Group of Diffeomorphisms of the Unite Circle As a Principle U(1)-Bundle

Home , Virasoro algebra

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle

Irina Markina, University of Bergen, Norway

Summer school Analysis - with Applications to Mathematical Physics Gottingen¨ August 29 - September 2, 2011

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 1/49 String

Minkowski space-time

moving in time

woldsheet string

string

Worldsheet as an imbedding of a cylinder C into the Minkowski space-time with the induced metric g.

2 Nambu-Gotô action SNG = −T dσ | det gαβ| ZC p T is the string tension.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 2/49 Polyakov action

Change to the imbedding independent metric h on worldsheet.

2 αβ Polyakov action SP = −T dσ | det hαβ|h ∂αx∂βx, ZC p

0 1 δSP α, β = 0, 1, x = x(σ ,σ ). Motion satisﬁed δhαβ = 0.

−1 δSP Energy-momentum tensor Tαβ = αβ T | det hαβ| δh p Tαβ = 0 and SP = SNG, whereas in general SP ≥ SNG

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 3/49 Gauges

The metric h has 3 degrees of freedom - gauges that one need to ﬁx. • Global Poincaré symmetries - invariance under Poincare group in Minkowski space • Local invariance under the reparametrizaition by 2D-diffeomorphisms dσ˜2 | det h˜| = dσ2 | det h| • Local Weyl rescaling p p α β ρ(σ0,σ1) α β hαβdσ dσ → e hαβdσ dσ

ρ(σ0,σ1) 2 αβ ν hαβ = e ηαβ, SP = −T dσ η ην∂αx ∂βx ZC

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 4/49 Virasoro constraints

−1 δSP Tαβ = αβ = 0 T | det hαβ| δh p T00 = T11 = 0 are Virasoro constraints. Introducing the complex variables on the worldsheet

Tzz = T00 + iT10 is analytic function and

Ln Tzz = , [Lm,Ln] = i(n − m)Ln m zn+2 + Xn∈Z

Ln are Virasoro generators that form the Witt algebra

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 5/49 Diﬀ S1 as a manifold

• Diﬀ S1 is the set of orientation preserving diffeomorphisms f : S1 → S1

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 6/49 Diﬀ S1 as a manifold

• Diﬀ S1 is the set of orientation preserving diffeomorphisms f : S1 → S1 • C∞(S1) is the space of C∞-functions ϕ: S1 → S1 with Frechét topology given by seminorms (m) 1 ϕm = sup{|ϕ (θ)| | θ ∈ S }, m = 0, 1,...

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 6/49 Diﬀ S1 as a manifold

• (Diﬀ S1, ◦) is a group, where f ◦ φ = f(φ) is the composition, f(θ) = θ is the identity, f −1 is the inverse element

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 6/49 Diﬀ S1 as a Lie-Frechét group Model vector space is the Frechet vector space

Vect(S1) ∼= C∞(S1, R) 1 v(θ)∂θ ∼ v : S → R

1 V0(0) = {v ∈ Vect(S ) | v≤ π}

∞ 1 1 1 U0(id) = {f ∈ C (S , S ) | f(θ) = −θ, for all θ ∈ S },

1 1 ψ : V0 → U0, ψv : S → S

v(θ) l(arc) = v(θ)

b θ b b ψ (θ) ψ v

v(θ) b

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 7/49 Diﬀ S1 as a Lie-Frechét group

Choose open U ∈ U0(id) consisting of diffeomorphisms.

