Group of diffeomorphisms of the unite circle as a principle …Group of diffeomorphisms of the unite circle as a principle U(1)-bundle Irina Markina, University of Bergen, Norway Summer

Group of diffeomorphisms of the unite circleas 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

string

string

moving in time

woldsheet

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

Nambu-Gotô action SNG = −T

∫

C

dσ2√| det gαβ|

T is the string tension.


Polyakov action

Change to the imbedding independent metric h onworldsheet.

Polyakov action SP = −T

∫

C

dσ2√| det hαβ |h

αβ∂αx∂βx,

α, β = 0, 1, x = x(σ0, σ1). Motion satisfied δSP

δhαβ = 0.

Energy-momentum tensor Tαβ =−1

T√| det hαβ |

δSPδhαβ

Tαβ = 0 and SP = SNG,

whereas in general SP ≥ SNG


Gauges

The metric h has 3 degrees of freedom - gauges thatone need to fix.• Global Poincaré symmetries - invariance under

Poincare group in Minkowski space• Local invariance under the reparametrizaition by

2D-diffeomorphisms dσ2√| det h| = dσ2

√| det h|

• Local Weyl rescalinghαβdσ

αdσβ 7→ eρ(σ0,σ1)hαβdσ

αdσβ

hαβ = eρ(σ0,σ1)ηαβ , SP = −T

∫

C

dσ2ηαβηµν∂αxµ∂βx

ν


Virasoro constraints

Tαβ =−1

T√| det hαβ |

δSPδhαβ

= 0

T00 = T11 = 0 are Virasoro constraints.

Introducing the complex variables on the worldsheet

Tzz = T00 + iT10 is analytic function and

Tzz =∑

n∈Z

Ln

zn+2, [Lm, Ln] = i(n−m)Ln+m

Ln are Virasoro generators that form the Witt algebra


Diff S1 as a manifold

• Diff S1 is the set of orientation preservingdiffeomorphisms f : S1 → S1




• C∞(S1) is the space of C∞-functions ϕ : S1 → S1

with Frechét topology given by seminorms‖ϕ‖m = sup|ϕ(m)(θ)| | θ ∈ S1, m = 0, 1, . . .






• Diff S1 ⊂ C∞(S1) is an open subset and by this itinherits the Frechét topology






• Diff S1 ⊂ C∞(S1) is an open subset and by this itinherits the Frechét topology

• (Diff S1, ) is a group, where f φ = f(φ) is thecomposition, f(θ) = θ is the identity, f−1 is theinverse element


Diff S1 as a Lie-Frechét group

Model vector space is the Frechet vector space

Vect(S1) ∼= C∞(S1,R)

v(θ)∂θ ∼ v : S1 → R

V0(0) = v ∈ Vect(S1) | ‖v‖ ≤ π

U0(id) = f ∈ C∞(S1, S1) | f(θ) 6= −θ, for all θ ∈ S1,

ψ : V0 → U0, ψv : S1 → S1

b

b

b

b

v(θ)

v(θ)

ψ

θ

ψv(θ)

l(arc) = ‖v(θ)‖



Choose open U ∈ U0(id) consisting ofdiffeomorphisms.

Then ψ−1(U) = V ∈ V0(0) is open

and(U,ψ−1) is the chart around id

(Uf , ψ−1), (Uf ) = f.U is the chart around f



diffS1 = (TidDiff S1, [·, ·]) is the Lie-Frechét algebra

diffS1 ∼= (VectS1,−[·, ·])

µ : Diff S1 × S1 → S1

f.θ 7→ f(θ).

Left action of Diff S1 on S1 produces Vect(S1) as rightinvariant (under the action of Diff S1) vector fields onS1.

This explains the opposite sign.


Exponential map

exp: VectS1 → Diff S1 sends the vectors

tv(θ) ∈ VectS1 → γ(t, θ) ∈ Diff S1

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

γ(t1 + t2, θ) = γ(t1, θ) γ(t2, θ), t ∈ R, θ ∈ S1

dγ(t, θ)

dt

∣∣∣t=0

= v(θ), γ(0, θ) = θ

γ(t, θ) = exp(tv(θ))


Exponential map

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


Exponential map


• exp: VectS1 → Diff S1 is neither injective norsurjective in any nb. of the origin


Exponential map


• exp: VectS1 → Diff S1 is neither injective norsurjective in any nb. of the origin

• f(θ) = θ + πn + ε sin2(nθ). If f has no fixed points

then it is conjugate to rotations.

