D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane
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1 D-brane K 1, 2 sugimoto@yukawa.kyoto-u.ac.jp ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3] D-brane D-brane /9/ ( SFR No. 4 1
2 D-brane K-homology [1] Witten D-brane Hilbert D-brane D-brane K K-homology K K-homology K K-homology K K-homology K D-brane 2 D-brane 3 D-brane K Witten [1] 4 D-brane K-homology [4] D-brane K 2 D-brane X X X I S 1 I string (open string) S 1 string (closed string) 1 open string closed string 1: D-brane 2
3 D-brane 1995 D-brane [5] 2.1 D-brane D-brane open string X M ϕ : M X ϕ(m) M M string M M D-brane 2 M p + 1 Dp-brane p p + 1 p 1 open string D-brane 2: D-brane D-brane string D-brane M M open string M X = R 10 M = R p+1 1 type IIA type IIB fermion D-brane BPS D-brane non-bps D-brane p odd even BPS D-brane RR D-brane non-bps D-brane 3
4 p M D-brane type IIA even A, Φ m BPS Dp-brane type IIA odd T, A, Φ m non-bps Dp-brane type IIB even T, A, Φ m non-bps Dp-brane type IIB odd A, Φ m BPS Dp-brane 1: D-brane D-brane 1 A M U(1) M G = U(N) M M = i I U i M N g ij : U i U j G ( s.t. g ij (x) 1 = g ji (x), g ij (x)g jk (x)g ki (x) = 1 on x U i U j U k ) U(N) U i N N 1-form A i U j A j U i U j A i (x) = g ij (x)a j (x)g 1 ij (x) + g ij (x)dg 1 ij (x), x U i U j (2.1) M M 1 T Φ m M Φ m 9 p m = p + 1, p + 2,..., 9 M x = (x 0, x 1,..., x p ) Φ m (x) D-brane x m D-brane M X M x = (x 0, x 1,..., x p ) (x, Φ p+1 (x),..., Φ 9 (x)) X (2.2) Φ m (2.2) M = R p+1 X = R 10 (2.2) non-bps D-brane T 3 V (T ) T 3 T = 1 4
5 V (T ) T 1 3: T = 1 V (T ) = D-brane D-brane D-brane D-brane N 4 open string D-brane N 2 1 D-brane N N D-brane N {}}{ 4: N D-brane A U(N) U(N) U(1) D-brane N U(1) N U(N) U(n) n D-brane U(m) m D-brane U(n + m) Φ m T N N U(N) adjoint Φ m D-brane Φ m D-brane x m D-brane N N N Φ m D-brane 1 string 5
6 {Φ m } 9 p {Φ m } D-brane 4.2 BPS D-brane 2.1 BPS D-brane D-brane D-brane N N D-brane D-brane ( anti D-brane D-brane ) Ñ 5 D-brane anti D-brane D-brane open string anti D-brane D-brane anti D-brane open string Anti D-brane N {}}{ Ñ {}}{ m m A, Φ Ã, Φ D-brane T anti D-brane 5: D-brane - anti D-brane open string D-brane U(Ñ) Ã U(Ñ) adjoint Φ m D-brane anti D-brane open string T N Ñ U(N) U(Ñ) bi-fundamental (2.1) M N Ñ M = i I U i U(N) U(Ñ) g ij : U i U j U(N) g ij : U i U j U(Ñ) U i A i, Ãi Φ m i, Φ m i T i U i U j (2.1) A i = g ij A j g 1 ij Φ m i + g ij dg 1 ij, Ãi = g ijãj g 1 ij + g ij d g ij 1, (2.3) = g ij Φ m j g 1 ij, Φ m i = g ij Φ m j g 1 ij, (2.4) T i = g ij T j g 1 ij (2.5) 6