−1 Then ψ (U) = V ∈ V0(0) is open and (U, ψ−1) is the chart around id −1 (Uf ,ψ ), (Uf ) = f.U is the chart around f

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 8/49 Diﬀ S1 as a Lie-Frechét group

1 1 diffS =(Tid Diﬀ S , [, ]) is the Lie-Frechét algebra

diffS1 ∼= (Vect S1, −[, ])

µ: Diff S1 × S1 → S1 f.θ → f(θ). Left action of Diff S1 on S1 produces Vect(S1) as right invariant (under the action of Diff S1) vector fields on S1. This explains the opposite sign.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 9/49 Exponential map

exp: Vect S1 → Diﬀ S1 sends the vectors

tv(θ) ∈ Vect S1 → γ(t, θ) ∈ Diﬀ S1 one parameter subgroups γ(t, θ): R × S1 → Diﬀ S1

1 γ(t1 + t2, θ) = γ(t1, θ) ◦ γ(t2, θ), t ∈ R, θ ∈ S

dγ(t, θ) = v(θ), γ(0, θ) = θ dt t=0

γ (t, θ) = exp(tv(θ))

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 10/49 Exponential map

• For ﬁnite dimensional Lie groups the exponential map is always diffeomorphism near the origin.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 11/49 Exponential map

• For ﬁnite dimensional Lie groups the exponential map is always diffeomorphism near the origin.

• exp: Vect S1 → Diﬀ S1 is neither injective nor surjective in any nb. of the origin

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 11/49 Exponential map

• For ﬁnite dimensional Lie groups the exponential map is always diffeomorphism near the origin.

• exp: Vect S1 → Diﬀ S1 is neither injective nor surjective in any nb. of the origin • π 2 f(θ) = θ + n + ε sin (nθ). If f has no ﬁxed points then it is conjugate to rotations.

π 2 θ2 θ1 3π π 4 4 θ θ 0 π 3 0 = 2π

θ7 θ4 5π 7π 4 θ6 4 3π θ5 2

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 11/49 exp is not injective

2π 1 1 Let fn be a rotation on n . Then fn ∈ S ⊂ Diff(S ). Let 2π 2π H = {φ ∈ Diff(S1) | φ θ + = φ(θ) + } n n be the subgroup of Diff S1 of all periodic 2π diffeomorphisms with period n .

fn commutes with H. Then 1 −1 1 −1 S ∋ fn = φfnφ ∈ φS φ , φ ∈ H fn belongs to all one-parametric subgroups from φS1φ−1, φ ∈ H.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 12/49 Central extension of Vect(S1)

The central extension g of a Lie algebra g by the Lie algebra (R, +) is e (g × R, [(ξ, a)(η,b)]ge) ξ,η ∈ g, a,b ∈ R satisfying the axioms of the Lie algebra.

The simplest trivial example is the direct product

g × R with the Lie brackets deﬁned by

[(ξ, a)(η,b)]eg := ([ξ,η]g, ab − ba)=([ξ,η]g, 0).

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 13/49 Central extension of Vect(S1)

The central extension g of g by (R, +) is

(g × R, [(ξ, a)(η,b)]ge)e [(ξ, a)(η,b)]ge) = [ξ,η], ω(ξ,η) satisfying the axioms of the Lie algebra: bi-linearity, skew symmetry, Jacobi identity, that gives cocycle condition,

ω([ξ,η], ζ) + ω([η, ζ], ξ) + ω([ζ, ξ],η).

The form ω is called 2-cocycle or Gelfand-Fuchs co-cycle

2π 2π ′ ′′ ′ ′′′ ω v(θ)∂θ, u(θ)∂θ = v (θ)u (θ) dθ = (v + v )u dθ Z Z 0 0 Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 14/49 Central extension of Vect(S1)

Central extension of Vect(S1) is called Virasoro algebra vir (v(θ)∂θ, a) ∈ vir Central extension of Vect(S1)

Central extension of Vect(S1) is called Virasoro algebra vir (v(θ)∂θ, a) ∈ vir

Is there a Lie group that has Virasoro algebra as it Lie algebra? Central extension of Vect(S1)

Central extension of Vect(S1) is called Virasoro algebra vir (v(θ)∂θ, a) ∈ vir

Is there a Lie group that has Virasoro algebra as it Lie algebra?