0 = 2π

π4

π2

3π4

π

5π4

3π2

7π4

θ0

θ1θ2

θ3

θ4

θ5

θ6

θ7


exp is not injective

Let fn be a rotation on 2πn . Then fn ∈ S

1 ⊂ Diff(S1). Let

H = φ ∈ Diff(S1) | φ(θ +

2π

n

)= φ(θ) +

2π

n

be the subgroup of Diff S1 of all periodicdiffeomorphisms with period 2π

n .

fn commutes with H.

ThenS1 ∋ fn = φfnφ

−1 ∈ φS1φ−1, φ ∈ H

fn belongs to all one-parametric subgroups fromφS1φ−1, φ ∈ H.


Central extension of Vect(S1)

The central extension g of a Lie algebra g by the Liealgebra (R,+) is

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



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

(g×R, [(ξ, a)(η, b)]g) [(ξ, a)(η, b)]g) =([ξ, η], ω(ξ, η)

)

satisfying the axioms of the Lie algebra: bi-linearity,skew symmetry, Jacobi identity, that gives cocyclecondition,

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

The form ω is called 2-cocycle or Gelfand-Fuchsco-cycle

ω(v(θ)∂θ, u(θ)∂θ

)=

∫ 2π

0

v′(θ)u′′(θ) dθ =

∫ 2π

0

(v′ + v′′′)u dθ



Central extension of Vect(S1) is called Virasoroalgebra vir

(v(θ)∂θ, a) ∈ vir




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




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

The correct group is the central extension

Vir of Diff S1 by (R,+)

that received the name Virasoro – Bott group.


Virasoro – Bott group

Central extension Vir of (Diff S1, ) by R

1F0−→ R

F1−→ VirF2−→ Diff S1 F3−→ 1.

imFi = kerFi+1, F1(R) is the center in Vir

Virasoro – Bott group

Central extension Vir of (Diff S1, ) by R

1F0−→ R

F1−→ VirF2−→ Diff S1 F3−→ 1.

imFi = kerFi+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

w(f, g) =

∫ 2π

0

log(f g)′ d log g′


Vir and KdV equation

Find the geodesic equation on Vir endowed with

((v1(θ)∂θ, a1), (v2(θ)∂θ, a2)

)L2 =

∫

S1

v1(θ)v2(θ) dθ + a1a2.

• (v(θ)∂θ, b) ∈ vir (u(θ)(dθ)2, a) ∈ vir∗




((v1(θ)∂θ, a1), (v2(θ)∂θ, a2)

)L2 =

∫

S1

v1(θ)v2(θ) dθ + a1a2.


• 〈adη(ξ), ω〉 = −〈ξ, ad∗

η(ω)〉, η, ξ ∈ g, ω ∈ g∗




((v1(θ)∂θ, a1), (v2(θ)∂θ, a2)

)L2 =

∫

S1

v1(θ)v2(θ) dθ + a1a2.


• 〈adη(ξ), ω〉 = −〈ξ, ad∗

η(ω)〉, η, ξ ∈ g, ω ∈ g∗

• 〈(u(θ)(dθ)2, a

),(v(θ)∂θ, b

)〉 =

∫S1 v(θ)u(θ) dθ + ab.

ad∗(v(θ)∂θ,b

) (u(θ)(dθ)2, a)=((−2v′u−vu′−av′′′)(dθ)2, 0

).




((v1(θ)∂θ, a1), (v2(θ)∂θ, a2)

)L2 =

∫

S1

v1(θ)v2(θ) dθ + a1a2.


• 〈adη(ξ), ω〉 = −〈ξ, ad∗

η(ω)〉, η, ξ ∈ g, ω ∈ g∗

• 〈(u(θ)(dθ)2, a

),(v(θ)∂θ, b

)〉 =

∫S1 v(θ)u(θ) dθ + ab.

ad∗(v(θ)∂θ,b

) (u(θ)(dθ)2, a)=((−2v′u−vu′−av′′′)(dθ)2, 0

).

