H I L I G. We wish to show that N H is trivial. Let γ and δ be two loops whose supports have disjoint interiors. Write S\{ p} as an increasing union of closed intervals I i ⊂ S, and write is trivial in L I i G, and therefore in colim I L I G. By uniqueness of limits (this is where we use This show that N H is the trivial group, and that the first map in (28) is an isomorphism.
Proposition 2 is stated in the category of topological groups, but it also holds in the category of Hausdorff topological groups (with identical proof: just replace every occurrence of colim by colim H ). The first isomorphism in (15) is an isomorphism of topological groups. The proof that this map is an isomorphism is identical to that of the second isomorphism in (15).

Diffeomorphism groups.
The material in this section is largely parallel to the one in the previous section, with one notable difference. Whereas conjugating by a loop never increases the support, conjugating by a diffeomorphism does typically increase supports. This introduces a number of small subtleties. Recall that we write S 1 := {z ∈ C : |z| = 1} for the standard circle, and S for a manifold diffeomorphic to S 1 . All our circles are assumed oriented.

2.2.1.
Diff(S 1 ) and its universal cover Given an interval I , we write Diff 0 (I ) ⊂ Diff(I ) for the group of diffeomorphisms of I that are tangent up to infinite order to the identity map at the two boundary points. If I is a subinterval of a circle S, then this group can be equivalently described as the subgroup Diff 0 (I ) ⊂ Diff + (S) of diffeomorphisms with support in I .

Theorem 11. Let S be a circle, and letDiff + (S) be the universal cover of the group of orientation preserving diffeomorphisms of S. Then the natural map
is an isomorphism of topological groups.
The Lie algebra X(S) of smooth vector fields on S has a well known central extension by iR, constructed as follows. Upon identifying S with S 1 , it can be described as the central extension associated to the 2- In terms of the topological basis n := −z n+1 ∂ ∂z of the complexified Lie algebra, this cocycle can also be described by the formula 4 : The corresponding central extension of X(S 1 ) is a universal central extension in the category of topological Lie algebras [GF68]. Since X(S) and X(S 1 ) are isomorphic as topological Lie algebras, the former also admits a universal central extension by iR (universal central extensions are well defined up to unique isomorphism). Finally, the universal central extension of X(S) integrates to a central extension 0 is an isomorphism of topological groups.
Remark 13. The central extension Diff R + (S 1 ) → Diff + (S 1 ) is non-trivial not only as an extension of Lie groups, but also as an extension of abstract groups. To see this, one can argue as follows: Let PSL(2, R) (n) be the subgroup of Diff + (S 1 ) corresponding to the subalgebra Span R {i 0 , n − −n , i n + i −n } of X(S 1 ). That Lie algebra lifts to a Lie algebra . The latter integrates to a subgroup of the Virasoro-Bott group isomorphic to PSL(2, R) (n) : Moreover, since PSL(2, R) (n) has trivial abelianization, 5 the lift s n is unique (without any continuity assumptions). Assume by contradiction that there exists a section s : Diff + (S) → Diff R + (S) which is a group homomorphism, possibly discontinuous. By uniqueness of the lift (18), we would then have s| PSL(2,R) (n) = s n , from which it would that follow that (ii) Let I − , I + ⊂ S 1 be two subintervals that cover the standard circle. Assume that each connected component of I − ∩ I + has length 2d. Then there exist continuous maps ( ) − , ( ) + : Diff <d (S 1 ) → Diff <d (S 1 ) such that, for every ϕ ∈ Diff <d (S 1 ), we have: Proof. We only prove the fist part of the lemma (the second part is completely analogous).
Lemma 17. (i) Let I be an interval and let I = {I i } be a finite collection of subintervals whose interiors cover that of I . Then the subgroups Diff 0 (I i ) generate Diff 0 (I ).
(ii) Let S be a circle and let I = {I i } be a collection of subintervals whose interiors cover S. Then the subgroups Diff 0 (I i ) generateD iff + (S).