The correct group is the central extension

Vir of Diﬀ S1 by (R, +) that received the name Virasoro – Bott group.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 15/49 Virasoro – Bott group

Central extension Vir of (Diﬀ S1, ◦) by R

F F F F 1 −→0 R −→1 Vir −→2 Diﬀ S1 −→3 1.

im Fi = ker Fi+1, F1(R) is the center in Vir Virasoro – Bott group

Central extension Vir of (Diﬀ S1, ◦) by R

F F F F 1 −→0 R −→1 Vir −→2 Diﬀ S1 −→3 1.

im Fi = ker Fi+1, F1(R) is the center in Vir Vir = (Diﬀ S1 × R) and the multiplication

(f, a)(g,b)=(f ◦ g,ab + w(f,g)),

w : Diﬀ S1 × Diﬀ S1 → R such that the product becomes associative

2π ′ ′ w(f,g) = log(f ◦ g) d log g Z0

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 16/49 Vir and KdV equation

Find the geodesic equation on Vir endowed with

(v1(θ)∂θ, a1), (v2(θ)∂θ, a2) L2 = v1(θ)v2(θ) dθ + a1a2. Z 1 S 2 ∗ • (v(θ)∂θ,b) ∈ vir (u(θ)(dθ) , a) ∈ vir

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 17/49 Vir and KdV equation

Find the geodesic equation on Vir endowed with

(v1(θ)∂θ, a1), (v2(θ)∂θ, a2) L2 = v1(θ)v2(θ) dθ + a1a2. Z 1 S 2 ∗ • (v(θ)∂θ,b) ∈ vir (u(θ)(dθ) , a) ∈ vir ∗ ∗ • adη(ξ), ω = −ξ, adη(ω), η, ξ ∈ g, ω ∈ g

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 17/49 Vir and KdV equation

Find the geodesic equation on Vir endowed with

(v1(θ)∂θ, a1), (v2(θ)∂θ, a2) L2 = v1(θ)v2(θ) dθ + a1a2. Z 1 S 2 ∗ • (v(θ)∂θ,b) ∈ vir (u(θ)(dθ) , a) ∈ vir ∗ ∗ • adη(ξ), ω = −ξ, adη(ω), η, ξ ∈ g, ω ∈ g • 2 u(θ)(dθ) , a , v(θ)∂θ,b = S1 v(θ)u(θ) dθ + ab. ∗ R ad u(θ)(dθ)2, a = (−2v′u−vu′−av′′′)(dθ)2, 0 . v(θ)∂θ,b

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 17/49 Vir and KdV equation

Find the geodesic equation on Vir endowed with

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 17/49 Vir and KdV equation

∗ ω˙ (t) = ad (ω(t)) dω(t)H ∗ ∗ ∼ ∗ ∼ dω(t)H ∈ Tω(g ) = g = g d H =(u(θ)∂θ, a) u(θ)(dθ)2,a Vir and KdV equation

∗ ω˙ (t) = ad (ω(t)) dω(t)H ∗ ∗ ∼ ∗ ∼ dω(t)H ∈ Tω(g ) = g = g d H =(u(θ)∂θ, a) u(θ)(dθ)2,a ∗ ad u(θ)(dθ)2, a = (−2v′u − vu′ − av′′′)(dθ)2, 0 . v(θ)∂θ,b ⇓ d u(θ)2, a = (−3uu′ − au′′′)(dθ)2, 0 . dt

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 18/49 Vir and KdV equation

d u(θ)2, a = (−3uu′ − au′′′)(dθ)2, 0 . dt ⇓

′ ′′′ ut = −3uu − au , a˙ = 0. The ﬁrst equation is the Karteweg-de Vries (KdV) nonlinear evolution equation that describes traveling waves in a shallow canal.

The second equation is just saying that the parameter a is the real constant.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 19/49 Other interesting equations.

Weighted family of metrics (, ) 1 can be deﬁned Hα,β

′ ′ (1) (v, a), (u, b) H1 = αvu + βv u dθ + ab. α,β Z 1 S THEOREM The Euler equations for the right invariant metric (, ) 1 , α = 0 on the Virasoro – Bott group are Hα,β given by

′ ′′ ′ ′′ ′′′ ′′′ α(ut + 3uu ) − β ((u ))t + 2u u + uu + au = 0 at = 0, for (u(θ,t)∂θ, a(t)) ∈ Vir for each t ∈ I.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 20/49 Other interesting equations.

