Lecture 10 :Kac-Moody algebras

Transcription

1 Lecture 10 :Kc-Moody lgebrs 1 Non-liner sigm model The ction: where: S 0 = 1 4 d xtr ( µ g 1 µ g) - positive dimensionless coupling constnt g(x) - mtrix bosonic field living on the group mnifold G ssocited with the Lie lgebr g The ction is rel when g(x) is vlued in unitry representtion. The trce is tken in such representtion, the prime indictes representtion-independent normliztion: Tr (t t b ) = δ,b, where [t, t b ] = c if bc t c nd t is ny mtrix representtion of the Lie lgebr genertors Problems with this ction: it is conformlly invrint clssiclly, but t the quntum level it is effectively mssive. even clssiclly there is problem with conserved currents: there re not chirl (holomorphic nd ntiholomorphic) conserved currents (it follows from the eqution of motion for this theory) Wess-Zumino-Witten models An dditionl term in the ction (topologicl term): Γ = i d 3 ytr ( g 1 g) 3 4π B 1

2 where B is 3-dim mnifold whose boundry is the compctifiction of the originl - dim spce. The g is n extension of the field g to the 3-dim mnifold. This extension is not unique. The mbiguity in the definition of Γ is n integer multiple of πi. Thus ny coupling constnt multiplying the Γ term dded to the ction hs to be integer. The full ction is defined s: where k hs to be integer. The eqution of motion: S = S 0 + kγ µ (g 1 µ g) + ik 4π ɛ µν µ (g 1 ν g) = 0 In terms of complex vribles: ( ) ( ) 1 + k (g 1 g) + 1 k 4π 4π (g 1 g) = 0 For = 4π k we hve the conservtion lw: (g 1 g) = 0 for k > 0 (for k < 0 the second solution implies conservtion of the dul currents). The conserved currents: J(z) = k gg 1, J( z) = kg 1 g The vrition of the ction The seprte conservtion of the currents J z, J z implies the invrince of the ction under g(z, z) Ω(z)g(z, z) Ω 1 ( z) where Ω, Ω re two rbitrry mtrices vlued in G. Under the infinitesiml trnsformtion Ω(z) = 1 + ω(z), Ω( z) = 1 + ω( z) the field g trnsforms s δ ω g = ωg, δ ω g = g ω.

3 The vrition of the ction written in terms of J = J t, ω = ω t reds δ ω, ω S = 1 πi dz ω J + 1 πi d z ω J This implies the vrition of field in the form: δ ω, ω X = 1 dz ω J X + 1 πi πi Trnsformtion lw for the current: d z ω J X (1) the vrition of the current from the definition of the current: δ ω = k ( (δ ω g)g 1 gg 1 δ ω g 1) = k ( ωg + ω g) g 1 +k gg 1 ω = [ω, J] k ω It cn be written s δ ω J = b,c if bc ω b J c k ω compring this with the vrition of the current computed from (1) one gets 3 Current lgebr J (z)j b (w) kδ b + c if bc J c (w) OPE of the currents: J (z)j b (w) kδ b + c if bc J c (w) The modes: J (z) = n Z z n 1 J n stisfy the commuttion reltions: [J n, J b m] = c if bc J c n+m + k n δ b δ n+m,0 The Kc-Moody lgebr ĝ k is defined by these commuttion reltions, where f bc re structure constnts of the Lie lgebr g nd k is its centrl extension (it is clled the level of the Kc-Moody lgebr). 3

4 The sublgebr of zero modes form the ordinry Lie lgebr g (clled the horizontl Lie sublgebr of Kc-Moody lg.): [J 0, J b 0] = c if bc J c 0 4 The Sugwr construction Energy-momentum tensor cn be constructed from the currents: T (z) = 1 β dim g =1 : J (z)j (z) : The constnt β is fixed by requiring J (z) to trnsform s (1, 0) primry fields: T (z)j (w) J (w) + J (w) z w [L m, J n] = nj m+n This is condition for the Virsoro genertors written in terms of the current modes: L n = 1 : J β m+nj m : m= It implies β = (C g + k) where C g the dul Coxeter number of the Lie lgebr g ( for su(n): C su(n) = N ). The centrl chrge cn be determined from the condition c = 0 L L 0 where It reds L = 1 : J β 1J 1 :, Jn 0 = 0, forn 0 c = k dim g (C g + k) 5 Exmples 5.1 The su(n) ˆ k cses su() ˆ k dim su() k = 3, C su() = c = 3k k + 4

5 For k = 1 the theory with centrl chrge c = 1 cn be relized by the CFT of free boson. su(3) ˆ k dim su(3) k = 8, C su() = 3 c = 8k k + 3 For k = 1 the theory with centrl chrge c = cn be relized by two free bosons. su(n) ˆ k dim su(n) k = N 1, C su(n) = N c = (N 1)k k + N For k = 1 the theory with centrl chrge c = N 1 cn be relized by N 1 free bosons. 5. The su() ˆ 1 cse In the free boson theory there re 3 fields with conforml weight (, ) = (1, 0): j(z) = i φ(z), j ± (z) = e ±i φ(z) where φ(z) = iϕ 0 + ϕ < (z) + π 0 ln(z) ϕ > (z). The currents constructed from these fields stisfy the OPE reltions The corresponding modes: j 1 = 1 ( j + + j ), j (z)j b (w) = j (z) = m stisfy commuttion reltions j = i ( j + j ), δ b + i ɛ bc z w jc (w) + reg. jmz m 1, j m = dz πi zm j (z) j 3 = j [j m, j b n] = i ɛ bc j c n+m + m δ b δ m+n,0 ɛ bc re structure constnts of su() lgebr, thus these commuttion reltions define the Kc-Moody lgebr su() ˆ 1 with k = 1. One cn lso check tht the energy-momentum tensor following from the Sugwr construction is in this cse the tensor of free boson theory: T (z) = 1 : j (z)j (z) := 1 : φ(z) φ(z) : 6 5

6 5.3 The ŝo(n) 1 cse The lgebr ŝo(n) 1 cn be relized by N rel free fermions with the usul OPE: ψ i (z)ψ j (w) δ ij trnsforming in the vector representtion of so(n), with trnsformtion mtrices t ij. The index stnds for pir of integers (r, s) such tht 1 r < s N nd the explicit reliztion of these mtrices is given by The currents hve the form: t ij t (rs) ij T r(t t b ) = δ b, = i ( ) δi r δj s δj r δi s t ijt kl = δ ik δ jl + δ il δ jk J (z) = γ ij ψ i t ijψ j In order to check if the currents stisfy the Kc-Moody lgebr one cn clculte the OPE: J (z)j b (w) = γ c The first term sets the prmeter γ = 1 The centrl chrge is given by if bc J c (w) + γ T r(t t b ) + reg nd from the second term we cn see tht k = 1. c = 1N(N 1) (N ) + 1 = N. The energy-momentum tensor given by the Sugwr construction reds T (z) = = = 1 8(N 1) 1 8(N 1) (( ) ψi t ijψ j (ψk t klψ l ) ) i,j,k,l [ δ ik δ jl δ il δ jk ] ((ψ i ψ j ) (ψ k ψ l )) i,j,k,l 1 4(N 1) (ψ i ψ j ) = N (ψ i ψ j ) 8(N 1) i i 6