N = , 4 Introduction 3 1 ADHM Construction Notation Yang-Mills Theory

Transcription

1 N = , 4 Introduction 3 1 ADHM Construction Notation Yang-Mills Theory BPST Instanton ADHM Construction Instanton Instanton BPST instanton ADHM Collective Coordinate Integral Instanton Calculus in Supersymmetric Gauge Theory N = 1, 2, 4 Supersymmetric Gauge Theory Supersymmetric Instanton Supersymmetric Collective Coordinates Prepotential and Centered Partition Function Example: one-instanton contribution in N = 2 SU(N) theory Introduction 52 3 Seiberg-Witten Theory

2 N = Duality Seiberg-Witten ADE singularity and SW theory Multi-instanton Calculus and Localization Formula Equivariant cohomology and localization equivariant cohomology (localization) Hollowood s approach Multi-instanton calculus and localization, Nekrasov s formula

3 Introduction 2 O 1 O n = Dφ e is O 1 O n O operator Minkowski Euclid Wick Euclidean O 1 O n = Dφ e S E O 1 O n (x 0 = ix 4 ) lattice QCD g exp( A/g 2 ) g exp g fermion boson cancellation 1/g 2 exp duality duality instanton 70 Belavin-Polyakov- Schwartz-Tyupkin BPST instanton [1] BPST instanton topological charge 1 Yang-Mills path-integral t Hooft [2] topological charge k k Atiyah-Drinfeld-Hicthin-Manin [3] 80 supersymmetric instanton N = 2 N = 1 3

4 N = 1 supersymmetric gauge gluino fermion operator N = 1 Yang-Mills superpartner λ α condensate λ α λ α N = 1 Yang-Mills QCD scale parameter 2 Amati-Rossi-Veneziano factorization condensate [4] strong coupling analysis Novikov-Shifman-Vainstein-Zakharov [5] Affleck-Dine-Seiberg [6] weak coupling analysis constrained instanton 2 dynamical scale Λ bare coupling constant cutoff λ α λ α = C ARV,NSVZ Λ 3 C ARV C NSVZ puzzle 90 N = 2 Yang-Mills instanton N = 2 Yang-Mills prepotential Seiberg-Witten [7] 1-instanton 2-instanton Finnell-Pouliot [8] Ito-Sasakura [9] Dorey-Khoze-Mattis [10] Seiberg- Witten instanton multi-instanton measure puzzle weak coupling analysis (constrained instanton) 2000 Hollowood-Khoze- Lee-Mattis [11] N = 2 instanton Hollowood localization [12] Hollowood Nekrasov [13] multi-instanton topological string large N duality instanton instanton introduction ADHM construction N = 2 Seiberg-Witten theory instanton 4

5 1 ADHM Construction spinor BPST instanton instanton topological charge k 1 M1 ADHM construction k instanton background path integral super 1.1 Notation Wess-Bagger notation [14] : x µ = (x 0, x 1, x 2, x 3 ). (1.1.1) metric (, +, +, +) Gamma 2 2 Dirac pair : ( ) 0 σ µ γ µ =. (1.1.2) σ µ 0 Dirac σ µ = ( 1, τ 1, τ 2, τ 3 ), σ µ = ( 1, τ 1, τ 2, τ 3 ) (1.1.3) τ Pauli Minkowski Euclid x 0 ix 4 4 : x 0 ix 4, x 4 = ix 0, (1.1.4) x n = (x 1, x 2, x 3, x 4 ). (1.1.5) x 4 ix 0 4-vector metric (+, +, +, +) Dirac Wick i factor Euclid Dirac σ n = σ nα α (i τ, 1), σ n = σ αα n ( i τ, 1). (1.1.6) 5

6 Euclid Dirac Dirac spinor index Wess-Bagger α α 2 index spinor ε tensor ε tensor ψ α = ε αβ ψ β, ψ α = ε α βψ β. (1.1.7) ε 12 = ε 21 = 1, ε 21 = ε 12 = 1, ε αβ ε βγ = δ α γ. (1.1.8) ε αβ ε βγ β δ α γ σ σ σ αα n = ε α βε αβ σ nβ β, σ nα α σ ββ n = 2δαδ β β α, tr σ n σ m = 2δ nm. (1.1.9) i Wess-Bagger spinor 4-vector spinor Dirac x α α = x n σ nα α (1.1.10) x n 4 α α 4 ADHM instanton bispinor Pauli ( ) ix 3 + x 4 ix 1 + x 2 x α α =, (1.1.11) ix 1 x 2 ix 3 + x 4 ix 1 + x 2 pair ix 3 + x 4 pair z 1, z 2 z 1 = x 2 + ix 1, z 2 = x 4 + ix 3. (1.1.12) x α α = ( z 2 z 1 z 1 z 2 ) (1.1.13) σ n σ n x αα = x n σ αα n = ( ix 3 + x 4 ix 1 x 2 ix 1 + x 2 ix 3 + x 4 ) (1.1.14) 6

7 ( ) x αα z 2 z 1 =, (1.1.15) z 1 z 2 ADHM spinor Dirac Lorentz σ mn = 1 4 (σ m σ n σ n σ m ), (1.1.16) spinor Lorentz dotted spinor σ mn = 1 4 ( σ mσ n σ n σ m ) (1.1.17) bar Lorentz Euclidian Euclid σ mn σ mn self-dual anti-self-dual 4 4 tensor σ mn = 1 2 ε mnklσ kl. (1.1.18) self-dual instanton anti-self-dual minus σ mn = 1 2 ε mnkl σ kl. (1.1.19) ε 4 tensor ε ε 1234 = 1 (1.1.20) m = 1 n = 2 σ 12 = 1 4 (σ 1 σ 2 σ 2 σ 1 ) = 1 4 (τ 1τ 2 τ 2 τ 1 ) = i 2 τ 3. (1.1.21) σ 1 iτ 1 1 Pauli σ 2 i 2 Pauli i i cancel Pauli iτ 3 /2 σ 34 σ 34 = 1 4 (σ 3 σ 4 σ 4 σ 3 ) = 1 4 (iτ 31 1( iτ 3 )) = i 2 τ 3 (1.1.22) σ 4 σ 4 σ 3 = iτ 3 iτ 3 /2 self-dual ε 1, 2, 3, 4 σ mn m n k, l

8 1 σ 12 = σ 34 bar anti-self-dual σ 12 = 1 4 ( σ 1σ 2 σ 2 σ 1 ) = i 2 τ 3, σ 34 = 1 4 ( σ 3σ 4 σ 4 σ 3 ) = i 2 τ 3 (1.1.23) σ 12 = σ 34 (1.1.24) ADHM instanton t Hooft eta symbol σ nm σ nm traceless tr σ mn = tr 1 4 (σ m σ n σ n σ m ) = 1 4 (2δ mn 2δ nm ) = 0, (1.1.25) tr σ mn = 0 (1.1.26) traceless 2 2 traceless Pauli σ mn = 1 2 iηc mnτ c (1.1.27) ηmn c t Hooft eta symbol bar Pauli σ mn = 1 2 iηc mnτ c. (1.1.28) instanton spinor instanton 1.2 Yang-Mills Theory Yang-Mills instanton BPST instanton Yang-Mills gauge BPST SU(2) SU(N) gauge N N SU(N) Lie Hermite anti-hermite anti-hermite i Hermite minus A m = A m (1.2.1) 8

9 SU(2) generator τ a 2 Pauli i τ Hermite i anti-hermite gauge gauge S[A] = 1 2 d 4 x tr N F mn F mn iθg2 16π 2 d 4 xtr N F mn F mn (1.2.2) 1 2 theta term θ field strength F mn = m A n n A m + g[a m, A n ], (1.2.3) Hermite i anti-hermite star F Hodge dual operator tilde 4 tensor F mn 1 2 ɛ mnklf kl (1.2.4) gauge 2 field strength F mn self-dual anti-self-dual instanton antiinstanton F mn = F mn. (1.2.5) instanton minus anti-instanton F mn = F mn. (1.2.6) instanton instanton d 4 x tr N (F mn ± F mn ) 2 0. (1.2.7) F F plus minus 2 trace anti-hermitian F mn instanton anti-instanton d 4 x tr N (F mn ± F mn ) 2 = d 4 x tr N (F mn F mn ± 2F mn F mn + F mn F mn ) (1.2.8) 9

10 3 F F tensor 1 F mn F mn = F mn F mn. (1.2.9) d 4 x tr N F mn F mn d 4 x tr N F mn F mn (1.2.10) F F k = g2 16π 2 d 4 x tr N F mn F mn Z. (1.2.11) F F trace k F k action F mn d 4 x tr N F mn F mn 8π2 k (1.2.12) g2 k k k instanton anti-instanton S[A] = 8π2 k + ikθ = 2πikτ, k > 0. (1.2.13) g2 1 k 2 F F k ikθ k τ 2πikτ τ Pauli τ k S[A] = 8π2 k + ikθ = 2πik τ, k < 0 (1.2.14) g2 τ τ = 4πi g 2 + θ 2π (1.2.15) combination combination Seiberg-Witten Riemann coupling θ combination 10

11 instanton path integral fluctuation 2 Gauss (anti-) self-dual ±D m F mn = 0 (1.2.16) ±D m F mn = 0 (1.2.17) field strength D m = m + ga m (1.2.18) anti-hermite 2 i self-dual anti-self-dual D m F mn = ±D m F mn = 0 (1.2.19) D m F mn = 0 Bianchi identity Abelian field strength field strength m ε mnkl ( k A l l A k ) (1.2.20) ε contraction zero m n trivial non-abelian instanton super 1.3 BPST Instanton instanton anti-instanton Belavin Polyakov Schwartz Tyupkin [1] instanton pseudo-particle title gauge SU(2) gauge SU(2) gauge A n 2 2 traceless 2 2 traceless Lorentz σ mn A n = g 1 2(x X) mσ mn (x X) 2 + ρ 2. (1.3.1) 11