′ ′′ ′ ′′ ′′′ ′′′ α(ut + 3uu ) − β ((u ))t + 2u u + uu + au = 0 at = 0, for (u(θ,t)∂θ, a(t)) ∈ Vir for each t ∈ I.

α = 1, β = 0 KdV equation. α = β = 1 Camassa-Holm equation. α = 0, β = 1 Hunter-Saxton equation.

If α = 0, the metric (, ) 1 becomes homogeneous Hα,β degenerate (, )H˙ 1 metric and one has to pass to Diﬀ S1/S1.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 21/49 Diﬀ S1/S1

Let S1 ⊂ Diﬀ S1 be the closed subgroup of rotations. S1 acts on the right:

µ: Diﬀ S1 × S1 → Diﬀ S1 f.τ → f ◦ τ = f(τ).

Diﬀ S1/S1 has a manifold structure.

Since the group S1 is not a normal subgroup, then the manifold Diﬀ S1/S1 has no any group structure.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 22/49 Tangent space of Diﬀ S1/S1

For the Lie-Frechét group Diff S1 corresponds the Lie algebra Vect(S1). Denote by u(1) the Lie algebra of S1 = U(1). The space u(1) consists of constant vector fields on the circle. 1 1 1 Then Tid Diff S /S = Vect(S )/u(1). The latter space is the space of vector fields with vanishing mean value on the circle. By making use of the right action

µ: Diff S1/S1 × Diff S1 → Diff S1/S1 h.f → h ◦ f = h(f). we get the tangent space at each point f ∈ Diff S1/S1.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 23/49 Diﬀ S1 as sub-Riemannian manifold

π : Diﬀ S1 → Diﬀ S1/S1 is the principal U(1)-bundle,

The vertical distribution V = ker(dπ) consists of constant vector ﬁelds. Notice that Vf isomorphic to the Lie algebra of the group S1.

The Ehresmann connection D = Vect(S1)/u(1) is formed by vector ﬁelds v(θ)∂θ with vanishing mean value: 1 2π v(θ)dθ = 0 2π Z0

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 24/49 Diﬀ S1 as sub-Riemannian manifold

The family of Kähler metrics on Diff S1/S1 are defined by making use of the co-adjoint action of the group Diff S1 on the dual space vir∗. The orbit of the point α(dθ)2, β under the co-adjoint ∗ 1 1 action of AdDiff S1 is isomorphic to Diff S /S if n2 N α/β = − 2 , n ∈ . This leads to the existence of 2 parametric family of symplectic structures ωα,β. The almost complex structure J on Vect(S1)/u(1) is invariant under the action of Diff S1 and the symplectic forms ωα,β are compatible and generate the Kähler 1 metric gα,β on Vect(S )/u(1).

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 25/49 Diﬀ S1 as sub-Riemannian manifold

This metric for v, u ∈ Vect(S1)/u(1) at f = id ∈ Diﬀ S1 is

∞ 3 gα,β(v∂θ, u∂θ) = (αn + βn )an¯bn, Xn=1

∞ inθ ∞ inθ where v(θ) = n=1 ane , u(θ) = n=1 bne . P P 1 Extend gα,β to D = dr Vect(S )/u(1) by making use of right action. If β ≥ 0 and −α < β, then the metric is positively deﬁnite. We work with α = 1, β = 1 and denote the metric by gD.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 26/49 Diﬀ S1 as sub-Riemannian manifold

1 We get the sub-Riemannian manifold (Diff S ,D,gD). Define 1 2π η(v∂θ) = v(θ) dθ 2π Z0 the mean value of any vector fields v ∈ Vect(S1). Functional η measures a deviation of vector field v ∈ Vect(S1) from being horizontal. Then the metric gVect(S1)(v∂θ, u∂θ) := gD v−η(v) ∂θ, u−η(u) ∂θ +η(v∂θ)η(u∂θ). is of bi-invariant type on π : Diff S1 → Diff S1/S1.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 27/49 Normal geodesics

