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

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Group of Diffeomorphisms of the Unite Circle As a Principle U(1)-Bundle 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 µ ν hαβ = e ηαβ, SP = −T dσ η ηµν∂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 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,... • Diff S1 ⊂ C∞(S1) is an open subset and by this it inherits 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 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,... • Diff S1 ⊂ C∞(S1) is an open subset and by this it inherits the Frechét topology • (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.
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