12 instanton t Hooft eta symbol Lorentz SU(2) τ a A n = 2 g (x X) m ηmn a iτ a (x X) 2 + ρ 2 2 (1.3.2) instanton topological charge k 1 field strength trace k F mn self-dual field strength 1 overall 2 g factor m A n = 2 [ σ mn g (x X) 2 + ρ 2 (x X) ] pσ pn 2(x X) m ((x X) 2 + ρ 2 ) 2 (1.3.3) m n ga m A n ga m A n = 4 [ ] (x X)p (x X) q σ pm σ qn g ((x X) 2 + ρ 2 ) 2, (1.3.4) field strength F mn = m A n n A m + g[a m, A n ] = 4 [ σ mn g (x X) 2 + ρ 2 (x X) p(x X) m σ pn (x X) p (x X) n σ pm ((x X) 2 + ρ 2 ) 2 + (x X) ] p(x X) q [σ pm, σ qn ] ((x X) 2 + ρ 2 ) 2. (1.3.5) σ pm σ qn Lorentz Lorentz 4 [σ pm, σ qn ] = δ pq σ mn + δ mq σ pn + δ pn σ mq δ mn σ pq, (1.3.6) (x X) p (x X) q 4 σ p, q 2 field strength cancel 3 m n cancel 1 field strength F mn = 4 [ g 1 (x X) 2 + ρ 2 (x X) 2 ((x X) 2 + ρ 2 ) 2 ] σ mn = g 1 4ρ 2 σ mn ((x X) 2 + ρ 2 ) 2 (1.3.7) 12

13 F mn σ mn σ mn self-dual F mn self-dual instanton charge k = g2 16π 2 d 4 x tr F mn Fmn, (1.3.8) k F self-dual F mn F mn 2 k = g2 16π 2 d 4 x tr F mn F mn = 1 π 2 d 4 ρ 4 x tr σ mn σ mn ((x X) 2 + ρ 2 ) 4. (1.3.9) σ mn Pauli τ trace tr σ mn σ mn = 6 (1.3.10) Dirac k = 1 π 2 ( 6) d 4 ρ 4 x ((x X) 2 + ρ 2 ) 4, (1.3.11) 4 d 4 ρ 4 x ((x X) 2 + ρ 2 ) 4 = drr 3 ρ 4 dω 3 (r 2 + ρ 2 ) 4 = Vol(S 3 ) ρ4 2 = Vol(S 3 ) ρ4 2 = Vol(S 3 ) ρ4 2 = Vol(S 3 ) ρ4 2 t dt 0 (t + ρ 2 ) 4, (r2 = t) ( 1 dt 0 (t + ρ 2 ) 3 ρ 2 ) (t + ρ 2 ) 4 { (t + ρ 2 ) ρ (t + ρ 2 ) 3 { 1 2ρ 4 1 } 3ρ 4 = Vol(S3 ) 12 0 } (1.3.12) 3 Vol(S 3 ) 2π 2 12 k = 1 π 2 ( 6)π2 6 = 1. (1.3.13) 1 BPST instanton instanton 1 13

14 instanton technical x 1 x 1 x BPST gauge gauge singular gauge A n = g 1 2ρ 2 (x X) m σ mn (x X) 2 ((x X) 2 + ρ 2 ) 1 2 (1.3.14) instanton gauge U(x) = σ m(x X) m x X (1.3.15) gauge gauge x 1 x 3 path integral singular gauge instanton F mn 1 ((x X) 2 + ρ 2 ) 2 (1.3.16) field strength Euclid X ρ instanton X instanton ρ scale instanton size ρ X 1: instanton instanton k = 1 instanton instanton number k parameter X ρ section 1.4 ADHM Construction 14

15 instanton 1 BPST instanton instanton Atiyah-Drinfeld-Hitchin-Manin 3 [3] Dorey-Hollowood-Khoze-Mattis[15] supersymmetric [3] notation review ADHM : λi α = a λi α + b α λi x α α, λ = 1,..., N + 2k ; i = 1,..., k. (1.4.1) 2k {{( }} ){ = N+2k. (1.4.2) size x x 1 λ 1 N + 2k i instanton index 1 k α α 1 2 Hermite 2k (N + 2k) αλ i = ai αλ + x αα b λ iα, λ = 1,..., N + 2k ; i = 1,..., k. (1.4.3) N+2k {{( }} ){ = 2k. (1.4.4) a b (N + 2k) N SU(N) N k instanton k (N + 2k)-vector 2k (N + 2k) vector base N vector (N + 2k) vector N U (N + 2k) N U = 0, αλ i U λu = 0. (1.4.5) conjugate U Hermite U = 0, U λ u λi α = 0. (1.4.6) 15

16 vector U λ uu λv = δ uv. (1.4.7) U instanton solution (A n ) uv = g 1 U λ u n U λv, u, v = 1,..., N. (1.4.8) u v SU(N) gauge 1 N (N + 2k) λ k zero U N N pure gauge k non-zero non-trivial : αλ i λj β = δ α β (f 1 ) ij. (1.4.9) 2k 2k size : { }} ){ {( N+2k 2k 2k+N {}} ) { {( 2k 2k {{( }} ){ = 2k. (1.4.10) spinor instanton index invertible f f x k k Hermite vector f U (N + 2k) vector base U δ µ λ = U λuu uµ + λi α f ij αµ j (1.4.11) vector (U ) P P = UU = 1 f (1.4.12) U orthonormal condition P 2 = P : P µ λ U λuu uµ = δ µ λ λi αf ij αµ j. (1.4.13) field strength A field strength self-dual 16

17 F mn = m A n n A m + [A m, A n ] = m (U n U) n (U m U) + U m UU n U U n UU m U = m U n U n U m U m UUU n U + n UUU m U = m U(1 UU) n U n U(1 UU) m U = m U f n U n U f m U = U( m )f( n )U U( n )f( m )U = Ubσ m f σ n bu Ubσ n f σ m bu = 4Ubσ mn f bu. (1.4.14) A A m = g 1 U m U (1.4.15) field strength g (1.4.14) F mn = m A n n A m + g[a m, A n ] = m (U n U) n (U m U) +U( m U)U n U U( n U)U m U, (1.4.16) (1.4.14) 2 3 Ū U U 1 2 U (1.4.14) (1 UU) m Ū n U m n UU (1.4.12) projection operator (1 UU) null space (1.4.14) 5 6 U = 0, (1.4.17) U zero m U + U m = 0 m U = U m (1.4.18) pair m n 6 x 1 = a + bx, = a + xb (1.4.19) 17

18 x m m = bσ m. (1.4.20) x x α α = x n σ nα α Dirac bσ m σ m n f i, j instanton index spinor σ n σ m (1.4.14) 7 σ n σ m σ m σ n Lorentz generator 4 field strength (1.4.14) σ mn field strength self-dual instanton Instanton instanton selfdual F mn 2 trace n gauge index trace F mn (1.4.14) g2 16π 2 d 4 x tr N F mn F mn = 1 π 2 d 4 x tr N Ubσ mn fbuubσ mn fbu. (1.4.21) Osborn [41] 1 π 2 d 4 x tr N Ubσ mn fbuubσ mn fbu = 1 16π 2 d 4 x( n n ) 2 tr k log f (1.4.22) boundary x 2 k 1 16π 2 d 4 x( n n ) 2 tr k log f = k. (1.4.23) instanton number k instanton ansatz instanton αλ i λj β = δ α β (f 1 ) ij (1.4.24) 18

19 spinor a, b 1 : (a + xb)(a + bx) = aa + xba + abx + xbbx = f 1 k k. a αλ i a λj β = a αλ ( ) 1 2 aa δ α β, (1.4.25) ij i b β λj = b βλ i aλj α ( ), (1.4.26) b λ αib β 1 λj = 2 bb δα β (1.4.27) ij a b spinor index ADHM a b k-instanton a b X ρ 1-instanton U base redundancy unitary Λ U N + 2k 2k instanton index spinor index 2 instanton index γ f γ ADHM ansatz Λ γ 1, U ΛU, f γfγ (1.4.28) = (f 1 ) (1.4.29) (γ 1 ) Λ Λ (γ 1 ) (1.4.30) Λ unitarity 1 f 1 f (1.4.28) f 1 (γ ) 1 fγ 1 (1.4.31) ansatz (1.4.9) 2 2 (δ α β ). U = 0 (1.4.32) 19

20 unitary 2 unitary identity b x α, β instanton index ( ) b β λj = 0 bβ (u+iα)j = δαδ β, (1.4.33) ij ) b λ βj = b u+iα βj = (0, δαδ β ij, (1.4.34) ( ) a λj α = a (u+jα)j α = a αλ j a α α = a n σ nα α, w uj α (a α α ) ij, (1.4.35) = a α(u+iα) j = (w α ju, (a αα ) ji ) (1.4.36) ā αα = x n σ αα n. u = 1,..., N; j = 1,..., k; α, α, β, β = 1, 2; λ = 1,..., N + 2k a λ (N + 2k) 2 u gauge index instanton index α 2 index u N j, α double index j k α 2 2k (N + 2k) a (N + 2k) 2k N 2k 2k 2k block (1.4.35) w u N j, α 2k 2k 2k spinor index i, j 2k 2k Hermite conjugate b ADHM constraint (1.4.27) 3 a ADHM constraint τ c α βa βa ( α = τ c α β w β ) ju w ui α + (a βα ) jk (a α α) ki = 0 (1.4.37) a a traceless Pauli zero (1.4.27) 2 a b ADHM spinor index instanton index a (a n) = a n, (1.4.38) a n Hermite 2 ADHM constraint f f = 2(w α w α + (a n + x n 1 k k ) 2 ) 1 (1.4.39) 20