Proof. We only prove the second statement (the first one is completely analogous). Assume without loss of generality that S = S 1 . Let {J j } j = 1...n be a refinement of our cover such that each J j ∩ J j+1 has length 2d, for some constant d > 0, and the other intersections are empty (cyclic numbering). Write ϕ as a product ϕ 1 . . . ϕ m of diffeomorphisms of displacement smaller than d. Now apply Corollary 16 to each ϕ i to rewrite it as a product ϕ i = ϕ i,1 . . . ϕ i,n , with ϕ i, j ∈ Diff 0 (J j ).
Let I be a collection of intervals in S 1 whose interiors form a cover, and that is closed under taking subintervals. As in (3), for any I 1 , I 2 ∈ I, the diagram commutes. Given a diffeomorphism ϕ ∈ Diff + (S 1 ) with support in some interval I ∈ I, we also write ϕ for its image in colim I Diff 0 (I ). This element is well-defined by the commutativity of (19).
The following result is a strengthening of Theorem 3: Theorem 18. Let I be a collection of intervals in S 1 whose interiors form a cover, and that is closed under taking subintervals. Let N colim I Diff 0 (I ) be the normal subgroup generated by commutators of diffeomorphisms with disjoint supports. Then the natural map to the universal cover of Diff + (S 1 ) is an isomorphism of topological groups.
Before embarking in the proof, let us show how Theorems 11 and 12 follow from the above result.
Proof of Theorem 11. Without loss of generality, we take S = S 1 . Let I be the poset of all subintervals of S 1 . By Theorem 18, the map (colim I ⊂S 1 Diff 0 (I ))/N →Diff + (S 1 ) is an isomorphism. So it suffices to show that N is trivial. Given diffeomorphisms with disjoint support ϕ, ψ ∈ Diff + (S 1 ), there exists an interval I ⊂ S 1 such that both ϕ and ψ are in Diff 0 (I ). The commutator of ϕ and ψ is trivial in Diff 0 (I ). It is therefore also trivial in the colimit.
Proof of Theorem 18. Given a diffeomorphism ϕ ∈ Diff + (S 1 ) with support in some interval I ∈ I, we write [ϕ] for its image in the group (colim I Diff 0 (I ))/N . Let {J j } j = 1...n be a cover of S 1 such that each intersection J j ∩ J j+1 (cyclic numbering) has length 2d for some d, and the other intersections are empty. The J j are chosen so that each J j−1 ∪ J j ∪ J j+1 is in I (cyclic numbering) and the distance between J j and J j+2 is greater than 6d. By Corollary 16, any element ϕ ∈ Diff <d (S 1 ) can be factored as ϕ = ϕ 1 . . . ϕ n , with ϕ j ∈ Diff <d 0 (J j ). Moreover, that factorisation may be chosen to depend continuously on ϕ. After identifying Diff <d (S 1 ) with an open subset ofDiff + (S 1 ), this provides a local section of the map in (20): (colim I Diff 0 (I ))/NDiff + (S 1 ).
The map (20) is surjective by Lemma 17, and admits continuous local sections.
