The Conformal Algebra Dana Faiez June 14, 2017 Outline ... • Conformal Transformation/Generators • 2D Conformal Algebra • Global Conformal Algebra and Mobius Group • Conformal Field Theory • 2D Conformal Field Theory • Virasoro Algebra • Application in Statistical Mechanics • Massless Free Boson Example Conformal transformations Distance squared between two points with coordinates xµ and xµ + dxµ is 2 µ ν ds = gµν(x)dx dx Change of coordinates 0 x ) x (x) µ @x 0 dxµ = dx ρ @x 0ρ This coordinate transformation keeps distance between two nearby points ds invariant. µ ν @x @x 0 0 0 0 0 0 ds2 = g (x) dx ρdx σ = g (x )dx ρdx σ µν @x 0ρ @x 0σ ρσ Under a change of coordinates, the metric transforms according to µ ν 0 0 @x @x g (x ) = g (x) ρσ µν @x 0ρ @x 0σ Ex: Scale transformation (for λ real number) 0 xµ ! x µ = λxµ then g transforms like: 0 0 0 2 gρσ(x ) = gρσ(λx) = Ω (x)gµν such that: !x Ω(x) ≡ e 2 0 0 1 where w = −2lnλ ) gρσ(x ) = λ2 gµν(x) More generally, we could have 0 0 2 gρσ(x ) = Ω (x)gµν(x) 0 Transformations x ! x (x) that satisfy this condition are known as conformal transformations: Conformal Group Conformal transformations forms a group: For two conformal transformations Ω1 and Ω2; 2 0 g"ρσ(x) = Ω2(x)gρσ(x) 2 2 2 = Ω2(x)Ω1(x)gρσ(x) = (Ω2(x)Ω1(x)) gρσ(x): ) conformal transformation associated with Ω2Ω1: Infinitesimal Limit Consider the infinitesimal coordinate transformation: 0 x µ = xµ + ζµ(x) Expand Ω2(x 0 ) for ! 0 0 0 Ω2(x ) ' 1 + κ(x ) = 1 + κ(x) + O(2) with κ being some unknown function of the coordinates. Plug this 0 2 into gρσ(x) = Ω (x)gµν(x) µ ν λ gµσ@ρζ + gρν@σζ + ζ @λgρσ + κgρσ = 0 Conformal Killing Condition. Let's consider the simple case of flat spacetime where gµν = ηµν with η =diag(1; −1; −1; :::; −1), in d-dimension. Then the conformal Killing condition reduces to: @ρζσ + @σζρ + κηρσ = 0 Contracting with ηρσ we get: −2(@:ζ) = dκ plugging this into the reduced conformal Killing condition, we get: 2 @ ζ + @ ζ = (@.ζ)η ρ σ σ ρ d ρσ ρ ρσ 2 Act with @ = η @σ on @ρζσ + @σζρ = − d (@.ζ)ηρσ to get: 2 d@ ζσ = (2 − d)@σ(@.ζ) We can make two conclusions: 1. The case d = 2 is special.Any solutions of the generalized 2 Laplace equation @ ζσ = 0 yield a conformal transformation. (We'll talk about this case later) Generators 2. For d 6= 2 ! ζµ can depend on x at most quadratically. ) All solutions of the conformal Killing condition of Minkowski peacetime for d 6= 2 can be found: translation Lorentz scale Special conformal µ z}|{µ z µ}| ν{ z}|{µ z µν 2}| µ ν{ ζ = a + bν x + cx + dν(η x − 2x x ) Transformation Transformation Generator µ µ µ D x ! λx x @µ µ µ µ ν Pµ x ! a + bν x @µ Jµν (xµ@ν − xν@µ) µ µ µλ 2 µ λ µν 2 µ ν K x ! aλ(η x − 2x x ) (η x − 2x x )@ν [D; Pµ] = −Pµ [D; K µ] = +K µ [Jµν; Pλ] = −ηµλPν + ηνλPµ [Jµν; K λ] = −ηµλK ν + ηνλK µ [Jµν; Jλρ] = −ηµλJνρ − ηνρJµλ + ηνλJµρ + ηµρJνλ [K µ; Pν] = 2(Jµν + ηµνD) (All the ones not mentioned are zero.) Counting the number of generator (d > 2): Lorentz #of gen. of SO(d,2) translation z }| { scale Special conformal z }| { z}|{ d(d − 1) z}|{ z}|{ (d + 2)(d + 1) d + + 1 + d = 2 2 Gen. of conformal (d-D) jj Gen. of SO(d,2) (d+2-D): Denote generators of SO(2,d) by JMN with M; N = 0; 1; :::; d − 1; d; d + 1 and (µ, ν = 0; 1; :::; d − 1) Then we have the following commutation relation between Jµνs time-like space-like time-like z }| { z }| { z }| { with η00 = +1; ηij = −δij ; ηdd = −1; ηd+1;d+1 = +1 : [JMN ; JPQ ] = −ηMP JNQ − ηNQ JMP + ηNP JMQ + ηMQ JNP D = Jd;d+1 µ µ µ,d µ,d+1 P and K L:C: of J and J Guess: 1 Jµ,d = (K µ + Pµ) 2 1 Jµ,d+1 = (K µ − Pµ) 2 These can be checked using commutation relations of Js. By symmetry considerations, conformal algebra of d-D Minkowski spacetime with isometry group of SO(d-1,1), is isomorphic to SO(d, 2) algebra. 