21 instanton index Osborn f x 1 decay x 2 a, w instanton parametrise redundancy instanton index unitary U(k) ( ) 1 N N Λ =, γ = Ξ, Ξ U(k) (1.4.40) Ξ1 2 2 w α w α Ξ, a n Ξ a nξ (1.4.41) fix b Dorey Nekrasov standard notation ( ) parametrize λi α = a λi α + b α λi x α α ( = = w uj α ) δ ij x α α + (a α α ) ij I J z 2 B 2 z 1 B 1 (1.4.42) ( z 1 B 1 ) z 2 B 2 b a w a x α α quaternion z 1, z 2, z 1 z 2 a α, α z B 1 block k k size 2 2 w 2 α 1 I 2 J bar conjugate : αλ i = ai αλ + x αα b λ iα = ( wi αu δ ij x αα + (a αα ) ) ij = ( I z 2 B 2 (z 1 B 1 ) J z 1 B 1 z 2 B 2 ). (1.4.43) 21

22 I J B 1, B 2 standard B 1, B 2 k k J, I N k I, J k N ADHM constraint parametrization : ( ) I z 2 B I J 2 (z 1 B 1 ) = J z 1 B z 2 B 2 z 1 B 1 1 z 2 B 2 ( z 1 B 1 ) z 2 B 2 = ( f f 1 ), (1.4.44) II J J + [B 2, B 2] + [B 1, B 1 ] = 0, IJ + [B 2, B 1] = 0, f 1 = II + ( z 2 B 2 )(z 2 B 2 ) + (z 1 B 1 )( z 1 B 1 ). (1.4.45) standard ADHM constraint B 2 B 2 B 1 standard notation x z 1 z 1 z 2 z 2 z 2, z 1 z 1, B 2 B 2, B 1 B 1 (1.4.46) ADHM constraint Nekrasov standard ADHM constraint : II J J + [B 2, B 2 ] + [B 1, B 1 ] = 0, IJ + [B 1, B 2 ] = 0, f 1 = II + (z 2 B 2 )( z 2 B 2 ) + (z 1 B 1 )( z 1 B 1 ). (1.4.47) Instanton instanton parameter instanton moduli space a w ADHM constraint residual unitary a spinor 4 k 2 w N 2k α 2 4Nk ADHM constraint 3 constraint k 2 constraint 22

23 residual unitary 4Nk instanton moduli space 2 4k 2 + 4Nk 3k 2 k 2 = 4Nk (1.4.48) BPST instanton ADHM BPST instanton k 1 i j instanton index 1 k 1 spinor (N + 2) 2 a i, j 1 1 spinor w N 2 a Dirac Hermite real X n : a n = X n. (1.4.49) BPS instanton space-time instanton parameter ADHM constraint w a ( w β u w u α + 2a na nδ β ) α = 0 (1.4.50) τ c α β a Pauli traceless w ρ w α uw u β = ρ 2 δ α β. (1.4.51) f BPST instanton : f = 2(ρ 2 + (x n X n ) 2 ) 1. (1.4.52) w N 2 ( 1 2 w u α = ρu 0 ), w αu = (1 2, 0)U ρ, (1.4.53) parametrise ρ w 2 ρ 2 phase unitary U 2 SU(N) frame framed moduli space SU(N) 23

24 vector U U = ( U ), U SU(N 2) (1.4.54) (N + 2) N redundancy 3 regular gauge BPS instanton solution singular gauge A n = g 1 2w αρ 2 (x X) m σ mn α βw β (x X) 2 ((x X) 2 + ρ 2 ) (1.4.55) SU(N) SU(2) w BPST instanton redundancy 4 U 1 fix : ( ) A SU(2) n 0 A n =, (1.4.56) 0 0 A SU(2) n = g 1 2ρ 2 (x X) m σ mn (x X) 2 ((x X) 2 + ρ 2 ). SU(N) N N SU(2) BPST instanton (1.4.56) unitary (1.4.55) redundancy U(1) ( ) A SU(2) n 0 A n = U U SU(N), U 0 0 U(1) SU(N 2), (1.4.57) unitary 1- instanton parameter X 4 scale parameter 1 gauge gauge SU(N) dim SU(N) = N 2 1 (1.4.58) dim SU(N) dim SU(N 2) dim U(1) = N 2 1 ((N 2) 2 1) 1 = 4N 5 (1.4.59) 3 redundancy w A µ instanton moduli 4 redundancy global instanton moduli 24

25 (4N 5) = 4N (1.4.60) 4N parameter (1.4.48) k = 1 gauge 1-instanton X ρ unitary gauge SU(N) k instanton parameter instanton a trace : X n = 1 k tr ka n. (1.4.61) ADHM construction 1.5 Collective Coordinate Integral ADHM construction Yang- Mills action path integral gauge instanton : A n (x) = A n (x; X) + δa n (x; X). (1.5.1) X redundancy 5 instanton 4Nk parameter moduli collective coordinate instanton fluctuation δa n gauge fluctuation gauge instanton background covariant derivative δa n zero D n δa n = 0 (1.5.2) field strength fluctuation δf mn = m δa n m δa n + g[δa m, A n ] + g[a m, δa n ] = D m δa n D n δa m, (1.5.3) fluctuation 2 energy massive mode 5 global 25

26 zero-mode path-integral zero mode δf mn self-duality D m δa n D n δa m = ɛ mnkl D k δa l (1.5.4) spinor δa n = 1 2 σ αα δa α α, D m = 1 2 D ββ σmβ β (1.5.5) A (α, α)-bi-spinor D m spinor x ββ self-duality 1 ( ) σ 4 mβ β σ n αα σ nβ β σ m αα ɛ mnkl σ kβ β σ l αα D ββ δaα α = 0 (1.5.6) Wess-Bagger APPENDIX B 9 σ mβ β σ αα n σ nβ β σ αα m = 2((σ nm ) β α δ α β + ( σ mn ) α βδ α β ) (1.5.7) self-dual part anti-self-dual part ɛ mnkl σ kβ β σ l αα = 1 ( ) 2 ɛ mnkl σ kβ β σ l αα σ lβ β σ k αα ) = ɛ mnkl ((σ lk ) α β δ α β + ( σ lk ) α βδ β α ( ) 1 = 2 2 ɛ nmlk(σ lk α ) β δ α β + 2 ( 1 2 ɛ nmlk( σ lk ) α β ) δ α β = 2(σ nm ) β α δ α β 2( σ nm ) α βδ α β (1.5.8) self-dual zero mode ( σ mn ) α βd βα δaα α = 0 (1.5.9) σ mn t Hooft eta symbol Pauli non-zero covariant derivative Pauli τ α β D βα δaα α = 0, (1.5.10) zero mode gauge D n δa n = D βα δa α β = 0 D βα δa α α = 0 ( β, α ). α D αα Λ α (C) = 0 (1.5.11) 26

27 α spinor Dirac α massless Dirac Λ α (C) = UCfb α U Ub α fcu (1.5.12) ADHM constraint U U C C C iλ a λj α = a i αλ C λj C iλ b α λj = bα iλc λj (1.5.13) constraint (1.5.12) Dirac Dirac operator : J Λ = UJU. (1.5.14) J = Cfb α b α fc. (1.5.15) C x U J J f D αα Λ α (C) = 2Ub α f( α C + C α )fb α U (1.5.16) zero Dirac α C + C α = 0 (1.5.17) (1.5.13) C zero mode parameter C (N + 2k) k 2 constraint redundancy 2kN 2k(N + 2k) 4k 2 = 2kN 2kN instanton background Weyl fermion massless mode zero mode zero mode Dirac zero mode δa m δa m = δx n δa m δx n + δ A m F mn + δf mn (1.5.18) 27

28 zero mode massive mode δ A m zero mode X m moduli parameter gauge moduli self-duality massive mode zero mode ADHM constraint zero mode consistent A m ADHM F µν self-duality check step check A n X µ = X µ (U nu) = U X µ U nu + U n X µ = U X µ nu + n ( = n U U X µ ( = D n U U X µ = D n ( U U X µ ( U U X µ ) + U ) ) n U U X µ X µ (UU + f ) nu n U(UU + f ) U X µ + U X µ f nu n U f U X µ ) + U a X µ f σ nbu Ubσ n f a X µ U (1.5.19) ( A α α X µ = D α α U U ) ( ) a α X µ + 2Λ α X µ (1.5.20) spinor index gauge Λ α Dirac a α moduli constraint zero mode constraint ADHM constraint X zero mode path-integral : X : X moduli parameter : X x : µ 1 4kN k 28

29 1 X µ SU(2) instanton instanton size gauge parameter parameter instanton size gauge X : instanton zero mode : moduli parameter ( ) a α δ µ A α α = 2Λ α X µ (1.5.21) instanton zero mode zero mode massive mode path integral instanton background : S = 2πikτ 1 d 4 x tr N δa αα (+) α β δa β α +. (1.5.22) 2 2πikτ Laplacian : (+) = D/D/ = D 2 gf mn σ mn (1.5.23) spinor spinor Fluctuation zero mode non-zero mode δa α α = δx µ δ µ A α α + Ãα α. (1.5.24) Zero mode action Laplacian zero string( ) zero mode measure zero mode zero mode determinant g metric g µν (X) = 2g 2 d 4 x tr N δ µ A n (x; X)δ ν A n (x; X) (1.5.25) 29

30 DA n = g 4kN µ dx µ 2π detg(x)d à n (1.5.26) moduli parameter X formal Gaussian massive mode DA n e S = e2πikτ g 4kN = e2πikτ g 4kN µ M k ω dx µ 2π detg(x) 1 det (+) 1 det (+) (1.5.27) Laplacian determinant det zero mode zero mode ω = µ dx µ 2π detg(x) (1.5.28) instanton moduli space volume form hyper-kähler hyper-kähler moduli ADHM path integral 30

31 2 Instanton Calculus in Supersymmetric Gauge Theory 2.1 N = 1, 2, 4 Supersymmetric Gauge Theory instanton path-integral measure N = 2 Seiberg-Witten prepotential 1-instanton supersymmetry instanton framework N supersymmetry gauge fermion 2 N N chiral anti-chiral scalar 2(N 1) N 1 on-shell superfield N = 4 off-shell on-shell N = 1 gauge 1 λ fermion 2 N = 2 gauge A m λ 1 λ 2 real scalar 2 N = 4 4 fermion 6 real scalar N = 1 A m λ, λ N = 2 A m λ 1 λ 2 λ 1 λ 2 φ 1 φ 2 N = 4 A m λ a λ a (a = ) φ 1, φ 6 adjoint Minkowski S = d 4 x tr N { 1 2 F 2 mn + iθg2 16π 2 F mn F mn + 2iD n λ A σ n λ A D n φ a D n φ a + gλ A Σ AB a [φ a, λ B ] + gλ A Σ aab [φ a, λ B ] g2 [φ a, φ b ] 2} (2.1.1) gauge Yang-Mills action fermion kinetic term scalar fermion bilinear scalar couple scalar potential 31