To write the normal geodesic on Diff S1 we define the u(1)-valued connection form A. Let γ : I → Diff S1, γ˙ (t, θ) ∈ Vect(S1). The value

γ˙ (t, θ) ξ(t, θ) = γ′(t, θ) is left logarithmic derivative of γ(t, θ). The left logarithmic derivative is the analogous of −1 1 1 σ : Tγ(t) Diﬀ S ∋ γ˙ (t) → ξ(t) ∈ Vect S . Then we project ξ(t) on the vertical distribution by η(ξ(t)). η(ξ(t)) does not depend on θ and it generates rotations in Diﬀ S1.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 28/49 Normal geodesics

1 All normal sub-Riemannian geodesics γsR on Diﬀ S are given by the formula

γsR(t) = φ(t)expu1 − tη(ξ(t)) , φ is the Riemannian geodesic with respect to gVect(S1) and ξ(t) is the left logarithmic derivative of φ.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 29/49 Geometrical objects

• The group Diﬀ S1 of orientation preserving diffeomorphism of the unit circle S1;

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 30/49 Geometrical objects

• The group Diﬀ S1 of orientation preserving diffeomorphism of the unit circle S1;

• Its central extension Vir = Diﬀ S1 ⊕ R – Virasoro-Bott group;

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 30/49 Geometrical objects

• The group Diﬀ S1 of orientation preserving diffeomorphism of the unit circle S1;

• Its central extension Vir = Diﬀ S1 ⊕ R – Virasoro-Bott group;

• Homogeneous space Diff S1/S1 – Kirillov’s manifold; • Groups Diff S1 and Vir and the homogeneous manifold Diff S1/S1 are modeled on Fréchet spaces.

... and their inﬁnitesimal representations.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 30/49 Geometrical objects

• Vir = Diﬀ S1 ⊕ R −→ vir;

• Diﬀ S1 −→ Vect(S1);

• Diﬀ S1/S1 −→ Vect(S1)/u(1).

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 31/49 Geometrical objects

Complexifcation: • (Vir, vir(1,0))

vir(1,0) ⊕ vir(0,1) = vir ⊗ C

• (Diﬀ S1, h(1,0))

(1,0) (0,1) 1 h ⊕ h = corank1(Vect(S ) ⊗ C)

• (Diﬀ S1/S1, h(1,0))

h(1,0) ⊕ h(0,1) = Vect(S1)/u(1) ⊗ C

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 31/49 Complexiﬁcation of real manifold

• TqM ⊗ C that is

(TqM × TqM), (a, b)(v1,v2)=(av1 − bv2, av2 + bv1)

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 32/49 Complexiﬁcation of real manifold

• TqM ⊗ C that is

(TqM × TqM), (a, b)(v1,v2)=(av1 − bv2, av2 + bv1)

2 • If J : TqM → TqM is such that J = −Id then

(1,0) Tq = {v − iJ(v) ∈ TqM ⊗ C | q ∈ M, v ∈ TqM},

(0,1) Tq = {v + iJ(v) ∈ TqM ⊗ C | q ∈ M, v ∈ TqM},

(1,0) (0,1) TqM ∼= Tq M ∼= Tq M

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 32/49 Complexiﬁcation of real manifold

• TqM ⊗ C that is

(TqM × TqM), (a, b)(v1,v2)=(av1 − bv2, av2 + bv1)

2 • If J : TqM → TqM is such that J = −Id then

(1,0) Tq = {v − iJ(v) ∈ TqM ⊗ C | q ∈ M, v ∈ TqM},

(0,1) Tq = {v + iJ(v) ∈ TqM ⊗ C | q ∈ M, v ∈ TqM},

(1,0) (0,1) TqM ∼= Tq M ∼= Tq M (1,0) (0,1) • TqM ⊗ C = Tq M ⊕ Tq M,

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 32/49 Integrable almost complex structure

(1,0) (1,0) (1,0) If [Tq M,Tq M] ∈ Tq M then the pair

(M,T (1,0)M) complex manifold Integrable almost complex structure

(1,0) (1,0) (1,0) If [Tq M,Tq M] ∈ Tq M then the pair

(M,T (1,0)M) complex manifold

Lie group G and Lie algebra g = TeG • g ⊗ C • g ⊗ C = g(1,0) ⊕ g(0,1) • g ⊗ C is integrable if g(1,0) is sub-algebra of g ⊗ C:

[g(1,0), g(1,0)] ⊂ g(1,0)

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 33/49 Cauchy-Riemann structure

Let N be a manifold.

