1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half

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1 Wess-Zumino-Witten Wess-Zumino-Witten., Knizhnik-Zamolodchikov-Bernard,,. 1 Affine Lie Affine Lie WZW 4 3 Knizhnik-Zamolodchikov-Bernard Hw Ward-Takahashi identities E α w Ward-Takahashi identities Weyl-Kac denominator = q, h T w Ward-Takahashi identity T w Ward-Takahashi identity KZB KZB Calogero-Gaudin Hamiltonians Hamiltonians

2 1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half sum of positive roots, strange formula 24ρ ρ = 2h dim g. Lie g affine Lie ĝ : ĝ = Cξ, ξ 1 g Cˆk., Lie : ˆk, ĝ = 0, ξ m A, ξ n B = ξ m+n A, B + ˆkA Bmδ m+n,0 for m, n Z, A, B g, A g, m Z, Am = ξ m A. A g current operator Az = m Z z m 1 Am., A g A0 ĝ, g ĝ. g M level k C M ˆk k., ˆκ = ˆk + h, κ = k + h. ˆκ M, level h, M critical level. V g, k C, V ĝ ĝ + = Cξ g Cˆk Amv +aˆkv = δ m,0 Av +akv A g, m Z, a C. V ĝ level k Weyl module, W k V : W k V = Uĝ Uĝ+ V. Lie a universal enveloping algebra Ua., v V 1 v W k V, V W k V g. g Verma module Weyl module ĝ Verma module. g V, W k V ĝ, V k W k V., V k W k V. g g = n + h n +, highest root θ. k, P k = { λ λ dominant integral λ θ k }., λ P k, highest weight λ g V λ, Weyl module W k,λ = W k V λ irreducible quotient L k,λ ĝ, highest weight ĝ.,, V λ highest weight vector v λ, g highest root vector E θ, L k,λ E θ 1 k λ θ+1 v λ W k,λ quotient. 2

3 1.2 Sz : Sz = 1 2 dim g p=1 J pzj p z., J p, J p g, A, B g, m, n Z, normal product : AmBn = { AmBn if m < 0, BnAm if m 0. M g, v M, m, A g, Amv = 0., Weyl module., ĝ,., Sz = m Z z m 2 Sm, Sm M well-defined, A g, m, n Z, : Sm, An = ˆκnAm + n, Sm, Sn = ˆκ m nsm + n + m3 m δ m+n,0ˆk dim g. 12, : ˆκ M, T m = ˆκ 1 Sm, T m M Virasoro. M level k, Virasoro central charge c k = k dim g/κ., T z = m Z z m 2 T m = ˆκ 1 Sz, T z energy-momentum tensor. M critical level i.e., ˆκ = 0 on M, M, Sm affine Lie g Sn. Sz g Casimir element C g = 1 2 p J pj p affine. g highest weight λ V λ C g λ λ + 2ρ/2. V λ Weyl module W k.λ, V λ S0, C g : S0 = C g = λ λ + 2ρ/2 on V λ., m > 0, SmV λ = {0}., λ = 0, V λ = V 0 g 1, S 1 : S 1V 0 = {0}., V 0 = C1 basis 1, S 21 = 1 2 p J p 1J p 11., : A, B g, AzBw = ka B A, Bw + z w 2 z w + AzBw, 3

4 Sw = 1 2 p J pwj p w = 1 2 lim J p zj p w k dim g. 1.1 z w z w 2 p 2 WZW,, Wess-Zumino-Witten WZW.., z 1,..., z N C, z = z 1,..., z N., V 1,..., V N g, M 0, M ĝ level k., V = N V i. M ĝ vam = A mv v M, A g, m Z., ˆk M k., Sm M vsm = S mv v M, m Z. Φ : M V M 0 C A g, m Z, v M, v V, v 0 M 0, : Φv Am v v 0 Φv v Amv 0 = zi m ρ i AΦv v v , level critical, : Φv T m v v 0 Φv v T mv 0 = z m+1 i + m + 1zi m ρ i κ 1 C g Φv v v z i, ρ i a a Ug v i a., v = v 1 v N, ρ i aφv v v 0 = Φv v 1 av i v N v 0. TK chiral vertex operator N + 2., v 0 M 0 Amv 0 = 0 A g, m 0, v M v Am = 0 A g, m < 0., w, w i C z i > w > 0, z i > w 1 > w 2 > 0. v = v 1 v N V A, B g : v z = v 1 z 1 v N z N = Φv v v 0, v zaw = Φv v Awv 0, v zaw 1 Bw 2 = Φv v Aw 1 Bw 2 v 0, 4