32 N = 2 N = 4 Σ : N = 2 Σ AB a = ɛ AB (i, 1), Σ aab = ɛ AB ( i, 1); N = 4 Σ a = (η 3, iη 3, η 2, iη 2, η 1, iη 1 ) Σ a = ( η 3, iη 3, η 2, iη 2, η 1, iη 1 ). η t Hooft η -symbol action on-shell SUSY δa n = ξ A σ n λ A ξ A σ n λ A, δλ A = iσ mn ξ A F mn igσ A abb ξb [φ a, φ b ] + Σ AB a σ n ξ B D n φ a, δλ A = i σ mn ξ A F mn ig σ B aba ξ b[φ a, φ b ] + Σ aab σ n ξ B D n φ a, δφ a = iξ A Σ aab λ B + iξ A Σ AB a λ B (2.1.2) on-shell 0 supersymmetry Minkowski Euclid x n = (x 0, x) x n = ( x, x 4 = ix 0 ) σ n = ( 1, τ) σ n = (i τ, 1) = i( τ, i) x 0 x 4 Dirac i Euclidean action i S Minkowski ( i) Euclidean action : S Euclidean = is Minkowski. (2.1.3) spinor Minkowski Lorentz SO(1 3) SL(2 C) α SL(2) doublet Hermite (λ A α ) = λ αa, (λ α A) = λ αa (2.1.4) Euclidean Lorentz SO(4) SU(2) SU(2) spinor λ α, λ α spinor spinor Minkowski Euclid action Minkowski Euclid Minkowski Euclidean action { S = d 4 x tr N 1 2 F mn 2 iθg2 16π 2 F mn F mn 2D n λ A σ n λ A + D n φ a D n φ a gλ A Σ AB a [φ a, λ B ] gλ A Σ aab [φ a, λ B ] 1 2 g2 [φ a, φ b ] 2}, (2.1.5) 32

33 Minkowski factor instanton D m F mn = 2g[φ a, D n φ a ] + 2g σ n {λ A, λ A }, /Dλ A = gσ AB a [φ a, λ B ], /Dλ A = gσ aab [φ a, λ B ], D 2 φ a = g 2 [φ b, [φ b, φ a ]] + gσ aab λ A λ B + gσ AB a λ A λ B. (2.1.6) Euclid version on-shell SUSY δa n = iξ A σ n λ A + iξ a σ n λ A, δλ A = iσ mn ξ A F mn igσ A abb ξb [φ a, φ b ] iσ AB a σ n ξ B D n φ a, δλ A = i σ mn ξ A F mn igσ aba B ξb [φ a, φ b ] iσ aab σ n ξ B D n φ a, δφ = iξ A Σ aab λ B + iξ A Σ AB a λ B. (2.1.7) N = 1 simple fermion gauge { S = d 4 x tr N 1 } 2 F mn 2 iθg2 16π 2 F mn F mn 2D n λ σ n λ (2.1.8) D m F mn = 2g σ αα n {λ α, λ α }, σ αα n D n λ α = 0, σ nα α D n λ α = 0 (2.1.9) supersymmetry δa n = iξσ n λ + iξ σ n λ, δλ = iσ mn ξf mn, δλ = i σ mn ξf mn. (2.1.10) N = 2 scalar 2 φ = φ 1 iφ 2, φ = φ 1 + iφ 2 (2.1.11) N = 2 N = 1 matter N = 1 scalar ψ gauge λ gauge N = 1 { S = d 4 x tr N 1 2 F mn 2 iθg2 16π 2 F mn F mn 2D n λ σ n λ 2D n ψ σ n ψ + D n φ D n φ + 2igψ[φ, λ] + 2ig[φ, λ]ψ g2 [φ, φ ] 2}, (2.1.12) 33

34 coupling scalar potential D m F mn = g[φ, D n φ] + g[φ, D n φ ] + 2g σ αα n {λ A α, λ A α }, /Dλ A = ig[φ, λ A ], /Dλ A = ig[φ, λ A ], (2.1.13) D 2 φ = g 2 [φ, [φ, φ]] 2igɛ AB λ A λ B, D 2 φ = g 2 [φ, [φ, φ ]] + 2igɛ AB λ A λ B σ instanton supersymmetric super δa n = iξ A σ n λ A + iξ A σ n λ A, δλ A = iσ mn ξ A F mn 1 2 igξa [φ, φ ] + σ n ξ A D n φ, δλ A = i σ mn ξ A F mn 1 2 igξ A[φ, φ] σ n ξ A D n φ, (2.1.14) δφ = 2ξ A λ A, δφ = 2ξ A λ A 2.2 Supersymmetric Instanton supersymmetric instanton super-instanton configuration minimize equations of motion gauge ADHM solution A m : ADHM solution, λ A = λ A = 0, φ a = 0. instanton background fermion /Dλ A = 0, /D λ A = 0, (2.2.1) D 2 φ a = 0. 34

35 N = 2 version ADHM φ λ Weyl (2.1.13) φ coupling constant order g modify supersymmetry coupling constant g order g super-instanton 6 fermion /D /D ( ) /D /D λ A = ( ) λa = D 2 λa + σ mn F mn λa (2.2.2) scalar 2 σ F self-dual σ mn F mn = 0 scalar D 2 λa = /D /D λ A = 0 (2.2.3) positive-definite 0 0 anti-chiral g linear order 0 λ 0 /D /Dλ A (+) λ A = D 2 λ A + F mn σ mn λ A = 0 (2.2.4) Dirac λ α = g 1/2 Λ α (M) = g 1/2 (ŪMf b α U Ūb αf MU) (2.2.5) C λ Grassmann Grassmann valued M M ADHM constraints fermionic M λ i a λj α = ā i αλ M λj, M iλ b α λj = b α iλ M λj C ( ) M λj = M (u+iα)j = µ uj (M α) ij (2.2.6) (2.2.7) 6 (2.1.12) (2.1.13) 35

36 (2.2.6) M α = M α (2.2.8) Ma α + ā α M µw α + αµ + [M α, a α α ] = 0 (2.2.9) fermion supersymmetric fermionic moduli M fermionic ADHM constraints fermion moduli BPST gauge supersymmetric zero mode superconformal zero mode super super δλ A = iσ mn ξ A F mn, δ λ A = i σ mn ξa F mn = 0 (2.2.10) F mn Lorentz generator λ F mn self-dual λ : λ A a = iσ mn ξ A F mn = 4i(σ mn ξ A ) α Ūbσ mn bfu = 4iŪ(bξA f b α b α fξ A b)u = Λ α ( 4ibξ A ). (2.2.11) fermion (2.2.5) M M A λi = 4iξA α b α λi, MλA i = 4iξ αa bλ αi (2.2.12) M 2 N super transformation superconformal ξα A (x) = ξα A x α α η αa, ξα A (x) = ξ A α + ηα Ȧ x αα (2.2.13) λ A a = Λ α ( 4ia η A ) = 4iŪ(a ηa f b α b α f η A ā)u = 4iŪ(bx ηa f b α b α f η A x b)u = 4i(σ mn x η A ) α Ūbσ mn bfu (2.2.14) M M A λi = 4ia λi α η A α, MλA i = 4i η Ȧ α ā αλ i (2.2.15) 36

37 scalar field modify D 2 φ a = g Σ aab λ (0)A λ (0)B. (2.2.16) scalar check review appendix ( ) φ a = 1 4 Σ aab ŪM A f M B U + Ū 0 N 0 U (2.2.17) 0 ϕ a 1 2 (2.2.16) ϕ a ϕ a = 1 4 Σ aab L 1 ( M A M B ) (2.2.18) L L : L Ω = 1 2 { w α w α, Ω} + 1 2ā αα a α αω ā αα Ωa α α Ωā αα a α α = 1 2 { w α w α, Ω} + [a n, [a n, Ω]] (2.2.19) order A λ λ g A m = g 1 A (0) m + ga (1) m + g 3 A (2) m +, λ A = g 1/2 λ (0)A + g 3/2 λ (1)A +, λ A = g 1/2 λ(0) A + g5/2 λ(1) A +, φ a = g 0 φ (0) a + g 2 φ (1) a + (2.2.20) A (0) m = Ū mu, λ (0)A = Λ(M A ) (2.2.21) exact g lower order N = 1 leading order A m = g 1 A (0) m, λ A = g 1/2 λ (0)A, λa = 0 (2.2.22) exact N = 2 scalar 0 A m = g 1 A (0) m. λ A = g 1/2 λ (0)A, λa = 0, φ a = g 0 φ (0) a (2.2.23) exact solution Seiberg-Witten scalar 37

38 scalar instanton exact size-dependence zero-size Derrick instanton action minimize minimize Affleck [18] mass operator insert Fadeev-Popov insert instanton size minimize operator operator insert modify instanton constrained instanton size instanton operator decouple size instanton instanton order g higher order modify operator scalar effective coupling operator instanton Affleck coupling Seiberg-Witten 2.3 Supersymmetric Collective Coordinates super collective coordinate supersymmetric collective coordinate ψ ia (i = 1, 2kN) λ A (x) = g 1/2 λ (0)A (x; X, ψ) + λ A (x; X, ψ) (2.3.1) volume form bosonic collective coordinate fermionic collective coordinate bosonic coordinate + fermionic fermion N N { [Dλ A ][Dλ A 2kN ] = g 4kNN dψ (Pfaff ia 1 ) } 1 2 Ω(X) [D λ A ][Dλ A ] A=1 A=1 i=1 (2.3.2) 38

