1 What is CFT? 1. 3 Strange duality conjecture (G) Geometric strange duality conjecture... 5

Σχετικά έγγραφα
Higher spin gauge theories and their CFT duals

g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King

( ) 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K.

On the Galois Group of Linear Difference-Differential Equations

Table 1. morphism U P 1 dominant (MMP) 2. dim = 3 (MMP) 3. (cf. [Ii77], [Miy01]) (Table 1) 3.

J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5

Discriminantal arrangement

IUTeich. [Pano] (2) IUTeich

Wishart α-determinant, α-hafnian

N. P. Mozhey Belarusian State University of Informatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS

The form factor program - a review and new results - the nested SU(N)-off-shell Bethe ansatz - 1/N expansion

Diderot (Paris VII) les caractères des groupes de Lie résolubles

11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))

Space-Time Symmetries

ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ Ä Œμ Ìμ. ±É- É Ê ± μ Ê É Ò Ê É É, ±É- É Ê, μ Ö

Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics

Example Sheet 3 Solutions

( ) Kähler X ( ),. Floer -Oh- - [6]. X Fano *, X ( = (C ) N ) W : X C ( ) (X,W). X = P, W (y) =y + Q/y. Q P. Φ:X R N, Δ=Φ(X). u Int Δ, Lagrange L(u) =

Single-value extension property for anti-diagonal operator matrices and their square

Reminders: linear functions

Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Heisenberg Uniqueness pairs

Global nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]

: 1. 10:20 12:40. 12:50 13:50 14:00 14:50 15:00 16:30 Selberg ( ) 18:45 20:00 20:15 21:45 Selberg ( ) 7:00 9:00

The q-commutators of braided groups

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

Jordan Form of a Square Matrix

J. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n

Lecture 13 - Root Space Decomposition II

THE BIGRADED RUMIN COMPLEX. 1. Introduction

([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-

arxiv: v1 [math.ra] 19 Dec 2017

Fractional Colorings and Zykov Products of graphs

ADE. 1 Introduction. (Ryo Fujita) Lie. U q (Lg) U(Lg) Dynkin. Dynkin. Dynkin. 4 A n (n Z 1 ), B n (n Z 2 ), C n (n Z 2 ), D n (n Z 4 )

Lecture 10 - Representation Theory III: Theory of Weights

Vol. 37 ( 2017 ) No. 3. J. of Math. (PRC) : A : (2017) k=1. ,, f. f + u = f φ, x 1. x n : ( ).

EE512: Error Control Coding

L p approach to free boundary problems of the Navier-Stokes equation

ECE Spring Prof. David R. Jackson ECE Dept. Notes 2

Symmetric Stress-Energy Tensor

Parametrized Surfaces

Abstract Storage Devices

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

PHASE TRANSITIONS IN QED THROUGH THE SCHWINGER DYSON FORMALISM

The ε-pseudospectrum of a Matrix

n+1 v x x 3 u2 1 + u2 2 1 ) + 1 (u 1, u 2 ) = 1 v2 1 ) (v 1, v 2 ) =

Partial Differential Equations in Biology The boundary element method. March 26, 2013

arxiv: v1 [math-ph] 4 Jun 2016

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

Wavelet based matrix compression for boundary integral equations on complex geometries

Βιογραφικό Σημείωμα. Γεωργίου Κ. Ελευθεράκη ΓΕΝΙΚΑ ΣΤΟΙΧΕΙΑ EKΠΑΙΔΕΥΣΗ

Markov chains model reduction

New bounds for spherical two-distance sets and equiangular lines

Congruence Classes of Invertible Matrices of Order 3 over F 2

Minimal Surfaces PDE as a Monge Ampère Type Equation

Affine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik

Aspects of the BMS/CFT correspondence

The Properties of Fuzzy Relations

ADVANCES IN MECHANICS Jan. 25, Newton ( ) ,., Newton. , Euler, d Alembert. Lagrange,, , Hamilton ( )

The Spiral of Theodorus, Numerical Analysis, and Special Functions

Lecture 16 - Weyl s Character Formula I: The Weyl Function and the Kostant Partition Function

Laplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago

CFT approach to the q-painlevé VI equation

Uniform Convergence of Fourier Series Michael Taylor

ACTA MATHEMATICAE APPLICATAE SINICA Nov., ( µ ) ( (

Other Test Constructions: Likelihood Ratio & Bayes Tests

ON THE STRUCTURE OF PRINCIPAL SUBSPACES OF STANDARD MODULES FOR AFFINE LIE ALGEBRAS OF TYPE A

Second Order Partial Differential Equations

The circle theorem and related theorems for Gauss-type quadrature rules

Finite size Emptiness Formation Probability for the XXZ spin chain at = 1/2

Note: Please use the actual date you accessed this material in your citation.

Non-Abelian Gauge Fields

Min-max Theory, Willmore conjecture, and Energy of links

Lie Algebras Representations- Bibliography

Distances in Sierpiński Triangle Graphs

AdS black disk model for small-x DIS

A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings

D Alembert s Solution to the Wave Equation

The k-α-exponential Function

ΒΙΟΓΡΑΦΙΚΟ ΣΗΜΕΙΩΜΑ. Dr. ΗΛΙΑΣ A. ΖΑΦΕΙΡΗΣ

Horizontal and Vertical Recurrence Relations for Exponential Riordan Matrices and Their Applications

ERG for a Yukawa Theory in 3 Dimensions

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

Adachi-Tamura [4] [5] Gérard- Laba Adachi [1] 1

RS306 High Speed Fuse for Semiconductor Protection

Lie Algebras Representations- Bibliography

Homomorphism in Intuitionistic Fuzzy Automata

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ

1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint

2 Composition. Invertible Mappings

第 61 回トポロジーシンポジウム講演集 2014 年 7 月於東北大学 Kontsevich-Kuperberg-Thurston ( ) Kontsevich-Kuperberg-Thurston Kontsevich Chern-Simons E. Witten Chern-

Bundle Adjustment for 3-D Reconstruction: Implementation and Evaluation

Kostka functions associated to complex reflection groups

Αναπαραστάσεις οµάδων και Αλγεβρες Τελεστών

4.6 Autoregressive Moving Average Model ARMA(1,1)

AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION

Iterated trilinear fourier integrals with arbitrary symbols

Transcript:

1 1994 9 6 1 What is CFT? 1 2 Wess-Zumino-Witten model 2 2.1 (R Representation theoretic formulation of WZW model.......... 2 2.2 (G Geometric formulation of WZW model.................. 4 2.3 (R=(G..................................... 4 3 Strange duality conjecture 5 3.1 (G Geometric strange duality conjecture................... 5 3.2 (R................... 5 3.2.1 (R local version ( p X............. 6 3.2.2 (R P 1 N conformal blocks.............. 6 4 Coset construction 7 4.1 GKO coset construction of unitary representations.............. 7 4.2 Coset construction of conformal blocks.................... 8 1 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, 1994 9 5 ( 7 (, ( 15 9 2010 4 29 L A TEX.

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,

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 1 + + 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 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 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 3. Strange duality conjecture 3.2.1 (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 2 + + 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 3.2.2 (R P 1 N conformal blocks Affine Lie algebra ĝ WZW model conformal blocks Lĝ, k, λ;x.

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 + 12 1, 4(k + 2(k + 3 V c,h = (cenral charge c, h.w. h h.w. irreducible representation of Vir.

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 [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, 1995. [BL] Beauville, Arnaud and Laszlo, Yves: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994, no. 2, 385 419. [BNR] Beauville, Arnaud, Narasimhan, M. S., and Ramanan, S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989, 169 179. [DT] Donagi, Ron and Tu, Loring W.: Theta functions for SL(n versus GL(n. Math. Res. Lett. 1 (1994, no. 3, 345 357. [FQS] Friedan, Daniel, Qiu, Zongan, and Shenker, Stephen: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52 (1984, no. 18, 1575 1578. [GKO] Goddard, P., and Kent, A., and Olive, D.: Virasoro algebras and coset space models. Phys. Lett. B 152 (1985, no. 1-2, 88 92. [I] [L] Ikeda, Takeshi: Coset constructions of conformal blocks. Dissertation, Tohoku University, Sendai, 1996. Tohoku Mathematical Publications, 3. Tohoku University, Mathematical Institute, Sendai, 1996. ii+55 pp. Langlands, Robert P.: On unitary representations of the Virasoro algebra. Infinitedimensional Lie algebras and their applications (Montreal, PQ, 1986, 141 159, World Sci. Publ., Teaneck, NJ, 1988. [NT] Nakanishi, Tomoki, and Tsuchiya, Akihiro: Level-rank duality of WZW models in conformal field theory. Comm. Math. Phys. 144 (1992, no. 2, 351 372. [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, 459 566, Adv. Stud. Pure Math., 19, Academic Press, Boston, MA, 1989.

2024 © DocPlayer.gr Πολιτική Απορρήτου | Όροι Χρήσης | Σχόλια