' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1
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- ひでより なかきむら
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1 , ,,., (KEK), ( ) ( )..,.,,,
2 ' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing
3 [ ] 1,,.., ( )., (complex structure), 0, (symplectic structure)., Yang-Mills, Einstein., ( ),.,.. de-rham,. de-rham,.?, Einstein..,., Cech,,.,,,. de-rham.,,,, Cech. de-rham, Cech, (1.1) 3
4 ,. Cech..,,,,.., 0,,.. m. g, f i (' 1 ' m ) i =1 n (1.2) f i j i (1.4).,,,.,, (1.3),(1.4), (1.4).,,,,,,.,, Yang-Mills (1.4),,,,, (1.4). (, 0,,,, n ( R n ),. A 1 A n n m m (1.5) 4
5 r i + A i (1.6) r i i + A i ' ' m (1.7)., (1.6), (1.4),., r i r j = r j r i (1.8) j i + A i A j ; A j A i (1.9).,,., (, R n ),,,., A i,.,,.,,,,,., (1.9),,..,., (1.9),., (1.8), r i ' =0 i =1 n (1.10),., (1.10) '(0).,, (1.10),, (1.8)., ', m ' ' 1 ' m (1.11) 5
6 , (1.10), ' j (0) = C A j (1.12)., (1.12) m, m m, g =(' 1 ' m ) (1.13).,., A,. A i 7;! g + g ;1 A i g i (1.14) (1.10),, g A 7;! 0 (1.15). (1.10), ' i base,., g (1.15)..,,,,., 1.,., Cech. 2 Cech Cech,, (1.15), (1.9),.,.,,,.,,,. 6
7 , g?,..., Cech, X.,. X = [ i2i U i X : U i : (2.1) G.,., Cech., ( ).,.,.,.,. G : ( : ( ) : +( ) (2.2),, C k C k (V G) 3 g i0 i k i 0 i k 2 I (2.3) V : [ U i (2.4) U i0 \\U ik ;! i0 i =0 (2.5), g U G,.,., k Cech. Cech,,,,. 7
8 (). g i0 i k 2 C k 7;! (g) i0 i k+1 (2.6) 2 C k+1, g (g) i0 i k+1 = X j (;1) j g i0 ^i j i k+1 (2.7) ( ^ij i j ). (g) 1234 = g 234 ; g g 124 ; g 123 (2.8).,.,,. C 1 C 2. C 1 ;! C 2 (2.9) C 1.,., 2.,, (g) 123 = g 23 g ;1 13 g 12 (2.10) =0 (2.11) ( g) 123 = (g) 12 ; (g) 13 +(g) 23 = g 2 ; g 1 ; (g 3 ; g 1 )+g 3 ; g 2 = 0 (2.12).,.,,.,,, =1 C k! C k+1,. 30, J.Giroux, Cohomologie non abelienneds, Lect. Note in Math. Springer
9 t U 1 U 2 S 1 1:, C 0 C 1 C 2 (2.10) 3,.,.,. X = S 1 = U 1 [ U 2 (2.13)., Cech. U 1 U 2 1. t S 1. U 1 U 2., C 1., g 12 G G. C 1 (S 1 G)=G G 3 g 12 : U 1 \ U 2 ;! =0 g 12 2 G g 12 2 G (2.14), C 0. C 0 (S 1 G) 3 g 1 : U 1 ;! G constant 3 g 2 : U 2 ;! G constant (2.15), (g) 12 = g 2 g ;1 1 9
10 . Cech H k (X G), (g) 12 =(g) 12 = g 2g ;1 1 (2.16) g 2 C k g =0 g ; g 0 = h g =0 g 0 = (g ; h) =0 =0 (2.17). g 12 = h () g 12 = g 12 (2.18).,. S 1. A, n n G. + A (2.19) r i r j ;r j r i =0 (2.20), 1,.,, ( ). (2.13) U 1 g 1, A g 1. U 2. S 1 = U 1 [ U 2 U 1 g 1! g 1 A U 2 g 2! g 2 A, S 1 A g, g.. (, ) U 1 \ U 2 g 1 = g 2 (2.21).,,. g 12 g 1 g 2 g 12 = g 2 g ;1 1 (2.22) 10
11 . (2.21) g 12 =1 (2.23). (2.21) 1, 1.,. =0 (2.24), g 12., g ;1 1 A g 2 A. (2.24). Cech. g 12 =1 (2.25) 2 g 1, g 2,,., A,,. h g 1 7;! hg 1, g 12. g 12 7;! g 2 g ;1 1 h ;1 (2.26) S 1 at G = H 1 (S 1 G) (2.27).?, (2.20). 11
12 ,,.,.,,.,., Cech,, Cech, H 1., Cech de-rham de-rham,, Cech.,., Cech. H 1. de-rham,. G. de-rham,.,,. H 1, H 0., A ;! A (2.28). H k k 3. H 0, H 1, H k,,.,., Cech ; (2.29) 12
13 , H 1.. H 1 : U 1 \ U 2 ;! G (2.30) de-rham ; (2.31),,.,. r i =. i + A i r i r j ;r j r i =0 j i +[A i A j ]=0 (2.33) M at G = (2:33) (,. F A =0, at) = f g 1 (2.34),,,,.,, Riemann 19, Riemann, Riemann.,. 3,,,.,. 13
14 U1 U2 M 2:. M = [U i U i C n (3.1), M ( 2).,,,. ( at U1 \ U 2! G ( ), ' i : U i ;! V i C n ' j ' ;1 i : C n ;! C n (3.2) (3.3) ' i,.,, ( ) ' j ' ;1 i.,., ' j ' ;1 i.,,. r i r j = r j r i r i + A i (3.4) 14
15 .,,. dx 1 dx n (3.5).., dx i ^ dx j = ;dx j ^ dx i (3.6).,. df = i dx i (3.7). 2,.,. ddf =0 (3.8) ddf = i dx i ) 2 f = dx j ^ dx i i j = 0 2 j = i (3.10) r i r j = r j r i (3.11) 15
16 . d A ' = X i = X i dx i + A i 'dx i i (r i ')dx i (3.12). (3.11).. d d 2 j = d A d A =0 (3.13) d A d A = d A ((r i ')dx i ) = (r j r i ')dx j ^ dx i (3.14) 2 j ;! d A ;! r i r j i ;! r i r j ' = r j r i ' (3.15)., ij, d A 2., (3.13) d A d A =0 A =0, Cech.,,. J =0 (3.16).. (3.13),, J ( ).,, 16
17 ,,.,, (3.13), A,, Cech., (3.16).,,,..,., R 2n, J x,. J x : 2n 2n (x 2 R 2n ) (3.17) J x J x = ;1 (3.18). R 2n C n, J x! (3.19) J x 0 I ;I 0. C n, J x p ;1.,,, p ;1. p ;1,. R 2n C n,.., J (3.16). J x x. C n R 2n z 1 z n (3.20) x 1 x n y 1 y n (3.21). C n R 2n p z i = x i + ;1 y i (3.22). dz i = dx i + p ;1 dy i dz i = dx i ; p ;1 dy i (3.23) 17
18 .,. dx 1 ^ dy 2 = 1 4 p ;1 (dz 1 + dz 1 ) ^ (dz 2 ; dz 2 ) (3.24) R 2n dx i dy j dz i dz i.,.,, i dz i dz i = i 2 ; p i =0 () f = i dz i dz 1 ^ dz 2 ^ dz 3 )= i dz i ^ dz 1 ^ dz 2 ^ dz 3 (3.28).,.,., = i dz i (3.29) d (3.30) 18
19 (3.16), J = 0 I ;I 0! (3.16)., 2 j i (3.32)., (3.16), J. (3.16), 2., d 2 =0 d 2 =0 () 8 >< =0 =0 (3.33)., J,,., (3.19),,,,.,., dz dz.. p. J p (dx i ) J p (dy i ). p =(x 1 y n ) (3.34) J p (dx i ) = X J p ki dx k + X J p k+n i dy k J p (dy i ) = X J p ki+n dx k + X J p k+n i+n dy k (3.35) 19
20 J ki ki? (2n 2n)., (3.19). J(dx i )=dy i J(dy i )=;dx i (3.36) Jdz i = J(dx i + p ;1 dy i )= p ;1 dz i Jdz i = ; p ;1 dz i (3.37). dz i dz i, J. (3.37), dz i J p ;1, dz i J ; p ;1., (3.37),.,, J 2 ;1, fdx 1 dy n g = (J p ;1 ) + (J ; p ;1 ) (3.38)., J p ;1 10, J ; p ;1 01., J (3.35),, J dz i dz i, (3.38). df J f J f (3.39), 2 J = )+(J 2 ) (3.40)., d 2 A = da + A ^ A (3.41).. J, 2 J =0 (3.42) 20
21 .,,. d 2 A =0 =) d A ' =0 J ' =0 (3.44) d A ' =0 (3.45), d 2 A =0 (3.46). (3.45), (3.46)., (3.44), (3.42). (3.44), J. (3.43) A =0,, J = 0 I ;I 0.,, (3.46) (3.42). J,?! ' J () Jd'= p ;1 d' (3.47).,,. (3.44), J. J + B 2 J =0 (3.48) 21