TN ⊗ C → H ⊂ TN ⊗ C of corank 1 Cauchy-Riemann structure

Let N be a manifold.

TN ⊗ C → H ⊂ TN ⊗ C of corank 1

H = H(1,0) ⊕ H(0,1), [H(1,0),H(1,0)] ⊂ H(1,0) Cauchy-Riemann structure

Let N be a manifold.

TN ⊗ C → H ⊂ TN ⊗ C of corank 1

H = H(1,0) ⊕ H(0,1), [H(1,0),H(1,0)] ⊂ H(1,0)

CR-structure (N,H(1,0)) is strongly pseudoconvex if

(1,0) (0,1) (1,0) [X, X¯]q ∈/ Hq ⊕ Hq , ∀ X ∈ H , Xq = 0

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 34/49 Left invariant C-R structure

Let G be a Lie group.

g ⊗ C → h ⊂ g ⊗ C of corank 1 Left invariant C-R structure

Let G be a Lie group.

g ⊗ C → h ⊂ g ⊗ C of corank 1

h = h(1,0) ⊕ h(0,1), [h(1,0), h(1,0)] ⊂ h(1,0) Left invariant C-R structure

Let G be a Lie group.

g ⊗ C → h ⊂ g ⊗ C of corank 1

h = h(1,0) ⊕ h(0,1), [h(1,0), h(1,0)] ⊂ h(1,0)

CR-structure (G, h(1,0)) is strongly pseudoconvex if

(1,0) (0,1) (1,0) [X, X¯]q ∈/ hq ⊕ hq , ∀ X ∈ h , Xq = 0

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 35/49 Complexiﬁed geometric structures

Vect(S1) = span{cos(nθ), sin(nθ)}, n = 0, 1, 2,...

1 inθ Vect(S ) ⊗ C = span{en = −ie }, n ∈ Z

[em,en]=(n − m)em+n, Witt algebra Vect(S1) ⊗ C does not correspond to any Lie group.

Any smooth complex vector ﬁeld on S1 can be integrated to a curve in the space of maps C∞(S1, C), but the last one does not form a group with respect to composition.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 36/49 Complexiﬁed geometric structures

• If v ∈ Vect(S1)/s1 then

∞

v(θ) = an cos(nθ) + bn sin(nθ) Xn=1

• The map J : Vect(S1)/u(1) → Vect(S1)/u(1):

∞

J(v) = bn cos(nθ) − an sin(nθ) Xn=1 ∞ • (1,0) inθ H = {v − iJ(v) = n=1 cne }, cn = an − ibn, (0,1) ∞ −inθ H = {v + iJ(v) = Pn=1 c¯ne } P Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 37/49 Complexiﬁed geometric structures

(Diﬀ S1/S1,H(0,1)) is the complex manifold

(Diﬀ S1, h(0,1)) is the CR manifold h0,1 = H0,1 = span{einθ, n = 1, 2,...} Moreover ∞ ∞ ∞ inθ −inθ i(n−k)θ cne ∂θ, c¯ne ∂θ = i (k + n)cnc¯ke ∂θ h Xn=1 Xn=1 i k,nX=1

(1,0) (1,0) is not in h ⊕ h unless all cn = 0.

(Diﬀ S1, h(1,0)) is strongly pseudoconvex.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 38/49 Complexiﬁcation of vir

inθ vir ⊗ C = spanC{e , c}, n ∈ Z.

