1 What is CFT? 1. 3 Strange duality conjecture (G) Geometric strange duality conjecture... 5
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1 What is CFT? 1 2 Wess-Zumino-Witten model (R Representation theoretic formulation of WZW model (G Geometric formulation of WZW model (R=(G Strange duality conjecture (G Geometric strange duality conjecture (R (R local version ( p X (R P 1 N conformal blocks Coset construction GKO coset construction of unitary representations Coset construction of conformal blocks What is CFT? (conformal field theory, CFT Riemann conformal blocks, (various conformal field theories. Example. Belavin-Polyakov-Zamolodchikov (BPZ minimal models Wess-Zumino-Witten (WZW models W-algebras, ( 7 (, ( L A TEX.
2 2 2. Wess-Zumino-Witten model parafermions coset models Super Symmetry + ( conformal blocks. WZW models BPZ minimal models coset models. : (R Representation theoretic approach (G Geometric approach. 2 Wess-Zumino-Witten model Wess-Zumino-Witten (WZW model Lie G Riemann.,. X : compact Riemann surface, G : semisimple Lie group over C, g := Lie G = (Lie algebra of G. WZW model conformal blocks (R (G : (R affine Lie algebra ĝ, highest weight integrable representations, adelic formulation on X (G moduli space of principal G-bundles on X, determinant line bundle ( G = SL n (C (R, G = SL n (C (G. 2.1 (R Representation theoretic formulation of WZW model. K := C(X = (the field of rational functions on X, K p := (the completion of K at p X = C((z p, Ô p = C[[zp ]], A := K p X p, adelic ring (, Ô := p X Ôp, maximal compact,
3 2.1. (R Representation theoretic formulation of WZW model 3 Ô A K, ĝ := g C((z CC, affine Lie algebra, ĝ A := g A CC, adele of affine Lie algebras, g Ô ĝ A g K, k = 0, 1, 2,... (level, {L k,λ } λ Pk = {h.w. integrable representations of ĝ with level k}, p 1,..., p N X, λ 1,..., λ N P k, λ p := λ i if p = p i, λ p := 0 otherwise, λ = (λp p X, L k, λ := p X L k,λ p (. ( : Riemann X p 1,..., p N λ 1,..., λ N. WZW model adele : L k, λ;x := L k, λ /(g KL k, λ, coinvariant quotient space, L k, λ;x = [L k, λ ]g K, invariant subspace. Definition. L k, λ;x conformal blocks. L k;x ( λ1,...,λ N p 1,..., p N := [( N i=1 L k,λ i ] g H0 (X,O X ( p p N, invariant subspace. ( Lemma. L k, λ;x = L λ1,...,λ N k;x p 1,..., p N (. Conformal blocks adele, p 1,..., p N. Theorem (WZW model, [TUY]. (1 dim L k, = dim ( λ;x L λ1,...,λ N k;x p 1,..., p N <. ( (2 dim L λ1,...,λ N k;x p 1,..., p N Riemann (X; p1,..., p N. stable curve. ( Exists projectively flat connection + (3 (3 (X; p 1,..., p N stable curve, ordinary double point q, q q, q stable curve ( X; p 1,..., p N, q, q. : L k;x ( λ1,...,λ N p 1,..., p N = µ P k L k; X ( λ1,...,λ N, µ,µ p 1,..., p N,q,q. (2+(3 conformal blocks factorization property.
4 4 2. Wess-Zumino-Witten model 2.2 (G Geometric formulation of WZW model, G = SL n (C,. SU X (n := (moduli sp. of semistable vector bundles on X of rank n with trivial det., θ n := (determinant line bundle on SU X (n. Definition. H 0 (SU X (n, θ k n generalized theta functions. 2.3 (R=(G Theorem ((R=(G, [BL]. λ = 0 := (0 p X ( Riemann highest weight 0 H 0 (SU X (n, θ k n = L k, 0;X.. generalized theta functions conformal blocks. Remark. λ 0 parabolic structure (level structure vector bundles moduli space,. Problem. WZW models CFTs geometric. [BL]. trivialization vector bundles t = (t X, (t p p X, E : vector bundle of rank n with trivial det., T 0 := (E, t t p : Ôn p E p Op Ô p trivialization at p, t X : K n H 0 (X, E OX K trivialization at generic point. (E, t T 0, p X γ p γ p = ˆt 1 X,p t p : K n p E p Op Kp K n p / =.. ˆt X,p t X : K n H 0 (X, E OX K E p Op Kp. : K n p T 0 SL n (A = p X SL n( K p, (E, t (γ p p X. SU X (n SL n (K\SL n (A/SL n (Ô =: SL X(n. SL X (n stack well-defined. Lemma. Grassmann SL n (A/SL n (Ô determinant line bundle θ n, : L k, 0 = H 0 (SL n (A/SL n (Ô, θ k n. Lemma, generalized theta functions affine Lie algebra highrst weight integrable representations, (R=(G.