• Hamiltonian equation on g∗ is ω(t) = ad∗dω(t)H(ω(t))



ω(t) = ad∗dω(t)H(ω(t))

dω(t)H ∈ T∗

ω(g∗) ∼= g∗ ∼= g

d(u(θ)(dθ)2,a

)H = (u(θ)∂θ, a)


ω(t) = ad∗dω(t)H(ω(t))

dω(t)H ∈ T∗

ω(g∗) ∼= g∗ ∼= g

d(u(θ)(dθ)2,a

)H = (u(θ)∂θ, a)

ad∗(v(θ)∂θ,b

) (u(θ)(dθ)2, a)=((−2v′u− vu′ − av′′′)(dθ)2, 0

).

⇓

d

dt

(u(θ)2, a

)=((−3uu′ − au′′′)(dθ)2, 0

).



d

dt

(u(θ)2, a

)=((−3uu′ − au′′′)(dθ)2, 0

).

⇓

ut = −3uu′ − au′′′,

a = 0.

The first equation is the Karteweg-de Vries (KdV)nonlinear evolution equation that describes travelingwaves in a shallow canal.

The second equation is just saying that the parametera is the real constant.


Other interesting equations.

Weighted family of metrics (·, ·)H1α,β

can be defined

((v, a), (u, b)

)H1

α,β

=

∫

S1

(αvu+ βv′u′

)dθ + ab. (1)

THEOREM The Euler equations for the right invariantmetric (·, ·)H1

α,β, α 6= 0 on the Virasoro – Bott group are

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.


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 (·, ·)H1α,β

becomes homogeneousdegenerate (·, ·)H1 metric and one has to pass toDiff S1/S1.


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 7→ f τ = f(τ).

Diff S1/S1 has a manifold structure.

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


Tangent space of Diff S1/S1

For the Lie-Frechét group Diff S1 corresponds the Liealgebra Vect(S1).

Denote by u(1) the Lie algebra of S1 = U(1). The spaceu(1) consists of constant vector fields on the circle.

Then TidDiff S1/S1 = Vect(S1)/u(1). The latter space isthe space of vector fields with vanishing mean valueon the circle.

By making use of the right action

µ : Diff S1/S1 × Diff S1 → Diff S1/S1

h.f 7→ h f = h(f).

we get the tangent space at each point f ∈ Diff S1/S1.


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 ofconstant vector fields. Notice that Vf isomorphic to theLie algebra of the group S1.

The Ehresmann connection D = Vect(S1)/u(1) isformed by vector fields v(θ)∂θ with vanishing meanvalue:

1

2π

∫ 2π

0

v(θ)dθ = 0



The family of Kähler metrics on Diff S1/S1 are definedby making use of the co-adjoint action of the groupDiff S1 on the dual space vir∗.

The orbit of the point(α(dθ)2, β

)under the co-adjoint

action of Ad∗Diff S1 is isomorphic to Diff S1/S1 ifα/β 6= −n2

2 , n ∈ N. This leads to the existence of 2parametric family of symplectic structures ωα,β.

The almost complex structure J on Vect(S1)/u(1) isinvariant under the action of Diff S1 and the symplecticforms ωα,β are compatible and generate the Kählermetric gα,β on Vect(S1)/u(1).



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

gα,β(v∂θ, u∂θ) =

∞∑

n=1

(αn+ βn3)anbn,

where v(θ) =∑

∞

n=1 aneinθ, u(θ) =

∑∞

n=1 bneinθ.

Extend gα,β to D = dr(Vect(S1)/u(1)

)by making use of

right action. If β ≥ 0 and −α < β, then the metric ispositively definite. We work with α = 1, β = 1 anddenote the metric by gD.



We get the sub-Riemannian manifold (Diff S1, D, gD).Define

η(v∂θ) =1

2π

∫ 2π

0

v(θ) dθ

the mean value of any vector fields v ∈ Vect(S1).Functional η measures a deviation of vector fieldv ∈ 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.