It remains to prove injectivity. Let g ∈ (colim I Diff 0 (I ))/N be in the kernel of the map toDiff + (S 1 ). By Lemma 17, we may rewrite it as a product [ By assumption, the relation holds inDiff + (S 1 ). Our goal is to show that the relation holds in (colim I Diff 0 (I ))/N . Any x ∈ Diff <d (S 1 ) can be factored as x = x 1 . . . x n with x j ∈ Diff <d 0 (J j ), so the set U := Diff <d 0 (J 1 ) . . . Diff <d 0 (J n ) = {x i . . . x n : x j ∈ Diff <d 0 (J j )} is a neighbourhood of e ∈Diff + (S 1 ). The set U is visibly path-connected. By Lemma 7, Eq. (5) is therefore a formal consequence of certain relations of length 3 between the elements of U: In order to prove that (22) holds, it is therefore enough to show that the relations hold in (colim I Diff 0 (I ))/N . Using that when 2n + 1 ≤ i ≤ 3n. As in (13), for any ϕ ∈ Diff 0 (J j ) and ψ ∈ Diff 0 (J k ), the following equation holds: We would like to replace [ψ] in (26) by an arbitrary word [ψ 1 ] . . . [ψ s ]: However, for general ψ i ∈ k Diff 0 (J k ) it is not clear that (27) should hold, because each time one conjugates ϕ by a diffeomorphism, its support grows. If we insist, however, that the ψ i have small displacement, so as to control the supports of ψ r ψ r +1 . . . ψ s ϕ ψ −1 s . . . ψ −1 r +1 ψ −1 r , then Eq. (27) will hold. The precise version of (27) that we will need is he following: Let ψ i ∈ k Diff <d 0 (J k ), and let ϕ ∈ Diff 0 (J j ). If there are at most three ψ i 's whose support is in J j−1 and not in some other J k , and at most three ψ i 's whose support J j+1 and not in some other J k , then Eq. (27) holds. The proof is an iteration of the argument used for (26), while keeping track of the size of the supports. (This only uses the fact that the distance between J j and J j+2 is greater than 3d. Later, we will use that it is greater than 6d.) Let σ ∈ S 3n be a permutation such that α σ (3k−2) , α σ (3k−1) , α σ (3k) ∈ Diff <d 0 (J k ).
i . Moreover, these words can be chosen so that each [α i ] appears at most once in each w i . By (27), we then have: Recall that our goal is to show that (25) holds. Let β i := α σ (i) . So far, we have: where J + k is obtained from J k by by enlarging it by 3d on each side. The crucial property of those slightly larger intervals is that J + k does not overlap with J + k+2 . The relation χ 1 χ 2 . . . χ n = e holds inDiff + (S 1 ) so the support of each χ k is contained in Finally, as in the proof of Theorem 6,

The based diffeomorphism group
Choose a base point p ∈ S, and let Diff * (S) ⊂ Diff(S) be the subgroup of diffeomorphisms that fix p and that are tangent to id S up to infinite order at that point. We call this group the based diffeomorphism group of S. Let Diff R * (S) be the restriction of the central extension by R to Diff * (S). The arguments of the previous section can be adapted without difficulty to prove the following variants: The proofs are identical to those in the previous section: replace every occurrence of Diff(S 1 ) by Diff * (S 1 ), and every occurrence of Diff 0 (I ) by Diff 0 (I ) ∩ Diff * (S 1 ). It is also possible to express the groups Diff * (S) and Diff R * (S) as colimits over the poset of subintervals whose interior does not contain p, provided one works in the category of Hausdorff topological groups: are isomorphisms of topological groups.
Proof. The proof is identical to that of Proposition 10. Let I be the poset of subintervals of S whose interior does not contain p, and let N colim I Diff 0 (I ) be the normal subgroup generated by commutators of diffeomorphisms whose supports have disjoint interiors. The proof of Theorem 18 applies verbatim (using a cover {J j } j=1...n for which the J j ∩ J j+1 have length 2d for some d, J 0 ∩ J n = {p}, and all other intersections are empty) and shows that the map (colim I Diff 0 (I )) N → Diff * (S) is an isomorphism of topological groups. Since Diff * (S) is Hausdorff, the natural map is an isomorphism, where N H denotes the image of N in colim H I Diff 0 (I ). The end of the proof consists in showing that N H is trivial. The argument is identical to the one in Proposition 10.

Application: Representations of Loop Group Conformal Nets
Fix a compact, simple, simply connected Lie group G, and let k > 0 be an integer. Let g be the complexified Lie algebra of G. There is a certain central extensionĝ of g[z, z −1 ] called the affine Lie algebra. There is also a conformal net A G,k associated to G and k, called the loop group conformal net (we will review these notions below).
It is well believed among experts that there should be an equivalence between the category of representations of A G,k and a certain category of representations ofĝ (see Conjecture 23 for a precise statement). In this section, leveraging Theorems 4 and 12, we prove one half of this conjecture. Namely, given a representation of the loop group conformal net, we construct a representation of the corresponding affine Lie algebra: Rep(ĝ).

Loop group conformal nets and affine
The latter only depends on I and not on the choice of circle S in which the interval is embedded.
