2017 : msjmeeting-2017sep-11i002 Ding-Iohara-Miki ( ) 1. Ding-Iohara-Miki - - [1, 2] [3, 4] gl(1) [5] 2 2 q- [6] q-w N [7, 8] [9] [10] W 1

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1 2017 : msjmeeting-2017sep-11i002 Ding-Iohara-Miki 1. Ding-Iohara-Miki - - [1, 2] [3, 4] gl1 [5] 2 2 q- [6] q-w N [7, 8] [9] [10] W [1, 2] [3, 4] z C D := z d dz z Dk z n = z n D + n k W 1+ 1 z n D k n Z, k Z 0 W W z n D k n Z, k Z 0 [W z n D k, W z m D l ] = W [z n D k, z m D l ] n ψn,m k,l j=1 jk n j l, n > 0, = 0, n = 0, m j=1 jl n j k, n < 0 + cψ k,l n,mδ n+m,0, 1 c C q- q C D k q D z q D z = qzq D D q D z W 1+ W z n q kd [W z n q kd, W z m q ld ] = q mk q nl W z n+m q k+ld q mk q nl + cδ n+m,0 3 1 q k+l : ,17K Mathematics Sbject Classification: 81R10,81T13,33D45,AGT,, awata@math.nagoya-.ac.jp web: W 2

2 k + l 0 3 k 0 W q kd W q kd new = W q kd + c 1 q k k + l = 0 ncq nk δ n+m,0 δ k+l,0 k + l = 0 k = l = 0 z q D n k W 1+ k < 0 2 W : W 2 1+ := W z n q kd n,k Z. [W z n q kd, W z m q ld ] = q mk q nl W z n+m q k+ld + q nk nc 1 + kc 2 δ m+n,0 δ k+l,0 4 2 c 1, c 2 C 2.3. t- - - N z = z 1, z 2,, z N D i := z i z i N z i q D i z j = q δ i,j z j q D i W z iq kd i N t C N x ± tz i z j n := W q ±1 z i n q ±D i 5 z i z j j [2] - - x + 0 k = ±1 k = ±1 W 2 1+ z i q D i z q D [2] z C x ± z = n Z x± n z n, ψ ± z = ±n N ψ± n z n q, t C F = Qq, t F = Qq 1/4, t 1/4 G ± z := 1 q ±1 z1 t 1 z1 q/t 1 z = z 3 G z 1, 6 gz := G+ z G z = g 1 z 1 Qq, t[[z]] 7 z < 1 G ± z = exp z n n q±n + t n + q/t n, 8 n>0 z n gz = exp n 1 qn 1 t n 1 q/t n. 9 n>0

3 - - U gγ 1 w/z gγ +1 w/z ψ+ zψ w = ψ wψ + z, [ψ ± z, ψ ± w] = 0, 10 gγ 1/2 w/z ±1 ψ + zx ± w = x ± wψ + z, 11 ψ zx ± w = x ± wψ zgγ 1/2 z/w ±1, 12 G ± w/zx ± zx ± w = x ± wx ± z w 3 G ± z/w, 13 z [x + z, x 1 q1 1/t w] = δγ 1 z/wψ + γ 1/2 w δγz/wψ γ 1/2 w 1 q/t 14 F δz = n Z zn γ ±1/2 ψ 0 ± M F U- α M γ 1/2 α = t/q l 1/4 α, ψ ψ 0 α = t/q l 2 α 15 M l 1, l 2 U [1] γ ±1/2 = γ ±1/2 γ ±1/2, 16 x + z = x + z 1 + ψ γ 1/2 1 z x+ γ 1 z, 17 x z = x γ 2 z ψ + γ 1/2 2 z + 1 x z, 18 ψ ± z = ψ ± γ ±1/2 2 z ψ ± γ 1/2 1 z 19 γ ±1/2 1 = γ ±1/2 1 γ ±1/2 2 = 1 γ ±1/2 3. Boson W Boson 3.1. Boson [a m, a n ] = m 1 q m 1 t m δ m+n,0 a 0 20 a n n Z F H 0 a n 0 = 0 n Z >0 λ = λ 1, λ 2,..., λ 1 λ a λ = a λ1 a λ2 0 a λ F 0 0 a n = 0 n Z <0 a λ = 0 a λ1 a λ2 a λ F , N F 1,N 0, 0 x + 0

4 [8] 1, 0 k = ±1 l 2 x + z x w δ δγ ±1 z/w γ ±1 l 2 x + z := exp x z := exp ψ ± z := exp 1 t n a n z n exp n 1 t n t/q n/2 a n z n exp n 1 t n n a nz n z N q/t N/2, 21 1 t n n t/qn/2 a n z n 1 z N q/t N/2, 22 1 t ±n 1 t/q n q/t n/4 a ±n z q/t n ±N/2 23 n C N Z F F U- [11, 10] γ 1/2 = t/q 1/4 1, N F 1,N 0, 1 [12, 13] 3.3. q-w N - - l 1 l 2 2 N q-w N q-w 2 = q- N N 2 N, 0 1, 0 N N i x +i z, ψ i z q-w N T i z := 1 j 1 < <j i N Λ j 1 zq/t 1 i 2 Λ j 2 zq/t 3 i 2 Λ j i zq/t i 1 2, 24 Λ i z := ψ 1 zψ 2 zq/t 1 2 ψ i 1 zq/t 2 i 2 x +i zq/t 1 i 2 25 q-w N T N z id - - q-w N q-w N T N z N N 1 q-w N q-w N 3.4. λ = λ 1, λ 2,... λ 1 λ 2 λ := λ i < λ λ lλ := λ 1, := 0, 0, i, j λ {i, j Z 2 1 i lλ, 1 j λ i }

5 F 1,0 0 x + z x + 0 x + 0 P λ = ε λ P λ, ε λ = 1 + t 1 l q λ i 1t i, 26 P λ = P λ a 1, a 2, 0 F 1, p = p 1, p 2,, p n C 1 t n a n Cp := exp 1 q n n p n n>0 28 n > 0 p n 0 Cp = 0 Cp a n, 29 n 1 qn 1 t n p n 0 Cp = 0 Cp a n 30 0 Cp a n, a n n N p p n, p n n N P λ/µ p P µ Cp P λ 31 P λ/µ p p n := xn i [14] 4. [10] 4.1. Φ 2 F l 1,l 2 F r 1,r 2 v F l 1+r 1,l 2 +r 2 +v l 1, l 2 = 1, N 0, 1 N Z,,, w C 3 Φ Φ : F z 0,1 F 1,N 1 F 1,N z, λ, 32 XΦ = Φ X X U, N N 1 33 F 0,1 P λ Φ Φ λ z Φ λ zα = ΦP λ α P λ α F 0,1 z X F 0,1 ad X, 33 F 1,N 1 34 ad ψ ± z, Φ λ w := ψ ± q/t ±1/4 zφ λ w ψ + q/t ±1/4 z 1, 35 ad x + z, Φ λ w := x + zφ λ w ad ψ z, Φ λ w x + z, 36 ad x z, Φ λ w := [ x q/t 1/2 z, Φ λ w ] ψ + q/t 1/4 z 1 37

6 Φ λ w 3637 δ δ w/z z, w Φ λ w ad x z, λ = Φ w δ = /z ad x + z, Φ λ w Φ λ w ad X, 0, 1 [ z, λ ] Φ λ z = Φ /z = P λ t ρ 1 2 x + q j 1 t 1 i z Φ z, 38 N N 1 Φ z = exp i,j λ 1 1 n 1 q a nz n 1 q n exp n n 1 q a nz n. 39 n x +, N F 1,N P λ t ρ 1 2 c C p n cq λ t ρ := c n q nλ i 1t n 1 2 i c n + 40 t n 2 t n 2 λ =, c = t 1 2 P λ ct ρ 1 2 := Pλ ct ρ 1 2 ; q, t = i,j λ cq j 1 1 q λ i j t λ i+1 x + q j 1 t 1 i z Φ z = q j t 2 i z N i,j λ i,j λ 1 1 t n exp n 1 q a nq/t n/2 z/q 1 n 2 n p n q λ t ρ 1 1 t n exp n 1 q a nq/t n/2 z/q 1 n 2 n p n q λ t ρ 28 C p C p p := p 1, p 2,, p := p 1, p 2, , 1 F 0,1 w 38 ad X, 0, 1 F w 0,1 [12, 13] ad ψ ± z, Φ λ w = q/t ±1/2 B ± λ w/z±1 Φ λ w, 43 ad x + z, Φ λ w = lλ+1 A + λ,i δqλ i t 1 i w/zφ λ+1i w, 44 ad x z, Φ λ w lλ = q/t 1/2 A λ,i δqλ i 1 t 1 i w/zφ λ 1i w 45