22 ( B (4.21) ),,,.,.,. 4 Index theorem deformation theory,,.,,,,,.,., index theorem deformation theory., 1. H 1, H k ,.,, Yang-Mills,, index theorem.., deformation theory index theorem.,. H 1 (M G) = M at G (4.1) G (4.1) H 1 (M G) ( ).,., G, M G, G, G (4.1).. G H.,,. 22
23 .,.., deformation theory. deformation theory,.,, d A d A =0 (4.2) A = A 0 + "A 1 + " 2 A 2 + (4.3). (4.3) (4.2),.,, 0 = d A d A = da + A ^ A., = da 0 + A 0 ^ A 0 + "da 1 + "A 0 ^ A 1 + "A 1 ^ A 0 + (4.4) d 2 A 0 =0 (4.5) da 0 + A 0 ^ A 0 =0 (4.6) 0=dA 1 + A 0 ^ A 1 + A 1 ^ A 0 (4.7).,, A 0, A 1 = A., d(a)+a 0 ^ A + A ^ A 0., = d A0 (A) =0 (4.8) d A0 (A) =0 (4.9)., A.,,., A 0,, A 0., (4.9).,. 23
24 ,, A(") =A 0 + "A + (4.10)., A(") A 0.,,,.,.,., g(") =1+"g 1 + (4.11) A(") = g(") ;1 A 0 = (1 + "g 1 ) ;1 d(1 + "g 1 ) + (1 + "g 1 ) ;1 A 0 (1 + "g 1 ) = "(dg 1 + A 0 g 1 ; g 1 A 0 )+ = d A0 g 1 (4.12)., (1+"g 1 ) ;1 =1; "g 1, 2.,,,., d A0 (A) =0 (4.13) A 0 + "A (4.14) A = d A0 g 1 (4.15), A 0 + "A., A = d A0 g 1 () A 0 + "A (4.16) H 1 DR(M A) = (d A u =0 ) u ; u 0 = d A v u u 0 (4.17) 24
25 . Cech, de-rham,. M..,.,, A(") () g(") ( A 0 g(") A(") ) (4.18).,.,, A 0 A(")... A(")..,, A 0. A 0 ) H 1 (M A) 2 J+"4J =0 J+"4J J + "B (4.21). J B =0 (4.22).,, B, J, p ;1, 1 = J (4.23) 25
26 (4.23) J B,., J B, B (4.22)., B. J B = X a ij i ^ dz j (4.24) 0 I ;I 0. (4.22)., B! (4.25) H 1 (M O(TM)) 3 [B] (4.26). H 1 1 (4.24) dz i 1. i. TM, O(TM). (4.26) M 1. H 1 Kodaira-Spencer index theorem,,, (tangent bundle) M 1,., (4.26)., 1, 1., 1,.,, H 1,.,, (4.23)., index theorem. H 1 (M A), H 1 (M A) Euler number X (M A) = (;1) k dim H k (M A) (4.27) k,,. 26
27 H 0, H 1., H 0, H 1, H 2.,,,., supersymmetry,. H 1?,., H k, H 1.. H 0, H 1,,.,.. (M A). (1) : H k (M A) =0 (k 2).., H 0 ; H 1., H 5,. 5,., H k (k 2).,., Riemann-Roch, Atiyah-Singer 50, Gauss-Bonnet.,., H k (k 2),. ( ) H k (k 2),, H 1..,. (2) : H k. 27
28 , H 1 (odd) ( ) H 2 (even) ( ), extended odd even,,.,, H 1, odd,.,.,, boson, fermion,., H 1 (M A) Euler number (M A), H 1,.,. m n. P 1 (x 1 x n ) P m (x 1 x n ) (4.28),, n m. M (3 p) =f (x 1 x n )jp i (x 1 x n )=0 i =1 m g (4.29) M p T p M p 2 M (JP i )(V )=0 i =1 m V 2 R n (4.30) (J Jacobi JP i $rp i ).,,., dp 1 dp m p., dp 1 dp m p M p.. 28
29 , H 1 V, H 2 dp a i, H 3 ( ) a i,. H 1 $fv jrp i (V )=0g H 2 $fa i j X a i dp i =0g H 3 $fa i g (4.31), H 2, ( ), H 2.., H 2, H 2 = H 3 = = 0, dp 1 dp m p,, H 1 dim H 1 = n ; m (4.32)., n m.,, dp dim H 1 n ; m.,.,.,, P i,., X k (;1) k dim H k = m ; n (4.33). H 3, dp, H 1 H 1 H 2 n ; m. H 1, H 2., (4.33)., index theorem.,.,? 29
30 ,,.,. P i, H 1 (M A) Euler number (M A), index theorem P i. (4.33) n m.,. ;,.,, H 1 $fvj rp i (V )=0g,, H 2 $fa i j P a i dp i =0g H 3 $fa i g,..,, index theorem,,.,, index (4.32),,.,, H 1 $fvjrp i (V )=0g (H 1 ),, H 1 $fv jrp i (V )=0g H 2 $fa i j P a i dp i = 0g m ; n.,.,,., (4.31). syzygy.,,, (4.31)., H 1,,,,., P 1 (x) = = P m (x) =0 rp 1 (V )= = rp m (V )=0, 30
31 , rp 1 (V ) rp m (V )., H 1 P 1 (x) = = P m (x) =0 rp 1 (V )= = rp m (V )=0,,,,.,.,,.,, 1. 1, 1, index theorem. Riemann. Riemann g : g = 3 3: ( 3) Riemann, = C 3g;3 (4.34). Riemann 1 k =0 1., l 1.,., X k H 0 ( g O(T g )) = 0 (;1) k dim H k ( g O(T g )) H k ( g O(T g )). 31
32 H 0 =0?,., genus 3 Riemann., g. X k (;1) k dim H k ( g O(T g )) Riemann-Roch,, g. Index theorem, index theorem, index theorem.., ( ), Yang-Mills, Einstein,. Yang-Mills d A F A =0 Einstein Ricci =0 (4.35) Yang-Mills,?. ( ) self-dual F A + F A =0 self-dual gravity W + =0 holomorphic curve ' : g! =0 (4.36) 32
33 ., (4.35), 2.., 4 d A 0-form(C 0 ) 4-form(C 4 ). C 0 d A! C 1 d A! C 2 d A! C 3 d A! C 4 (4.37) 0-form form 4, dx 1 dx 4. 2-form 6( = 4 C 2 ), dx i ^ dx j. 3-form 4, 4-form 1. n =4 d A : C k! C k+1 C 0 d! A C 1 d! A C 2 d! A C 3 d! A C form 1-form 2 - form 3 - form 4 - form ( ) : (4.38), 4+4=1+6+1., X C k = X C k (4.39).., de-rhame complex (4.38)... C 1 d A! C 2 (4.40), C 1 C 2 m n. d A f =0 (4.41),,.. 8 >< >: P 1 (f 1 f n ) P m (f 1 f n ) (4.42) 2 p.4.,. 33
34 n m. n = m., (4.40) 2. (4.42),. n = m (4.43), n m,. n>m. (4.42), m,, n ; m. n ; m.,.,.., m>n,. (n 6= m) n>m) 1 n<m) (4.44),,,,., (4.40) C 0. C 1, C 2, C 0. l n m, C 0! C 1! C 2 l n m ( ) ( ) ( ) ( (4.43)) (4.45) n = m + l (4.46).,.,,, (4.46). 34
35 , (4.45) C 3? C 3,, C 3,.,. Batalin-Vilkoviski formalism, ( ), C 3 C 4, (4.45),, (4.39).,,. Self duality -1, self dual. 4, 0! 1! 2! 3! (4.47) ( 4).., ( 3, 4 ).,,., self duality. 1 2,,. 0, 1, 2. 3,., 0, 1, 2 1, 4, 6, 6+16= 4. 3,,.., 4 d A d A =0 (4.48).,. 1, 2, 4,. (, )., (4.48) 2, 2 ( 2 ). 2 3-form. 2, 0 d! A 1 d! A (4.49) 35
36 , 1+1=2 2 at connection (= 1 + 6) 4. self duality. 2, 4 0 = ; (4.50) base 2 + base, dx 1 ^ dx 2 + dx 3 ^ dx 4 dx 1 ^ dx 3 ; dx 2 ^ dx 4 dx 1 ^ dx 4 + dx 2 ^ dx 4 (4.51) ^ = dx 1 ^ dx 2 ^ dx 3 ^ dx 4 = 2 4 (4.52). 4-form ; = ; (4.53) 3 3, self dual 0, 1, 2 + (4.54) 0! 1! (4.55) 1+3=4.,., 4., (4.50).., 0 d A! 1 d A! 2 d 2 A =0. 0 d A! 1 d A! 2 + d A 2 + part (, F A = da + A 2 +part, self dual 36
37 ., d 2 A =0 (d A 2 + part )., d 2 A =0 d A ' =0. d 2 A =0 d A ' =0., (d A 2 + part ) F A = F A + + F ; A F A + =0,. F A + =0. d 2 A =0 d A' =0,., 4, 3., 3,.,,, n de-rham j-complex, ( (4.55) 1+3=4).,., 4. [ ] 37
38 [ ],. Self duality -2 self dual geometry. self duality, 4,,,., 4, 4-form,, metric g ij.., tensor,, 1 4-form. : 4-form 1 (4.56) dx 1^dx 2^dx 3^dx 4.,,.,, k-form (n; k)-form. : k! n;k (4.57) Hodge (n = 4)., u u 2. u ^u = juj 2 (4.58)., n =4 u 2-form, Hodge g ij M. n =4 u : 2-form g ij! f(x)g ij (4.59) 4., Hodge 2-form invariant 4., Hodge 2 1,. 2-form, =1 (4.60) 2 + : u = u 2 ; : u = ;u (4.61) 38
39 , ; 3.,, 3, d A 2-form 0 d! A 1 d! A 2 + (2 ; ) (4.62) 1, 4, 3,.,, 1-form 1-form, 1-form. 2-form, Hodge form. 1 : 1-form 2 : 2-form u = u (4.63), 1. d 2 A =0, F A = da + A ^ A =0 (4.64), F A 2-form,, F A = F + A + F ; A (4.65). F + A =0 (4.66). self dual, Yang-Mills (2 ) 1., [(4.35),(4.36) ], 2 (4.62)., (4.66), deformation A = A 0 + "A (4.67), (4.66) d A A + d A A =0 (4.68). d + AA =0 (4.69). (4.62) 1, 4, 3. ( 1 ) A 4 dim G, G. ( 0 39
40 ), g(x), g(x) G. dim G. (4.66), dim G.,,,.,,,.,,., (4.62) supersymmetry, supersymmetric,.,, (4.66),, (4.64) 1.., 1,,., (4.66)., (4.66)..,,. (4.66)?,.,., (4.64),.,... F + A?, F + A A =0. 40
41 ,.,. 2-form 2 = (4.70). base, C 2 dz 1 ^ dz 2 dz i ^ dz j dz 1 ^ dz z 1 z 2 2 C 2 (4.71). (4.62), 2 = = ;.,, (4.62), 2 + = !! = dz 1 ^ dz 1 + dz 2 ^ dz 2 z i = dx i + idy i 2 ; (4.72). self dual., 2-form ( 2 = ) Riemann 2 = ; Kahler (4.72), F A =0 ) F A 20 02! (4.73).., F A 20 02,, de-rham d d d A. d A A A A ' d A ' 01 part.! F A 2 A : k l! k +1 l., F A : k l! + =0 k l 2 A =0 (4.75) 41
42 .,, (4.75) A. J, J. (4.62), 2 A =0 A' =0 local (4.76). 2, (4.75) A ' = 0., ',., d 2 A =0 ) g 01 =0 (4.77),, A 2 A =0 =0 (4.78)., holomorphic., g 01, M U i. M = [ i U i (4.79), U i \ U j, g ij, g ij : U i \ U j! GL(n C) (4.80)., M. g ij constant ) (4.81), (4.77)., g ij (4.58),. g ij ) (4.82), F A + =0 d 2 A =0,, holomorphic. 42
43 (4.77) A g., (4.78) A... A, A ' =0 '. base, A A (g') g., base, (g') =(@g)' + g@ A ' (4.83), F A + = 0.,. (4.82)., Serre GAGA 3,,,., compact, compact, GAGA. M compact g,,.,,. F + A.,, F A + =0 A =0 (4.84), ) (. 2 A =0 ) A F + A =0 (4.85)..., = !, F + A =0 3 Serre Geometry analitique et geometry algebrique,,. 43
44 !.,,,. (4.85). Seiberg-Witten. 5 Stability,. stability. Mumfold,, Hausdor. Hausdor? (5.1),, Hausdor?,., 2 3,. Hausdor,.. M = f g = (5.2), X=G : G compact Hausdor (5.3), G compact Hausdor. Lie compact (5.4). stability, stability,, 44
45 , Lie, Lie noncompact Hausdor., Lie compact?., C = X (5.5) C X = C=0 (5.6),. C X c C z, cz (= ) (5.7), C=C X =2 z 1 z 2 2 C z 1 = cz 2! (5.8) C, C X. 2.. Hausdor. Hausdor, z i 6=0 lim z i =0 (5.9). z i, limit p i =[z i ] 2 C=C X (5.10) lim[z i ] = [0] (5.11),.,. [z i ]=[z j ] (5.12),. lim p i (5.13) 0 p i unstable stable 45
46 ( ) unstable stable. unstable, stable,. fc 2 C j c 0=0g C X : noncompact (5.14) C X, C X. C X noncompact, noncompact. elementary,,, genus singular Riemann ( 4)., unstable.., X ( 5). X ", X singular, "., S 2 genus,,,. ( 5) S 2, S 2. X " "..., ( lim X" = X lim X " = X "0 " 0 6=0 X 6= X " ( X " " 6= 0 ) S 2 ε S 2 S 2 4: 5: 46