Normal geodesics

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

ξ(t, θ) =γ(t, θ)

γ′(t, θ)

is left logarithmic derivative of γ(t, θ).

The left logarithmic derivative is the analogous ofσ−1 : Tγ(t)Diff S1 ∋ γ(t) 7→ ξ(t) ∈ VectS1. Then weproject ξ(t) on the vertical distribution by η(ξ(t)).

η(ξ(t)) does not depend on θ and it generates rotationsin Diff S1.


Normal geodesics

All normal sub-Riemannian geodesics γsR on Diff S1

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.


Geometrical objects

• The group Diff S1 of orientation preservingdiffeomorphism of the unit circle S1;


Geometrical objects


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


Geometrical objects


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

• Homogeneous space Diff S1/S1 – Kirillov’smanifold;

• Groups Diff S1 and Vir and the homogeneousmanifold Diff S1/S1 are modeled on Fréchetspaces.

... 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).


Geometrical objects

Complexifcation:

• (Vir,vir(1,0))

vir(1,0) ⊕ vir(0,1) = vir⊗C

• (Diff S1,h(1,0))

h(1,0) ⊕ h(0,1) = corank1(Vect(S1)⊗C)

• (Diff S1/S1,h(1,0))

h(1,0) ⊕ h(0,1) = Vect(S1)/u(1)⊗C


Complexification of real manifold

• TqM ⊗C that is

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





• If J : TqM → TqM is such that J2 = −Id then

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

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

TqM ∼= T(1,0)q M ∼= T

(0,1)q M





• If J : TqM → TqM is such that J2 = −Id then

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

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

TqM ∼= T(1,0)q M ∼= T

(0,1)q M

• TqM ⊗C = T(1,0)q M ⊕ T

(0,1)q M ,


Integrable almost complex structure

If [T(1,0)q M,T

(1,0)q M ] ∈ T

(1,0)q M then the pair

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

Integrable almost complex structure

If [T(1,0)q M,T

(1,0)q M ] ∈ T

(1,0)q 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)


Cauchy-Riemann structure

Let N be a manifold.

TN ⊗C 7→ H ⊂ TN ⊗C of corank 1




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




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

[X, X ]q /∈ H(1,0)q ⊕H

(0,1)q , ∀ X ∈ H(1,0), Xq 6= 0


Left invariant C-R structure

Let G be a Lie group.

g⊗C 7→ h ⊂ g⊗C of corank 1




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




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

[X, X ]q /∈ h(1,0)q ⊕ h

(0,1)q , ∀ X ∈ h(1,0), Xq 6= 0


Complexified geometric structures

Vect(S1) = spancos(nθ), sin(nθ), n = 0, 1, 2, . . .

Vect(S1)⊗C = spanen = −ieinθ, 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 beintegrated to a curve in the space of maps C∞(S1,C),but the last one does not form a group with respect tocomposition.



• If v ∈ Vect(S1)/s1 then

v(θ) =

∞∑

n=1

an cos(nθ) + bn sin(nθ)

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

J(v) =

∞∑

n=1

bn cos(nθ)− an sin(nθ)

• H(1,0) = v − iJ(v) =∑

∞

n=1 cneinθ, cn = an − ibn,

H(0,1) = v + iJ(v) =∑

∞

n=1 cne−inθ



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

(Diff S1,h(0,1)) is the CR manifold

h0,1 = H0,1 = spaneinθ, n = 1, 2, . . .

Moreover

[ ∞∑

n=1

cneinθ∂θ,

∞∑

n=1

cne−inθ∂θ

]= i

∞∑

k,n=1

(k + n)cnckei(n−k)θ∂θ

is not in h(1,0) ⊕ h(1,0) 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

vir⊗C = spanCeinθ, 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 complexvalued 2-cocycle.

ωC(−ieimθ∂θ − ie

inθ∂θ) =

κ(n3 − n) if n+m = 0,

0 if n+m 6= 0,

where κ is a constant dependent on the undergroundphysical theory. The complexification vir⊗C of vir isalso called Virasoro algebra and it is more useful inphysics then the real Virasoro algebra.


Vir as a complex group

Virasoro-Bott group Vir admits a left invariant complexstructure. 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.

vir(1,0) =( ∞∑

n=0

aneinθ∂θ , αa0c

)∈ vir⊕C

and vir(0,1) = vir(1,0).

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



• (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) ofcomplex corank 1;

Infinite dimensional left invariant C-R structure





• (Diff S1,h(1,0)) h(1,0) ⊕ h(0,1) ⊂ (Vect(S1)⊗C) ofcomplex 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 Frechéthomogeneous manifold


Relation to analytic functions

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



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

• F = f ∈ A0 and univalent, f(z) = cz(1 +∞∑n=1

cnzn)

F ⊂ A0 is an open subset



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


cnzn)


• F1 = f ∈ F and |f ′(0)| = 1 f(z) = eiφz(1+∞∑n=1

cnzn))

F1 ⊂ F is a pseudoconvex surface



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


cnzn)


• F1 = f ∈ F and |f ′(0)| = 1 f(z) = eiφz(1+∞∑n=1

cnzn))

F1 ⊂ F is a pseudoconvex surface

• F0 = f ∈ F and f ′(0) = 1, f(z) = z(1 +∞∑n=1

cnzn))



•

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



•


•

(Diff S1,h(1,0))←→ (F1, T(1,0)F1) is C-R map



•


•

(Diff S1,h(1,0))←→ (F1, T(1,0)F1) is C-R map

•

(Diff S1/S1,h(1,0))←→ (F0, T(1,0)F0) is biholomorphic


Univalent Functions

• Realization Diff S1/S1 via conformal welding:


Univalent Functions


0

η

ξ

U

S1

1 0

y

x

Ω

Γ

z = f(ζ) = ζ + c1ζ2 + . . .

z = g(ζ) = a1ζ + a0 + a−11ζ+ . . .


Univalent Functions


0

η

ξ

U

S1

1 0

y

x

Ω

Γ

z = f(ζ) = ζ + c1ζ2 + . . .

z = g(ζ) = a1ζ + a0 + a−11ζ+ . . .

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


Principal bundles

1. The bundle π : Diff S1 → Diff S1/S1 is the principalU(1)-bundle

2. The bundle Π: Vir→ Diff S1/S1 is the trivialC∗-bundle.

C⋆

prby R

F

prby R

? _oo F // Vir

prby R

//oo C⋆

prby R

S1 F1

prF0

? _ooF1 // Diff S1oo

prDiff S1/S1

// S1

F0F0 // Diff S1/S1.oo


Infinitesimal action

Right action

µ : Diff S1/S1 × Diff S1 → Diff S1/S1

h.f 7→ h f = h(f).

This action is transferred to the right action over F0.The infinitesimal generator σf : Vect(S1)→ TfF0 isgiven by the variational formula of A. C. Schaeffer andD. C. Spencer

f2(ζ)

2π

∫

S1

(wf ′(w)

f(w)

)2v(w)dw

w(f(w)− f(z)),


Kirillov’s vector fields

Schaeffer and Spencer linear operator

f 2(ζ)

2π

∫

S1

(wf ′(w)

f(w)

)2v(w)dw

w(f(w)− f(z)),

that maps Vect(S1)⊗ C −→ TfF0 ⊗ C.




f 2(ζ)

2π

∫

S1

(wf ′(w)

f(w)

)2v(w)dw

w(f(w)− f(z)),


• Taking Fourier basis vk = −izk, k = 1, 2, . . . for T (1,0),we obtain

Lk[f ](z) = zk+1f ′(z).




f 2(ζ)

2π

∫

S1

(wf ′(w)

f(w)

)2v(w)dw

w(f(w)− f(z)),


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

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



• Virasoro commutation relation

[Lm, Ln]vir = (m− n)Lm+n +c

12n(n2 − 1)δn,−m,

c ∈ C. L0[f ](z) = zf ′(z)− f(z) corresponds to rotation.

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

Ln = ∂n +

∞∑

k=1

(k + 1)ck∂n+k, ∂k = ∂/∂ck,


The end

Thank you for your attention


Documents

Group of diffeomorphisms of the unite circle as a principle …Group of diffeomorphisms of the unite circle as a principle U(1)-bundle Irina Markina, University of Bergen, Norway Summer