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Virasoro and Kac-Moody Algebra Virasoro and Kac-Moody Algebra Di Xu UCSC Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 1 / 24 Outline Mathematical Description Conformal Symmetry in dimension d > 3 Conformal Symmetry in dimension d = 2 Quantum CFT Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 2 / 24 Kac-Moody algebra Suppose g is the Lie algebra of some ordinary finite-dimensional compact connected Lie group, G. We take G~ to be the set of (suitably smooth) maps from the circle S1, to G. We could represent S1 as the unit circle in the complex plane 1 S = fz 2 C : jzj= 1g (1) and denote a typical map by z ! γ(z) 2 G (2) The group operation is defined on G~ in the obvious way, given two 1 such maps γ1; γ2 : S ! G, the product of γ1 and γ2 is γ1 · γ2(z) = γ1(z)γ2(z) (3) This makes G~ into an infinite-dimensional Lie group. It is called the loop group of G. Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 3 / 24 To find the algebra of the loop algebra, start with a basis T a, 1 ≤ a ≤ dim g, with a b ab c [T ; T ] = ifc T (4) a A typical element of G is then of the form γ = exp[−iT θa] ~ A typical element of G can then be described by dimg functions θa(z) defined on the unit circle, a γ(z) = exp[−iT θa(z)] (5) P1 −n n We can make a Laurent expansion of θa(z) = n=−∞ θa z , and a a n introduce generators Tn = T z Use the infinitesimal version of elements near identity, and commutation relation, we see that G~ has the Lie algebra a b ab c [Tm; Tn ] = ifc Tm+n (6) which is called the untwisted affine Kac-Moody algebra g~, the algebra of the group of maps S1 ! G Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 4 / 24 Witt algebra To construct the infinite dimensional group corresponding to the Virasoro algebra, consider the group V~ of smooth one-to-one maps S1 ! S1 with the group multiplication now defined by composition ξ1 · ξ2(z) = ξ1(ξ2(z)) (7) To calculate the Lie algebra of V~ , consider its faithful representation defined by its action on functions f : S1 ! V where V is some vector space d D f (z) = f (ξ−1(z)) ≈ f (z) + i(z)z f (z) (8) ξ dz P1 n Making a Laurant expansion of (z) = n=−∞ −nz , then introduce n+1 d ~ generators Ln = −z dz , We find the Lie algebra v [Lm; Ln] = (m − n)Lm+n (9) which is called Witt algebra Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 5 / 24 Central Extensions In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra h Let g be a Lie algebra and Rp an abelian Lie algebra. A Lie algebra g0 is called a central extension of g by Rp if (i) Rp is (isomorphic to) an ideal contained in the center of g0 and (ii) g is isomorphic to g0=Rp. The central extension of Kac-Moody and Witt algebras are a b ab c ab [Tm; Tn ] = ifc Tm+n + kmδ δm+n;0 (10) c [L ; L ] = (m − n)L + m(m2 − 1)δ (11) m n m+n 12 m+n;0 Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 6 / 24 Highest weight representation If we consider the Virasoro algebra alone, there is only one vacuum state jhi in any given irreducible highest weight representation and it satisfies L0 jhi = h jhi ; Ln jhi = 0; n > 0 (12) In order for there to be a unitary representation of the Virasoro algebra corresponding to givin values of c and h, it is necessary that eithor c ≥ 1 and h ≥ 0 (13) or 6 c = 1 − (14) (m + 2)(m + 3) and [(m + 3)p − (m + 2)q]2 − 1 h = (15) 4(m + 2)(m + 3) where m = 0; 1; 2:::; p = 1; 2; :::; m + 1; q = 1; 2; :::; p Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 7 / 24 Conformal Symmetry in dimension d > 3 By definition, a conformal transformation of the coordinates is an invertible mapping x ! x0 which leaves the metric tensor invariant up to a scale: 0 g µν = Λ(x)gµν(x) (16) The set of conformal transformations manifestly forms a group, and it obviously has the Poincare group as a subgroup, which corresponds to the special case Λ(x) = 1 Now let’s investigate the infinitesimal coordinate transformation xµ ! xµ + µ, conformal symmetry requires that @µν + @νµ = f (x)gµν (17) where 2 f (x) = @ ρ (18) d ρ By applying an extra derivative, we get constrains on f (x) 2 (2 − d)@µ@νf = ηµν@ f (19) Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 8 / 24 The conditions require that the third derivatives of must vanish, so that it is at most quadratic in x. Here we list all the transformations # Finite Transformations Generators 0µ µ µ translation x = x + a Pµ = −i@µ 0µ µ dilation x = αx D = −ixµ@µ 0µ µ ν rigid rotation x = Mν x Lµν = i(xµ@ν − xν@µ) µ µ 2 0µ = x −b x = − ( ν@ − 2@ ) SCT x 1−2b·x+b2x2 Kµ i 2xµx ν x µ Table: Conformal Symmetry Transformations Manifestly, the special conformal transformation is a translation, preceded and followed by an inversion xµ ! xµ=x2 Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 9 / 24 Counting the total number of generators, we find d(d−1) (d+2)(d+1) N = d + 1 + 2 + d = 2 And if we define the following generators 1 J = L J = (P − K ) (20) µν µν −1,µ 2 µ µ 1 J = DJ = (P + K ) (21) −1;0 0,µ 2 µ µ where Jab = −Jba and a; b 2 {−1; 0; 1; :::; dg. These new generators obey the SO(d + 1; 1) commutation relations [Jab; Jcd ] = i(ηad Jbc + ηbcJad − ηacJbd − ηbd Jac) (22) This shows the isomorphism between the conformal group in d dimension (Euclidean Space) and the group SO(d + 1; 1) We could generalize this result, for the case of dimensions ; d = p + q ≥ 3, the conformal group of Rp q is SO(p + 1; q + 1) Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 10 / 24 Conformal Symmetry in dimension d = 2 The condition for invariance under infinitesimal conformal transformations in two dimensions reads as follows: @00 = @11;@01 = −@10 (23) which we recognise as the familiar Cauchy-Riemann equations appearing in complex analysis. A complex function whose real and imaginary parts satisfy above conditions is a holomorphic function. So we can introduce complex variables in the following way 1 z = x0 + ix1; = 0 + i1;@ = (@ − i@ ); (24) z 2 0 1 0 1 0 1 1 z¯ = x − ix ; ¯ = − i ;@¯ = (@ + i@ ); (25) z 2 0 1 Since (z) is holomorphic, so is f (z) = z + (z) which gives rise to an infinitesimal two-dimensional conformal transformation z ! f (z) Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 11 / 24 Since has to be holomorphic, we can perform a Laurent expansion of . 0 X n+1 z = z + (z) = z + nz (26) n2Z 0 X n+1 z¯ = z¯ + ¯(z¯) = z¯ + ¯nz¯ (27) n2Z The generators corresponding to a transformation for a particular n are n+1 ¯ n+1 ln = −z @z ln = −z¯ @z¯ (28) These generators obey the following commutation relations: [ln; lm] = (n − m)ln+m (29) ¯ ¯ ¯ [ln; lm] = (n − m)ln+m (30) ¯ [ln; lm] = 0 (31) Thus the conformal algebra is the direct sum of two isomorphic algebras, which called Witt algebra. Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 12 / 24 Global Conformal Transformation in two dimension 2 Even on the Riemann sphere S ' C [ f1g, ln; n ≥ −1 are non-singular at z = 0 while ln; n ≤ 1 are non-singular at z = 1. We arrive the conclusion that Globally defined conformal transformations on the Riemann sphere S2 are generated by fl−1; l0; l1g The operator l−1 generates translations z ! z + b In order to get a geometric intuition of l0, we perform the change of variables z = reiφ to find ¯ ¯ l0 + l0 = −r@r ; i(l0 − l0) = −@φ; (32) ¯ so l0 + l0 is the generator for two-dimensional dilations and ¯ i(l0 − l0) is the generators for rotations The operator l1 corresponds to special conformal transformations 1 which are translations for the variable w = − z Di Xu (UCSC) Virasoro and Kac-Moody Algebra 2015/06/11 13 / 24 In summary we have argued that the operators fl−1; l0; l1g generates transformations of the form az + b z ! with ad − bc = 1; a; b; c; d 2 (33) cz + d C which is the Mobius group SL(2; C)=Z2 = SO(3; 1) . The Z2 is due to the fact that the transformation is unaffected by taking all of a; b; c; d to minus themselves. The global conformal algebra is also useful for characterizing properties of physical states. In the basis of eigenstates of the two ¯ ¯ operators l0 and l0 are denote eigenvalues by h; h. The scaling dimension ∆ and the spin s of the state are given by ∆ = h + h¯ and s = h − h¯.
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