group of diffeomorphisms of the unite circle as a principle …group of diffeomorphisms of the unite...
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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.
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 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
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 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
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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
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 preservingdiffeomorphisms 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 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, . . .
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 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
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 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, ) is a group, where f φ = f(φ) is thecomposition, f(θ) = θ is the identity, f−1 is theinverse 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)
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)‖
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 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
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 8/49
Diff S1 as a Lie-Frechét group
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.
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 9/49
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))
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 10/49
Exponential map
• For finite dimensional Lie groups the exponentialmap 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 exponentialmap is always diffeomorphism near the origin.
• exp: VectS1 → Diff S1 is neither injective norsurjective 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 exponentialmap is always diffeomorphism near the origin.
• 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
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 11/49
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.
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 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).
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)]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
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 Virasoroalgebra vir
(v(θ)∂θ, a) ∈ vir
Central extension of Vect(S1)
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?
Central extension of Vect(S1)
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?
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
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′
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 =
∫
S1
v1(θ)v2(θ) dθ + a1a2.
• (v(θ)∂θ, b) ∈ vir (u(θ)(dθ)2, 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 =
∫
S1
v1(θ)v2(θ) dθ + a1a2.
• (v(θ)∂θ, b) ∈ vir (u(θ)(dθ)2, a) ∈ vir∗
• 〈adη(ξ), ω〉 = −〈ξ, ad∗
η(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 =
∫
S1
v1(θ)v2(θ) dθ + a1a2.
• (v(θ)∂θ, b) ∈ vir (u(θ)(dθ)2, a) ∈ vir∗
• 〈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
).
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 =
∫
S1
v1(θ)v2(θ) dθ + a1a2.
• (v(θ)∂θ, b) ∈ vir (u(θ)(dθ)2, a) ∈ vir∗
• 〈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))
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 17/49
Vir and KdV equation
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)
Vir and KdV equation
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
).
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 18/49
Vir and KdV equation
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.
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 19/49
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.
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 (·, ·)H1α,β
becomes homogeneousdegenerate (·, ·)H1 metric and one has to pass toDiff 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 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.
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 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.
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 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
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 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).
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
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.
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 26/49
Diff S1 as sub-Riemannian manifold
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.
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 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.
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 28/49
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.
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 29/49
Geometrical objects
• The group Diff S1 of orientation preservingdiffeomorphism 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 preservingdiffeomorphism 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 preservingdiffeomorphism of the unit circle S1;
• 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).
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))
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
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)
• 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
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)
• 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 ,
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 32/49
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)
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 7→ H ⊂ TN ⊗C of corank 1
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)
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)
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
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 7→ h ⊂ g⊗C of corank 1
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)
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)
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
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 35/49
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.
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) =
∞∑
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
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 = 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.
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 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.
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) ofcomplex 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) 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
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
• F = f ∈ A0 and univalent, f(z) = cz(1 +∞∑n=1
cnzn)
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
• F = f ∈ A0 and univalent, f(z) = cz(1 +∞∑n=1
cnzn)
F ⊂ A0 is an open subset
• F1 = f ∈ F and |f ′(0)| = 1 f(z) = eiφz(1+∞∑n=1
cnzn))
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
• F = f ∈ A0 and univalent, f(z) = cz(1 +∞∑n=1
cnzn)
F ⊂ A0 is an open subset
• 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))
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
•
(Diff S1,h(1,0))←→ (F1, T(1,0)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
•
(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
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:
0
η
ξ
U
S1
1 0
y
x
Ω
Γ
z = f(ζ) = ζ + c1ζ2 + . . .
z = g(ζ) = a1ζ + a0 + a−11ζ+ . . .
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:
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.
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 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
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 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)),
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)
2π
∫
S1
(wf ′(w)
f(w)
)2v(w)dw
w(f(w)− f(z)),
that maps Vect(S1)⊗ C −→ TfF0 ⊗ 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)
2π
∫
S1
(wf ′(w)
f(w)
)2v(w)dw
w(f(w)− f(z)),
that maps Vect(S1)⊗ C −→ TfF0 ⊗ C.
• Taking Fourier basis vk = −izk, k = 1, 2, . . . for T (1,0),we obtain
Lk[f ](z) = zk+1f ′(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)
2π
∫
S1
(wf ′(w)
f(w)
)2v(w)dw
w(f(w)− f(z)),
that maps Vect(S1)⊗ C −→ TfF0 ⊗ C.
• Taking v−k = −iz−k, k = 1, 2, . . . for T (0,1), 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
[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,
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 48/49
The end
Thank you for your attention
Group of diffeomorphisms of the unite circle as a principle U(1)-bundle – p. 49/49