47 ", " singular., ", ". limit Hausdor,,. S 2, Riemann, essential.,, S 2 S 2? holo. : S 2! S 2 1 PSL(2 C) noncompact (5.15) 1 PSL(2 C) noncompact. Mumfold, noncompact.. stable = compact (5.16) Riemann. ( ' ' :! holo. 1 : 1 ) compact (5.17) Mumfold unstable, stable. stable, (5.18),,, unstable,,,., Hausdor,.., stable (5.19) ( 4), Riemann stable., S 2 [ S 2 X (5.20) 47
48 stable., Gromov-Witten invariant,,. g : genus g Riemann (g x ) (5.21) J : g (5.22) ' : g! (M J M ) (5.23) M :, ', ' J J M. Riemann J,, (Jacobi ') J = J M (Jacobi ') (5.24) ', J.,,,, u :! : dieo. : Diff() ' 7! ' u (5.25) J 7! u J (u J ) (5.26),,, dieo.,.,. stability, ( 6).,. S 2 S 2 S 2 1 6: 48
49 S 2,, S 2 S 2,.. (' ) ' :! M (5.27).,, u ' ( u :! ' u = ' =) (5.28) u holomorphic.,,. stability, stability.,. stable,, Hausdor, Mumfold. stability,,, stable, compact. Hausdor.,, Mumfold.... stability. 6 compact,, compact. compact, compact., A 0 (6.1) - local theory local theory, local theory, global theory, global 49
50 compact,, compact, M = n F A + =0o = (6.2) M = f(' )j J M ' = ' J g =Diff() (6.3) compact. compact, compact compact., singular object.. singular object,,. compact,,. compact 4, 2 A self dual gravity =0 (6.4) W + =0 (6.5) Riemann, W Weyl, W =0, fg ij f : (6.6) Weyl (W + ).,, 2. self-dual gravity. Atiyah-Singer,.,,, compact. compact, singular object. Riemann. compact.? compact singular, singular, singular. singular 50
51 ., singular. Riemann,,, singular., singular,,. singular Riemann,., ( 7)., ( 8). 7: X n 3 p (6.7) p 8: n n ; 1. X;p nonsingular (6.8) " ( ) n =2, 1 S 1.,, p, S 1,, ( 9). Riemann compact,, 1 S 1. 51
52 p 9: X n Riemann? n =2 Riemann. Riemann,,, compact, singular Riemann. singular,, compact, singular., Riemann, singular singular Riemann. Hausdor?., Hausdor, Riemann. ( 10) Hausdor 10: 52
53 , , 2, ( 11). p p + 1 (6.9).,,. +1 ;1 11:,, ( 12) Gauss : Bonnet. Riemann ( 11), 1, 2,. 53
54 p :, 4,, 3., 3 ( 13). Ricci > 0 (6.10) 3?.,,. Ricci.,, 3 3. conjecture,, Ricci Hamilton, Ricci > 0 =) S 3 = (6.11) conjecture., 1, S 3. S 3, 3. 3, 3..,. X " f " (z 1 z 2 z 3 )=0 (6.12) f 0 = gh (6.13) X 0 = fg =0[ h =0g (6.14) 54
55 2 g =0 ;1 14: h =0, ( 14). 6 4, 2.. S 2 ( 15), ( 16). S 2 S 2 S 2 S 2 15: 16: [ 14 ] Riemann,. compact, self-dual gravity.., d Ai F Ai =0 Z k FAi k 2,. lim F Ai? ( S ),. 55
56 ?.,. lim F Ai, A i A i F Ai codimension = 4 codimension? X S. dim X ; dim S =4 X 4 =) S (6.15),. 6, 2,.?..., F A. A = nx i=1 A i dx i n =4 (6.16) 4,, ( 17). 4 compact,. codimension 4, Z k FA k 2 Z F A ^ F A (6.17) 4. 56
57 17: +1 18:,,.,, ( 18)., 4-form 4. self-dual?, 4, 4 self-dual.,?., 4, codimension 4. F A ^ F A 4 () codim. 4 (6.18) 57