Recall that the group algebra C[G] of a group G is the set of finite linear combinations of elements of G. We write [g] ∈ C[G] for the image in the group algebra of an element g ∈ G. Given a central extension 0 Following [BDH15, §1.A], a conformal net is a functor from the category of intervals and embeddings to the category of von Neumann algebras and * -algebra homomorphisms (satisfying various axioms). Associated to each compact, simple, simply connected Lie group G and integer k ≥ 1, there is a conformal net A G,k called the loop group conformal net. The loop group conformal net sends an interval I to a certain von Neumann algebra A G,k (I ). The latter is a completion of the twisted group algebra of L I G associated to the central extension (29). In particular, there is a homomorphism We write Rep(A) for the category of representation of A, and Rep f (A) for the subcategory whose objects are finite direct sums of irreducible representations.

Affine Lie algebras
Let g R be the Lie algebra of G, and let g := g R ⊗ R C be its complexification. The affine Lie algebraĝ is the central extension of g[z, z −1 ] by the 2-cocycle ( f, g) → Res z=0 f, dg , where , denotes the basic inner product on g (c.f. Sect. 2.1.1). 6 The Kac-Moody algebra is the semi-direct product C ĝ associated to the derivation ( f, a) → (−z ∂ f ∂z , 0) ofĝ. We write 0 for the generator of C in the semi-direct product.
Definition 21 ([Kac90, Chapt. 3 and 10]). Let k ∈ N. A representation ρ ofĝ on a vector space V is called a level k integrable positive energy representation if: 1. It is the restriction of a representation of C ĝ for which the generator L 0 := ρ( 0 ) of C is diagonalizable, with positive spectrum. 2. For every nilpotent element X ∈ g and every n ≥ 0, the operator X z −n acts locally nilpotently on V (the operators X z n are automatically locally nilpotent for n > 0). 3. The central element 1 ∈ C ⊂ĝ acts by the scalar k.
We note that the choice of operator L 0 is not part of the data of an integrable positive energy representation.
We write Rep k (ĝ) for the category of level k integrable positive energy representations ofĝ, and write Rep k f (ĝ) for the subcategory whose objects are finite sums of irreducible representations.
It is well known that every object of Rep k (ĝ) can be equipped with a positive definitê g-invariant inner product [Kac90,Chapt. 11], and thus completed to a Hilbert space. The action ofĝ extends to an action on the Hilbert space by unbounded operators, and the real form g[z, We note that the action of S 1 ⊂ S 1 LG is not part of the data of a level k positive energy representation.
We write Rep k (LG) for the category of level k positive energy representations of LG, and Rep k f (LG) for the subcategory whose objects are finite direct sums of irreducible representations.

The comparison functors Rep(A G,k ) → Rep k (LG) and Rep k (LG) → Rep k (ĝ).
It has long been expected that there should be a one-to-one correspondence between representations of A G,k and level k integrable positive energy representations ofĝ. One way to state this is as an equivalence of categories: . We prefer the following statement, as it excludes the possibility of A G,k having representations which are not direct sums of irreducible ones:

Conjecture 23. Let Vec f be the category of finite dimensional vector spaces, and let
Hilb be the category of Hilbert spaces and bounded linear maps. Then there is a natural equivalence of categories: Here, the objects of Rep k f (ĝ)⊗ Vec f Hilb are formal expressions of the form n i=1 V i ⊗ H i with V i ∈ Rep k f (ĝ) and H i ∈ Hilb, and Hom Hom Hilb (H i , H j ). Theorems 29 and 30 below, together with Remark 31 (or Remark 24), prove one half of Conjecture 23. Namely, they combine to a fully faithful functor This proves, among other things, that every representation of A G,k is a direct sum of irreducible ones, and that there is an injective map (conjecturally a bijection) from the set of isomorphism classes of irreducible objects of Rep(A G,k ) to the set of isomorphism classes of irreducible objects of Rep k (ĝ).
Remark 25. An alternative proof of the above result can be found in the unpublished manuscript [CW16]. (specifically the direct integral argument) by assuming from the beginning that L 0 is diagonalizable. We have opted instead for a more self-contained exposition.
