0. Intro ( K CohFT etc CohFT 5.IKKT 6.

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1 0. Intro ( K CohFT etc CohFT 5.IKKT 6.

2 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν

3 vector x µ,y µ g µν x µ y ν g µν = x µ y µ (2) µ,ν x 2 = x x = x µ x µ (3) ( AB = BA) AB = BA ψ, χ, η φ :[A, B] AB BA {A, B} AB + BA δ ij : i = j i j δ(x y) :Dirac dxf(x)δ(x y) =f(y) ɛ µ ν :

4 τ l : l τ M: g µν : F: Map(l, M) x µ (τ) :F τ L : Map(F, C (M)) S l L(x µ(τ)) : S S x µ ( (Euler-Lagrange d dτ ( L (dx µ /dτ) δs ) = 0 L x µ = 0 (4)

5 0-2. : P µ L ( τ X µ ) H = P µ τ X µ L = P 2 2m + V (x) (5) E [X µ,p ν ]=X µ P ν P ν X µ = iδ µν P ν = i ν P ν X ν X µ P ν X µ P ν V (x) = kx 2 H = 1 2m p2 + kx 2 (6)

6 a kx + i p, a kx i p (7) 2m 2m [x, p] =i [a, a ]=1, [a, a] =[a,a ] = 0 (8) N = a a H = a a = N (9) N n N n = n n [a, a ]=1 a n = n +1 n +1, a n = n n 1 (10) N 0 a 0 =0 n = 1 a n 0 (11) n! n n E n = n ; n =0, 1, 2, (12)

7 1-2. n x A ( dx 1 dx n exp( x T π Ax) = n ) deta φ i (x) A ( Dφ exp( φ i (x)a ij 1 ) (x, y)φ j (y)) = C deta (13) Fermion dψψ =1, dψ =0 ψ i = A i j ψ j J = det A dψ 1 dψ n = (det A)dψ 1 dψ n

8 Fermion ψ i M ij dψ 1 dψ n exp( ψ i M ij ψ j )=2 n 2 det M (14)

9 1: 2: ( O Dφ i O exp ( S(φ i )) O (15) Z O =1 Z Dφ i exp ( S(φ i )) (16)

10 2. Cohomological F.T. etc Mathai-Quillen M : x M s a (x): TFT BRS ˆδ ˆδx µ = ψ µ dx µ, ˆδχa = H a, ˆδψµ = ˆδχ a = 0 (17) Cohomological Field Theory { } S = ˆδ 1 2 χ a(2s a (x)+a ab µ ψ µ χ b + b a ). (18) = 1 2 sa (x) χ aω ab µνψ µ ψ ν χ b i µ s a (ψ) µ χ a. s a (x) =0 : s a 2 =( µ s a δx µ ) 2 + (19) x 1/ det µ s a 2 ψ, χ det( µ s a ) (M 0 = {x s a =0} DxDψ 0 Dχ 0 e 1 2 χ a0ωab µν ψµ 0 ψν 0 χ b0 = Paff(Ω ab ) (20) M 0 M 0 Z = k ɛ k χ k (M 0 ) (21) k ɛ k = ± ( CohFT

11 Cohomological field theory iθ µν =[x µ,x ν ] θ θ = θ + δθ. (22) Topological Field Theory BRS ˆδδ = ±δ ˆδ, δ Z θ = DφDψDχDH δ ( ( = DφDψDχDH ˆδ dx DˆδV ) exp ( S θ ) ) δ V exp ( S θ )=0. θ-shift δ θ BRS θ-shift M θ δ θ θ µν = θ µν + δθ µν. (23) θ S θ det θdx D L( θ, 1 θ x ν ). (24) θ = exp2 i µ θ µν ν (25)

12 3.N.C.Cohomological Scalar model 3-1.Finite Matrix model with a connection M : N N Hermitian matrix V : N N φ ab : canonical coordinate of M : connection Γ(V ) Γ(T M V )=V Aji;mn(φ) : V kl e ij : local N 2 dim1 ij e kl = A ˆδ{ χ ij ([φ(1 φ)] ji + iχ mn A kl ji,mn(φ)ψ kl ih ij )} iχ ba { (ψ(φ 1) + φψ) ab ij;kl mn e mn Tr(φ(1 φ)) 2 (26) ijklmn ψ ij ψ kl F (ij, kl; ab, mn)χ mn } F (ij, kl; ab, mn) δ Akl;ab mn δ Aij;ab mn + i [A δφ ij δφ ij;aba cd kl;cd mn Akl;abA cd ij;cd mn ] kl (c,d)