39 fermionic measure Pfaffian inverse 7 Grassmann S[A m, λ A, λ A, φ a ] = S[g 1 A (0) m + Ãm, g 1/2 λ (0)A + λ A, λ A, φ a ] ( 4πi = 2πi g 2 + θ ) (2.3.3) k + S kin + S int 2π S kin = S int = { d 4 x tr N 1 A αα (+)β α 2 Ã β α 2D n λ A σ n λa + D n φ a D n φ a }, { d 4 x tr N λ (0)A Σ aab [φ a, λ (0)B ] 2g 1/2 [Ãn, λ A ] σ n λ (0)A } 2g 1/2 λ (0)A Σ aab [φ a, λ B ] + (2.3.4) instanton effective action e S eff = e 2πikτ [DÃ][Db][Dc][D λ][dλ][dφ] exp( S kin S int S gh ) (2.3.5) determinant supersymmetry cancel Dirac operator bilinear determinant det (+) det ( ) = µ 4Nk ( (+) /D /D, ( ) /D /D) (2.3.6) det spectrum fermion boson Pauli-Villars regularization regulator Pauli-Villars regulator mass spectrum µ super collective coordinate instanton effective action Z N k = M k ω N e e Seff = ( µ g ) 4kN(4 N ) 4kN e 2πikτ µ=1 dx µ N 2π 2kN A=1 i=1 dψ ia 7 : Phaffian 2n 2n M ij = M ji det g(x) (Pfaff Ω) N e e S eff (X,ψ), (2.3.7) PhaffM = ɛ i 1j 1 i 2 j 2 i n j n M i1 j 1 M i2 j 2 M inj N : det M = (PhaffM) 2. 39

40 instanton effective action φ S S = d 4 x tr N (D n φ (0) a D n φ (0) a λ (0) Σ aab [φ (0) a, λ (0)B ]) (2.3.8) bosonic collective coordinate fermionic collective coordinate ADHM (X, ψ) (a, M) (2.3.9) instanton effective action S = 4π 2 tr k { 1 2 Σ aabµ A φ 0 aµ B + w α φ 0 aφ 0 aw α ϕ a Lϕ a } { 1 = 4π 2 tr k 2 Σ aabµ A φ 0 aµ B + w α φ 0 aφ 0 aw α ( ) ( 1 4 Σ aabm A 1 M B + w α φ 0 aw α L 1 4 Σ acdm C M D + w βφ ) } 0 aw β (2.3.10) ω (N ) = c (N ) N k d 4k(N+k) a d 2k(N+k) M A det L 1 N M k vol U(k) A=1 k 2 { 3 ( 1 δ 2 tr k T r (τ c β α a βa ) α r=1 N c=1 A=1 α=1 2 δ (tr k T r (M A a α + a α M A) } (2.3.11) c (N ) k = 2 k(k 1)/2+kN(2 N ) π 2kN(1 N ). (2.3.12) a M ADHM constraint δ det L 1 N Jacobian factor hyper Kähler factor Seiberg-Witten 8 instanton moduli space bosonic ADHM instanton moduli fermionic Seiberg-Witten theory 40

41 moduli volume form ADHM constraint δ instanton moduli U(k) vol U(k) Jacobian factor δ duality 9 exp constraint : χ a : Hermitian k k, matrices (a = 1,, 2(N 1)), D : Hermitian k k matrices ψ α A : Grassmann k k matrices (A = 1,, N ). Grassmann ψ fermionic ADHM constraint couple ADHM constraint D Jacobian χ det L instanton measure k = 22(2 N ) π (2 3N ) c (N ) k d 4k(N+k) a d 3k2 D d 2(N 1)k2 χ vol U(k) (2.3.13) N d 2k(N+k) M A d 2k2 ψ A e S e S e L.M. Z (N ) A=1 S = 4π 2 tr k { w α χ a + φ 0 aw α 2 [χ a, a n] Σ aabµ A (µ B χ a + φ 0 aµ B ) Σ aabm A M B χa }, S L.M. = 4iπ 2 tr k { ψ α A(M A a α + a α M A ) + D τ c α βa βa α }. (2.3.14) L M Lagrange Multipliers S + S L.M. instanton moduli space effective action ADHM supersymmetry brane 9 :

42 δa α α = iξ αa M A α, (2.3.15) δm A α = 2iΣ AB a ξ α B[a α α, χ a ], (2.3.16) δw α = iξ αa µ A, (2.3.17) δµ A = 2iΣ AB a ξ α B(w α χ a + φ 0 aw α ). (2.3.18) δχ a = Σ AB a ξ αa ψ α B, (2.3.19) δψ α A = 2Σ aba B [χ a, χ b ]ξ α B i D τ α βξ β A, (2.3.20) δd = τ α βσ AB a ξ αb [ψa, β χ a ], (2.3.21) localization N = 2 action : S =4π 2 tr k { w α χ a + φ 0 aw α 2 [χ a, a n] 2 + i 2 µa (µ A χ + φ 0 µ A ) + i 2 M a M A χ χ =χ 1 iχ 2, φ 0 = φ 0 1 iφ 0 2. N f f=1 } κ f κ f (χ g 1 m f ), flavor effective action ADHM tr a X n = k 1 tr k a n (2.3.22) instanton super X n ξ A = i 4 tr k M A, (2.3.23) supertranslation instanton measure instanton M centered instanton moduli space M k = M k R 4 2N (2.3.24) Z N,N F k = ω (N,N F ) e S e S e L.M. (2.3.25) fm k 42

43 2.4 Prepotential and Centered Partition Function Yang-Mills ( ) N = 2 SU(N) Yang-Mills scalar field Cartan abelian N = 1 superfield W αu = (A mu, λ u ), Φ u = (φ u, ψ u ) (u = 1,, rank G) SU(N) U(1) superfield rank SU(N) = N 1 S eff = 1 { } 1 d 4 x Im 4π 2 τ uv(φ)wu α W vα + Φ Du (Φ)Φ u (2.4.1) θ 2 θ 2 θ 2 10 effective abelian vector multiplet chiral multiplet effective coupling τ uv dual Φ D prepotential 11 : Φ Du (Φ) = F Φ u, τ uv (Φ) = 2 F Φ u Φ v. (2.4.2) prepotential 1-loop perturbation instanton F = F pert + 1 2πi Λ k(2n N f ) F k (2.4.3) anti-chiral fermion 4 λ gauge multiplet (2.4.1) λ fermion ψ ψ τ uv (Φ) ψ ψ k=1 4 ψ τ 2 τ prepotential 4 4 Green prepotential λ λ Wick contraction contraction propagator instanton scale λ α u 1 (x 1 )λ β u 2 (x 2 )ψ γ u 3 (x 3 )ψ δ u 4 (x 4 ) = Λ k(2n N f ) 1 4 F k 2πi φ 0 u 1 φ 0 u 2 φ 0 u 3 φ 0 u 4 d 4 X S αα (x 1, X)Sα(x β 2, X)S γγ (x 3, X)Sγ(x δ 4, X) (2.4.4) 10, θ 2 Grassmann θ θ 2. R d 2 θ. θ 2 θ2 R d 2 θd 2 θ. 11 :

44 S(x, X) = 1 4π 2 / 1 (x X) 2 (2.4.5) 4 Green prepotential path-integral λ α u 1 (x 1 )λ β u 2 (x 2 )ψ γ u 3 (x 3 )ψ δ u 4 (x 4 ) ( ) µ k(2n Nf ) = e 2πikτ ω (N =2,N f ) e S e λ α u g 1 (x 1 )λ β u 2 (x 2 )ψ γ u 3 (x 3 )ψ δ u 4 (x 4 ) M k (2.4.6) Green massive mode zero mode ω N =2 N f instanton moduli volume form S instanton effective action fermion classical zero mode zero mode dependence supersymmetry supersymmetric zero mode supersymmetric δλ A = ig 1/2 Σ aab /Dφ a ξ B = g 1/2 /Dφ ξ A (2.4.7) ξ B scalar /Dφ φ D 2 φ = gχ χ (2.4.8) matter matter 0 φ 0 φ = U ( ) φ ϕ U (2.4.9) 1 2 ϕ ϕ = L 1 1 N F κ f κ f + w α φ 0 w α (2.4.10) 4 f=1 L 1 operator inverse instanton { ( ( /Dφ 1 ) uu = / (x X) 2 w u α φ 0 u 1 k + L 1 w βφ 0 w β 1 N F κ f κ f )}wu α (2.4.11) 4 f=1 44

45 (2.4.7) effective action antichiral fermion λ α u(x) = 2 gs αα (x, X)ɛ AB ξ A α S φ 0 u + (2.4.12) instanton effective action φ 0 propagator (2.4.6) effective action φ 0 propagator λ α u 1 (x 1 )λ β u 2 (x 2 )ψ γ u 3 (x 3 )ψ δ u 4 (x 4 ) = 1 ( ) µ k(2n Nf ) 4π 2 g2 e 2πikτ 4 g φ 0 u 1 φ 0 u 2 φ 0 u 3 φ 0 ω (N =2,N f ) e S e u 4 fm k d 4 XS αα (x 1, X)Sα(x β 2, X)S γγ (x 3, X)Sγ(x δ 4, X) (2.4.13) X instanton moduli fermionic moduli ξ path-integral fermionic partner supermoduli centered instanton moduli space prepotential 4 operator 12 up to 4π F k centered instanton partition function F k = g k(2n N f )+2 Ẑ (N =2,N f ) k, Λ 2N N f N f = µ 2N N f e 2πiτ (2.4.14) microscopic prepotential (2.4.12) (2.4.13) = 2 derivative OK g g constraint instanton 2.5 Example: one-instanton contribution in N = 2 SU(N) theory SU(N) 1-instanton microscopic 12 trφ 2 45