5 v zsw = Φv v Swv 0 = κφv v T wv 0., 2.1, : v zaw = v zaw 1 Bw 2 ρ i A w z i v z, 2.3 = ka B v z + v za, Bw 2 + w 1 w 2 2 w 1 w 2 ka B = w 1 w 2 + N 2 N ρ i A, B w 2 z i w 2 w 1 +, 1.1 : v zsw = 1 2 = dim g i,j=1 p=1 ρ i A v zbw 2 w 1 z i i,j=1 ρ i Aρ j B w 1 z i w 2 z j ρ i J p ρ j J p w z i w z 2 v z ρ i C g w z i 2 + j i r i,jz i z j w z i v z. 2.4 v z 2.5, r i,j z = p ρ ij p ρ j J p /z. C g,, Casimir element C g = 1 2 p J pj p., κ 0 level critical, 2.2 : v zsw = κ ρi κ 1 C g w z i w z i z i v z , 2.4, 2.5, 2.6 Ward-Takahashi identities., 2 Ward-Takahashi identities 2.5, 2.6., κ 0 : κ z i j i p ρ i J p ρ j J p z i z j v z = Knizhnik-Zamolodchikov KZ. κ = 0, Sw 1 Sw 2, Sw 1 Sw 2 Ward-Takahashi identities 2, w : G z; w = 1 2 dim g i,j=1 p=1 ρ i J p ρ j J p N w z i w z 2 = 5 ρ i C g w z i 2 + j i r i,jz i z j w z i. 2.8

6 , rational Gaudin Hamiltonians H i : H i = j i r i,j z i z j = j i G z; w Gaudin Hamiltonians. p ρ i J p ρ j J p z i z j i = 1,..., N Knizhnik-Zamolodchikov-Bernard Knizhnik-Zamolodchikov-Bernard KZB. 3.1 κ 0, 2.1, 2.2. ĝ level k M 0 : M 0 weight subspace., ε C, ν h, M 0 ε, ν = { v M 0 T 0v = εv, Hv = νhv H h }, M 0 = ε,ν M 0ε, ν. ĝ Verma module M k,λ, ε C, ν h, dim M 0 ε, ν < dim M k,λ ε, ν., M 0 ε, ν. M 0ε, ν = Hom C M 0 ε, ν, C, M 0 = ε,ν M 0ε, ν, M 0 ĝ. M = M 0., M M 0 = M 0 M 0 f trace : tr M0 f = ε,ν fc ε,ν., C ε,ν M 0ε, ν M 0 ε, ν canonical element. c = c k = k dim g/κ, q = e 2πiτ, q < 1, h h. w > z i > qw, w 1 > w 2 > z i > qw 1, A, B g, : v z q,h = v 1 z 1 v N z N = tr M0 Φ q T 0 c/24 e h v, 3.1 Awv z q,h = tr M0 Φ q T 0 c/24 e h Aw v, Aw 1 Bw 2 v z q,h = tr M0 Φ q T 0 c/24 e h Aw 1 Bw 2 v, T wv z q,h = tr M0 Φ q T 0 c/24 e h T w v., M 0 a, : av z q,h = tr M0 Φ q T 0 c/24 e h a v, v za q,h = tr M0 Φ q T 0 c/24 e h v a. 6

7 3.2 Hw Ward-Takahashi identities H a a = 1,..., g Cartan subalgebra h orthonomal basis. g roots, positive roots, +. α root vector E α, E α E α = 1 normalize. H α = E α, E α, H α h, h = h, H α = α : H α H β = αh β., H a, E α H a, E α g. 2.1, H h : Hmv z q,h = zi m ρ i H v z q,h, +q m Hmv z q,h H0v z q,h = Hv z q,h = H v z q,h., h h H H fh = fh + sh. s s=0 m = 0, h-invariance : ρ i H v z q,h = ,, Ward-Takahashi identity : w Hwv z q,h = H + Zz i /wρ i H v z q,h. 3.3, Zz 1 > z > q : Zz = m 0 z m 1 q m = 1 z m 0 zm 1, 1 = 1 1 z z 1 z, Zz = z 2 1 z + z m q m 1 q + z m m 1 q = z m 2 1 z + q m 1 q m zm z m 3.4 m>0 m<0 m>0, q 1 > z > q, z 1 Zz 1 1+z z, q 1 > z > 1 Zqz = Zz + 1., Zz C. Zz : Zz C {q m } m Z. Zqz = Zz + 1. Zz 1 1+z 2 1 z 0 as z 1. 7