13 (φ(1 φ)) = 0 Projection P : N k G k (N) Poincare P t (G k (N)) = (1 t 2 ) (1 t 2N ) (1 t 2 ) (1 t 2(N k) )(1 t 2 ) (1 t 2k ) N Z = P 1 (G k (N))( 1) k (27) Z = k=0 P ±1 (G k (N)) = N! k!(n k)! (28) N P 1 (G k (N))( 1) k =(1 1) N (29) k=0

14 3-2.N.C.Coh.F.T. [ ] S = dx D gl + S top (30) ( L = ˆδ 1 ( 2 χ 2(φ (1 φ) µ B µ ) + i d n zd n yψ(z)a(z; x, y)χ(y) ih) ). ( ) 1 +ˆδ 2 χµ ( µ φ + B µ ih µ ) Topological action (g (31) S top = gτ 2n (F,, F) (32) Connes s Chern character homomorphism ; ch 2n : K 0 (A) HC 2n (A) ch 2n (p) = τ 2n (f,,f) (33) n=0 where f ij =[p i p, p j p]. F ij =[φ i φ, φ j φ] (34) Cyclic cohomology

15 3-3. θ θ large θ, model S 0 = Trˆδ{ ˆχ( ˆφ(1 ˆφ) iĥ)} + Trˆδ{ ˆχ µ ( ˆB µ iĥµ)} (35) Tr{( ˆφ(1 ˆφ)) 2 +(ˆB µ ) 2 } (36) N ( ˆφ(1 ˆφ)) = 0 ˆB µ =0 Projection P GMS soliton P Finite θ θ

16 3-4. Moyal plane Fock space ( ˆφ( ˆφ a)) = 0 ˆB µ =0 GMS ap G k (N)(Moyal plane ) ( k P Moyal plane topological S top = gk θ Poincare P t (G k (N)) = (1 t 2 ) (1 t 2N ) (1 t 2 ) (1 t 2(N k) )(1 t 2 ) (1 t 2k ) N Z = P 1 (G k (N))e gk ( 1) k lim N g =0 k=0 = lim N (1 eg ) N (37) Z = 0 (38)

17 4.K CohFT 4-1.Topological Gauge Theory G G M A/G. ˆδ ˆδ 2 = δ g Yang-Mills ˆδA µ = iψ µ, ˆδψ µ = D µ θ = δ g A µ, (1) M A/G Poincare dual O(φ, dφ) = O(φ, dφ) e ˆδΨproj. (2) e ˆδΨproj A/G A A ˆδΨ proj S = (ˆδΨ+ˆδΨ proj ). (3) C C. θ η θ M δ g φ = Cθ. (4)

18 ˆδ θ = η, ˆδη = δg θ. (5) Ψ proj Ψ proj = C ψ, θ. (6) ˆδΨ p = (ˆδC )ψ + C Cθ, θ C ψ, η. (7) θ θ = 1 C C (ˆδC )ψ (8) Pf

19 M : n dim Riemannian Manifold v rank N φ : End(v) N N Hermitian matrix φ ab (x) and H ab (x) ψ ab (x) and χ ab (x), S 0 = S = S 0 + S pro (9) trˆδ{χ(φ(1 φ) H)} (10) M (φ(φ a)) 2 (11) (φ(φ 1)) = 0 P dimension k G k (N) = U(N) U(k) U(N k)

20 U(N k) U(k) G k,n θ = 1 [ψ, ψ] (12) C C M k,n = {φ M G k (N)}/G k,n Note: BRS ˆδφ = ψ G k,n V ect k (M) =[M,BU(k)] : BU(k) m=k+n+1 G k(m) Z = k V ect k (M) χ(m k,n ) (13) K (M) group : virtuarl dim K K(M) =Z K, M connected K = K K (M) = [M,BU( )] k > 1 2 dim M K (M) =[M,BU(k)] N K

21 4-4. K-theory Moyal plane τ 0 (P k )=k, τ 2 (P k )=k k K 0 N.C.torus SL(2,Z) θ K N.C.torus Morita K-group

22 5.N.C. Cohomological Yang-Mills Theory θ δ g,θ : θ δ 2 = δ g,θ (14) δ θ δ δδ θ δ θ δ = δ δ θ (15) δ 2 = δ g,θ+δθ (16) θ shift δ exact δ Z θ δ g,θ+δθ

23 N.C. Cohom. Yang-Mills Theory on 10-dim Moyal space IKKT ( 2,4,6,8 dimensional reduction N=4 Vafa-Witten theory U(1) Vafa-Witten theory Vanishing Theorem Moore-Nekrasov-Shatashvili Vanishing Theorem N.C. Cohomological Yang-Mills Theory on 4-dim Moyal space ADHM Matrix model D 1 D 3

24 1. K 2. ( 3. θ 4. N.C.

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T

.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977