46 Nekrasov 1-instanton instanton index ADHM spinor ( ) w a = a (2.5.1) w 2 N a 2 2 spinor instanton effective action trace S = 4π 2 { w u α χ + φ 0 w u α 2 + i 2 µa u (µ ua χ + φ 0 u µ ua ) N f f=1 S L.M. = 4iπ 2 { ψ α A(µ A u w u α + w u α µ A u + D τ α β w β u w u α } 13 } κ f κ f (χ m f ) + S L.M., (2.5.2) (2.5.3) µ Grassmann µ µ : α u µ A u µ A u 2w u α α u ψ αa, µ A u µ A u + 2w u α αu ψ αa. (2.5.4) α u = χ + φ 0 u, α u = χ + φ 0 u w u α ADHM φ 0 scalar χ { S =4π 2 w u α χ + φ 0 w u α 2 + i ( µ A u + 2w )( u α 2 αu ψ αa + 4π 2 i ( 4w ) u αw u β 2 αu ψ αa ψ β A + D τ α β w β u w u α. µ ua 2w u β α u )} ψ β A (2.5.5) (2.5.4) shift ( µ A u + 2w )( u α αu ψ αa µ ua 2w u β α u ) ψ β A (2.5.6) µ A u µ ua µ linear Grassmann N (2π 2 αu) 2 (2.5.7) u=1 13 (2.5.2) 2 m f N f. 46

47 shift ψ bi-linear (2.5.5) linear constraint ψ (2.5.6) matter ( N F ) N F d N F κd NF κ exp π 2 κ f κ f (χ m f ) = π 2N F (m f χ) (2.5.8) f=1 factor w w 2 Gauss : ) d 2N wd 2N w exp ( 4π 2 A u w uw α u α + 4iπ 2 Bu τ α β w β u w u α f=1 N = (2π) 2N 1 A 2 u + B. (2.5.9) u 2 w w spinor 2 quadratic spinor Pauli (2.5.5) A 2 u α u 2 B u D + Ξ u Ξ u ψ bi-linear u=1 Ξ u = (α u) 1 ψ Ȧ α τ α β ψ β A (2.5.10) w, w Ẑ (N =2,N F ) k = 1 (2π) 3 d 2 χd 3 D 2 d 2 ψ A A=1 N u=1 α 2 u α u 4 + ( D + Ξ u ) 2 N F (m f χ) (2.5.11) ψα Ȧ Grassmann Ξ ψ A A spinor 4 fermion Ξ 2 d 2 ψ A Ξ c uξ d v (2.5.12) 2 A=1 f=1 = 8 δcd α uα v (2.5.13) F F (Ξ) = N u=1 α 2 u α u 4 + ( D + Ξ u ) 2 (2.5.14) 47

48 F 2 Ξ Ξ 0 Taylor 2 (2.5.13) 2 d 2 ψ A F (Ξ) = 4 A=1 N u,v=1 1 2 F (Ξ) αuα v Ξ c u Ξ c. (2.5.15) v Ξ=0 χ conjugate 2 N u,v=1 1 2 F (Ξ) αuα v Ξ c u Ξ c = 1 v Ξ=0 D 2 (2.5.11) Ẑ (N =2,N F ) k = 1 (2π) 3 d 2 χ ( d 3 D 4 ) 2 F (Ξ = 0) N F D 2 χ 2 χ D D Ξ = 0 d 3 D D 2 N u=1 2 F (Ξ = 0) χ 2. (2.5.16) f=1 (m f χ) (2.5.17) α 2 u α u 4 + D 2 (2.5.18) : dddω 3 N u=1 α 2 u α u 4 + D 2 = 4π 0 = 2π dd dd N u=1 N u=1 α 2 u α u 4 + D 2 α 2 u α u 4 + D 2. (2.5.19) D 0 + D = ±i α u 2 (2.5.20) 2π dd N u=1 α 2 u α u 4 + D 2 = 4π2 i N u=1 = 2π 2 N u=1 ( N α 2 ) v Res α v 4 + D 2, D = i α u 2 α u α u 48 N v=1 ( u) v=1 α 2 v α v 4 α u 4. (2.5.21)

49 = π 1 Ẑ (N =2,N F ) k f 1 (χ, χ ) = N u=1 N F α u α u d 2 χ 2 χ 2 f 1(χ, χ )f 2 (χ) (2.5.22) N v u α 2 v α v 4 α u 4, (2.5.23) f 2 (χ) = (m f χ). (2.5.24) f=1 χ f 2 matter matter 1 χ Stokes singular 0 2 α u = 0 α v 4 α u u v singularity cancel singularity α u = 0 χ φ u singularity re iθ ( χ = eiθ r 2r r 1 ) i θ (2.5.25) 2 θ 2 e2iθ = χ 2 r r(2 + r r ) + θ ( ) (2.5.26) θ θ 0 2π r θ : ( r ) 2π dθe 2iθ f 1 (r, θ)f 2 (re iθ ). (2.5.27) 4π r 0 contour α u = 0 f 1 (r, θ) = N u=1 e 2iθ v u N F α 2 v α v 4 r 4, (2.5.28) f 2 (re iθ ) = (m f + φ 0 u re iθ ). (2.5.29) f=1 49

50 e iθ e iθ (2.5.29) re iθ r 0 r / r 0 α 2 v α v 4 φ u φ v 1-instanton F 1 Ẑ(N =2,N F ) = N u=1 v u 1 (φ 0 u φ 0 v) 2 N F (m f + φ 0 u) + S N F 1 (2.5.30) S N F 1 flavor 0 f=1 0 N F < 2N 2, α 1 N F = 2N 2, S N F 1 = α N 1 f=1 m f N F = 2N 1, α NF 1 f,f =1 (f<f ) m f m f + α 2 N u=1 (φ0 u) 2 N F = 2N. (2.5.31) instanton parametrization G [19] r+1 i=1 G = A r, F 1 (a) = r (a 1,, â i,, a r+1 ) 2 r+1, (2.5.32) (a 1,, a r+1 ) G = B r, F 1 (a) = 2 r i=1 Qr 1 (a 1,, â i,, a r ) 2Q r, (2.5.33) (a 1,, a r ) G = C r, F 1 (a) = 2r 2 r, (2.5.34) i=1 a2 i G = D r, F 1 (a) = 2 r i=1 a2 i Qr 1 (a 1,, â i,, a r ) Q r, (2.5.35) (a 1,, a r ) 9 G = G 2, F 1 (a) = 3 i=1 (a, β i). (2.5.36) 2 m m Vandermonde determinant m m (a 1,, a m ) (a k a l ) 2, (2.5.37) k<l m Q m (a 1,, a m ) (a 2 k a2 l )2. (2.5.38) F 4 E k<l 50

51 G = F 4 F 1 (a) = 25 {(a 2 1 a2 2 )2 (a 2 3 a2 4 )2 + (a 2 1 a2 3 )2 (a 2 2 a2 4 )2 + (a 2 1 a2 4 )2 (a 2 2 a2 3 )2 } 4 i<j (a2 i a2 j )2 (2.5.39) G = E r F 1 (a) = α + (G) α 0 + (G):(α,α 0 )=0 (a, α0 ) 2 α 1 1 (α) (a, α1 )(a, α 1 α) α + (G) (a, (2.5.40) α)2 E 6 G 2 Seiberg-Witten ADHM construction 1-instanton ADHM construction moduli parameter parametrize SU(N) 1-instanton parametrize SU(2) BPS instanton G 2-instanton 51

52 Introduction for Day 2 Seiberg-Witten measure 1 14 Seiberg-Witten 1 moduli Nekrasov Hollowood Hollowood equivalent equivariant equivariant cohomology Berline-Vergne Hollowood U(1) k Nekrasov : Seiberg-Witten SU(N) 1- Equivariant cohomology Berline-Vergne Hollowood U(1) Nekrasov 14 : 1 instanton

53 3 Seiberg-Witten Theory Seiberg-Witten N = 2 Yang-Mills 3.1 : N = 2 Supersymmetric Gauge Theory G : gauge A a µ ψ q λ a ψ a q q φ a N = 2 vector multiplet a = 1..., dimg ψ eq N = 2 hypermultiplet N f flavors G adjoint N = 2 vector multiplet A µ λ ψ φ a 1 dim G N = 2 vector multiplet N = 1 supersymmetry A µ λ 1 gauge multiplet φ ψ matter N = 2 q q q 2 superpartner ψ q ψ eq N = 2 : L = 1 ( 1 g 2 Tr 4 F mnf mn iλ α i σα m αd m λ α i (D m φ )(D m φ) 1 2 [φ, φ] 2 i φ ɛ ij [λ αi, λ j α] + i ) φɛ ij [λ αi, λ α j ] 2 2 (3.1.1) 1 2 hypermultiplet N = 2 53

54 supersymmetry 2 supercharge supercharge index A N = 2 A = 1 2 : N = 2 SUSY Q A α Q αa (A = 1 2) {Q A α, Q αb } = 2δBσ A α m αp m, (3.1.2) {Q A α, Q B β } = ɛab ɛ αβ Z, (3.1.3) {Q αa, Q βb } = ɛ AB ɛ α βz. (3.1.4) Z : central charge Q Q A B index N = 1 2 super supersymmetry supercharge 2 super N = 1 2 non trivial Q Q Q Z Z generator Q P Lorentz generator central charge Z U(1) Witten Olive [20] Z = φ (Q e + iq m ) (3.1.5) Higgs Z Higgs Q e Q m : Q e = 1 φ Q m = 1 φ d 3 x i (φ a F0i) a electric charge, (3.1.6) d 3 x i (φ a 1 ) 2 ɛ ijkfjk a magnetic charge. (3.1.7) Higgs F 0i Q e dual N = 2 central charge 54

55 massive P µ = (M, 0, 0, 0) (3.1.8) P µ M Minkowski Q 1 Q 2 a α b α : a α = 1 2 (Q 1 α + ɛ αβ (Q 2β ) ), (3.1.9) b α = 1 2 (Q 1 α ɛ αβ (Q 2β ) ). (3.1.10) anti-commuting operator SUSY : {a α, a β } = {b α, b β } = {a α, b β } = 0 (3.1.11) {a α, a β } = δ αβ(2m + Z), {b α, b β } = δ αβ(2m Z) (3.1.12) a α b α a α b α a α a β b α b β central charge N = 2 unitarity central charge 2 2M Z 2M + Z Z M 1 2 Z { 2M + Z 0 2M Z 0 M 1 Z (3.1.13) 2 Z M 1 2 φ Q 2 e + Q 2 m BPS (3.1.14) BPS [16, 17] BPS BPST BPS Bogomol nyi-prasad-sommerfield = 16 null SUSY 55

56 BPS BPS BPS Plank coupling constant coupling BPS BPS Seiberg-Witten M > 1 2 Z dim = 24 = 16 M = 1 2 Z dim = 22 = 4 BPS BPS R ( U(1) ) supercharge supercharge 2 unitary M R : ( Q 1 Q 2 ) M ( Q 1 Q 2 ), M U(2) R = SU(2) R U(1) R (3.1.15) supercharge M unitary unitary U(1) U(1) φ e 2iα φ ψ e iα ψ U(1) R λ e iα λ v µ v µ (α R) (3.1.16) superfield R superfield super component U(1) R 3.2 N = 2 SU(2) 0 56

57 G = SU(2) classical V (φ) = 1 2g 2 Tr[φ, φ ] 2 = 0 (3.2.1) Higgs flat direction φ = 1 2 ( a 0 0 a ), (a C) flat direction (3.2.2) φ Tr φ 2 φ 1 2 a2 u = tr φ 2 = 1 2 a2 (φ : Higgs field) (3.2.3) u a parametrize parametrize u moduli moduli u 0 u 0 u 0 Higgs Higgs mechanism mass SU(2) U(1) massless u 0 U(1) multiplet u = 0 Higgs 0 SU(2) R u 0 SU(2) U(1) u = 0 SU(2) U(2) R symmetry (unbroken) 2 u SU(2) U(1) classical

58 SU(2) u U(1) u = 0 2: classical moduli space R SU(2) R U(1) R SU(N c ) Z 4Nc SU(2) N c = 2 Z 8 Moduli space of vacua (quantum) SU(2) R (unbroken) U(1) R Z 4Nc SU(N c ) Tr φ 2 moduli parameter Z 8 φ ω 1 8 φ ω 2 u φ 2 ω 4 ω 4 1 u 15 φ ω 2 φ u ω 4 u = u (ω 8 = 1) (3.3.1) QCD Λ QCD Higgs QCD Higgs 2 2 Λ 2 u u Λ 2 u Higgs weak coupling instanton N = 2 coupling 15 Higgs φ, Z 2. 58

59 u QCD energy scale Λ QCD u = tr φ 2 u Λ 2 weak coupling (asymptotic free) u Λ 2 strong coupling u u Λ 2 u u Λ θ N = 2 Yang-Mills Yang-Mills massive mode mode : D(massive)e S Y M e S eff (3.3.2) massless u multiplet massive massless U(1) gauge multiplet massless generic u U(1) N = 2 vector multiplet multiplet U(1) N = 2 vector multiplet A m λ ψ a massive index U(1) 1 vector multiplet 59

60 N = 2 1 N = 2 superspace N = 2 superspace (x m θ A θ A ) Ψ = φ +... (3.3.3) 2 superspace coordinate Ψ N = 2 L eff = d 2 θ 1 d 2 θ 2 F(Ψ) (3.3.4) N = 2 superspace F term coordinate superspace N = 2 superspace N = 1 superspace 2 1 super N = 1 superspace : L eff = 1 [ 4π Im d 2 θd 2 θ F(A) A A + d 2 θ 1 2 ] F(A) 2 A 2 W α W α (3.3.5) A = (φ, ψ) W α = (λ α, A µ ) F(A) : prepotential A F (3.3.4) A N = 2 vector multiplet Higgs chiral multiplet W α vector multiplet F A F term chiral superfield Ψ Ψ 0 Ψ Ψ supersymmetric F Ψ F 1 F 2 Higgs φ a a D = F(a) a τ(a) = da D da = 2 F(a) a 2 = θ eff 2π + i 4π geff 2 dual field of a (3.3.6) effective coupling (3.3.7) F 1 a dual field duality dual F 2 60

61 (3.3.5) 2 W α 2 2 F µν F µν F 2 Higgs τ parametrize U(1) effective coupling U(1) effective θ induce F Seiberg-Witten a a coupling F tree level a dual a D a 1 1 classical tree level τ(a) = τ cl a D = τ cl a F cl (a) = 1 2 τ cla 2. (3.3.8) N = 2 1-loop exact 1-loop a β β E log E β β function : dg d log E = b 16π 2 g3 one-loop exact (3.3.9) 1-loop g 3 b matter N = 2 flavor N F color N C b = 2N c 4N f (3.3.10) a coupling g(a) 4π g(a) 2 = b 2π log a Λ (3.3.11) 61

62 QCD parameter Λ Λ µ evaluate Λ = µe 8π2 /(4g(µ) 2 ) (3.3.12) (3.3.11) θ τ(a) τ(a) = i π log a2 Λ 2 (3.3.13) 1 a D 1-loop a D = ia π a2 log Λ 2 F 1 loop (a) = i ( ) a 2 2π a2 log Λ 2 (3.3.14) classical 1-loop instanton Seiberg 1988 [21] exact (3.3.14) instanton F(a) = 1 2 τ 0a 2 + i ( a 2 2π a2 log Λ 2 ) + ( ) Λ 4k F k a 2. (3.3.15) a instanton Seiberg-Witten moduli space a a D moduli space u u a a D a D a a F(a) k=1 (a(u), a D (u)) a D (a) F(a) = a D (a) (3.3.16) a β Seiberg-Witten 62

63 generic coupling discrete u Z 2 symmetry u = Λ 2 u = Λ 2 2 Seiberg-Witten massless singularities u = ±Λ 2 τ 0 ( g 2 eff ) effective theory is ill-defined new massless field massless 1 N = 2 mass central charge Z BPS Z U(1) Z a a D n e n m central charge massless field BPS 0 a a D 0 2 Λ 2 monopole massless 1 0 monopole massless Λ 2 massless M 1 2 Z Z = n ea + n m a D massless field Z = 0 u = Λ 2 : monopole massless (n m n e ) = (1 0) u = Λ 2 : dyon massless (n m n e ) = (1 1) Seiberg- Witten monopole nonlocal monopole monopole dual monopole dual Duality S = d 4 x ( 1 4g 2 F mnf mn + θ ) 32π 2 F mn mn F = 1 32π Im d 4 x τ (F + i F ) 2 (3.3.17) 63

64 θ τ θ action Z = DA m e is = DF δ(ɛ mnpq n F pq )e is (3.3.18) path integral field strength field strength action A D action field strength dual S = d 4 x(a D ) m ɛ mnpq n F pq = 1 16π Im d 4 x(f D + i F D )(F + i F ) (3.3.19) Z Z = DF DA D e i(s+s ) = S D = 1 32π Im DA D e is D (3.3.20) d 4 x 1 τ (F D + i F D ) 2. (3.3.21) S S F 2 F A D A D path integral dual A = A D, a = a D, a D = a τ = τ D = 1 τ (3.3.22) dual coupling coupling τ 1 τ monopole massless u = +Λ 2 dual U(1) massless monopole hypermultiplet β dual a a D a D (3.3.14) a a D u = +Λ 2 : dual U(1) + monopole (hypermultiplet) τ D = i π log a D +, (3.3.23) a = i π a D log a D + (3.3.24) dyon dyon weak couple dual a a D : 64

65 u = a D (u) 2ia ( a ) π ln (3.3.25) Λ a(u) 2u (3.3.26) u = Λ 2 a D (u) c 0 (u Λ 2 ) + (3.3.27) a(u) a 0 + i π a D ln a D + (3.3.28) u = Λ 2 a D (u) a(u) c 0 (u + Λ 2 ) + (3.3.29) a(u) a 0 + i π (a D a) ln(a D a) (3.3.30) u a(u) u = a 2 /2 a D (3.3.14) u = Λ 2 a D u = Λ 2 Λ 2 monopole massless a D a u Λ 2 a a D Seiberg-Witten a a D u a a D Riemann surface : y 2 = (x 2 Λ 4 )(x u) (3.3.31) meromorphic 1 meromorphic differential (SW differential) 2 x u λ SW = dx (3.3.32) 2π y Seiberg-Witten 16 (3.3.31) ±Λ 2 u

66 Λ 2 Λ 2 u α 3: β a a D a D (u) = λ SW (3.3.33) β a(u) = λ SW (3.3.34) α u Λ 2 β 1 β 0 dyon u Λ 2 a D a 0 dyon massless meromorphic differential Seiberg-Witten Λ microscopic Seiberg-Witten monopole massless dyon massless Seiberg-Witten u Λ 2 dyon massless 2 α β α β : γ = mα + nβ m, n Z (3.3.35) m n α β m n 66

67 β α 4: dyon u Λ 2 γ 0 α β dyon massless Λ 2 Λ 2 u 5: α β 3.4 ADE singularity and SW theory Seiberg-Witten curve SU(2) SU(2) : y 2 = (x 2 u) 2 Λ 4 (3.4.1) SU(N) hyperelliptic curve [22, 23]: SU(N c ) : y 2 = (x N c u 1 x N c 2 u Nc 1) 2 Λ 2N c. (3.4.2) G [24, 25, 26] : G ( ) : y 2 = P R f G (x)2 Λ 2h x 2d 2h (3.4.3) d (x) = (x λ i a) : G R f (degree d ) P R f G i=1 h : the dual Coxeter number OK Seiberg-Witten curve Lie 67

