2

III ( Dirac ) ( ) ( ) 2001. 9.22

2

1 2 1.1... 3 1.2... 3 1.3 G P... 5 2 5 2.1... 6 2.2... 6 2.3 G P... 7 2.4... 7 3 8 3.1... 8 3.2... 9 3.3... 10 3.4... 11 3.5... 12 4 Dirac 13 4.1 Spin... 13 4.2 Spin Dirac... 14 4.3 Dirac... 15 1

1 1 G Lie GL n (R),GL n (C),O(n),U(n) g Lie V G R n, C n ρ : G GL(V ) G X X G P G X ρ P E = P G V P, E G X {U α } U α P P = α U α G P g αβ : U αβ G U αβ := U α U β P S = {s α } s α : U α G g αβ s β = s α on U αβ 1 2

P P E = α U α V E g V αβ : U αβ GL(V ) 1.1 1.1. G A = {A α } A α : U α Ω 1 (U α, End(V )) A β = g 1 αβ A αg αβ + g 1 αβ dg αβ (1.1) 1.2. A = {A v α} A V α : U α Ω 1 (U α, End(V )) A V β = gv αβ 1 A V α gv αβ + gv αβ 1 dg V αβ G = O(n), U(n) A V α ( ) 1.3. (1.1) g 1 αβ (d + A α)g αβ = d + A β 1.4 (A ). C P := α U α g A β g 1 αβ A αg αβ + g 1 αβ dg αβ C P / X C P / G P C P / 1.2 1.5. E X :Γ(E) Γ(T X E) f C (X),s Γ(X, E) (fs)=df s + f s 3

G = U(n), O(n) d(s 0,s 1 )=( s 0,s 1 )+(s 0, s 1 ) E Uα = U α V, s Uα : U α V s 1.6. : s Uα = ds + A V α s d :Ω k (E) Ω k+1 (E) Ω k (E) = Γ(Λ k T X E) k =0 d = d (ω s) =d ω s +( 1) deg ω ω d s ω Ω l (E), s Ω k (E) d :Ω k (End (E)) Ω k+1 (End (E)) d (α s) =(d α) s +( 1) deg ω ω d s tr : End (C n ) C tr : End (E) C Ω k (End (E)) d Ω k+1 (End (E)) tr Ω l d tr Ω l+1 4

1.3 G P G π : P X G 0 T fiber P TP π TX G G A P T fiber P G P g (trivial) A P : TP g A P Ω 1 (P, g) A P Ω 1 (P, g) G P Lie G g Ω 1 (G, g) canonical Mauer-Cartan g 1 dg A P : TP g A P Mauer-Cartan {A α } A P = g 1 A α g + g 1 dg (1.2) s α s αa P = A α E s Γ(E) π : P V s π s P X v P ṽ π v, s = ṽ, dπ s 2 G P E 5

2.1 G A = {A α },A α Ω 1 (U α, g) A F (A) :={F (A α )} F (A α ):=da α + 1 2 [A α A α ] F (A V α ):=da V α + A V α A V α 2.1 (F (A) ). F (A β )=g 1 αβ F (A)g αβ g P = U α g F (A β )=g 1 αβ F (A)g αβ g P X F (A) 2.2 d d :Ω k (E) Ω k+2 (E) d d :Γ(E) Ω 2 (E) 2.2. F ( ) Ω 2 (End(E)) d 2 s = F ( ) s F ( ) 6

G ρ ρ :Ω 2 (g P ) Ω 2 (End(V )) F (A) F ( ) F (A) F ( ) G = U(n),O(n) End(E) u(e), o(e) 2.3 (Bianci ).. E s d F ( ) =0 0=d 3 s d 3 s = d F ( )s F ( )(d s) =(d F ( ))s 2.3 G P G P A P Ω 1 (P, g) G canonical A P F (A P ) da P () ()- v 0,v 1 X ṽ 0, ṽ 1 P π F (A P )(v 0,v 1 )=da P (ṽ 0, ṽ 1 ) 2.4 1. 2 p, q p 7

p p p q 2. 3 R Chern Chern 3.1 X CW E X C n E Chern c k (E) H 2k (X, Z) Chern ch k (E) H 2k (X, Q) c k (E) H 2k (X, Z) ch k (E) H 2k (X, Q) : f : X X c k (f E)=f c k (E) ch k (f E)=f ch k (E) c := c 0 + c 1 + c 2 + ch := ch 0 + ch 1 + ch 2 8

c(e 0 E 1 )=c(e 0 )c(e 1 ) ch(e 0 E 1 )=ch(e 0 )+ch(e 1 ) ch(e 0 E 1 )=ch(e 0 )ch(e 1 ) c 0 =1 ch 0 (E) = rank E c 1 = ch 1 L X ch(l) =e c 1(L) normalization E Chern Chern Normalization P 2 (C) P 1 (C) : ξ X H 2 (P 1 (C)) = Zα c 1 (ξ) =α normalize normalization c n (E) =e(e) n = rank E e(e) Euler Thom Euler 3.2 X (Chern-Weil ) 9

n E X A E F A Ω 2 (End(E)) exp:ω 2 (End(E)) k Ω 2k (End(E)) Taylor ch(a) :=tr ( 1 ) e 2π F A ch(a) = k ch k(a) 3.1. ch k (A). Bianki d F A =0 0=trd F A = d(trf A ) 1 trf A e 2π F A det : End (E) C det:(ω 0 Ω 2 )(End(E)) k Ω 2k (End(E)) k ( ) 1 c k λ k := det λid E + 2π F A c k (A) ch k (A) c k (E) :=[c k (A)] A 3.3 Grassmann : BU n = Gr(n) ={V C ; dim V = n} C n : ξ n Gr(n) C := lim(c C 2 C 3 ) 10

3.2. X C n E X f : X BU n E = f ξ n f up to homotopy 3.3. H (BU n, Z) =Z[c 1,c 2, c n ] H (BU n, Q) =Z[ch 1,ch 2, ch n ] c 1,c 2, c n ch 1,ch 2, ch n C n : n i=1 π i ξ 1 CP CP }{{} n CP = BU(1) π i i f : CP CP BU(n) H (BU(n)) H (CP CP )= n H (CP )=Z[x 1,...,x n ] H (CP )=Z[x i ] x i H 2 (CP ) H (CP ) Z x 1,...,x n k c k Q n k=1 xk i /k! ch k C n E X f : X BU(n) c k (E) :=f c k ch k (E) :=f ch k 3.4 X n E X Chern c n (E) E c n (E) H 2n (X, π 2n 1 (S(C n ))) 11

S(C n ) 2n 1 i <2n 1 π 2n 1 (S(C n )) = Z π i (S(C n ))=0 Euler E s 1,s 2 c n 1 (E) c n 1 (E) H 2n 2 (X, π 2n 3 (V 2 (C n ))) V 2 (C n ):={C n } π 2n 3 (V 2 (C n )) = Z i <2n 3 π i (V 2 (C n )) = 0 3.5 Grothendieck Grothandieck Chern L c 1 (L) n E X ξ E C P(E) X x := c 1 (ξ E ) H 2 (P(E)) X 3.4. H (P(E)) = H (x) H (X) x H (X) x n 1 Chen c k (E) x n k 12

Milnor-Stasheff n E X c n (E) =e(e) π k : V k (E) X πk = C } C {{} E n k k rank E n k = n k : H 2(n k) (X) H 2(n k) (V k (E)) c n k (E) Euler c n k (E n k ) 4 Dirac 4.1 Spin SO(2m) Spin(2m) SO(2m) maximal torus: S0(2) SO(2) }{{} m T SO(2) = U(1) z SO(2m) z i U(1) z =(z 1,z 2,...z m ) z lift z 4.1. Spin(2m) : 1. trace ( z + )= 2m trace ( z )= 2m 2m = + 2m 2m 1/2 1/2 z ± 1 2 1 z ± 1 2 2...z ± 1 2 m z ± 1 2 1 z ± 1 2 2...z ± 1 2 m z, zn 1/2 (z 1,...,z m z 1 = = z m =1 z =1,z ±1/2 k =1 13

2. Clifford c : R 2m End( ) c : R 2m Hom ( +, ) Hom (, + ) c(v) 2 = v 2 v 0 v 1 = c(v 0 )c(v 1 )+c(v 1 )c(v 0 )=0 4.2 Spin Dirac 2m Riemann X X Spin TX SO(2m) Spin(2m) P Spin X X Spin : S ± := P Spin Spin(2m) ± X S ± Levi-Civita S ± Γ(S ± ) S ± Γ(T X S ± ) c Γ(S ± ) c S ± Dirac TX Dirac E X E S ± E := 1 S ± + E 1 Γ(S ± E) Γ(T X S ± E) c Γ(S ± E) Dirac c E Dirac 14

4.3 Dirac Λ k R 2m Γ(S S) = Ω k E = S = S + S Dirac d, d E = S = S + S Dirac D :Γ(S + S) Γ(S S) singnature X = dim(ker D) dim(coker D) D :Γ(S + S + ) Γ(S S + ) d + d :Ω even Ω odd χ(x) = dim(ker(d + d )) dim(coker (d + d )) 15

[1] J.W.Milnor, J.D.Stasheff, Characteristic classes, Princeton University Press, 1974 [2] J.W.,, 1998 [3],,, 1997 [4],,, 1998 16