8 : z = e u, 1 1+z = 2 1 z u 1 + Ou., Zz : Zz = m>0 1 1 zq m z 2 1 z zq. m m<0, H h : 1 q m w 2 HmHw 2 v z q,h = kh Hmw m 2 v z q,h + zi m ρ i H v zhw 2 q,h, H0Hw 2 v z q,h = H Hw 2 v z q,h., Ward-Takahashi identity : w 1 w 2 Hw 1 Hw 2 v z q,h = kh HZ w 2 /w 1 + H + Zz i /w 1 ρ i H H + Zz j /w 2 ρ j H v z q,h 3.5 i=j, Z z = z z Zz. Z z Z qz = Z z., 3.4, q 1 > z > q, Z z : : z = e u, z 1 z 2 Z z = z 1 z 2 + m>0 = u 2 + O1. mq m 1 q m zm + z m E α w Ward-Takahashi identities α : 1 q m e αh E α mv z q,h =, : w E α wv z q,h = zi m ρ i E α v z q,h. σe αh ; z i /wρ i E α v z q,h., σx; z 1 > z > q : σx; z = m Z 8 z m 1 q m x.

9 1/1 z = m 0 zm σx; z = m>0 z m q m x 1 q m x z + 1 = 1 1 z x 1 + m>0 z m 1 x q m x m<0 z m q m x 1 q m x z m q m x 1 1 q m x 1 3.7, q 1 > z > q, q 1 > z > 1 σx; qz = x 1 σx; z., σx; z z C. 0 x q m m Z, σx; z z : σx; z C {q m } m Z. σx; qz = x 1 σx; z. σx; z = 1/1 z + regular at z = 1., σx; z 1 = σx 1 ; z, σx; z 1 > x > q : σx; z = m Z x m 1 q m z., σx; z = σz; x., α : 1 q m e αh w 2 E α me α w 2 v z q,h = kmw m 2 v z q,h + w m 2 w 2 H α w 2 v z q,h + zi m ρ i E α E α w 2 v z q,h., : w 1 w 2 E α w 1 E α w 2 v z q,h = kσ e αh ; w 2 /w 1 + σe αh ; w 2 /w 1 + σe αh ; z i /w 1 ρ i E α Hα + Zz i /w 2 ρ i H α σe αh ; z j /w 2 ρ j E α v z q,h 3.8 j=1, σ x; z := z σx; z. z 3.7, q 1 > z > q, σ x; z : σ z x, z = 1 z + mz m q m x 2 1 q m x + mz m q m x q m x 1 m>0 9

10 3.4 Weyl-Kac denominator = q, h ĝ Weyl-Kac denominator = q, h : = q, h =q ρ ρ/2h 1 q m m>0 e ρh 1 e αh m>01 q m e αh 1 q m e αh α + Strange formula ρ ρ/2h = dim g/24. = q, h : H h, H = ρh + e αh αh 1 e + q m e αh αh 1 q m e qm e αh, 3.11 αh 1 q m e αh α + m>0 q ρ ρ q 2h = m>0 mqm 1 q m + α + q m e αh 1 q m e + qm e αh αh 1 q m e αh 3.5 T w Ward-Takahashi identity 1 Ward-Takahashi identity, H = H a a = 1,..., 3.5 α 3.8, w1w2 w 1 w 2 = w2/w1 2 1 w2/w1 2 k dim g, w 1, w 2 w., 3.6, 3.9, 3.12, 1 2 lim Z z + σ e αh ; z z dim g q ρ ρ q 2h = 3.13 z 1 1 z 2 α, 3.7, 3.11, 1 2 lim z 1 α αh σe αh ; z σe αh ; z = H, H α = a αh ah a. : 3.14, κw 2 T wv z q,h q ρ ρ q 2h = k Ha H a z; w H a z; wh a z; w + α H a z; w = Ha + E α z; we α z; w v z q,h Zz i /wρ i H a,