68 [27, 28] G Lax A = B = r b i H i + a i (E αi + E αi ) + za 0 E α0 + z 1 a 0 E α0 (3.4.4) i=1 r b i H i + a i (E αi E αi ) + za 0 E α0 z 1 a 0 E α0 (3.4.5) i=1 da dt = [A B] H i G E α α G spectral curve Σ G R P R G (x u 1, u r z) det(x1 d A) = 0 d = dimr Martinec-Warner spectral curve Seiberg-Witten curve identify Σ = Σ G R λ SW = x dz z Toda spectral curve Seiberg-Witten curve hyperelliptic curve ADE List of Toda spectral curves (I) : Simply-Laced Case (h = h ) A r ( A (1) r :r + 1 ) h = r + 1 {1, 2,, r} x r+1 u 1 x r 1 u r ) (z + µ2 = 0 z D r ( D r (1) :2r ) h = 2r 2 {1 3, 2r 3 r 1} ( ) x 2r u 1 x 2r 2 u r 2 x 4 u r x 2 u 2 r 1 x 2 z + µ2 z E 6 ( E (1) 6 :27 ) (Lerche-Warner,[29]) h = 12 { } ) 2 ) 1 (z 2 x3 + µ2 z + u 6 q 1 (x) (z + µ2 z + u 6 + q 2 (x) = 0 = 0 68

69 q 1 = 270x u 1 x u 2 1x u 2 x 10 + (26u u 3 )x 9 162u 1 u 2 x 8 + (6u 1 u 3 27u 4 )x 7 (30u 2 1u 2 36u 5 )x 6 + (27u 2 2 9u 1 u 4 )x 5 (3u 2 u 3 6u 1 u 5 )x 4 3u 1 u 2 2x 3 3u 2 u 5 x u 3 2, q 2 = 1 2x 3 (q2 1 p 2 1p 2 ), p 1 = 78x u 1 x u 2 1x 6 33u 2 x 5 + 2u 3 x 4 5u 1 u 2 x 3 u 4 x 2 u 5 x u 2 2, p 2 = 12x u 1 x 8 + 4u 2 1x 6 12u 2 x 5 + u 3 x 4 4u 1 u 2 x 3 2u 4 x 2 + 4u 5 x + u 2 2 (3.4.6) E 7 ( E (1) 7 :56 ) h = 18 {1, 5, 7, 9, 11, 13, 17} P 56 (x u 1, u 6 u 7 + z + µ2 z ) = 0 E 8 ( E (1) 8 :248 ) h = 30 {1, 7, 11, 13, 17, 19, 23, 29} P 248 (x u 1, u 7 u 8 + z + µ2 z ) = 0 For G = ADE P R G (x; u 1 u r + z + µ2 z ) = 0 ADE Dynkin simply-laced B C SO Sp F 4 G 2 hyper elliptic ADE spectral curve List of spectral curves (II) : Non Simply-Laced Case ( h h ) B r ( A (2) 2r 1 :2r ) h = 2r h = 2r 1 {1, 3,, 2r 1} P 2r A 2r 1 (x; u 1, u 3,, u 2r 1, u 2 = = u 2r 4 = 0, u 2r 2 = z + µ2 z ) = 0 ) = x 2r u 1 x 2r 2 u 2r 1 x (z + µ2 = 0 z C r ( D (2) r+1 :2r + 2 ) h = 2r h = r + 1 {1, 3,, 2r 1} P 2r D r+1 (x; u 1,, u r = z µ2 z, u r+1) = 0 = x 2r+2 u 1 x 2r u r 1 x 4 u r+1 x 2 ) 2 (z µ2 = 0 z F 4 ( E (2) 6 :27 ) h = 12 h = 9 {1, 5, 7, 11} ) PE 27 6 (x; u 1 u 2 = 0 u 3 u 4 u 5 = 6 (z + µ2, u 6 ) = 0 z 69

70 G 2 ( D (3) 4 :8 ) h = 6 h = 4 {1, 5} ) PD 8 4 (x; u 1 = 2u, u 2 = u 2 (z + µ2, u 3 = 3 z = 3 (z µ2 z ) 2 x 8 + 2ux 6 [ u 2 + (z µ2 (z + µ2 z ), u 4 = v + 2u z )] x 4 + [ v + 2u ) (z + µ2 ) = 0 z )] x 2 = 0. (z + µ2 z (E, F, G), spectral curve SW [26, 25] 2 SW prepotential 1 microscopic [19] spectral curve G 2, E 6 [30, 31] 70

71 4 Multi-instanton Calculus and Localization Formula Hollowood 4 3 Nekrasov U(1) Hollowood 4.1 Equivariant cohomology and localization Berline-Getzler-Vergne [32] [33] equivariant cohomology n C M ADHM G g G g M (g ϕ)(x) = ϕ(g 1 x) (4.1.1) X g M X M (X M ϕ)(x) = d dɛ ϕ(e ɛx x) (4.1.2) ɛ=0 M A(M) C[g] A(M) C[g] g C[g] A(M) G A G (M) g α(x) = α(ad(g)x), for any g G, X g (4.1.3) M g α M X g α L X α(x) = 0 (4.1.4) 71

72 A G (M) (equivariant differential form) C[g] A(M) (equivariant exterior differential) d g d g α(x) = dα(x) i(x)(α(x)) (4.1.5) i(x) contraction graded i(x)ω = ω(x), forω: 1-form i(x)(ω ω ) = i(x)ω ω + ( 1) p ω ω, forω: p -form, ω : q -form.(4.1.6) i(x) p -form (p 1) -form d g d 2 gα(x) = L(X)α(X) (4.1.7) L(X) = di(x) + i(x)d G A G (M) d 2 g = 0 equivariant cohomology d g α = 0 α A G (M) (equivariantly closed form) α = d g β (equivariantly exact form) α(x) α(x) = α [0] (X) + α [1] (X) + (4.1.8) i -form α [i] (X) d g α(x) = 0 i(x)α(x) [i] = dα(x) [i 2] (4.1.9) M α(x) = α(x) [n] (4.1.10) M α β α = d g β top Form M α(x) [n] = dβ(x) [n 1] (4.1.11) exact α(x) = d g β(x) = 0 (4.1.12) 0 α α 72

73 4.1.2 (localization) M G M metric g(x Y ) G α(x) M X M 0 M M 0 (X) singular Lemma (Poincaré Equivariant version) : M 0 (X) α(x) [n] X g Proof: D X λ = α M M 0 (X) λ D X = d i(x) M global U(1) - metric metric g = g µν dx µ dx ν (4.1.13) G - L X g = 0 X Killing g: T M T M β β g(x, ) = g µν X ν dx µ. (4.1.14) 0 = D 2 X β = L Xβ L X β = 0 β D X β = K X + Ω X K X = g µν (x)x µ X ν, global C (4.1.15) Ω X = dβ = (Ω X ) µν dx µ dx ν (Ω X ) µν = g µλ ν X λ g νλ µ X λ (4.1.16) K X M 0 nonzero D X β M M 0 (1 + x) 1 = ( 1) k x k k=0 ξ = β(d X β) 1 D X ξ = 1, L X ξ = 0. (4.1.17) α λ = ξα α = 1 α = (D X ξ)α = D X (ξα) = D X (λ). (4.1.18) Z(s) αe sdxβ, (s 0) (4.1.19) M 73

74 Z(s) s s 0: Z(0) = M α s : Z(s) M 0 M sharp Gaussian peak s R + αe sdxβ α equivariantly cohomologous d ds Z(s) = α(d X β)e sd Xβ M { } = D X (αβe sdxβ ) + βd X (αe sdxβ ) = s βαdxβe 2 sdxβ = 0 (4.1.20) M Z(s) s α = lim αe M s M sd Xβ (4.1.21) G Theorem (Berline-Vergne ) : α X g X M α(m) = ( 2π) l α(x) [0] (p) (4.1.22) M det 1 2 L p p M 0 (X) l = dim M/2 L p p L(X)ξ = [X M ξ] Proof: p M 0 G - metric exponential map p X M = λ 1 (x 2 1 x 1 2 ) + + λ l (x n n 1 x n 1 n ) (4.1.23) x 1 x 2, x n det 1/2 L p = λ 1 λ l θ p p θ p = λ 1 1 (x 2dx 1 x 1 dx 2 ) + λ 1 l (x n dx n 1 x n 1 dx n ) (4.1.24) L(X M )θ p = 0 (4.1.25) 74

75 n θ p (X M ) = x 2 i = x 2. (4.1.26) M 0 (X) x 2 0 D X θ p ( ) θ α(x) α(x) [n] = d D X θ i=1 [n 1] (4.1.27) p Bɛ p = { x; x 2 ɛ } Sɛ p = { x; x 2 = ɛ } α(x) = lim α(x) M S p ɛ ɛ 0 M pb p ɛ = p M 0 (X) S p ɛ θ α(x) D X θ p x ɛ 1 2 x S ɛ S 1 θ α(x) θ α ɛ (X) = D X θ S 1 D X θ (4.1.28) (4.1.29) α ɛ (x, dx) = α(x)(ɛ 1 2 x, ɛ 1 2 dx) α(x) [0] (p) (4.1.30) θ = S 1 D X θ θ(1 dθ) 1 = θ(dθ) l 1 = (dθ) l = ( 2) l l!(λ 1 λ l ) B 1 dx 1 dx n B 1 = πl l! ( 2)l l!(λ 1 λ l ) (4.1.31) 4.2 Hollowood s approach Berline-Vergne Hollowood measure smooth 0 ρ 0 Euclid 75

76 point-like Uhlenbeck M k,n = M k,n M k 1,N R 4 M k 2,N Sym 2 (R 4 ) Sym k (R 4 ) (4.2.1) R 4 point-like Sym i (R 4 ) R 4 M k i N Sym i (R 4 ) k i point-like k i moduli space M k i N i point-like instanton Sym i R 4 smooth ADHM : M k,n M (ζ) k,n (4.2.2) τ c α β(w βw α + a βα a β α ) = ζc 1 k (4.2.3) µ A w α + w α µ A + [M αa, a α α] = 0. (4.2.4) ζ = 0 (1.4.37) Kronheimer- [34] [35] U(N) Hollowood [36] ωe S e = e S e M k,n M ζ k,n (4.2.5) S = S + S LM (4.2.6) S LM = 4iπ 2{ ψ α A(µ A u w u α + w u α µ A u ) + D c ( τ c α β w β u w u α ζ c)} (4.2.7) ADHM Lagrange multiplier Grassmann ψ ia i = 1, 2kN 2 2kN = 4kN M k N dim = 4kN isomorphism A = α ψ ia h i α, 1-form (4.2.8) 76