58 ,, singular 4-form codimension 4., singular 4-form =) codim. 4 (6.19),.?,?,, 1.,..., Z k FAi k 2 < C S C=1 mass (6.20) 7 Donaldson Donaldson.. topological eld theory,. topological eld theory invariant, Seiberg-Witten, n F + A =0 o = 58
59 Riemann ', ' holomorphic. f(' )j ' holo.g =Diff() ' :! M dieo..,.,, f g = invariant (7.1) invariant., X = f g = 1 f g = (7.2) X (7.3),. topological eld theory. X ( ) 1 2 k (7.4) topological eldtheory.,. Z 1 ^ 2 ^^ k = h 1 2 k i (7.5) invariant invariant Donaldson..,,,. M2X M : (7.6) : F + A =0 ; M metric g M =0 ; M J M (7.8) ' :! M (7.9) 59
60 ,, (7.5). ) g 0 m g 1 m., gt m [0-1]., t. t. t (7.10) ( 19)., M t ( t ) [ M t = M c (7.10) t=0 M 1 M 0 M t 19: Z cm d ( 1 ^ 2 ^^ k )=0 (7.11) Stokes, 1 ^ 2 ^^ k =0 (7.12) cm Z c M = M 1 ;M 0 (7.13) M 1 1 ^ 2 ^^ k ; Z M 0 1 ^ 2 ^^ k =0 (7.14),,.. 60
61 , ( compact )..,. J M. ' :! M ' holomorphic complex structure J J 2 M =1 (7.15)., '... Z Z 1 ^^ k = M(M JM 0 ) M(M J 1 M ) 1 ^^ k (7.16) (Gromov-Witten invariant, ) J M almost complex structure,.,. = f(j ')g Diff() (7.17). M. mix,, Gromov-Witten invariant (coupled gravity). almost complex structure, Gromov. Witten,,..... (7.16).. dl dt = X X : S2 l : S 2 l(;1) =p l(+1) =q (7.18), S 1.. S 1?. S 1. 61
62 p p X0 X1 q 20: q 21: X 0 ( 20) X 1 ( 21). X 0 X 1. X 0 =) X 1 "X 1 +(1; ")X 0 (0 " 1) (7.19),. " =1, dl dt = X 1 l(;1) =p l(+1) =q (7.20),.,.., ( 22). (p) (q),.,,. compact., f l X 0 f<0, ( ). f, 22.. ' :;! M 62
63 1 1 ε 22: 8 Symplectic structure symplectic structure, 2-form. d! =0! 2-form (8.1), J M!,. S 1, S 1 ( 23). S 1 l(0) S 1 l(1) 23: ' : S 1 [0 1] ;! M (8.2) ' holomorphic.,, J M! compatible (J M! tame ).,, (8.2)( 23), ', Z '!>0 (8.3), tame. Kahler Kahler, Kahler form!. J M,!. 63
64 (8.3)... Kahler... Riemann Kahler. M. M almost complex structure. almost,?, M almost,! J M,!(JX X) > 0 (8.4). (8.3)., f f, (8.3),,, f(l(1)) ; f(l(0)) > 0 (8.5).,, S 2,, 2. (8.5),.,,, 24:,. 64
65 1,.,, (8.5),.,, ( R '!<const.),.,, ', R '!<const., compact,. 24 noncompact,, symplectic structure, (8.4) J M, Donaldson. Z M 1 ^^ k tame J M J M : (8.6) tame J M (8.6) J M. Gromov Gromov-Witten,., almost complex structure, symplectic structure.!. almost complex structure,., complex structure,, almost complex structure, 1,,. 1, almost complex structure,, ;1..., symplectic structure, (8.1),,.!,. almost complex structure J 2 = ;1.. (8.1). symplectic structure,. almost complex structure 65
66 . almost complex structure. almost structure.., compact.,., Gromov-Witten, Kahler, J!, symplectic geometry!. 9 Wall crossing wall crossing,. Donaldson. open topological string., C n L i (L i ' R n i =1 k) (9.1)! = X dz i ^ dz i L i j! =0 (9.2)! L i. Lagrangian submanifold. C n R n, 2 ( 25). L 3 L 4 C n L 1 L 2 25: ' : D 2 ;! C n (' holomorphic) (9.3) '. '.. f ' j '(@D 2 ) L 1 [ L 2 [[L k g = M(L 1 L k ) (9.4) M,. L i., 66
67 Donaldson, wall crossing (9.4)., L?. topological open string. L D-brane, D-brane state,.. Donaldson, compact,. M(L 1 (t) L k (t)) M t (9.5) 1 + ( ) M t : ( 26),,. t, +1, +1. ( ). Donaldson.?,.,. ( ),.,,,. ( 27) compact,.,,., 67
68 : :. ( 28)., n =1. C 4 ( 29). L 4. ( (9.3), n =1), 1. Riemann 4.., ( 31). ( 33) , 31. ( 27),, ( 28).. 4 D z,, D w j w j< 1. 68
69 L 3 t<t0 C n=1 p p L 1 L 2 L 4 p 1 t<t 0 p 4 29: 30: L 3 t=t 0 p p 2 3 p 3 p2 ( ) 1 t=t 0 L 1 L 2 L 4 p 1 t=t 0 p 4 p 1 t>t 0 p 4 31: 32: L 3 t>t 0 p 3 p 2 L 1 L 2 L 4 p 1 t>t0 p 4 33: 34: 69
70 .,, ' : p i ;! L i \ L i+1 (9.6) D 2, 4., 31 t = t 0. t t 0, p 3 p 2. Riemann,,, ( 32). 31, 32, 32 ( ). ( ), Riemann.,., Riemann Riemann,,. 34 Riemann _. _, fd 2 4 g (9.7),. [0 1). compact, 34. [0 1]. 34.,,...,,.., Donaldson.. Donaldson. S 2 S 1, 2 S S 1 S 1 35:,. S 1. S 2 =S 1.,, 70
71 . 28.,, Donaldson well dendness. wall crossing. S 1,,.? Donaldson Riemann,. Seiberg-Witten d A d A =0 da + A ^ A =0 A 2 (10.2),, 3. 2,, 2.. (10.1) 2. 3, (10.1) 3. da + A ^ A + A ^ A ^ A =0-3-form. A 1-form da 2-form, A ^ A 2-form A ^ A ^ A 3-form,. de-rham,., 0! 1! 2! (10.3) de-rham complex : d 2 =0 (10.4) 71
72 d A d + A d A (10.5) d 2 A =0 de-rham deformation.. A 1-form, deform. odd *) even (10.6),,. (10.6) deform., A 1-form odd form. A odd form? d! d + A (10.7).,. (10.6) deform,. (d + A)(u) = du + A ^ u (10.8) (d + A) 2 =0 () da + A ^ A =0. (10.7), d,. 2.. Lie, L 1 -algebra m k (k =1 2 3 ) (10.9) m k (u 1 u k ) m 1 = d m 2 =[ ],. m k u 1 u k k,, m 1, m 2 Lie.. X l X 2S n m l (u (1) u (l;1) ) m n;l+1 (u (l) u (n) ) =0 (10.10) 72 S n : n
73 m 3? m 1, m 2 (10.9), m 3 (10.10)., m 1 m 1 =0! d 2 =0 n =2 (10.11) m 2 (u 1 m 2 (u 2 u 3 )) + m 2 (u 2 m 2 (u 3 u 1 )) + m 2 (u 3 m 2 (u 1 u 2 )) = 0 (10.12) n =3 (m 1 =0 ) m 1 =0, n =3, m 2 Lie Jacobi identity. [u 1 [u 2 u 3 ]] + [u 2 [u 3 u 1 ]] + [u 3 [u 1 u 2 ]] = 0 (10.13) m 2 [ ] u., Jacobi,.,. Lie,., a i i, deform, (d + A)(u) =du +[A u] (10.14) a X a i a i 2 i (10.15) d a =(d + a)(u) =du + X k2 m k (a a u) (10.16), d a., 2 d 2 a =0 (10.10),, m 3, da + X m k (a a)=0 (10.17) da +[a a] =0 (10.18) 73
74 ., open string. H k, a i. 2..? ( 36),,, S 2 interact. string. S 2 N N N N :,,?,,. index theorem.. Atiyah-Singer, Atiyah Singer, 3. de-rham Riemann Dirac, 10.,.,,. 74
75 ,,,??,,, Einstein, 2?,,.,.. Yang Mills Yang-Mills,.,,,,,.,..... ( ),,.... ( )?.... ( ) 75
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1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
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