コホモロジー的AGT対応とK群類似

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1 AGT K ( ) Encounter with Mathematics October 29, 2016

2 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv: (6 ) 4 N = 2 SU(2) = 2 Liouville ( 2 )Nekrasov ( ) = (Virasoro ) ( ) Virasoro( W)

3 Nekrasov Virasoro Whittaker ( )AGT K ( q) K K Nekrasov Virasoro Whittaker K gl 1

4 Nekrasov 1 = [M(r, d)] Nekrasov /Liouville c = 1 Painlevé τ W Maulik-Okounkov K Nekrasov version Liouville q gl 1 Fock 3

5 1 2 Nekrasov Gieseker Nekrasov 3 AGT Virasoro Whittker AGT rk 4 K K Nekrasov Virasoro Whittaker K AGT 5 Macdonald W Whittaker Macdonald

6 Nekrasov, Adv. Theor. Math. Phys. 7 (2003), Invent. math. 162 (2005), Lectures on Instanton Counting

7 2.1 Gieseker M(r, d): P 2 (:=CP 2 ) r, c 2 = n torsion free (Gieseker ) M(r, d) = (E, φ) = E : P 2 torsion free, l, rk(e) = r, c 2 (E) = d, φ : E l O r l l := {[0 : z 1 : z 2 ]} P 2. /, torsion free M(r, d) dim C = 2rd. φ c 1 (E) = 0. M(1, d) (C 2 ) [d] = Hilb d (C 2 ). (E I Z, dim Z = 0, length(z) = d, Z P 2 \ l C 2.) M(r, d)

8 M reg 0 (r, d) M(r, d): = Donaldson (1984) M d SU(r) : S4 (= R 4 { }) SU(r) A : SU(r) P / { = (A, ϕ) SU(r) γ γ = id ϕ : P SU(r) Donaldson G }

9 Uhlenbeck U d SU(r) : Md SU(r) Uhlenbeck U d SU(r) = d k=0 M k SU(r) Sd k R 4. U d SU(r) ADHM π : M(r, d) U d SU(r) (E, φ) ((E, φ), Supp(E /E)) X torsion free E E E := H(E, O X ) 0 E E A 0 A

10 ADHM U(r, d) = (B 1, B 2, j, k) B 1, B 2 End(C d ), j Hom(C r, C d ), k Hom(C d, C r ) (1)[B 1, B 2 ] + jk = 0 // GL d (C) GIT g (B 1, B 2, j, k) := (gb 1 g 1, gb 2 g 1, gj, gk) // (1) GLd (C) GL d (C) M(r, d) {(B 1, B 2, j, k) (1), (2)} / GL d (C) (2) B α (S) S(α = 1, 2) Im j S S C n.

11 T := (C ) 2 T, T GL r (C): (t 1, t 2, e 1,..., e r ) = ((t 1, t 2 ), diag(e 1,..., e r )) T T M(r, d) (t 1, t 2, e 1,..., e r ) (E, φ) := ((F 1 t 1,t 2 ) E, φ ) P 2 (C ) 2 : F t1,t 2 : [z 0 : z 1 : z 2 ] [z 0 : t 1 z 1 : t 2 z 2 ] T O r l (s 1,..., s r ) (e 1 s 1,..., e r s r ) (φ : E l O r l ) (φ : (F 1 t 1,t 2 ) E l O r l ). ADHM (B 1, B 2, j, k) (t 1 B 1, t 2 B 2, je, t 1 t 2 ek) U d SU(r) T

12 (E, φ) M(r, d) T iff E = I 1 I r with (1) I α = I Zα, dim Z α = 0, Z α P 2 \ l, (2) φ(i α l ) = α-th factor of O r l, (3) I α (C ) 2 I α C[x, y] I α I α x i 1 y j 1 j (i, j) xy 2 xy 3 Young Y α 1 1 M(r, d) T I = x 5, x 4 y, xy 2, xy 3, y 4 i x 3 y x 4 Y(r, d) := {r Young (Y 1,..., Y r ) rα=1 Y α = d} Y

13 e α 1 T (t 1, t 2, e 1,..., e r ) e α t i T R(T) R(T) Z[t ±1 1, t±1 2, e±1 1,..., e±1 r ]. T [Ellingsrud-Göttsche 1998] Y Yr (E, φ) M(r, d) T Y T Y M(r, d) T R(T) T Y M(r, d) = N α,β (t 1, t 2 ) := e β e 1 α r α,β=1 ( N α,β (t 1, t 2 ), t lyβ ( ) 1 t ay α ( ) Y α t lyα ( )+1 1 t ay α ( ) 2 Y β ).

14 K K T ( ): T Grothendieck K T ( ) K T (pt) K T (pt) R(T). π : M(r, d) U d SU(r) T π : K T (M(r, d)) K T (U d SU(r) ) (π := i ( 1)i R i π ) K T ( ) loc := K T ( ) R(T) R, R := Frac(R(T)). ι : M(r, d) T M(r, d). U d SU(r) T d[0] Sd C 2 U d SU(r) (U d SU(r) )T = {d[0]} (1 ) ι ι 0 : (U d SU(r) )T U d SU(r).

15 Thomason : ι 0 ι : K T (M(r, d) T ) loc K T (M(r, d)) loc. ι 1 ι Y : { Y } M(r, d) ι 1 ( ) = ι ( ) Y Y Y(n,r) 1 T M(r, d) Y K T (M(r, d)) loc π K T (U d SU(r) ) loc ι 1 ι 1 0 K T (M(r, d) T ) loc = Y R Y K T ((U d SU(r) )T ) loc = R

16 (Borel-Moore) H T ( ) = HT (, Q): T Borel-Moore {U n } n 1 : T ET BT (, H i (U n, Q) = 0 for 1 i n, U n U n /T: T ) H T n (X) := H n 2 dim T+2 dim Un (X T U n ). H T d (X) = 0 for d > 2 dim X. deg [M(r, d)] = 2 dim C M(r, d) = 4rd. H T ( ) H T (pt) H T (pt) S(T) := Sym((Lie T) ) H T ( ) loc := H T ( ) S(T) S, : ι 0 ι : H T (M(r, d)t ) loc S := Frac(S(T)). H T (M(r, d)) loc.

17 H T (M(r, d)) loc π H T (Ud SU(r) ) loc ι 1 ι 1 0 H T (M(r, d)t ) loc = Y S Y H T ((Ud SU(r) )T ) loc = S Y ι Y ( ) e T (T Y ) = (ι 0 ) 1 π ( )

18 Nekrasov ε 1, ε 2, a = (a 1,..., a r ) t 1 = e ε1, t 2 = e ε2, e α = e aα R(T). S(T) := Sym((Lie T) ) = Z[ε 1, ε 2, a ] S = Frac(S(T)) = Q(ε 1, ε 2, a ). Z(q; ε 1, ε 2, a ) = q d Z d (ε 1, ε 2, a ) := d=0 q d (ι 0 ) 1 π [M(r, d)] S[[q]]. d=0

19 Y ι Y ( )/e T (T Y ) = (ι 0 ) 1 π ( ) Nekrasov Z d (ε 1, ε 2, a ) = n Y α,β (ε 1, ε 2, a ) := 1 α,β r n Y α,β (ε 1, ε 2,, a ) 1 Y Y(r,d) Y α [ l Yβ ( )ε 1 + (a Yα ( ) + 1)ε 2 + a β a α ] Y β [(l Yα ( ) + 1)ε 1 a Yβ ( )ε 2 + a β a α ].

20 Z Y (ε 1, ε 2, a ) := [ 1 α,β r 1 rk = 1 Z Y = n Y 11 (ε 1, ε 2 ) n Y α,β (ε 1, ε 2, a )] 1. n Y 11 = Y [ l Y( )ε 1 + (a Y ( ) + 1)ε 2 ][(l Y ( ) + 1)ε 1 a Y ( )ε 2 ] Z 0 = Z ( ) = 1, Z 1 = Z (1) = 1 ε 1 ε Z 2 = Z (2) + Z (1,1) = + = 1 2ε 2 (ε 1 ε 2 )ε 2 ε 1 ε 2 ε 1 (ε 2 ε 1 )2ε 1 2ε 2 1 ε2 2 Z 3 = Z (3) + Z (2,1) + Z (1,1,1) = = 1 6ε 3 1 ε3 2 rk = 1 Z rk=1 (q; ε 1, ε 2 ) = q d Z d (ε 1, ε 2 ) = exp( q ) ε d 0 1 ε 2

21 Virasoro Virasoro Lie Vir : L n (n Z) : [L n, L m ] = (n m)l n+m + n(n2 1) 12 δ n+m,0 c. : Vir + := L n (n > 0), Vir 0 := L 0, 1, Vir := L n (n < 0). Verma M( ) (c, : generic) Vir + = 0, L 0 =. L 0 : M( ) = n 0 M( ) n, M( ) n := {v M( ) L 0 v = ( + n)v}. Verma M( ) Vir = 0, L 0 =. Shapovalov : M( ) M( ) C, ul n v = u L n v (u M( ), v M( )) = 1.

22 Whittker AGT (P 1 4, 1 ) ( )Nekrasov AGT Gaiotto arxiv: (M( ) Whittaker ) w(ξ) = + n 1 ξ n/2 w n, w n M( ) n L 1 w n = w n 1, L k w n = 0 (k 2). Whittaker w (ξ) = + n 1 ξ n/2 w n, w n M( ) n w n L 1 = w n 1, w n L k = 0 (k 2).

23 AGT w (ξ) w(ξ) = Z rk=2 (q; ε 1, ε 2, a ) Virasoro Nekrasov c (ε 1 /ε 2 + ε 2 /ε 1 ) (ε 1 /ε 2 + ε 2 /ε 1 + 2)/4 (a 2 a 1 ) 2 /ε 1 ε 2 ξ q/(ε 1 ε 2 ) 2

24 Z rk=1 Heisenberg Whittaker Z rk=1 (q; ε 1, ε 2 ) = exp(q/ε 1 ε 2 ). H: Heisenberg, [a n, a m ] = nδ n+m,0. π λ = C[a 1, a 2,...] λ : Fock, a 0 λ = λ λ. deg a n = n, deg 0 = 0. Whittaker w(ξ) = 0 + n 1 ξn/2 w n, deg(w n ) = n, a 1 w n = w n 1, a k w n = 0 (k > 1). w(ξ) = 0 +ξ 1/2 a 1 0 +ξ 2/2 1 2 a = exp(a 1ξ 1/2 ) 0. w (ξ) = 0 exp(a 1ξ 1/2 ). w (ξ) w(ξ) = 0 exp(a 1ξ 1/2 ) exp(a 1ξ 1/2 ) 0 = exp([a 1a 1]ξ) = exp(ξ). w (ξ) w(ξ) = Z rk=1 (q; ε 1, ε 2 ), ξ = q/ε 1 ε 2

25 rk W(sl r ) H r := d 0 H T (M(r, d)) loc H r Verma Shapovalov d 0 qd/2 [M(r, d)] Whittaker Z r = Whittaker

26 Nekrasov Virasoro q K Nekrasov Nekrasov (2003) Transform. Groups 10 (2005) Virasoro (1996) AGT q (2010), arxiv:

27 K Nekrasov T K K T (M(r, d)) loc π K T (U d SU(r) ) loc (ι ) 1 (ι 0 ) 1 K T (M(r, d) T ) loc = Y R Y K T ((U d SU(r) )T ) loc = R t 1 = e ε 1, t 2 = e ε 2, e α = e a α R(T). Z K (q; t 1, t 2, e ) = d 0 q d Z K d (t 1, t 2, e ) := d 0 q d (ι 0 ) 1 π O M(r,d).

28 Z K d (t 1, t 2, e ) = N Y α,β := 1, Y Y(r,d) 1 α,β r N Y α,β Y α (1 exp[l Yβ ( )ε 1 (a Yα ( ) + 1)ε 2 a β + a α )] Y β (1 exp[ (l Yβ ( + 1)ε 1 + a Yα ( )ε 2 a β + a α ]) (q, t i, e α ) = (ħ 2r q, e ħεi, e ħaα ) ħ 0 Z K (q; t 1, t 2, e ) Z(q ; ε 1, ε 2, a ) (ι 0 ) 1 = ch T T Z K d (t 1, t 2, a ) = i ( 1) i ch T H i (M(r, d), O M(r,d) ).

29 rk = 1 ( Z K rk=1 (q; t q n ) 1, t 2 ) = exp n(1 t n 1 )(1 tn 2 ) n 1 rk = 2 AGT (2009). Virasoro K q q

30 Virasoro Macdonald (1996) q, t C, generic, p := qt 1 Vir q,t : : T n (n Z) : [T n, T m ] = k=0 n=1 f l (T n l T m+l T m l T n+l ) l=1 (1 q)(1 t 1 ) (p n p n )δ n+m,0 1 p ( f k z k (1 q n )(1 t n ) z n ) = exp 1 + p n n : T(z) = n T nz n f(z) = k 0 f kz k f(w/z)t(z)t(w) f(z/w)t(w)t(z) = c(δ(pw/z) δ(w/pz)) f(z)

31 Vir q t = e βħ, q = e ħ, ħ 0 limit T(z) = n T nz n T(z) = 2 + βħ 2 (1 β)2 (L(z) + ) + O(ħ 4 ) 4β L(z) = L n z n {L n } Vir c. c = 13 6(β + 1/β).

32 Whittaker h C generic M(h): Vir q,t Verma T n h = 0 (n > 0), T 0 h = h h h Vir q,t deg T n = n M(h) = n 0 M(h) n. Verma M(h) h Shapovalov : M(h) M(h) C (M(h) Whitteker ) w q,t (ξ) = h + n 1 ξn/2 w n, w n M(h) n, T 1 w n = w n 1, T k w n = 0 (k 2) w q,t (ξ)

33 K AGT w q,t (ξ) w q,t (ξ) = Z K rk=2 (q; t 1, t 2, e ) Vir q,t Nekrasov q t 1 1 t t 2 h e 1 e e 1 ξ q 1 e 2

34 rk W(sl r ) q W(sl r ) d K T (M(r, d)) loc Whittaker correspondence correspondence d K T (M(1, d)) loc Schiffmann-Vasserot Feigin-Tsymbaliuk AGT gl 1 Ding

35 gl 1 Burban-Schiffmann (arxiv:math/ ) Hall Drinfeld (elliptic Hall algebra) (2007 ) (q, γ)-analog of W Schiffmann-Vasserot, arxiv: B.Feigin-Tsymbaliuk, arxiv: B.Feigin Y, arxiv: , Ding B.Feigin-E.Feigin Mukhin (2010) quantum continuous gl

36 (C 2 ) [n] correspondence Z n,n+i = {(I, J) (C 2 ) [n] (C 2 ) [n+i] J I}. K i = ±1, 0 τ n,n±1 : Z n,n±1 (C 2 ) [n] (C 2 ) [n±1]. τ n : (C 2 ) [n] τ n,n :τ n (C 2 ) [n] (C 2 ) [n] (C 2 ) [n] K := d 0 K T ((C 2 ) [d] ) loc u 1,k := t 1/2 1 u 1,k := t 1/2 2 n 0 τ k 1 n,n+1, n 0 τ k n+1,n (k Z), u 0,l := n 0 l τ n,n (1 t l 1 ) 1 (1 t l 2 ) 1 (l > 0), u 0, l := n 0 l τ n,n + (1 tl 1 ) 1 (1 t l 2 ) 1 (l > 0). U t1,t 2 (gl 1 ) := u 1,, u 0,, u 1, C(t 1/2 1,t 1/2 2 )

37 U t1,t 2 (gl 1 ) U t1,t 2 (gl 1 ) = u i,j (i, j) Z 2 \ {(0, 0)} Z 2 SL(2, Z) j i u 1,j u 0,j u 1,j

38 Macdonald Heisenberg Fock π α = Q[a 1, a 2,...] α a n p n = x n 1 + xn 2 + (n ) Λ Q a n n pn. {P λ (q, t) λ : }: Macdonald : Λ Q(q,t) : E 1 P λ (q, t) = P λ (q, t) ( q λ i t i ) E 1 E 1 = (η 0 1)/(t 1) with η(z) = η n z n, ( η(z) := exp n>0 1 t n n a n z n) ( exp n>0 1 q n a n z n) n

39 : η(z) η(z) η(z)η(w) = (1 w/z)(1 qt 1 w/z) (1 t 1 w/z)(1 qw/z) η(z)η(w) G (w/z)η(z)η(w) = G + (z/w)η(w)η(z), G ± (x) := (1 q ±1 x)(1 t 1 x)(1 q 1 t ±1 x) Ding Hopf

40 ([FHHSY] ) G ± (x) := (1 q ±1 x)(1 t 1 x)(1 q 1 t ±1 x), g(x) := G + (x)/g (x). x ± (z) = x ± n z n, ψ ± (z) = ψ m ± z m, n Z ±m 0 γ ±1/2 (central) ψ ± (z)ψ ± (w) = ψ ± (w)ψ ± (z), ψ + (z)ψ (w) = g(γ+1 w/z) g(γ 1 w/z) ψ (w)ψ + (z), ψ + (z)x ± (w) = g(γ 1 2 w/z) 1 x ± (w)ψ + (z), ψ (z)x ± (w) = g(γ 1 2 w/z) ±1 x ± (w)ψ (z), [x + (z), x (1 q)(1 1/t) (w)] = 1 p ( ) δ(γ 1 z/w)ψ + (γ 1/2 w) δ(γz/w)ψ (γ 1/2 w) (p := q/t), G (w/z)x ± (z)x ± (w) = G ± (z/w)x ± (w)x ± (z).

41 ([Feigin-Hasizume-Hoshino-Shiraishi-Y.]) Fock π 1 U q 1,t(gl 1 ) x + (z) η(z) ([Schiffmann-Vasserot, Feigin-Tsymbaliuk]) x ± n, ψ± m n Z, m 0 (γ±1/2 = (t/q) ±1/4 ) U q 1,t(gl 1 ) x ± n u n,±1, ψ ± n u ±n,0 U q 1,t(gl 1 ) π 1 K

42 W U q,t 1(gl 1 ) n (n) {b n n Z} ψ ± : ψ ± (z) = ψ ± 0 exp ( ± n>0 b nγ n/2 z n),. [b m, b n ] = 1 m (1 q m )(1 t m )(1 (q/t) m )(γ m γ m )γ m δ m+n,0. t(z) := α(z)x + (z)β(z). ( α(z) := exp b n z n ) ( b n z n ), β(z) := exp. γ n γ n γ n γ n n>0 Q(q, t). ( f k (z) := exp n=1 n>0 (1 q n )(1 t n )(1 (q/t) (k 1)n ) 1 (q/t) kn z n).

43 (n) (t(z)) = n i=1 Λ i(z) W q,t (sl n ). 1 i = j, f n (w/z)λ i (z)λ j (w) = Λ i(z)λ j (w) γ + (w/z; q, t) i < j, γ (w/z; q, t) i > j. γ ± (x; q, t) := (1 q 1 x)(1 qt 1 x). (1 x)(1 t 1 x)

44 Whittaker Macdonald K AGT w q,t (ξ) Virasoro T(z) = Λ 1 (z) + Λ 2 (z), Λ 1 (z) = p 1/2 1 t n a n t exp[ 1 + p n n t n p n/2 z n ] n=1 exp[ n=1 (1 q n ) a n n pn/2 z n ] n=1 Λ 2 (z) = p 1/2 t 1 1 t n a n exp[ 1 + p n n t n p n/2 z n ] exp[ (1 t n ) a n n p n/2 z n ] n=1

45 Fock π α Vir q,t generic q, t, h M(h) π α, h = p 1/2 q α + p 1/2 q α w q,t (ξ) M(h)[[ξ 1/2 ]] π α [[ξ 1/2 ]] Λ[[ξ 1/2 ]] (Y.) w q,t (ξ) = λ ξ λ /2 P λ (q, t) (i,j) λ (q/t) 1/2 t α 1 q j+1 t 2α i 1 q λi j 1 q λ i j+1 t λ j i U q,t 1(gl 1 ) Vir q,t

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib ( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson