Hitchin-Chatterjee T. (U, (g α0 α 1 )): U = {U α } α A, (M ) g αβ : U αβ T, (U αβ ) g αβ g βγ = g αγ. (U αβγ ) T T, T g αβ : U αβ T P αβ U αβ, Hitchin

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1 Gerbes, II : Hitchin-Chatterjee Murray gerbe. Hitchin-Chatterjee Murray 1

2 Hitchin-Chatterjee T. (U, (g α0 α 1 )): U = {U α } α A, (M ) g αβ : U αβ T, (U αβ ) g αβ g βγ = g αγ. (U αβγ ) T T, T g αβ : U αβ T P αβ U αβ, Hitchin-Chatterjee. 2

3 (Hitchin-Chatterjee). M gerbe data (U,P αβ,s αβγ ) (a) M U = {U α } α A, (b) U αβ TP αβ, (c) U αβγ s αβγ : U αβγ (δp) αβγ, (d) U αβγδ (δs) αβγδ =1., (δp) αβγ = P βγ P 1 αγ P αβ, (δs) αβγδ = s βγδ s 1 αγδ s αβδ s 1 αβγ. U αβγ Tδ(δ(P )) αβγβ := (δp) βγδ (δp) 1 αγδ (δp) αβδ (δp) 1 αβγ U αβγδ T. (P P 1 = U T.) 1 gerbe data Brylinski. 2. 3

4 M = S 3 gerbe data S 3 = R 3 { }, U 0 = S 3 { }, U = S 3 {0}, U = {U 0,U }. U 0 U = S 2 R, H 2 (S 2, Z) = Z 1 T, U 0 U TP 0. : P 0 = P 1 0, P 00 = P =. U αβγ, (δp) αβγ, s αβγ =1., (δs) αβγδ =1. gerbe data, S 3 canonical gerbe. (1 H 3 (S 3, Z) = Z.) 4

5 Murray T, U = {U α } Y = α U α. Hitchin-Chatterjee (gerbe data)murray(bundle gerbe). (Gerbe data,,, bundle gerbe.) T: Y π M T g Y [2] T Y [3]. δg=1 T T, g : Y [2] T Y. 5

6 M Y, π : Y M, (surjective submersion). ( : x M, x U M s : U Y U.) p: { Y [p] = {(y 1,...,y p ) π(y 1 )= = π(y p )} π i : Y [p] Y [p 1] (i) M π Y π 1 π 2 Y [2] Y [3] Y [4] Y [p] TQ, Y [p+1] TδQ δq =(π 1 Q) (π 2 Q) 1 (π p+1 Q) ( 1)p. δ(δq)y [p+2]. () (). 6

7 :. (q 0). 0 A q (M) π A q (Y ) δ A q (Y [2] ) δ Proof.. φ : M Y. φ p : Y [p] Y [p+1], φ p ( y) =( y, φ(π( y)))., H : A q (Y [p+1] ) A q (Y [p] ), Hω = φ pω, δh ± Hδ =1.. {U α }φ α : U α Y Uα. α Y α H α, 1{ρ α }, H : A q (Y [p+1] ) A q (Y [p] ), Hω = α ρ α H α ω, δh ± Hδ =1. 7

8 Bundle gerbe (Murray). M bundle gerbe (Y,P,s) (a) π : Y M. (b) P Y [2] T. (c) s : Y [3] δp δs =1. Y π M P δp s Y [2] Y [3] 1 δs=1 Y [4] T. 8

9 Gerbe data bundle gerbe Gerbe data (U = {U α },P αβ,s αβγ ) bundle gerbe. Y = α U α, π : Y M, Y [p] = α 1,...,α p U α1 α p. P P = α 1 α 2 P α1 α 2, δp = P α2 α 3 Pα 1 1 α 3 P α1 α 2. α 1 α 2 α 3 s : Y [3] δp s Uα1 α 2 α 3 = s α1 α 2 α 3, δs =1. (Y bundle gerbe, gerbe data.) 9

10 Bundle gerbe. (Y,P,s) given. (a) (Y,P,s)T(pseudo T-bundle) (R, v). { TR Y. v : Y [2] δr 1 P δv = s. (b) T(R, v)(r,v ) w : Y R 1 R δw = v v 1. R Y π M δr 1 P Y [2] v δp δv=s Y [3] Bundle gerbe T. TQ M (R π Q, v)t. 10

11 Bundle gerbe. M bundle gerbe (Y,P,s) φ : M Y T. Proof. T(R, v). R Y π M δr 1 P Y [2] v δp δv=s Y [3] φ p : Y [p] Y [p+1], φ p ( y) =( y, φ(π( y))), : π i φ p = { φp 1 π i, (i =1,...,p) id. (i = p +1), R = φ 1 P 1 φ 2 δp = δr 1 P, v = φ 2 s φ 3 δs = δv s 1., T. 11

12 . M bundle gerbe (Y,P,s)T (R, v)(r,v ). φ : M Y, (R, v) (R,v ). Proof. Q = R 1 R, t = v v 1, Q Y π M δq t Y [2] 1 δt=1 Y [3] φ : M Y, w : Y Q δw = t. φ p : Y [p] Y [p+1], φ p ( y) =( y, φ(π( y))) { φ1 π π i φ 2 = i, (i =1, 2) id, (i =3), δ(φ 1 t)=φ 2 (δt) t 1. 12

13 Dixmier-Douady M U, T: R α Y Uα π U α δrα 1 Y [2] Uα P v α δp Y [3] Uα δv α =s R α R 1 β Y Uαβ π U αβ w αβ δr 1 α δr β Y [2] Uαβ δw αβ =vα 1 v β., f α : U αβγ T, π f αβγ = w βγ w 1 αγ w αβ. ( f : Y T,δ f =1 f : M T,π f = f.) (ˇδf) αβγδ =1. 13

14 {(R α,v αβ ),w αβ }, (f αβγ ) Z 2 (U, T). (π f αβγ = w βγ w 1 αγ w αβ.) {(R α,v αβ ),w αβ }, ρ α : Y Uα R α R 1 α, δρ α = v α v α., π k αβ = ρ α ρ 1 β w 1 αβ w αβ (k αβ ) C 1 (U, T), f αβγ f αβγ =(ˇδk) αβγ. δ U H 2 (U, T)., UV, H 2 (U, T) H 2 (V, T), δ U δ V, bundle gerbe G =(Y,P,s), δ(g) H 2 (M,T) = H 3 (M,Z). (Dixmier-Douady.) 14

15 . G : δ(g) =0. Proof. ( ) T(R, v) (R α,v α ), f αβγ =1. ( ) {(R α,v α ),w αβ }δ(g). f αβγ =1. w αβ (R α,v α ) T. 15

16 Bundle gerbe. (a) Bundle gerbe G =(Y,P,s)G = (Y,P,s ) G G =(Y π Y,P P,s s ). (b) Bundle gerbe G =(Y,P,s) G 1 =(Y,P 1,s 1 ).. M bundle gerbe G G, δ(g G )=δ(g)+δ(g ), δ(g 1 )= δ(g). Proof. δ(g G ), δ(g)δ(g ),. δ(g 1 ). 16

17 Bundle gerbe. G =(Y,P,s), G =(Y,P,s ) given. Bundle gerbe (ϕ, ϕ) :G G (a) ϕ : Y Y : Y ϕ Y M M. (b) ϕ : P P Tss : P ϕ P δp s δ ϕ δp s Y [2] ϕ [2] Y [2], Y [3] ϕ [3] Y [3].. Bundle gerbe G G G G 1 T. : G G δ(g) =δ(g ).,. 17

18 Bundle gerbe G δ(g): {M bundle gerbe}/s-iso H 2 (M,T) (Murray-Stevenson). {M bundle gerbe}/s-iso = H 2 (M,T) = H 3 (M,Z). Proof.. Čech (f αβγ ) G =(Y,P,s), Y = α U α, P = Y [2] T, s : Y [3] ( ) 1 δp, s Uαβγ =id fαβγ., (R α,v α )w αβ R α = Y Uα T = ᾱ Uᾱα T, v α (: Y [2] Uα δr 1 ) P, 1 v α =id f Uᾱ β ᾱ βα w αβ : Y Uαβ δr α δrβ 1 ( wαβ Uᾱ = fᾱαβ 1 ), (f αβγ ). 18

19 Lifting bundle gerbe() : G: Y π G :, 1 T Ĝ q G 1. M (e.g. G = SO(n), Ĝ = Spinc (n).) Lifting bundle gerbe G =(Y,P,s): P Ĝ Y [2] q τ G y 1 τ(y 1,y 2 )=y 2. δp (y1,y 2,y 3 ) = Ĝτ(y 2,y 3 ) Ĝ 1 τ(y 1,y 3 ) Ĝτ(y 1,y 2 ), s(y 1,y 2,y 3 )=ĝ 23 (ĝ 12 ĝ 23 ) 1 ĝ 12. (ĝ ij Ĝ, q(ĝ ij) τ(y i,y j )) Ĝassociative δs =1. 19

20 GY M Ĝ (Ŷ,ˆq). Ŷ ˆq M M (a) Ŷ M Ĝ. (b) ˆq : Ŷ M M, ˆq(ŷĝ) =ˆq(ŷ)q(ĝ). (e.g. Ĝ = Spin c (n) Spin c.) Y Y GT., δ(g) H 2 (M,Z) Y. : 1 T Ĝ G 1. : H 1 (M,Ĝ) H1 (M,G) β H 2 (M,T). 20

21 H : Hilbert. Ĝ = U(H), G = PU(H) =U(H)/T K(Z, 2), Y = EPU(H), M = BPU(H) K(Z, 3). H 3 (K(Z, 3), Z) = Z. K: Lie, Q X: K. G = LK = C (S 1,K), Ĝ = LK, Y = LQ, M = LX. δ(g), transgression map: τ L : H 4 (X, Z) H 3 (LX, Z), Qc(Q) H 4 (X, Z). (String class. [Killingback]) C (S 1,X) S 1 pr 1 ev X C (S 1,X) 21

22 Bundle gerbe. G =(Y,P,s) : M bundle gerbe. (a) G 1A 1 (P ) TP s (δ ) =0. (b) curving B 1A 2 (Y ) δb = F ( ). (c) B 3-curvature H(B) 1A 3 (M) π H(B) =db. Y π M P δp s Y [2] Y [3] 11: 1 δs=1 Y [4] {G } A 1 (Y )/π A 1 (M), { curving} A 2 (M)., 3-curvature B. ( 0 A q (M) π A q (Y ) δ A q (Y [2] ) δ. ) 22

23 T. G =(Y,P,s), given. T(R, v)a 1A 1 (R) TR, v ((δa) 1 )=0. R Y π M δr 1 P Y [2] v δp δv=s Y [3] {T(R, v)} A 1 (M). (G,,B)T((R, v),a), F (A) =B. 23

24 bundle gerbe (G,,B), Dixmier-Douady. U = {U α } :, (R α,v α ) : (Y,P,s) Uα T, w αβ : (R β,v β )(R α,v α ), A α : T(R α,v α ). (f αβγ,θαβ 1,θ2 α) C 2 (U, T A 1 A 2 ), π f αβγ = w βγ wαγ 1 w αβ, π θαβ 1 = 1 2π 1 w αβ (A α A 1 β ), π θα 2 = 1 2π 1 (F (A α) B). Deligne. ˆδ(G,,B) H 2 (M,D 2 ) = H 3 (M,Z(3) D ). 24

25 (G,,B) ˆδ(G,,B)=0. Bundle gerbe, ˆδ((G,,B) 1 )= ˆδ(G,,B), ˆδ((G,,B) (G,,B )) = ˆδ(G,,B)+ˆδ(G,,B ). (G,,B)(G,,B ) (G,,B) 1 (G,,B ). ˆδ(G,,B)=ˆδ(G,,B )., : {(G,,B)}/s-iso H 2 (M, D 2 ) = H 3 (M,Z(3) D ). 25

26 bundle gerbe (Murray-Stevenson). { bundle gerbe (G,,B)}/s-iso = H 2 (M, D 2 ) = H 3 (M,Z(3) D ). Proof., (f αβγ,θ 1 αβ,θ2 α) (G,,B). (f αβγ )G, B = α,β( 2π 1θ 1 αβ + u 1 du) 1A 1 (P ), B = α ( 2π 1θ 2 α) 1A 2 (Y ). (R α,v α )w αβ A α = ᾱ (2π 1θ 1 ᾱα + u 1 du),. 26

27 {G}/s-iso = H 2 (M,T) = H 3 (M,Z), {(G,,B)}/s-iso = H 2 (M, D 2 ) = H 3 (M,Z(3) D ). {(G, )}/s-iso = H 2 (M,D 1 ) = H 2 (M,T)., (G, ) (G, ). Deligne: A 2 /A 2 Z H 2 (M,T) inj H 2 (M,D 2 ) surj H 3 (M,Z), ˆδ(G,, B) δ(g) inj H 2 (M,D 2 ) surj A 3 (M) Z. ˆδ(G,, B) 1 2π 1 H B ((G,,B) : flat H B = 0.) 27

28 Lifting bundle gerbe : 1 T Ĝ q G 1, 0 q 1R ĝ g 0. G Ĝĝ = g 1R. Z : G Hom(g, 1R), Z(g) X =Ad g (X 0) (Ad g X) 0. Ĝ, G Maurer-Cartan ˆθ, θ, ν = ˆθ (q θ) 0Tq : Ĝ G. GY Θ A 1 (Y,g), = τ ν + Z(τ 1 ) π 2 Θ 1A 1 (P ) lifting bundle gerbe G =(Y,P,s). P Ĝ Y π 2 Y [2] q τ G y 1 τ(y 1,y 2 )=y 2. 28

29 Gπ : Y M M 1R Y Ad ĝ Y Ad g L : Y Hom(ĝ, 1R). Gπ : Y M ΘL, B = L F (Θ 0) 1A 2 (Y ) curving. F (Θ 0) A 2 (Y,ĝ) F (Θ 0) = d(θ 0) [Θ 0, Θ 0] ĝ. L Γ(Y Ad Hom(ĝ, 1R)), d Θ L = dl+ad Θ L A 1 (Y Ad Hom(g, 1R)). bundle gerbe (G,,B), H(B) = d Θ L F (Θ) 1A 3 (M). 29

30 Lifting bundle gerbe (G,,B) ˆδ(G,,B)=0, (Ŷ,ˆq) Ĝ Ŷ M Θ q Θ ˆq Θ=0, L F ( Θ) =0. Ŷ M ˆq Y M 1 T Ĝ q G 1, 0 q 1R ĝ g 0. L A 0 (M, Y Ad Hom(ĝ, 1R)), F ( Θ) A 2 (M, Y Ad ĝ). 30

第86回日本感染症学会総会学術集会後抄録（I）

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