2D Conformal Algebra The Reduced conformal killing condition becomes the Cauchy-Riemann equation in 2D: @0ζ0 = +@1ζ1 ;@0ζ1 = −@1ζ0 with gµν = δµν: Complex function satisfying above eq. Holomorphic For a holomorphic ζ(z), function f (z) = z + ζ(z) gives rise to an infinitesimal 2D conformal transformation z ! f (z): Generators of 2D Conformal Algebra In search of 2D Conformal generators, expand ζ(z) around z=0 using a Laurent expansion: 0 X n+1 z = z + ζ(z) = z + ζn(−z ); n2Z 0 X n+1 z = z + ζ(z) = z + ζn(−z ); n2Z with infinitesimal parameters ζn and ζn constants. Generators corresponding to a transformation for a particular n are: n+1 n+1 ln = −z @z ; ln = −z @z ; with n 2 Z: Commutators are as follows: [lm; ln] = (m − n)lm+n [lm; ln] = (m − n)lm+n [lm; ln] = 0 This is known as the Witt algebra. Number of independent infinitesimal conformal transformations in 2D is infinite. Global Conformal Algebra Examining the ambiguity at x=0 and x ! 1 and work on 2 Riemann sphere S ' C [ 1: n+1 • ln = −z @z non-singular at z=0 only for n≥ −1: 1 1 n−1 • use conformal transformation z = − w ln = −(− w ) @w non-singular at w = 0 only for n≤ +1: 2 Globally defined conformal transformations on S ' C [ 1 are generated by l−1; l0; l+1: Determining the global conformal group generated by these generators: • l−1 = −@z (z ! z + b) : Translation. • l0 = −z@z (z ! az) (remember z 2 C) After Change of variable z = reiφ, we find: 1 i 1 i l = − r@ + @ ; l = − r@ − @ 0 2 r 2 φ 0 2 r 2 φ The linear combinations l0 + l0 = −r@r and i(l0 − l0) = −@φ are the generators of dilation and rotation, respectively. 2 • l+1 = −z @z Special conformal transformation and translation 1 for w = − z : 2 z c l1 z = −cz is the infinitesimal version of z ! cz+1 : Operators l−1; l0; l+1 generate transformations of the form: az+b z ! cz+d with a; b; c; d 2 C requiring ad − bc 6= 0 The conformal group of S2 ' [ 1 is the Mobius group SL(2;C) : C Z2 Conformal Field Theory A field theory that is invariant under conformal transformations. ) The physics of the theory looks the same at all length scales (theory can't have any preferred length scale. There can not be anything like a mass, Compton wavelength etc.) 2 Consider the fields φ of our theory on C : φ(z; z): Define: Chiral fields as φ(z) and anti-Chiral fields as φ(z): If a field transforms under conformal transformation z ! f (z) according to: 0 @f h @f h φ(z; z) ! φ (z; z) = ( @z ) ( @z ) φ(f (z); f (z)) It is called primary a field of conformal dimensions (h; h). If this holds for f 2 SL(2; C)=Z2, i.e. only for global conformal transformations, then φ is called a quasi-primary field. **Note: Not all fields in CFT are primary (or quasi-primary). The energy-momentum tensor: Consider scale invariance by transformation xµ ! λxµ which is part of global conformal group; ν The associated current is : Jµ = x Tµν: Applying Noether's theorem: µ Tµ = 0 This is the key feature of a conformal field theory in any dimension. 2D Conformal Field Theory 2D CFTs have 1-D symmetry ! severe restrictions on CFTs. (In some cases) can be solved exactly, using the conformal bootstrap method. Ex of 2D CFTs: massless free bosonic theories, and certain sigma models. Consider 2D Euclidean: @xα @xβ Tµν transforms as Tµν = @xµ @xν Tαβ: 0 1 1 1 By a quick change of variable: x = 2 (z + z) and x = 2i (z − z) we find: @z Tzz = 0 and @z Tzz = 0 The two non-vanishing components of the energy-momentum tensors are a chiral and anti-chiral field: Tzz (z; z) =: T (z) ; Tzz (z; z) =: T (z): Conformal transformations of the energy{momentum tensor: Under conformal transformation z ! f (z), T changes as follows: @f c T 0(z) = ( )2T (f (z)) + S(f (z); z) @z 12 c is called the central charge and S(w, z) denotes the Schwarzian derivative. 1 3 3 2 2 S(w; z) = 2 ((@z w)(@ w) − (@ w) ): (@z w) z 2 z • For non-vanishing central charges, T (z) is not a primary field. • Schwarzian derivative S(w; z) vanishes for SL(2; C) transformations w = f (z) ) T (z) is a quasi-primary field. Central Charge Introducing Virasoro Algebra: Remember Witt Algebra: [lm; ln] = (m − n)lm+n translation dilation&rotation special conformal z}|{ z}|{ z}|{ with l1 ; l0 ; l−1 This algebra has a central extension i.e. Group A is abelian and its image is in the center of E, for an extension of G by A, given by exact sequence of group homomorphism: 1 ! A ! E ! G ! 1: If Lie algebra of G is g, that of A is a, and that of E is e, then e is a central Lie algebra extension of g by a.