7 := q λ i t i i 1 A + λ,i := 1 t 1 t x j 1 q t x j, 46 j=1 1 x j 1 q x j 1 t x j A λ,i := 1 1 q x j t t 1 j=i+1 1 x j 1 q x j = 1 t 1 1 t x l+1 l 1 t x j 1 q x j t 1 q x, 47 l+1 j=i+1 1 x j 1 q x j B ± λ z := 1 zx ±1 1 z t2 q i ±1 1 zt ±1 1 z t x 48 q i ±1 λ i = λ i 1 A + λ,i = 0, λ i = λ i+1 A λ,i = 0 λ i < λ i 1, i- λ + 1 i = λ 1, λ 2,..., λ i 1, λ i + 1, λ i+1,... λ i > λ i+1, i- λ 1 i = λ 1, λ 2,..., λ i 1, λ i 1, λ i+1, Φ Φ Φ λ Φ : F 1,N+1 F 1,N F z 0,1, N N+1 49 XΦ = Φ X X U z, λ 50 Φ α = λ Φ λzα P λ / P λ P λ α F 1,N+1 51 P λ P λ P λ P λ := P λ, P λ q,t = 1 q λi j+1 t λ i 1 q λ i j t λ i+1 i,j λ 52 X F 0,1 ad X, 50 ad ψ ± z, Φ λw := ψ ± q/t 1/4 z 1 Φ λ wψ ± q/t 1/4 z, 53 ad x + z, Φ λw := ψ q/t 1/4 z 1 [ x + q/t 1/2 z, Φ λw ], 54 ad x z, Φ λw := x zad ψ + z, Φ λw Φ λwx z 55

8 x + z λ = Φ w = z [ ] z Φ λz = Φ N N+1 = P λ qt ρ 1 2 x q j 1 t 1 i z Φ z, 56 z, λ i,j λ Φ 1 1 z = exp n 1 q n t/qn/2 a n z n 1 q n exp n 1 q n t/qn/2 a n z n 57 x, N F 1,N i,j λ exp x q j 1 t 1 i z Φ z = i,j λ q j 1 2 t 1 2 i z N t n n 1 q a nz/q 1 n 2 n p n q λ t ρ 1 1 t n exp n 1 q a nz/q 1 n 2 n p n q λ t ρ 28 Cp C p ad X, 4.4. ad ψ ± z, Φ λw = q/t ±1/2 B ± λ w/z±1 Φ λ, 59 ad x + z, Φ λw lλ = q A λ,i δqλ i 1 t 1 i w/zφ λ 1 i w, 60 ad x z, Φ λw lλ+1 = q/t 1/2 q 1 A + λ,i δqλ i t 1 i w/zφ λ+1 i w 61 F 0,1 [ w, µ z, λ ] [ w, µ ] [ z, λ ] Φ zw z N+2 N+1 N := Φ zw z N+2 N+1 Φ z = Φ λ wφ λ z 62 2 F 1,N [ ] v/z v v/z v M 1 M Φ z z := λ Φ 1 P λ P λ [ Φ M 1 z, λ z, λ M z λ ] = λ 1 P λ P λ Φ λ z Φ λ z 63

9 4.5. v/z v M 1 M Φ z z { Sx := exp 1 1 t n 1 + n 1 q n n 1 α n = i 1 Sq λ i t ρ i q 1/2 z, 64 } { q n x n α n exp t n t n 1 + n 1 q n q t n x n α n }, 65 1 a1 1 + q/t n n q/t n /2 a 2 n 66 a 1 n = a n 1 a 2 n = 1 a n Sz q- - - X U v/z v M 1 M X, Φ z = 0 67 z [15] 5. AGT 5.1. ι ιp n = p n, ιa n = a n n > 0 [10] [ z, λ ] ιp µ Φ z ιp ν C µλ νq, t, 68 [ ] z ιp ν Φ N N+1 ιp µ C ν µλ q, t 69 z, λ C µλ ν q, t = P λ t ρ ; q, t σ ιp µ /σ tλ q ρ ; t, qp ν/σ q λ t ρ ; q, tq 1/2 /t 1/2 σ ν f ν q, t 1, C µλ νq, t = 1 λ + µ + ν C µ λ ν t, q 71 [16] f λ q, t := i,j λ 1qj 1 2 t 1 2 i 28 Cp, C p 31 Iqbal-Kozcaz-Vafa [17] [18] 70

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