5 5 3 Strange duality conjecture,. 3.1 (G Geometric strange duality conjecture. SU X (n := (moduli sp. of semistable vector bundles on X of rank n with trivial det., U X (n, d := (moduli sp. of semistable vector bundles on X of rank n and of degree d, UX (n := U X(n, n(g 1 (g := (genus of X, τ r,l : SU X (r U (l U (rl, (E, F E OX F, Θ n := (divisor { E U X (n H0 (X, E 0 } U (n line bundle, θ n := (divisor { E SU X (n H 0 (X, E OX L 0 } SU (n line bundle. L J g 1 (X = Pic g 1 (X. Lemma. τ r,l (Θ rl = θ l r Θ r l. Lemma. dim H 0 (U X (n, Θ n = 1. : C = H 0 (UX(rl, Θ rl H 0 (SU X (r UX(l, τ r,l (Θ rl = H 0 (SU X (r, θr l H 0 (UX(l, Θ r l. : ν r,l : H 0 (SU X (r, θr l H 0 (UX(l, Θ r l. Conjecture ([B],[DT]. ν r,l. Beauville [B] strange duality. strange duality conjecture. Remark (. Beauville-Narasimhan-Ramanan [BNR] proved the case of l = 1. Verlinde formula = dim H 0 (SU X (r, θr l = dim H 0 (UX (l, Θ r l. Donagi-Tu [DT] generalized the conjecture to an arbitrary degree. 3.2 (R WZW model? : local version P 1 N conformal blocks.
6 6 3. Strange duality conjecture (R local version ( p X Affine Lie algebra ĝ level k highest weight integrable representation Lĝk λ. λ ĝ highest weight g-part., sl n (C sl(n. r level l Young diagrams : Y r,l = { Y = (y i r i=1 Z r y 1 y r, y 1 y r l }. Young diagram Y = (y i r i=1 Y r,l Y : Y = y 1 + y y r Young diagram Y Y r,l Maya diagram M = M(Y : { 1 (j yi M = M(Y = (m ij, m ij = 0 (j > y i. Maya diagram M = (m ij t M = ( t m ij : t m j,νr+i = m i,νl+j (ν Z, i = 1,..., r, j = 1,..., l. t M(Y Y l,r Y, t Y. Young diagrams Y r,l : Y r,l = { Y = (ȳ i r i=1 Y r,l ȳ r = 0 }. Young diagram Y Y r,l Young diagram Y Yr, l : Y = (ȳ i r i=1, ȳ i = y i y r. Y r,l ŝl(r level l highest weight integrable representation highest weight sl(r-part. Y Y r,l ŝl(r level l highest weight integrable representation Lŝl(r,l Y : F = p Z Lŝl(rl 1 Λ p Lĝl(1 rl p = Y =p Lŝl(rl Y Y Y r,l. Lŝl(l r t Y Lĝl(1 rl Y F Fermion Fock space. : Lŝl(rl 1 = Λ Lŝl(r l p Y Lŝl(lr. t Y (R P 1 N conformal blocks Affine Lie algebra ĝ WZW model conformal blocks Lĝ, k, λ;x.
7 7 Theorem ([NT]. : C = Lŝl(rl 1, λ;p 1 µ Y(λ Lŝl(r l, µ;p 1 Lŝl(lr, t µ;p 1. λ = (λ p p X, λ pi = Λ mi (i = 1,..., N, λ p 0, Y(λ = { (µ p p X µ pi Y r,l, µ pi = m i (i = 1,..., N, µ p 0 }, t µ = ( t µ p p X. Lŝl(r l µ;p 1 : Lŝl(r l µ;p 1 = Lŝl(l r, t µ;p 1. Lŝl(lr, t µ;p 1 Remark. P 1 N p 1,..., p N [NT]. conformal blocks factorization property Knizhnik- Zamolodchikov (KZ equation braid monodromy. Nakanishi-Tsuchiya [NT] Riemann, Beauville- Laszlo [BL] (R=(G, strange duality conjecture. Nakanishi-Tsuchiya [NT] Riemann conformal blocks factorization property., strange duality KZ equation braid monodromy,. 4 Coset construction, Goddard-Kent-Olive [GKO] Virasoro algebra coset construction. 4.1 GKO coset construction of unitary representations. g = sl 2 (C, ĝ = sl 2 (C C[t, t 1 ] CC, affine Lie algebra, k = 1, 2, 3,... (level, C, P k = {0, 1, 2,..., k} λ, L k,λ = (level k, highest weight λ highest weight integrable representation of ĝ, Vir = C[t, t 1 ] d dt CC, the Virasoro algebra, c := 1 6 (k + 2(k + 3 (central charge, C, h λ,µ = ((k + 3(λ + 1 (k + 2(µ , 4(k + 2(k + 3 V c,h = (cenral charge c, h.w. h h.w. irreducible representation of Vir.
8 8 4. Coset construction Theorem ([GKO]. λ P k, ε P 1, : L k,λ L 1,ε = L k+1,µ V c,hλ,µ µ P k+1, λ+µ+ε 0 mod 2 Remark. [GKO] V c,hλ,µ Vir unitary representation. central charge c 1 Vir unitary representations ([FQS], [L]. 4.2 Coset construction of conformal blocks Vir WZW model conformal blocks. WZW model : V c, h;x = V c;x ( h1,...,h N p 1,...,p N. [GKO] : L k, λ;x L 1, ε;x L k+1, µ;x V c,. h λ, µ ;X : µ:λ p +µ p +ε p 0 mod 2 φ : L k+1, λ;x L k, ε;x V c, h λ, µ ;X L 1, ε;x. Conjecture. φ. Remark. k = 1, 2,..., c = 1 6/((k + 2(k + 3 R c R c = { h λ,µ λ P k, µ P k+1 }. h k λ,k+1 µ = h λ,µ R p (k + 1(k + 2/2. : P k P k+1 = Rc P 1, (λ, µ (h, ε. λ + µ + ε 0 (mod 2, h = h λ,µ. Remark. X = P 1 3 conformal blocks φ. WZW model Virasoro algebra CFT factorization properties : (1 φ. (2 P 1 3 conformal blocks. Example. k = 1, c = 1/2 R c = {0, 1/2, 1/16}. L 1,0, L 1,1, V 1/2,0, V 1/2,1/2, V 1/2,1/16 Fermion. P 1 3 conformal blocks k = 1. [I].
9 9 [B] Beauville, Arnaud: Vector bundles on curves and generalized theta functions: recent results and open problems. Current topics in complex algebraic geometry (Berkeley, CA, 1992/93, 17 33, Math. Sci. Res. Inst. Publ., 28, Cambridge Univ. Press, Cambridge, [BL] Beauville, Arnaud and Laszlo, Yves: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994, no. 2, [BNR] Beauville, Arnaud, Narasimhan, M. S., and Ramanan, S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989, [DT] Donagi, Ron and Tu, Loring W.: Theta functions for SL(n versus GL(n. Math. Res. Lett. 1 (1994, no. 3, [FQS] Friedan, Daniel, Qiu, Zongan, and Shenker, Stephen: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52 (1984, no. 18, [GKO] Goddard, P., and Kent, A., and Olive, D.: Virasoro algebras and coset space models. Phys. Lett. B 152 (1985, no. 1-2, [I] [L] Ikeda, Takeshi: Coset constructions of conformal blocks. Dissertation, Tohoku University, Sendai, Tohoku Mathematical Publications, 3. Tohoku University, Mathematical Institute, Sendai, ii+55 pp. Langlands, Robert P.: On unitary representations of the Virasoro algebra. Infinitedimensional Lie algebras and their applications (Montreal, PQ, 1986, , World Sci. Publ., Teaneck, NJ, [NT] Nakanishi, Tomoki, and Tsuchiya, Akihiro: Level-rank duality of WZW models in conformal field theory. Comm. Math. Phys. 144 (1992, no. 2, [TUY] Tsuchiya, Akihiro, Ueno, Kenji, and Yamada, Yasuhiko: Conformal field theory on universal family of stable curves with gauge symmetries. Integrable systems in quantum field theory and statistical mechanics, , Adv. Stud. Pure Math., 19, Academic Press, Boston, MA, 1989.
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