7 3 D-brane K-theory 3.1 D-brane non-bps D-brane D-brane anti D-brane D-brane D-brane 3 T = 0 T = 0 T = 0 T = 1 D-brane Non-BPS D-brane D-brane - anti D-brane D-brane T = 1 D-brane T = 0 D-brane Sen 3 V (T ) = 0 T = 1 D-brane D-brane D-brane Dp-brane anti Dp-brane 2.2 T (r, θ) = f(r) e iθ. (3.1) (r, θ) Dp-brane (p + 1) R p+1 (p 1) f(r) 6 r f(r) = 0 r f(r) = 1 r Dp-brane anti Dp-brane (r, θ) (p 1) 7
8 f(r) 1 r 6: f(r) D(p 2)-brane D(p 2)-brane D(p 2)-brane 1 (3.1) r 1 T = 1 T = constant R 1 r = R S 1 S 1 U(1) ( T = 1 T ) π 1 (U(1)) = Z π 1 (U(1)) D(p 2)-brane (3.1) D-brane Dp-brane anti Dp-brane D(p 2)-brane Dp-brane anti Dp-brane N D(p n)-brane 2.2 T N N T T = T T = 1 π n 1 (U(N)) 1 [6] 8
9 π n 1 (U(N)) N N π n 1 (U(N)) 0 (n : odd) = Z (n : even) (3.2) n Z D(p n)- brane 1 BPS Dp-brane p type IIA type IIB D-brane D-brane D-brane descent relation Dp-brane - anti Dpbrane N brane R p+1 R n R p+1 n R p+1 n R n { } S n S n S n = U U U = R n = S n \{ } U U T U T U U (2.5) g : U U U(N) g : U U U(N) T (x) = g(x)t (x) g 1 (x), x U U (3.3) Dp-brane - anti Dp-brane U T = 1 ( ) U U T = g g 1 U(N) U U U(N) U U S n 1 I (I ) π n 1 (U(N)) g g S n (3.3) g = g U T = 1 Dp-brane - anti Dp-brane Dp-brane - anti Dp-brane 3.2 K D-brane K Dp-brane - anti Dp-brane D-brane R n S n 9
10 Dp-brane - anti Dp-brane (p + 1) D-brane p = 9 10 D-brane 1 BPS D9-brane type IIB type IIB 2.2 D9-brane N anti D9-brane Ñ N = Ñ N = Ñ D9-brane anti D9-brane 10 X D9-brane A anti D9-brane à X N Ñ E Ẽ K 0 (X) K 0 (X) := { (E, Ẽ) E, Ẽ X }/ (3.4) (E, Ẽ) (E, Ẽ ) def E H = E H, Ẽ H = Ẽ H (3.5) X H, H = K 0 (X) K K 0 (X) (E, Ẽ), (F, F ) (E F, Ẽ F ) I (I, I) (E, Ẽ) (Ẽ, E) K K 1 K 0 (X) D-brane (3.4) E D9-brane Ẽ anti D9-brane E Ẽ N Ñ (3.4) E Ẽ (3.5) (H, H) (H, H ) (H, H) X = S n D9-brane anti D9-brane (3.5) D9-brane anti 1 K [7] 10
11 D9-brane D9-brane - anti D9-brane K 0 (X) K 2 open string D-brane X = R n D-brane K open string D-brane K 0 (X) type IIB D-brane D9-brane anti D9-brane D9-brane anti D9-brane K 0 (X) (3.4) rank E = rank Ẽ K 0 (X) K 0 (X) K 0 (X) = Z K 0 (X) Z E Ẽ D9-brane X X X Ẋ K0 0 (X) := K (Ẋ) K K (3.4), (3.5) X X K Z D-brane D-brane anti D-brane 3.3 K 3.2 [8, 9] K [10] type IIB D9-brane anti D9-brane T (x) x X Hilbert H Hilbert D9-brane Hilbert Hilbert H 11
12 D9-brane Hilbert H anti D9-brane Hilbert H T (x) : H H, T (x) : H H, (3.6) x X Hilbert H {e n } n 1 H n=1 c n e n c n C X f n (x) C 0 (X) n=1 f n e n 1 C 0 (X) Hilbert H C0 (X) C 0 (X) X X 0 H C0 (X) H C0 (X) C 0 (X)-module T T B(H C0 (X), H C0 (X)) H C0 (X) H C0 (X) B(H C0 (X)) H H B(H, H) x X (3.6) T B(H C0 (X), H C0 (X)), T B( H C0 (X), H C0 (X)) 3.1 T T = T T = 1 D9-brane anti D9-brane T T T T 1 T T T T 1 1 D9-brane - anti D9-brane T T T T 1 K 0 (X) := T B(H C 0 (X), H C0 (X)) T T 1 C 0 (X) K(H) T T 1 C 0 (X) K( H) / (3.7) K(H) H K K(H) def H {ψ n} n 1, {φ n } n 1 lim n µ n = 0 {µ n } n 1 (3.8) K = n=1 µ n ψ n φ n 1 n=1 f nf n C 0 (X) 12
13 ψ n φ n H (, ) v H v ψ n (φ n, v) K = K (3.8) ψ n = φ n T T 1 C 0 (X) K(H) T (x)t (x) 1 0 C 0 (X) X D9-brane anti D9-brane (3.7) (a) (b) (c) (a) (b) D9-brane - anti D9-brane (c) (3.7) K 0 (X) (3.4) K (3.4) (3.7) D-brane type IIB type IIA D9-brane D-brane 1 type IIA D9-brane non-bps D9-brane D-brane non-bps D9-brane 2.2 non-bps D-brane N N N N = H (3.7) T = T K 1 (X) := { T B(H C0 (X)) T = T, T 2 1 C 0 (X) K(H) } / (3.9) type IIA D-brane K 1 (X) K n (X) n Z K K n (X) = K n+2 (X) (Bott ) K 0 (X) K 1 (X) 2 K type IIB type IIA 2 D-brane 13
14 3.4 X = R n R 10 R n R 10 n R 10 n R n D9-brane D-brane type IIB K 0 (R n ) type IIA K 1 (R n ) K 0 (R n ) = 0 (n : odd) Z (n : even), K1 (R n ) = Z (n : odd) 0 (n : even) (3.10) D-brane R 10 n D(9 n)-brane D(9 n)-brane 1 Dp-brane (BPS Dp-brane) type IIB p type IIA p 4 K-homology K K-homology D-brane K-homology K-homology K-homology D-brane [9, 4] K-homology D-brane [4] K-homology [11] [10] 4.1 K-homology D-brane K-homology D-brane X smooth K-homology K i (X) := { (M, E, ϕ) } /, (i = 0, 1). (4.1) (M, E, ϕ) M : Spin c dim M i (mod 2) E : M ϕ : M X (4.2) 14
15 Spin c U(1) fermion 1 (M, E, ϕ) K-cycle M E M K-cycle (M 0, E 0, ϕ 0 ), (M 1, E 1, ϕ 1 ) (M 0, E 0, ϕ 0 ) (M 1, E 1, ϕ 1 ) (a) (c) (a) K-cycle (M 0, E 0, ϕ 0 ), (M 1, E 1, ϕ 1 ) ( W, E W, ϕ W ) = (M 0, E 0, ϕ 0 ) ( M 1, E 1, ϕ 1 ) (4.3) (W, E, ϕ) (M 0, E 0, ϕ 0 ) (M 1, E 1, ϕ 1 ) (W, E, ϕ) (4.2) W Spin c (4.3) W W M 1 M 1 7 (M 0, E 0, ϕ 0 ) (M 1, E 1, ϕ 1 ) (W, E, ϕ) M 0 M 1 W 7: (b) (M, E 1, ϕ) (M, E 2, ϕ) (M, E 1 E 2, ϕ) (4.4) (c) (M, E, ϕ) ( M, Ĥ ρ (E), ϕ ρ) (4.5) 1 M w ij : U i U j SO(n) SO(n) Spin(n) M p : Spin(n) SO(n) = Spin(n)/Z 2 p w ij = w ij w ij : U i U j Spin(n) Spin(n) w ij U i U j U k w ij w jk w ki = 1 Spin g ij : U i U j U(1) w ij w ij = g ij w ij Spin c 15
16 M ρ : M M x M ρ 1 (x) M Ĥ M ρ 1 (x) = S 2k Ĥ S 2k K 0 (S 2k ) = Z X D-brane M BPS D-brane ϕ X 1 type IIA type IIB BPS D-brane K 0 (X) type IIB K 1 (X) type IIA D-brane (4.2) M 2.1 M X M {Φ m } ϕ 2.2 {Φ m } X {Φ m } E D-brane M (2.1) (a) D-brane (b) 2.2 D-brane (c) 3.1 D-brane descent relation M Dp-brane M M 8 D(p + 2k)-brane anti D(p + 2k)-brane K 0 (S 2k ) 3.1 D(p+2k)-brane - anti D(p+2k)-brane Dp-brane (c) D(p+2k)-brane anti D(p+2k)-brane Dp-brane (p + 1) M Dp-brane D-brane anti D-brane 8: D-brane - anti D-brane D-brane 4.2 K-homology 3 K D-brane 3.1 D-brane decent relation D9-brane - anti D9-brane non-bps D9-brane D-brane D-brane 16
17 D-brane D-brane D-brane 0 D-brane D( 1)-brane 3.3 type IIB D( 1)-brane - anti D( 1)-brane type IIA non-bps D( 1)-brane 3.3 Hilbert H 0 brane type IIA non-bps D( 1)-brane D( 1)-brane 0 1 T 10 Φ m (m = 0, 1,..., 9) non-bps D( 1)-brane Hilbert H D-brane non-bps D( 1)-brane (p + 1) Dp-brane T non-bps D( 1)-brane K-homology K 1 (X) := { (H, φ, T ) } / (4.6) H : Hilbert φ : C 0 (X) B(H) (4.7) T : T B(H) T = T T 2 1 K(H), [T, φ(a)] K(H) for a C 0 (X) 3.3 B(H) Hilbert H K(H) 3.8 H T T 2 1 K(H) 3.3 non-bps D( 1)-brane φ D-brane {Φ m } X = R 10 {Φ m } H Φ m x m Φ m φ : C 0 (X) f(x 0, x 1,..., x 9 ) f(φ 0, Φ 1,..., Φ 9 ) B(H) (4.8) 17
18 φ C 0 (X) B(H) C {Φ m } φ 1 C φ φ(c 0 (X)) M φ(c 0 (X)) = C 0 (M) M D-brane φ : C 0 (X) C 0 (M) M X 2 ϕ : M X ϕ : C 0 (X) f f ϕ C 0 (M) φ D-brane M X ϕ [T, φ(a)] K(H) for a C 0 (X) T [T, Φ m ] 2 (4.6) 3.3 (a) (b) non-bps D( 1)-brane (c) non-bps D( 1)-brane T K-homology type IIA D-brane K-homology 4.1 D-brane K-homology K-theory D-brane type IIB D( 1)-brane - anti D( 1)-brane D( 1)-brane Hilbert H anti D( 1)-brane Hilbert H {Φ m : H H} { Φ m : H H} T : H H T T 2 = 1 T T = T T = 1 [T, Φ m ] = 0 T Φ m Φ m T = 0 K 0 (X) := { (H, H, φ, φ, T ) } / (4.9) 1 2 X K ϕ 1 (K) M D-brane X 18
19 H, H : Hilbert φ : C 0 (X) B(H) φ : C 0 (X) B( H) T : T B(H, H) T T 1 K(H), T T 1 K( H), T φ(a) φ(a) T K(H, H) for a C 0 (X) (4.10) (4.6) (4.9) X C 0 (X) C 0 (X) C C X D-brane Φ m Φ m 4.3 K K-homology K K-homology K K i (X) K-homology K i (X) i = 0, 1 H i (X; Z) H i (X; Z) i K K-homology Q K 0 (X) Q = H i (X; Q), K 1 (X) Q = H i (X; Q) (4.11) i: even K 0 (X) Q = H i (X; Q), i: odd K 1 (X) Q = i: even i: odd H i (X; Q) (4.12) Q Z n K K-homology X n Poincaré H i (X; Z) = H n i (X; Z) K i (X) = K n i (X) (4.13) X 10 type IIB D9-brane - anti D9-brane D-brane K K 0 (X) D( 1)-brane - anti D( 1)-brane D-brane K-homology K 0 (X) type IIA non-bps D9-brane D-brane K K 1 (X) non-bps D( 1)-brane D-brane K-homology K 1 (X) 19
20 X 9 10 R X R type IIB D9-brane - anti D9-brane type IIA non-bps D9-brane D-brane R K 0 (X) K 1 (X) R D0-brane R 4.2 D-brane type IIB non-bps D0-brane type IIA D0-brane - anti D0-brane K 1 (X) K 0 (X) (4.13) X 9 K 0 (X) = K 1 (X), K 1 (X) = K 0 (X) 5 K K-homology D-brane type I [1, 12, 13, 6] Type I type IIB string D9-brane anti D9-brane type IIB D9-brane N O(N) D-brane K (3.4) KO 0 (X) 3.4 Z (n 0, 4 (mod 8)) KO 0 (R n ) = Z 2 (n 1, 2 (mod 8)) 0 ( ) (5.1) Z D1-brane, D5-brane, D9-brane Z 2 D( 1)-brane, D0-brane, D7-brane, D8-brane Z 2 D-brane K type I D-brane D-brane 2 open string KO KO n (X) (n 0, 1, 2, 3, 4, 5, 6, 7 mod 8) 8 20
21 type I 8 D-brane KO n (X) 3 4 type I D-brane 4.3 K Z n orientifold [3] K orientifold K K D- brane K K K K-homology [1] E. Witten, D-branes and K-theory, JHEP 9812, 019 (1998) [arxiv:hep-th/ ]. [2] D. E. Diaconescu, G. W. Moore and E. Witten, E 8 gauge theory, and a derivation of K-theory from M-theory, Adv. Theor. Math. Phys. 6, 1031 (2003) [arxiv:hep-th/ ]. [3] O. Bergman, E. G. Gimon and S. Sugimoto, Orientifolds, RR torsion, and K-theory, JHEP 0105, 047 (2001) [arxiv:hep-th/ ]. 21
22 [4] T. Asakawa, S. Sugimoto and S. Terashima, D-branes, matrix theory and K-homology, JHEP 0203, 034 (2002) [arxiv:hep-th/ ]. [5] J. Polchinski, S. Chaudhuri and C. V. Johnson, Notes on D-Branes, arxiv:hep-th/ [6] T. Asakawa, S. Sugimoto and S. Terashima, Exact description of D-branes via tachyon condensation, JHEP 0302, 011 (2003) [arxiv:hep-th/ ]. [7] M. Karoubi, K-Theory, an introduction, Springer-Verlag. [8] E. Witten, Overview of K-theory applied to strings, Int. J. Mod. Phys. A 16, 693 (2001) [arxiv:hep-th/ ]. [9] J. A. Harvey and G. W. Moore, Noncommutative tachyons and K-theory, J. Math. Phys. 42, 2765 (2001) [arxiv:hep-th/ ]. [10] B. Blackadar, K-Theory for Operator Algebra, Cambridge University Press. [11] P. Baum and R. G. Douglas, K-homology and Index Theory Proc. of Symposia in Pure Mathematics, 38 (1982) 117. [12] O. Bergman, Tachyon condensation in unstable type I D-brane systems, JHEP 0011, 015 (2000) [arxiv:hep-th/ ]. [13] T. Asakawa, S. Sugimoto and S. Terashima, D-branes and KK-theory in type I string theory, JHEP 0205, 007 (2002) [arxiv:hep-th/ ]. 22
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