[(v,αc), (u, βc)] = [v, u], ωC(v, u)c , [v, c] = 0, v, u ∈ Vect(S1) ⊗ C, α,β ∈ C and ωC is the complex valued 2-cocycle.

3 imθ inθ κ(n − n) if n + m = 0, ωC(−ie ∂θ − ie ∂θ) = 0 if n + m = 0, where κ is a constant dependent on the underground physical theory. The complexiﬁcation vir ⊗ C of vir is also called Virasoro algebra and it is more useful in physics then the real Virasoro algebra. Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 39/49 Vir as a complex group

Virasoro-Bott group Vir admits a left invariant complex structure. It means that vir ⊕ C admits

vir ⊕ C = vir(1,0) ⊕ vir(0,1) and the manifold (Vir, vir(1,0)) is the complex group.

∞ (1,0) inθ vir = ane ∂θ , αa0c ∈ vir ⊕ C n Xn=0 o and vir(0,1) = vir(1,0).

(0,1) (0,1) vir = h for a0 = 0.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 40/49 Complexiﬁed geometric structures

• (Vir, vir(1,0)) vir(1,0) ⊕ vir(0,1) = vir ⊗ C; Inﬁnite dimensional complex Lie-Frechét group

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 41/49 Complexiﬁed geometric structures

• (Vir, vir(1,0)) vir(1,0) ⊕ vir(0,1) = vir ⊗ C; Inﬁnite dimensional complex Lie-Frechét group

• (Diﬀ S1, h(1,0)) h(1,0) ⊕ h(0,1) ⊂ (Vect(S1) ⊗ C) of complex corank 1; Inﬁnite dimensional left invariant C-R structure

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 41/49 Complexiﬁed geometric structures

• (Vir, vir(1,0)) vir(1,0) ⊕ vir(0,1) = vir ⊗ C; Inﬁnite dimensional complex Lie-Frechét group

• (Diﬀ S1, h(1,0)) h(1,0) ⊕ h(0,1) ⊂ (Vect(S1) ⊗ C) of complex corank 1; Inﬁnite dimensional left invariant C-R structure

• (Diﬀ S1/S1, h(1,0))

h(1,0) ⊕ h(1,0) = Vect(S1)/u(1) ⊗ C Inﬁnite dimensional complex Frechét homogeneous manifold Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 41/49 Relation to analytic functions

∞ • A0 = {f : D → C | f ∈ C (D), f ∈ Hol(D), f(0) = 0}

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 42/49 Relation to analytic functions

∞ • A0 = {f : D → C | f ∈ C (D), f ∈ Hol(D), f(0) = 0}

∞ n • F = {f ∈A0 and univalent}, f(z) = cz(1 + cnz ) nP=1

F⊂A0 is an open subset

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 42/49 Relation to analytic functions

∞ • A0 = {f : D → C | f ∈ C (D), f ∈ Hol(D), f(0) = 0}

∞ n • F = {f ∈A0 and univalent}, f(z) = cz(1 + cnz ) nP=1

F⊂A0 is an open subset

∞ ′ iφ n • F1 = {f ∈F and |f (0)| = 1} f(z) = e z(1 + cnz )) nP=1

F1 ⊂F is a pseudoconvex surface

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 42/49 Relation to analytic functions

∞ • A0 = {f : D → C | f ∈ C (D), f ∈ Hol(D), f(0) = 0}

∞ n • F = {f ∈A0 and univalent}, f(z) = cz(1 + cnz ) nP=1

F⊂A0 is an open subset

∞ ′ iφ n • F1 = {f ∈F and |f (0)| = 1} f(z) = e z(1 + cnz )) nP=1

F1 ⊂F is a pseudoconvex surface ∞ ′ n • F0 = {f ∈F and f (0) = 1}, f(z) = z(1 + cnz )) nP=1

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 42/49 Relation to analytic functions

•

(Vir, vir(1,0)) ←→ (F,T (1,0)F) is biholomorphic

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 43/49 Relation to analytic functions

•

(Vir, vir(1,0)) ←→ (F,T (1,0)F) is biholomorphic

•

1 (1,0) (1,0) (Diﬀ S , h ) ←→ (F1,T F1) is C-R map

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 43/49 Relation to analytic functions

•

(Vir, vir(1,0)) ←→ (F,T (1,0)F) is biholomorphic

•

1 (1,0) (1,0) (Diﬀ S , h ) ←→ (F1,T F1) is C-R map

•

1 1 (1,0) (1,0) (Diﬀ S /S , h ) ←→ (F0,T F0) is biholomorphic

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 43/49 Univalent Functions

• Realization Diﬀ S1/S1 via conformal welding:

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 44/49 Univalent Functions

• Realization Diﬀ S1/S1 via conformal welding:

η y

2 S1 z = f(ζ) = ζ + c1ζ + ... Γ ξ x 0 1 0 U Ω

1 z = g(ζ) = a1ζ + a0 + a−1 ζ + ...

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 44/49 Univalent Functions

• Realization Diﬀ S1/S1 via conformal welding:

η y

2 S1 z = f(ζ) = ζ + c1ζ + ... Γ ξ x 0 1 0 U Ω

1 z = g(ζ) = a1ζ + a0 + a−1 ζ + ...

−1 1 1 1 1 • γ = f ◦ g|S1 ∈ Diﬀ S /S , f ∈F0 ⇆ γ ∈ Diﬀ S /S .

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 44/49 Principal bundles

1. The bundle π : Diff S1 → Diff S1/S1 is the principal U(1)-bundle 2. The bundle Π: Vir → Diff S1/S1 is the trivial C∗-bundle.

C⋆ o ? _F o F / Vir / C⋆ R R R R prby prby prby prby 1 o _ o F1 / 1 / 1 S ? F1 Diﬀ S S

prF prDiﬀ S1/S1 0 o F0 / 1 1 F0 Diﬀ S /S .

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 45/49 Inﬁnitesimal action

Right action

µ: Diff S1/S1 × Diff S1 → Diff S1/S1 h.f → h ◦ f = h(f).

This action is transferred to the right action over F0. 1 The inﬁnitesimal generator σf : Vect(S ) → Tf F0 is given by the variational formula of A. C. Schaeffer and D. C. Spencer

2 f 2(ζ) wf ′(w) v(w)dw , 2π Z f(w) w(f(w) − f(z)) S1

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 46/49 Kirillov’s vector ﬁelds

Schaeffer and Spencer linear operator

f 2(ζ) wf ′(w) 2 v(w)dw , 2π Z f(w) w(f(w) − f(z)) S1

1 that maps Vect(S ) ⊗ C −→ Tf F0 ⊗ C.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 47/49 Kirillov’s vector ﬁelds

Schaeffer and Spencer linear operator

f 2(ζ) wf ′(w) 2 v(w)dw , 2π Z f(w) w(f(w) − f(z)) S1

1 that maps Vect(S ) ⊗ C −→ Tf F0 ⊗ C.

k (1,0) • Taking Fourier basis vk = −iz , k = 1, 2,... for T , we obtain k+1 ′ Lk[f](z) = z f (z).

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 47/49 Kirillov’s vector ﬁelds

Schaeffer and Spencer linear operator

f 2(ζ) wf ′(w) 2 v(w)dw , 2π Z f(w) w(f(w) − f(z)) S1

1 that maps Vect(S ) ⊗ C −→ Tf F0 ⊗ C.

−k (0,1) • Taking v−k = −iz , k = 1, 2,... for T , we obtain

L−k[f](ζ) = very difﬁcult expressions.

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 47/49 Kirillov’s vector ﬁelds

• Virasoro commutation relation

c 2 [L ,L ] =(m − n)L + n(n − 1)δ − , m n vir m+n 12 n, m ′ c ∈ C. L0[f](z) = zf (z) − f(z) corresponds to rotation.

• In afﬁne coordinates (c1,c2,..., ) we get Kirillov’s operators for n = 1, 2,... :

∞

Ln = ∂n + (k + 1)ck∂n+k, ∂k = ∂/∂ck, Xk=1

Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 48/49 The end

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Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 49/49