Remark 28. The corresponding questions for the Virasoro conformal net have been studied by a number of people. Carpi [Car04], based on results by Loke [Lok94] and D'Antoni and Köster [DFK04], proved that every irreducible positive energy representation of a Virasoro conformal net comes from a positive energy representation of the Virasoro algebra with same central charge. The converse (local normality) was shown to hold by Weiner [Wei17] (with partial results by Buchholz and Schulz-Mirbach [BSM90]), and the positive energy condition was removed in [Wei06].
Precomposing by the quotient map LG → LG k , we get an action of Diff R×Z LG such that every λ ∈ U (1) ⊂ LG acts by scalar multiplication by λ k . In particular, we get an action of ( The main result of [Wei06] shows that the generator L 0 of (S 1 ) Z has positive spectrum.
So far, we have constructed a representation of LG on H that satisfies all the conditions of a level k positive energy representation, except that the S 1 is replaced by its universal cover (S 1 ) Z . In order to show that H is a positive energy representation (Definition 22), we need to modify the action of (S 1 ) Z so that it descends to an action of S 1 .
Decompose H as a direct integral according to the characters of the central Z ⊂ (S 1 ) Z : (This direct integral will turn out to be a mere direct sum, but don't know this at the moment.) Direct integrals for loop group representations are tricky, because disintegration theory only applies to separable locally compact groups, and LG is not locally compact. So we proceed with care. In particular, we never make the claim that the Hilbert spaces H θ carry actions of LG.
For each θ ∈ U (1), extend the character n → θ n of Z to a character z → z log(θ)/2πi of (S 1 ) Z (principal branch of the logarithm). Let C θ denote the vector space C, equipped with the action of (S 1 ) Z given by the above character. Then the representation H := ⊕ θ∈U (1) H θ ⊗ C θ of (S 1 ) Z descends to a representation of S 1 whose generator has positive spectrum (the spectrum of L 0 has been modified by a bounded amount). As mere vector spaces, we have C θ ∼ = C, and therefore H ∼ = H . Use this isomorphism to equip H with an action of LG. We wish to show that the actions of S 1 and of LG on H assemble to an action of S 1 LG.
Pick a countable dense subgroup (S 1 ) Z ⊂ (S 1 ) Z that contains the central Z, and let LG , the Hilbert spaces H θ carry representations of (S 1 ) Z LG for almost all θ . By construction, on almost each H θ ⊗ C θ , the action of (S 1 ) Z LG descends to an action of S 1 LG . The actions of S 1 and LG on H therefore assemble to an action of S 1 LG . At last, since S 1 LG is dense in S 1 LG and since the actions of S 1 and LG on H are strongly continuous, these two actions assemble to an action of S 1 LG. This finishes the proof that H , and hence H , is a positive energy representation of LG.
Positive energy representations of S 1 LG come with no smoothness assumptions. It is therefore not clear, a priori, that it should be possible to differentiate them. Zellner showed that, in such a representation, the set of smooth vectors is always dense [Zel15, Thm 2.16]. One can therefore differentiate it to a representation of the corresponding Kac-Moody Lie algebra. We present an alternative proof of that same result. (Our proof does not cover the case G = SU (2): it relies on the fact that every rank 2 sub diagram of the affine Dynkin diagram of G is of finite type, something which holds for all groups except for SU (2).) Proof. An integration functor Rep k f (ĝ) → Rep k (LG) was constructed in [GW84] and [TL99]. The functor sends irreducible representations ofĝ to irreducible representations of LG. It is therefore visibly fully faithful.
The category Rep k (LG) is tensored over the category of Hilbert spaces (i.e., the tensor product of a positive energy LG representation with a Hilbert space is again a positive energy LG representation). So the above functor extends to a functor which is again visibly fully faithful. In order to show that the functor (34) is an equivalence of categories, we need to show that it is essentially surjective. Let T G , T LG , T S 1 LG be the maximal tori of G, LG, and S 1 LG, so that Let G , LG , S 1 LG be the character lattices of T G , T LG , and T S 1 LG , and let (an affine sublattice canonically isomorphic to Z× G ). We write + k ⊂ k for the set of possible highest weights of irreducible highest weight level k integrable representations of C ĝ [Kac90, Chapt. 10]. Let π : k → G be the projection map, and let A k := π( + k ) ⊂ G . The finite set A k parametrizes the isomorphism classes of irreducible objects of Rep k (ĝ).
Let H be a level k positive energy representation of LG. By definition, the action of LG extends (in a non-unique way) to an action of S 1 LG such that the generator of S 1 has positive spectrum. Pick such as extension of the action. Then H decomposes as the Hilbert space direct sum of its weight spaces: Let P := {λ ∈ k | H λ = 0}. The affine Weyl group W = N T S 1 LG /T S 1 LG acts on k , and preserves P. By the positive energy condition, P is contained in the "half-space" Z ≥0 × G ⊂ Z × G = k . Combining this with its W -invariance, we learn that P is contained in a paraboloid.
Let D be the affine Dynkin diagram associated to G. Every node a ∈ D corresponds to an embedding SU (2) → LG. We write SU (2) a for the subgroup of LG which is the image of that embedding. Let T a ⊂ T S 1 LG be the subgroup of the torus which centralizes SU (2) a , and let a be the corresponding quotient of S 1 LG , with projection map p a : S 1 LG → a ( a is the character lattice of T a ). The kernel of p a has rank one. Let us also define k,a := p a ( k ).
Since P is contained inside a paraboloid, for each σ ∈ k,a , the set (σ ) := {λ ∈ P | p a (λ) = σ } is finite. It follows that, for every σ , the representation of SU (2) a on λ∈ (σ ) H λ contains only finitely many isomorphism classes of irreducible SU (2) a representations. In particular, the action of su(2) a on λ∈ (σ ) H λ is by bounded operators (which are in particular everywhere defined). In this way, we obtain actions of the Lie algebras su(2) a on the algebraic direct sum Those Lie algebras contain all the generators {E a , F a , H a } of the Serre presentation ofĝ.
To check that the above generators satisfy the Serre relations, we consider rank two subgroups of LG. For every pairs of vertices a, b ∈ D, the subgroup G ab ⊂ LG generated by SU (2) a and SU (2) b is compact as it correspnds to the sub-Dynkin diagram of D on the two vertices, and the latter is either A 1 A 1 , A 2 , B 2 , or G 2 -this is where we use that G is not SU (2). Applying the same arguments as above, we see that there are actions of the corresponding Lie algebras g ab onȞ . Every Serre relation is detected in one of the Lie algebras g ab . So the generators {E a , F a , H a } satisfy all the relations and we get an action ofĝ onȞ . Finally, we can use the action of T S 1 LG on H (and thus onȞ ) to extend the action ofĝ onȞ to a action of C ĝ.
Let L μ be the irreducible highest weight representation of C ĝ with highest weight μ ∈ + k . We write H (μ) := Hom C ĝ (L μ ,Ȟ ) for the multiplicity space of L μ insidě H , so thatȞ = (Ȟ is a representation of C ĝ satisfying the three conditions listed in Definition 21, and the category of such representations is semi-simple in the sense that every object is a direct sum of irreducible ones [Kac90, Chapt. 9, 10].) The multiplicity space H (μ) can also be described as the joint kernel of the lowering operators F a acting H μ . By this second description, we see that H (μ) is a closed subspace of H μ , and thus a Hilbert space in its own right. LettingL μ be the Hilbert space completion of L μ , we can then upgrade the isomorphism (35) to an isomorphism of Hilbert spaces: (where ⊗ now denotes the Hilbert space tensor product).
Recall the projection π : + k → A k . Two representations L μ and L μ of C ĝ are isomorphic as representations ofĝ if and only if π(μ) = π(μ ). For λ ∈ A k , let H [λ] := 2 π(μ)=λ H (μ). The decomposition (36) then induces a direct sum decomposition This finishes the proof that H is in the essential image of the functor (34).

The based loop group and its representations. Let Rep k
l.n. (LG) be the essential image of the functor (32). We call it the category of locally normal representations of LG at level k. By Theorem 30 (and Remark 31), Conjecture 23 is equivalent to the statement that Rep k l.n. (LG) = Rep k (LG) (the latter was defined in Definition 22). In [Hen15], we introduced the category of locally normal representations 7 of G at level k (Definition 32). We denote it here by Rep k l.n. ( G). In that same preprint, we announced that the Drinfel'd center of the category of locally normal representations of the based loop group at level k is equivalent to the category of locally normal representations of the free loop group at level k, where the latter is equipped with the fusion and braiding inherited from Rep(A G,k ): In the more recent preprint [Hen17], we considered the category T A G,k of solitons of the conformal net A G,k , and proved that Z (T A G,k ) = Rep k l.n. (LG).
In order to complete our proof of (37), we need to identify Rep k l.n. ( G) with T A G,k . This is the main result of the present section. There is an obvious fully faithful functor

Solitons and representations of G
which takes a locally normal representation of G and only remembers the actions of the von Neumann algebras A G,k (I ), p ∈I . Clearly, any λ ∈ U (1) ⊂ G k acts by scalar multiplication by λ. Precomposing by the quotient map G → G k , we get an action of G such that each λ ∈ U (1) ⊂ G acts by λ k . By construction, this is a locally normal representation. By Propositions 10 and 19, these assemble to a strongly continuous action Based on results of Carpi and Weiner [CW05,Wei06], it was proved in [DVIT18, App. A] that the maps Diff R 0 (I ) → A G,k (I ) extend to a certain larger group involving non-smooth diffeomorphisms. Given an interval I , let Diff 1,ps (I ) be the group of orientation preserving piecewise smooth C 1 diffeomorphisms of I whose derivative is 1 at the boundary points. And let Diff 1,ps (S 1 ) be the group of orientation preserving piecewise smooth C 1 diffeomorphisms of S 1 that fix the base point p, and whose derivative is 1 at that point. Let Diff R 1,ps (I ) and Diff R 1,ps (S 1 ) be the corresponding central extensions by R, constructed by using the same cocycle that was used to construct the central extensions of Diff 0 (I ) and of Diff * (I ).
By [DVIT18, App. A], the maps Diff R 0 (I ) → A G,k (I ) extend to the larger group Diff R 1,ps (I ). The proofs in Sect. 2.2 go through with the groups Diff 1,ps (I ), Diff R 1,ps (I ) and Diff R 1,ps (S 1 ) in place of Diff 0 (I ), Diff R 0 (I ) and Diff R * (S 1 ). In particular, the homomorphism (39) extends to a homomorphism Diff R 1,ps (S 1 ) G → U (H ).
Let R ⊂ Diff R 1,ps (S 1 ) be (the canonical lift of) the subgroup of Möbius transformations that fixes p. Upon mapping S 1 to the real line via the stereographic projection that sends p = 1 to ∞, this group gets identified with the group of translations of the real line. We write P for the infinitesimal generator, and call it the energy-momentum operator (if the Hilbert space has an action of Diff R (S 1 ), then the energy-momentum operator is given by P = −L −1 + 2L 0 − L 1 ).

Conjecture 35.
For every locally normal representation H ∈ Rep k l.n. ( G) of the based loop group, the energy-momentum operator P has positive spectrum.
The above conjecture has been recently proven by Del Vecchio, Iovieno, and Tanimoto [DVIT18, Thm 3.4] (and is thus no longer a conjecture).
We define a positive energy representation of the based loop group to be a representation whose energy-momentum operator has positive spectrum: Definition 36. A level k positive energy representation of the based loop group is a continuous representation ρ : G → U (H ) satisfying: 1. ρ is the restriction of a representation R G → U (H ) such that the infinitesimal generator P of the group R has positive spectrum. 2. λ ∈ U (1) ⊂ G acts by scalar multiplication by λ k .
We note that the action of R is not part of the data of a positive energy representation.
The following is a strengthening of Conjecture 35: