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On the Classification of the Coadjoint Orbits of the Sobolev Bott–Virasoro

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On the Classification of the Coadjoint Orbits of the Sobolev Bott–Virasoro On the classification of the coadjoint orbits of the Sobolev Bott–Virasoro group Fran¸coisGay-Balmaz1 AMS Classification: 81R10, 22E65, 58D05, 17B68 Keywords: Bott-Virasoro group, Virasoro algebra, coadjoint orbit, Hilbert manifold, diffeomorphism group. Abstract The purpose of this paper is to give the classification of the Bott-Virasoro coadjoint orbits, with nonzero central charge, in the functional analytic setting of smooth Hilbert manifolds. The central object of the paper is thus the completion of the Bott-Virasoro group with respect to a Sobolev topology, giving rise to a smooth Hilbert manifold and topological group, called the Sobolev Bott–Virasoro group. As a consequence of this approach, analytic and geometric properties of the coadjoint orbits are studied. 1 Introduction The classification of the coadjoint orbits of the Virasoro algebra in the smooth category can be found in different formulations in Lazutkin and Pankratova [1975], Segal [1981], Kirillov [1982], Witten [1988] and others. Balog, Feh´erand Palla [1998] reformulate and complete these results; see also Dai [2000] and Dai and Pickrell [2003] for another approach. In this paper the classification problem of the coadjoint orbits of the Vira- soro algebra is studied in the setting of smooth infinite dimensional Hilbert manifolds. The techniques are extensions of those in Balog, Feh´erand Palla [1998] to the Sobolev manifold category. The classical approach to the Bott–Virasoro group consists in seeing it as a one- C∞ 1 dimensional central extension of the group D+ (S ) of smooth orientation preserving diffeomorphisms of the circle. This group can be endowed with the structure of a smooth infinite dimensional Fr´echet manifold, making it into a regular Lie group. Analysis on Fr´echet manifolds can be made rigorous by using the convenient calculus of Kriegl and Michor [1997], for example. See Guieu and Roger [2007] for an extensive study of the Bott–Virasoro group. Unfortunately, many fundamental results such as the implicit function theorem, valid on finite dimensional manifolds, do not generalize to the case of Fr´echet manifolds, 1Section de Math´ematiques, Ecole´ Polytechnique F´ed´erale de Lausanne. CH–1015 Lausanne. Switzerland. Partially supported by a Swiss NSF grant. [email protected] 1 without appeal to the Nash-Moser theorem. Consequently, it has been of fundamental interest to consider completions of diffeomorphism groups, or more generally, of mani- folds of maps, with respect to Ck, H¨older,or Sobolev Hs topologies, in order to obtain smooth Banach or Hilbert manifolds. This approach was successfully used, for example, in Ebin [1968], Ebin and Marsden [1970], Isenberg and Marsden [1982], Fischer Tromba [1984], Gay-Balmaz and Ratiu [2007], Gay-Balmaz, Marsden and Ratiu [2008] to address different problems in mathematical physics and geometry. The present paper is one more step in this direction. It studies the classification and geometric properties of the Bott–Virasoro coadjoint orbits, in the setting of smooth Hilbert manifolds. More precisely, we consider the Sobolev Bott–Virasoro group, which is the Hr-completion of the smooth Bott–Virasoro group, acting via the coadjoint action on the Hs-completion of the regular part of its dual Lie algebra. We will see that the coadjoint action is well-defined provided r ≥ s + 3 and s > 1/2. When classifying the coadjoint orbits, the natural choice will appear to be r = s + 3. Using this rigorous functional analytic approach new results concerning coadjoint orbits are found. 2 Sobolev vector fields and diffeomorphisms on S1 Consider the circle S1 and its universal covering map x ∈ R 7−→ s = ei2πx ∈ S1. We denote by ∂ the smooth vector field on S1 defined by i2πx i2πx 1 ∂(e ) := i2πe ∈ Tei2πx S . Any vector field u ∈ X(S1) on the circle can be identified with the Z-periodic function u : R −→ R, through the relation u(ei2πx) = u(x)∂(ei2πx). For any real number s ≥ 0 we have the equivalence u ∈ Xs(S1) ⇐⇒ u ∈ Hs( ), Z R where Xs(S1) denotes the Hilbert space of Sobolev class Hs vector fields on S1 and Hs( ) is defined by Z R Hs( ) := {u ∈ Hs ( ) | u is -periodic} . Z R loc R Z s s s Here Hloc(R) denotes the functions which are locally in H , that is, u ∈ Hloc(R) if and only if for each open bounded subset U ⊂ R, there exists v ∈ Hs(U) such that u = v on U. Equivalently we have s s ∞ Hloc(R) = {u : R −→ R | ϕu ∈ H (R), ∀ ϕ ∈ C0 (R)}. In terms of Fourier series, the space Hs( ) is characterized by Z R ( ) X X Hs( ) = a ei2πx |n|2s|a |2 < ∞, a = a . Z R n n n −n n∈Z n∈Z 2 Recall that for k ≥ 0 and s > 1/2 + k, the Sobolev embedding theorem states that we have the continuous inclusions Hs(U) ⊂ Ck(U) and Xs(S1) ⊂ XCk (S1), k for any bounded open subset U of R, where XC (S1) denotes the space of Ck vector fields on S1. Recall also that for s > 1/2 and r ≥ −s, pointwise multiplication on C∞(U) extends to a continuous bilinear map Hr(U) × Hs(U) −→ Hmin{r,s}(U), (u, v) 7−→ uv; (2.1) see Corollary 9.7 of Palais [1968] for a proof. C1 1 1 Denote by D+ (S ) the group of C orientation preserving diffeomorphisms of the C1 1 circle. Any g ∈ D+ (S ) determines, up to an additive constant m ∈ Z, a strictly increasing C1 function g : R −→ R such that g(x + 1) = g(x) + 1 and g(ei2πx) = ei2πg(x). r 1 r For all r > 3/2, we denote by D+(S ) the set of Sobolev class H orientation preserving diffeomorphisms of S1. The equality r 1 C1 1 r 1 1 D+(S ) = D+ (S ) ∩ H (S ,S ), (2.2) proven in Ebin [1968], implies the equivalence g ∈ Dr (S1) ⇐⇒ g ∈ Dr ( ), + + Z R where Dr ( ) := {g ∈ Hr ( ) | g0 > 0, g(x + 1) = g(x) + 1} . + Z R loc R The relation between the derivatives of g and g is given by the equality T g(∂(ei2πx)) = g0(x)∂(g(ei2πx)). r 1 s 1 From now on, we will use the symbols D+(S ), r > 3/2, and X (S ), s > 1/2, but their elements will be denoted by g and u as if they would belong to Dr ( ) and Hs( ). + Z R Z R r 1 Recall that for all r > 3/2, the set D+(S ) can be endowed with the structure of a r 1 r 1 smooth Hilbert manifold modeled on the Hilbert space X (S ). More precisely, D+(S ) is an open subset of the smooth Hilbert manifold Hr(S1,S1). With respect to this r 1 manifold structure, D+(S ) is a topological group with smooth right translation. For r > 1/2 + k, k ≥ 1 we have the smooth Sobolev embedding r 1 Ck 1 D+(S ) ⊂ D+ (S ), Ck 1 k where D+ (S ) denote the smooth Banach manifold of class C orientation preserving diffeomorphisms of S1. The reader can consult Ebin [1968] and Ebin and Marsden [1970] for proofs and details concerning the Hilbert manifold structure of the group of Sobolev diffeomorphisms of a compact manifold. 3 3 The Sobolev Bott–Virasoro group The completion of the Bott–Virasoro group in the Sobolev topology Hs is presented in this section. The Bott–Virasoro group is, up to isomorphism, the unique nontrivial central exten- sion of the diffeomorphism group of the circle by R. This group, denoted by BVir is, as a set, the Cartesian product C∞ 1 BVir = D+ (S ) × R The group multiplication on BVir is defined by (g, α)(h, β) = (g ◦ h, α + β + B(g, h)), C∞ 1 C∞ 1 where B : D+ (S ) × D+ (S ) −→ R is the Bott-Thurston two-cocycle defined by 1 Z B(g, h) := log(g0 ◦ h)d log h0, 2 S1 The Lie algebra vir of BVir is called the Virasoro algebra and is given by the central extension of X(S1) associated to the Gelfand-Fuchs two-cocycle Z 1 1 0 00 C : X(S ) × X(S ) −→ R,C(v, w) = v w . S1 Consequently, the Lie bracket is given by [(v, a), (w, b)] = (v0w − vw0,C(v, w)) . Identifying the dual Lie algebra of vir with itself (regular dual) through the L2 pairing, the coadjoint action reads † 0 2 Ad(g,α)(u, c) = (g ) (u ◦ g) + c S(g), c , where S is the Schwarzian derivative defined by g000 3 g00 2 S(g) := − . g0 2 g0 We now show that for all r > 3/2 it is possible to complete the Bott–Virasoro group with respect to the Sobolev Hr topology. To see this, it suffices to observe that for all r > 3/2, the Bott-Thurston two-cocycle is well defined as a map r 1 r 1 B : D+(S ) × D+(S ) −→ R. r 1 0 r−1 0 r−2 Indeed, for g, h ∈ D+(S ), we have log(g ◦ h) ∈ H and d log h ∈ H . Using the condition r > 3/2, we obtain that the L2-pairing involved in the definition of B makes sense. Thus the Hr completion of the Bott–Virasoro group is given by r r 1 BVir := D+(S ) × R 4 and is called the Sobolev Bott–Virasoro group. It is a smooth Hilbert manifold modeled r r 1 r 1 on the Hilbert space vir := X (S ) × R. Note that D+(S ) is a codimension one r r 1 submanifold of BVir . Using the properties of D+(S ) and Lemma 3.1 below, it is r readily seen that BVir is a topological group with smooth right translations R(g,α).
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