11 E α z; w = σe αh ; z i /wρ i E α. 3.17, H a z; w, = Ha, κ w 2 T wv z q,h = kq q kρ ρ 2h H a z; wh a z; w + α Ha Ha v z q,h E α z; we α z; w v z q,h T w Ward-Takahashi identity 2 T w Ward-Takahashi identity. 2.2, 1 q m T mv z q,h = z m+1 i + m + 1zi m ρ i κ 1 C g v z q,h, z i T 0v z q,h = q q + c v z q,h m = 0, translation invariance : z i + i v z q,h = z i, i = ρ i κ 1 C g. v 1 z 1 v N z N q.h dz 1 1 dz N N, a C z 1,..., z N az 1,..., az N.,, : w 2 T wv z q,h = T z; w + c v z q,h , T z; w = Z z i /w i + Zz i /w z i + i + q z i q. 3.22,, w 2 T wv z q,h = q q c v z q,h + T z; w v z q,h

12 3.7 KZB c Strange formula c = k dim g/κ, = kρ ρ, T w 24 2κh 2 Ward-Takahashi identities 3.18, 3.23, : T z; w 1 H a z; wh a z; w + E α z; we α z; w v z q,h 2κ α = h κ q q 1 2h Ha Ha v z q,h 3.24 N = 0, v z q,h = 1 q,h M 0 character., M 0 g highest weight ρ Verma module level k = 0 Weyl module ĝ highest weight 0, ρ Verma module, 1 q,h = 1., 3.24., : q q 1 2h Ha Ha = , : B.,. Theorem 3.1 KZB v v q,h 3.1, = q, h 3.10, F = v v q,h : ρ i HF = 0 for H h h-invariance, 3.26 z i + i F = 0 translation invariance, 3.27 z i κt z; w G z; w F = , i = ρ i κ 1 C g, G z; w = 1 2 H a z; wh a z; w + α E α z; we α z; w T z; w, H a z; w, E α z; w 3.22, 3.16, KZB w. 12

13 3.8 KZB 3.28 w C, w qw, 2 q m z i i = 1,..., N, m Z., : Z z i /wp i + Zz i /wq i + R., P i, Q i, R w, w qw, N Q i = 0., 3.28 P i = 0, Q i = 0, R = 0. P i, Q i z i 3.28 Laurent, R min{ q 1 z i } > w > max{ z i } w Laurent. Zz = O1 z, 1 z 2 Zz2 = Z z + regular at z = 1. σx 1 ; zσx; z z qz {q m } m Z, z = 1 Laurent z/1 z 2 + regular at z = 1., σx 1 ; zσx; z = Z z + const.., i = 1,..., N,, G z; w = + α Zz i /w ρ i H a Ha + Zz j /z i ρ i H a ρ j H a j i Z z i /wρ i H a ρ i H a Zz i /w j i σe αh ; z j /z i ρ i E α ρ j E α Z z i /wρ i E α ρ i E α + regular at w = z i = Z z i /wρ i C g + Zz i /w ρ i H a Ha + j iρ i ρ j rz j /z i + regular at w = z i. rz = Zz H a H a + α σe αh ; ze α E α , P i = 0 κq i = κ z i + i ρ i H a Ha + i ρ j rz j /z i F 3.31 z i j iρ 13

14 ., i = ρ i κ 1 C g. R. σx, z = x σx, z. 3.4, 3.6, 3.7, 1 > z > q x : Zz 2 Z z = 2 z m 1 q m 1 q m q m 1 q m + 1 4, m 0 m σx, z = z m 1 q m x1 q m x 1. m Z 3.33, q 1 > z > q, z = 1. h-invariance 3.2, , :, κ T 0v z q,h = k q q ρ ρ 2h + Ha Ha + sz = Zz 2 Z z ρ i ρ j A B = ρ i Aρ j B. Ha H a ρ i ρ j sz j /z i v z q,h i,j=1 H a H a α σe αh, ze α E α, 3.35 i = j ρ i ρ j A B., 3.18, :, 3.19, κ T 0v z q,h = kq q kρ ρ 2h T 0v z q,h = Ha Ha + i,j=1 3.36, 3.37, : κr = κq q 1 Ha Ha + 2 Ha Ha ρ i ρ j sz j /z i v z q,h v z q,h q q c v z q,h + q 24 q v z q,h ρ i ρ j sz j /z i F i,j=1

15 , N ρh i, 3.29 G z; w, : G z; w = Z z i /wρ i C g Zz i /w ρ i H a Ha + j iρ i ρ j rz j /z i Ha Ha + ρ i ρ j sz j /z i i,j=1 Theorem 3.2 KZB v v q,h 3.1, = q, h 3.10, F = v v q,h h-invariance 3.26, translation invariance 3.27 : κ z i + i F = H i F, 3.40 z i κq q F = H 0F. 3.41, i = ρ i κ 1 C g, H i = ρ i ρ j rz j /z i i = 1,..., N, 3.42 ρ i H a Ha + j i H 0 = 1 Ha Ha + ρ i ρ j sz j /z i i,j=1 rz, sz 3.30, KZB. N Q i = 0, N H i = 0. FV. 4 Calogero-Gaudin Hamiltonians 3.42, 3.43 H i i = 0,..., N 2 Calogero-Gaudin Hamiltonians.,., critical level, v z q,h trace T 0,, v z q,h. KT2., twisted 15

16 KT1., 2 Calogero-Gaudin Hamiltonians. N = 1., g V 1 weight zero subspace V 1 0 dual space h Hamiltonians, H 0 : H 0 = 1 2 Ha Ha + 1 pe αh E α E α , px = σx, 1. pqx = px px {q m } m Z, px = x + regular at x = 1 1 x 2., x = e 2πiu, 4π 2 px = u + η 1., u Weierstrass, η 1 q. h h h = 2πiu, h x a h = 2πi a u ah a., Ha = 1 2πi u a., H 0 4π 2 η 1 C g : H CM = 1 2 u a α αue α E α. 4.2, g = sl n C, Cartan subalgebra h = {2πi diagx 1,..., x n i x i = 0}, x i h. 2 i x i h h. Ct ±1 1,..., t ±1 n t 1 t n β gl n C E i,j t i t j E i,j. V 1 v β = t 1 t n β g. β, V 1 g vector representation C n S βn C n., V 1 weight zero subspace Cv β, i j, E ij E ji = t i t i t j t j + 1 v β ββ + 1., : H CM = 1 2 h + ββ + 1 α 1 i<j n, Calogero-Moser Hamiltonian OP. x i x j Hamiltonians Hamiltonians., 2 S 2 w Ug Casimir element C 2 = C g affine, Ug center Zg C d d C d affine, S d w. A r, B r, C r Hay., S 2 w 2 Hamiltonians G 2 w = G z; w, S d w d Hamiltonians G d w. G 2 w w = z i 2, w z i 2 ρ i C 2, w z i 1 N 1 Hamiltonians, w 1 Hamiltonian 16

17 , N 2 Hamiltonians. G d w w = z i d, w z i d ρ i C d, Nd 1 d Hamiltonians., A n 1 g = sl n+1 C d 2, 3, 4,..., n., d 1 g maximal nilpotent subalgebra dim n., N dim n Hamiltonians. z i flag variety g, Hamiltonians N flag varieties h. N dim n +, h-invariance, Hamiltonians N dim n., Calogero-Gaudin., g 2 compact Riemann X. G d w Nd 1 d Hamiltonians, G d w dim H 0 X, Ω d = g 12d Hamiltonians. N dim n + g 1 dim g Hamiltonians., X N stable parabolic G-bundles g = Lie G moduli space., Hamiltonians. B Bernard, D.: On the Wess-Zumino-Witten models on the torus. Nucl. Phys. B303, FV Felder, G., Varchenko, A.: Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations. Internat. Math. Res. Notices 5, Hay Hayashi, T.: Sugawara operators and Kac-Kazhdan conjecture. Invent. math. 94, KT1 KT2 OP TK Kuroki, G., Takebe, T.: Twisted Wess-Zumino-Witten models on elliptic curves. Comm. Math. Phys. 190, Kuroki, G., Takebe, T.: Bosonization and integral representation of solutions of the Knizhnik-Zamolodchikov-Bernard equations. preprint math.qa/ to appear in Comm. Math. Phys. Olshanetsky, M., Perelomov, A.: Quantum integrable systems related to Lie algebras. Phys. Rep. 94, Tsuchiya, A., Kanie, Y.: Vertex operators in conformal field theory on P 1 and monodromy representations of braid group. In: Conformal field theory and solvable 17

18 lattice models Kyoto, 1986, Adv. Stud. Pure Math. 16, ; Errata. In: Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math., 19,

R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15

R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15 (Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec