( ) 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 [Mo] T. Mochizuki, Donaldson type invariants for algebraic surfaces. Transition of moduli stacks, Lecture Notes in Mathematics, 1972. Springer-Verlag, Berlin, 2009. xxiv+383 pp [NY] H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math. 162 (2005), no. 2, 313 355 ( :19340006) 2010 Mathematics Subject Classification: Primary 14D21; Secondary 57R57, 81T13, 81T60,,,, 606-8502 e-mail: nakajima@kurims.kyoto-u.ac.jp web: http://www.kurims.kyoto-u.ac.jp/~nakajima
Donaldson Seiberg-Witten [GNY] 1. 1.1. 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 Seiberg-Witten 1.2. 2 Σ X ( ) 1 KdA = e(σ) = Index p X 2πi Σ p Σ
M T M M T M T α = α p e(t p M) M p M T e(t p M) p M T p M T p M t T ( ) log t M = P 1 T = S 1 [z 0 : z 1 ] [z 0 : tz 1 ] p 0 = [1 : 0], p = [0 : 1] t, t 1 α M L c 1 (L) P 1 C 2 1 L L = { ([z 0 : z 1 ], v 0, v 1 ) P 1 C 2 λ C (v 0, v 1 ) = λ(z 0, z 1 ) }. T L ([z 0 : z 1 ], v 0, v 1 ) ([z 0 : tz 1 ], v 0, tv 1 ) L α p L p t T log t p 0 t 0 = 1, p t (1) P 1 c 1 (L) = 0 1 + 1 1 = 1 1 n t = diag(t 1,..., t n ) log t = diag(ε 1,..., ε n ) (1) α p, e(t p M) ε 1,..., ε n P n [z 0 : z 1 : : z n ] [z 0 : t 1 z 1 : : t n z n ] [1 : 0 : : 0],..., [0 : : 0 : 1] n + 1 c 1 (L) n P n 0 n = + ε 1 ε n ε n 1 ( ε 1 )(ε 2 ε 1 ) (ε n ε 1 ) + + ε n n ( ε n )(ε 2 ε n ) (ε n 1 ε n ) (1)
( 1) n c 1 (L) n c 1 (L) d d n 0 ( 1) n h d n (ε 1,..., ε n ) h d n (ε 1,..., ε n ) ε 1,..., ε n (d n) d < n ( ) 0 d > n 0 0 P n P n 2n Gromov- Witten Donaldson (Ellingsrud- Göttsche ) Nekrasov Donaldson Gromov-Witten 1.3. (1) M M T
M = C 2, T = S 1 S 1 (x, y) (t 1 x, t 2 y) t 1, t 2 α = 1 (1) 1 = 1 C ε 2 1 ε 2 Ĉ2 C 2 0 P 1 ( 1 ) Ĉ2 C 2 Ĉ2 p 2 0 p 1 1: C 2 T p 1 p 2 t 1, t 2 /t 1 t 1 /t 2 t 2 Ĉ 2 1 = 1 ε 1 (ε 2 ε 1 ) + 1 = 1 (ε 1 ε 2 )ε 2 ε 1 ε 2 C 2 1 Ĉ2 1 Ĉ2 C 2 C 2 1 ( [Ĉ2 ] [C 2 ] ) Ĉ2 C 2 C 2 1, Ĉ 2 1 1 1 Nekrasov
1.4. C 2 Nekrasov C 2 C 2 n S n (C 2 ) C 2n n S n C 2n /S n ( ) S n (C 2 ) 1 = 1 n! C 2n 1 T = S 1 S 1 C 2n n (0,..., 0) (ε 1 ε 2 ) n q n n 1 (ε 1 ε 2 ) n n! S n (C 2 ) 1 = exp( q ε 1 ε 2 ) q n S n (C 2 ) = exp(qc 2 ) n exp 1/ε 1 ε 2 C 2 1 C 2 n Hilb n (C 2 ) {I C[x, y] I dim C[x, y]/i = n} C 2 n I Hilb n (C 2 ) Hilb n (C 2 ) I n = 2 p L T p C 2 I = {f C[x, y] f(p) = 0, df L = 0} Hilb 2 (C 2 ) L S n (C 2 ) Hilb n (C 2 )
n π : Hilb n (C 2 ) S n (C 2 ) S n (C 2 ) Hilb n (C 2 ) 1 Hilb n (C 2 ) π Hilb n (C 2 ) S n (C 2 ) 1 = 1 Hilb n (C 2 ) S n (C 2 ) ( ) Hilb n (C 2 ) S n (C 2 ) 1 1 T Hilb n (C 2 ) C2 Hilb n (C 2 ) ( ) xy 2 y 3 y 2 y xy s 1 x x 2 2: Y Hilb n (C 2 ) Hilb n (C 2 ) 1 = Y e(t Y Hilb n (C 2 )) 1 e(t Y Hilb n (C 2 )) ( l Y (s)ε 1 + (a Y (s) + 1)ε 2 )((l Y (s) + 1)ε 1 a Y (s)ε 2 ) (2) s Y
s Y l Y, a Y leg length, arm length 2 1 ( l Y (s)ε 1 + (a Y (s) + 1)ε 2 )((l Y (s) + 1)ε 1 a Y (s)ε 2 ) = 1 (ε 1 ε 2 ) n n! Y s Y (Jack ) 2. Donaldson 2.1. Donaldson ( D ) 1989 4 X ( ) U(2)- A F (A) F A = da + 1 [A A] 2 2 F A = F A Uhlenbeck ξ = c 1 (P ), n = c 2 (P ) n n
1994 Kronheimer Mrowka D ( ) D 4 ( ) Witten Seiberg Seiberg-Witten ( SW ) ( ) Spin c - A ψ D + A ψ = 0, F + A + µ(ψ, ψ) = 0 Spin c A U(1)- µ(ψ, ψ) ψ SW 0 0 4 ( simple type ) SW Witten D SW Witten Seiberg D 4 R 4 R 4 D Seiberg-Witten 2.2. Witten Witten X χ(x) σ(x) (KX) 2 def. = 2χ(X) + 3σ(X), χ h (X) def. = χ(x) + σ(x) 4 X 4
ξ, n M(ξ, n) µ(α) k µ(p) l M(ξ,n) α, p 4 X α H 2 (X) p µ X M(ξ, n) µ(α), µ(p) D z, x, Λ D ξ (exp(αz + px)) def. = µ(α) k µ(p) l zk x l ) 3χ h (X) n,k,l M(ξ,n) k!l! Λ4n (ξ2 X KM-simple type 2 x 2 Dξ = 4Λ 4 D ξ ξ, α D ξ (α) def. = ( 1 D ξ,n (α k ) + 1 ) k! 2 Dξ,n (α k p) n,k D ξ (exp(αz + px)) Witten D ξ (α) = 2 (K2 X ) χ h(x)+2 ( 1) χ h(x) e (α2 )/2 s SW(s)( 1) (ξ,ξ+c 1(s))/2 e (c 1(s),α) (3) (, ) X, (α 2 ) = (α, α) SW(s) c structure s SW c 1 (s) = c 1 (S + ) H 2 (X, Z) s X KM-simple type SW-simple type SW(s) 0 c 1 (s) 2 = (KX 2 ) 2.3. Seiberg-Witten Witten D SW Pidstrigach-Tyurin Feehan-Leness (B.Chen ) Feehan-Leness D ξ (exp(αz + px)) = s f(z, x; χ h (X), (K 2 X), s, ξ, α, s 0 ) SW(s),
f χ h (X), (KX 2 ) s, ξ, α, s 0 z, x U(2) U(2)- U(1)- U(2)- X c s 0 s 0 Witten f(z, x) Feehan-Leness X D SW f(z, x) X Res da (4) a= n 1,n 2 Hilb n 1 (X) Hilb n 2 (X) Nekrasov Feehan-Leness f(z, x) χ h (X), (KX 2 ) s, ξ, α, s 0 4.1 SW(s) f(z, x) unique f(z, x) Feehan-Leness a (= U(2)- )
3. Seiberg-Witten 2002 Nekrasov R 4 D Nekrasov Seiberg-Witten Seiberg-Witten Nekrasov Nekrasov Nekrasov-Okounkov Braverman-Etingof Nekrasov 3.1. Nekrasov M 0 (r, n) R 4 SU(r)- Uhlenbeck( ) n R 4 S 4 E E E = C r Uhlenbeck S 4 R 4 M 0 (r, n) C 2 = R 4 T 2 T r ε 1, ε 2 a 1,..., a r a Nekrasov Z inst (ε 1, ε 2, a, Λ) def. = n Λ 2rn M 0 (r,n) M 0 (r, n) M 0 (r,n) π : M(r, n) M 0 (r, n) 1
M(r,n) M(r, n) C 2 r torsion free sheaf M 0 (r, n) S n (C 2 ) Hilb n (C 2 ) M(r, n) M 0 (r, n) M(r, n) M(r, n) T 2 T r r (Y 1,..., Y r ) n (2) Z inst r (4) M(r, n) T 2 T r T r M(r, n) T r = r M(1, n α ) n α n = n 1 + n r n α M(1, n α ) n α α=1 Z inst (ε 1, ε 2, a, Λ) = Λ n α n α Hilb n 1 (C 2 ) Hilb nr (C 2 ) 1 e(n) N Hilb n 1 (C 2 ) Hilb n r (C 2 ) M(r, n) normal bundle e(n) (4) 1/e(N) a ( r = 2 a a ) 3.2. Seiberg-Witten r 2 T r M 0 (r, n) T r T r 1 a 1 + + a r = 0 r = 2 a = a 2 = a 1 Seiberg-Witten
Nekrasov log Z inst (ε 1, ε 2, a, Λ) ε 1, ε 2 log Z inst (ε 1, ε 2, a, Λ) = 1 ε 1 ε 2 ( F inst ( a, Λ) + O(ε 1, ε 2 ) ) F inst ( ) log Z inst mild Z inst mild F inst Seiberg-Witten F inst u Λ y 2 = (z 2 + u) 2 4Λ 4 = (z 2 + u 2Λ 2 )(z 2 + u + 2Λ 2 ) (5) A, B- a = A ds = 1 π z 2 dz y ds, a D = u Λ da du = 1 dz 2π y 0 u a a D a ( Λ) a D = 2π 1 F a F F Λ ( ) 2 1a 4a 2 log 3a 2 Λ A B ds
Λ F inst u = a 2 1 4 Λ Λ F inst (6) a u Seiberg-Witten 3.3. Seiberg-Witten D Fintushel-Stern 4 X X#P 2 D ( u (6) ) Nekrasov R 4 = C 2 Ĉ2 U(r)- Uhlenbeck( ) M 0 (r, k, n) k = c 1 (E), [C] C 2 n M 0 (r, k, n) M 0 (r, n ) (k = 0 n = n ) M 0 (r, k, n) M 0 (r, n ) 1 = 1 M 0 (r,0,n) M 0 (r,n) Ĉ2 M(r, k, n) torsino free sheaf M 0 (r, k, n) M(r, k, n) M 0 (r, k, n) M 0 (r, k, n) M(r, k, n) M(r, k, n) (Y 1 1,..., Y 1 r ), (Y 2 1,..., Y 2 r ) r k 1,..., k r kα = k, Yα 1 + Yα 2 + 1 (k α k β ) 2 = n 2r α α<β
2r Ĉ2 (Y 1 1,..., Y 1 r ), (Y 2 1,..., Y 2 r ) C 2 M(r, k, n) M(r, n) (k = 0 k ) e(t Y 1 Hilb Y 1 (C 2 )) ε1 ε 1 e(t Y 2 Hilb Y 2 (C 2 )) ε 2 ε 2 ε 1 ε1 ε 1 ε 2 ε 2 ε 2 C 2 C 2 Ĉ2 (1) M 0 (r, k, n) M 0 (r, n ) (2) Z inst Z inst ε 1, ε 2 0 F inst contact term Seiberg-Witten contact term F inst F inst F ( ) 4. Witten [GNY] 4.1. 2.3 D SW ( ) f [EGL]
X f ( Nekrasov ) f(z, x) exp (F (z, x)) F (z, x) ξ, s, s 0, α, p, c 1 (X), c 2 (X) ( ) X 1 q n n Hilb n (C 2 ) 1 = exp(q 1) C 2 α 3, α 4 0 F (z, x) f(z, x) (α 2 ), (ξ, α), (c 1 (X), α), (s, α) X Feehan-Leness f(z, x) χ h (X), (K 2 X ) s, ξ, α, s 0 (1) c 1 (X) 4 (2) log f(z, x) (2) (1) ( ) 4.2. Nekrasov F (z, x) X X Hilb n (X) Hilb n (X) X Hilb n (X) T = ( ) ( ) }{{} n X T
f(z, x) exp F (z, x) F (z, x) = p X T F p (z, x) X T F p (z, x) p ε 1 (p), ε 2 (p) α p, ξ p,... F p (z, x) X = C 2 log ε 1, ε 2 α p ε 1 (p), ε 2 (p), α p X = C 2 exp F p (z, x) Nekrasov r = 2 M 0 (2, n) 1 ASD A Ker D + A V e(v) Z inst (ε 1, ε 2, a, m, Λ) = n Λ 3n M(2,n) e(v) Uhlenbeck M 0 (2, n) M(2, n) V M(2, n) m V S 1 m = a U(2)- Seiberg-Witten 4.3. F p (z, x) Z inst (ε 1, ε 2, a, m, Λ) log Z inst log Z inst = 1 ( ) F inst + (ε 1 + ε 2 )H inst + ε 1 ε 2 A inst + ε2 1 + ε 2 2 B inst + ε 1 ε 2 3
F inst Seiberg-Witten A inst, B inst H inst 0 a = F inst a Seiberg-Witten u ( u ) a u A, B u global a u u a u u = ±2Λ 2 (5) Witten a = Witten (5) Nekrasov e(v) Seiberg-Witten Witten 0 0 c (3) 0 SW Witten 7. 4 SW-simple type X superconformal simple type (KX 2 ) χ h(x) 3
( 1) ( w 2(X), w 2 (X)+c 1 (s))/2 SW(s)(c 1 (s), α) n = 0 (8) s w 2 (X) w 2 (X) 0 n χ h (X) (KX 2 ) 4 [MMP] Seiberg-Witten 4 D SW [MMP] X superconformal simple type D ξ mod 2 ( ) superconformal simple type Witten [EGL] G. Ellingsrud, L. Göttsche, M. Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), 81 100; arxiv:math.ag/9904095. [MMP] M. Marino, G. Moore and G. Peradze, Superconformal invariance and the geography of four-manifolds, Commun.Math.Phys. 205 (1999) 691-735; arxiv:hep-th/9812055.