kougiroku7_26.dvi

Transcription

1 2005 : D-brane tachyon : ( ) :,,,,,,,

2 Introduction Second Revolution (1994 ) Type II bosonic string Light-cone gauge physical states Superstring physical states D-brane D-brane D open string D-brane D-brane Tachyon potential D-brane :Type IIA :Type IIB

3 1 1.1 Introduction part tachyon computer overview D-brane tachyonic D-brane soliton overview scale 2

4 ( 1) 1: 1 quark lepton graviton 1 graviton graviton 1 2: 1loop Feynman graph ( 2 ) 1loop Feynman graph Feynman graph loop loop momentum momentum loop loop 0 pinch ( 2 ) graph ( 3) amplitude 3 loop torus torus 3

5 3: 1-loop Feynman ( 4 ) 4: loop 4 ( 5 ) 5 5: invariant 5 line 5 consistent 1 4

6 4 input Minkowski 1 5 Type Type I Type IIA Type IIB Het SO(32) Het E 8 E 8 O(32) U(1) SO(32) E 8 E 8 open+closed closed closed closed closed N=1 N=2 N=2 N=1 N= I Type IIA Type IIB Heterotic SO(32) Heterotic E 8 E 8 5 gauge SO(32) E 8 E 8 open open string closed string 6: open string closed string string closed string 6 open string string closed string string Type I open closed closed string boson fermion boson fermion spin N =1 N =2 N =1 1 N = compact 10 5

7 10 overview introduction string firtst revolution second revolution quark lepton 7: u, d, c, s, t, b 6 quark lepton muon tauon neutrino gauge photon W boson Z boson gluon gauge 6

8 SU(5) SO(10) E 6 (ψ 10,ψ 5,ψ 1 ) 3 ψ 16 3 ψ 27 3 SU(5) GUT SO(10) GUT E 6 GUT SU(5) 1 10 ψ 10 ψ SO(10) 7 SO(10) 16 3 E 6 GUT SO(10) SO(10) E gauge U(1) SU(2) SU(3) 1 3 gauge Heterotic E 8 E scale matter 1 4 minimal N =1 supersymmetry symmetry compact N = 1 supersymmetry supersymmetry 4 minimal 6 7

9 Calabi-Yau Calabi-Yau orbifold 6 Calabi-Yau Holonomy SU(3) gψ ψ g SU(3) 8: Holonomy SU(3) Holonomy 8 vector 1 SU(3) SU(3) 2 E 6 GUT GUT Grand Unified Theory G U T E 6 GUT consistency 1 anomaly cancellation Tr (R R) = Tr (F F ) (1) Σ R R =(R μν a b ) dx μ dx ν curvature 2-form F F = F μν dx μ dx ν gauge field strength 6 4 Calabi-Yau curvature non-zero non-zero non-zero non-zero gauge non-zero gauge gauge (1) R curvature F gauge Σ 8

10 field strength (1) R F (2) 1 Calabi-Yau Holonomy SU(3) curvature curvature SU(3) R F gauge SU(3) gauge SU(3) E 8 gauge E 8 SU(3) E 6 E 6 (3) SU(3) Holonomy (2) E 6 E 6 index gauge 27 Euler E 6 gauge 3 point Heterotic gauge consistent gauge consistent gauge consistent GUT gauge E N =1 2 E 6 4 N =1 gauge E 6 gauge graviton 1 1 9

11 Calabi-Yau 9: potential potential ( 9 ) potential 1 ( 9 ) 1 Hetero E 8 E

12 1.3 Second Revolution (1994 ) second revolution second revolution keyword duality( ) D-brane M 3 keyword keyword (duality) 2 Type IIA Type IIB T-duality duality Type IIA R S 1 compact R S 1 SR 1 Type IIB S 1 compact 1/R S 1 compact Type IIA on S 1 R Type IIB on S1 1/R (4) 2 S S 1 9 S 1 compact Type IIA on S 1 R R 10: T-duality Type IIB on S 1 1/R S 1 S 1 momentum state 10 Type IIB 1/R S 1 compact S 1 ( 10 ) S 1 momentum 10 2 string Type IIB S-duality Type IIB dilaton scalar φ 11 1/R

13 e φ coupling constant g s = e φ (5) g s Type IIB graviton 2 tensor 2 B μν C μν g s 1/g s B μν C μν g s 1/g s B μν C μν (6) C μν B μν S-duality g s g s S-duality 1/g s duality duality S-duality T-duality Type IIA Type IIB T-duality Type IIB S-duality duality Type I Heterotic SO(32) S-duality Heterotic SO(32) Heterotic E 8 E 8 2 T-duality T Type IIA Type IIB S Type I S Het SO(32) T Het E 8 E 8 2 keyword D-brane Type IIB B μν 2 tensor charge S-duality S-duality 12

14 B μν C μν B μν charge S-duality C μν charge D1-brane D-brane T-duality D1-brane 1+p Dp-brane Polchinski D-brane Polchinski D-brane 1+p open string ( 11) open open string (1 + p) D Dp-brane [Polchinski 95] 11: Dp-brane string D-brane Type I D1 D5 D9-brane Type IIA D0 D2 D4... Type IIB Hetero Type I Type I Type IIA Type IIB Het SO(32) Het E 8 E 8 D1, D5, D9 D0, D2, D4, D6, D8 D( 1), D1, D3, D5, D7, (D9) Type IIA Type IIB 13

15 3 keyword M Type IIA Type IIA graviton g μν C μ φ B μν C μνρ (μ, ν, ρ =0,...,9) } g μ 10 {{ g } C μν 10 }{{ } g MN C MNP (M,N,P =0,...,9, 10) 10 massless boson 5 field g μν metric graviton C μ gauge φ dilaton scalar B μν Type IIB 2 tensor C μνρ 3 tensor μνρ C μ g μ 10 φ g B μν C μν 10 g μν g μ 10 g g MN M N 0 10 C μν10 C μνρ MNP tensor M X momentum 10 Type IIA D-brane D0-brane Witten M M membrane M membrane M X

16 M membrane M mother M M MonS 1 MonS 1 S 1 0 Type IIA Type IIB Het SO(32) Type I MonI Het E 8 E 8 MonI S 1 MonS 1 I Type IIA M 11 1 compact 10 Type IIA Type IIB Type IIA T-duality 1 S 1 compact S 1 S 1 torus compact zero Type IIB T-duality Heterotic E 8 E 8 M I = S 1 /Z 2 [0, 1] compact Heterotic E 8 E 8 Heterotic SO(32) S 1 compact Type I S 1 5 M compact zero M miracle magic mystery 1 S-duality M Type IIB S-duality Type IIB MonS 1 S 1 (7) Type IIB M S 1 S 1 compact (7) 2 S 1 2 S 1 S-duality M ((7) ) S 1 S 1 Type IIB nontrivial strong 15

17 coupling weak coupling duality Type I Heterotic S-duality ((8) (9) ) Type I MonS 1 I (8) M ((8) ) M S 1 compact M S 1 compact (8) S 1 I S-duality Het SO(32) MonI S 1 (9) (9) M S-duality nontrivial M miracle 1 M matrix model M matrix M M W W Witten initial M Witten M 11 Witten Witten leader M Witten M second revolution ( 12) Type IIA Type IIB T-duality Type IIB S-duality Type I Heterotic S-duality Hetero SO(32) E 8 E 8 T-duality 11 M ( 13) 11 M low energy effective theory impact 16

18 12: (second revolution ) 13: (1995 ) Dp-brane p 1+p ( 14) open string open string open string mode gauge p +1 gauge p +1 gauge gauge M gauge gauge D-brane 17

19 Open string gauge = (1 + p) gauge 14: gauge 14 gauge gauge gauge gauge keyword MQCD(M QCD) string junction string AdS/CFT duality impact D-brane D-brane event horizon black hole S =logw S = A/4 D-brane 15: D-brane black hole ( 15) black hole black hole entropy Hawking radiation entropy black hole entropy Bekenstein-Hawking black hole horizon 18

20 entropy S = A/4 (10) D-brane entropy D-brane log S =logw (11) 2 black hole impact Type IIA D0-brane D0-brane 0 line D0-brane ( 16) D0-brane effective 16: D0-brane action action 1 S m (ẋk ) 2 dt (k =1,...,9) (12) x x k (t) D0-brane open string open string scalar D0-brane 1 D0-brane N ( 17) open string D-brane N open string 2 N 2 open string open string x k (t) x k (t) N N 19

21 t open string D0-brane 17: D0-brane N BFSS IKKT Banks-Fisher-Shenker-Susskind M IKKT Type IIB string M x k tachyonic D-brane D9-brane D9-brane brane gauge tachyon scalar D-brane Type I Type IIA Type IIB O(N + 32) O(N) U(N) U(1) U(N) U(N) (, ) adjoint (, ) Type I Type IIA Type IIB 3 D9-brane N Type I gauge O(32) 20

22 O(N + 32) O(N) Type IIA U(1) U(N) U(1) Type IIB gauge U(N) U(N) gauge N gauge tachyon scalar tachyon 18 potential scalar tachyon higgs V (T ) 18: tachyon potential tachyon tachyon potential minimum D9-brane potential minimum D9-brane potential minimum D-brane soliton 1 tachyon inflation 1 inflation inflation overview T ( )... ( ) 21

23 momentum R compact momentum /R ( ) ( ) 1 11 ( ) tensor tensor ( ) ( ) nontrivial ( ) ( ) ( ) consistent maximum free parameter 10 Type IIB Type IIA low energy effective theory 10 Type IIA 11 1 compact Type IIA low energy effective theory effective theory non trivial nontrivial ( ) ( ) matrix model ( ) x k ( ) ( ) 22

24 ( ) brane ( ) ( ) ( ) (12) x k (t) 15 D0-brane open string ( ) openstring 1 N N N N N D-brane... ( ) ( ) open string x k (t)? x k (t) ( ) D-brane D-brane D-brane scalar ( ) D-brane ( ) ( ) N N ( ) x k k x k N N ( ) ( ) ( ) ( ) ( ) ( ) mass mass mass zero massive mass model Planck scale mass ( ) ( ) ( ) ( ) tachyon tachyon mass spectrum unitarity ( ) potential minimum potential minimum 23

25 tachyon higgs tachyonic potential minimum ( ) ( ) consistent scalar ( ) ( ) ( ) ( ) potential minimum D-brane soliton ( ) ( ) D-brane (18 15) black hole mass D-brane brane ( ) brane soliton ( ) soliton ( ) QCD quark pion bound state ( ) brane + ( ) tachyon D9-brane 10 brane D-brane open string 10 open string D9-brane open string open string gauge D-brane D-brane soliton ( ) ( ) ( ) 24

26 ( ) N 2 ( 17) 1 D0 1 brane ( ) brane ( : ) time slice ( ) ( ) M 1 2 x 10 x 10 string ( ) string ( ) x 10 string massive string ( ) 11 string 11 string string 11 X 10 S 1 compact string 11 mode massive massive D0-brane D0-brane mass mass ( ) M object ( ) membrane membrane M matrix ( ) p brane S 1 compact p 1 brane D0-brane M object 11 S 1 D0-brane ( ) M 1+2 x string x 10 x 10 momentum 10 25

27 D0-brane ( ) 1+2 object 11 ( ) M 1+2 object ( ) ( ) 2 Type II bosonic string superstring bosonic string 2.1 bosonic string ( 19) t "world line" x 19: world-line line world-line action S = dt m 2 26 ( ) 2 d x (13) dt

28 (13) (13) ( ) 2 d x S = m dt 1 (14) dt speed Taylor leading term (13) dt 1 ( ) d x 2 dt dt 2 (d x) 2 ηmn dx M dx N (15) η η MN = (16) D x M (t, x) line element ds 2 world-line t "world line" x 20: world-line world-line ( 20) line element ds 2 = η MN dx M dx N = η MN dx M 27 dτ dx N dτ dτ 2 (17)

29 h = η MN dx M (14) action h S = m dτ h (19) h induced metric action action worldline line element metric action world-line 1 S τ dτ h η MN dx M dx N τ reparameterization invariance S world-line dτ S τ dx N dτ (18) world sheet 21: world-sheet world-line ( 19) ( 21) tube tube world-sheet τ 28

30 world-sheet σ τ (τ,σ) (σ 0,σ 1 ) world-sheet induced metric ds 2 ds 2 = η MN dx M dx N x M x N = η MN dσ α dσ β (20) } σ {{ α σ β } h αβ x σ τ ((20)2 ) α β 0, 1 σ α σ τ ( ) x η M x N MN h σ α σ β h αβ action S NG = T d 2 σ det(h αβ ) (21) d 2 σ = dσ 0 dσ 1 = dτdσ (22) T = 1 2πα (23) α = ls 2, l s : string length (24) d 2 σ dσ 0 dσ 1 σ τ dσdτ action Nambu-Goto action string 1 T string tension α mass α l 2 s 2 scale l s l 2 s l s string length string length scale σ closed string string σ 2π 0 2π 0 σ 2π, σ σ +2π (25) (21) action action (19) world-line 29

31 Nambu-Goto det world-sheet 1 Nambu-Goto action reparameterization invariance (τ,σ) S NG world-sheet S NG (τ,σ) (21) action action action g αβ determinant nonzero world-sheet metric tensor determinant g αβ det(g αβ ) < 0) S P = T d 2 σ gg αβ h αβ (26) 2 g g =det(g αβ ) (27) g αβ g αβ g metric notation (26) action (21) action S P Polyakov action Polyakov action g αβ metric (τ,σ) ( g) determinant 1 g αβ g αβ (τ,σ) e 2w(τ,σ) g αβ (τ,σ) w (τ,σ) g αβ scalar g determinant e 2w(τ,σ) factor 2 squareroot factor 1 (26) g αβ 1 cancel Weyl g αβ (τ,σ) e 2w(τ,σ) g αβ (τ,σ) Weyl action 30

32 2.1.3 Polyakov action Nambu-Goto action Polyakov action EOM equation of motion g αβ g αβ (21) action Polyakov action g αβ g δ g = 2 1 ggαβ δg αβ (28) g αβ g αβ (26) g determinant g g αβ g factor g αβ h αβ g 1g 2 αβ ( δs P δg = T g h αβ αβ 1 ) 2 2 (gγδ h γδ )g αβ = 0 (29) 0 h αβ = 1 2 (gγδ h γδ )g αβ (30) det root det h αβ 1 2 (gγδ h γδ ) factor det 2 root det hαβ = 1 2 (gγδ h γδ ) g (31) (31) (26) (21) Nambu-Goto action Nambu-Goto action Polyakov action g g S NG = S P (32) X M δs P δx M = T α ( gg αβ β X M ) = 0 (33) 31

33 (29) 1 (33) 2 { 1 hαβ 1 2 (gγδ h γδ )g αβ =0 2 α ( (34) gg αβ β X M )= Light-cone gauge Polyakov action system 1 light-cone gauge gauge 1 gauge light-cone gauge τ σ reparameterization invariance Weyl Weyl action a X + (τ,σ)=τ b σ g σσ =0 (35) c g =(detg αβ )= 1 d g τσ σ=0 =0, (g τσ = g 01 ) X ± X ± = 1 2 (X 0 ± X 1 ) (36) X ± X + τ σ τ ((35) a ) g αβ α, β 0 1 σ 1 τ,σ σ σ σ 1 g 11 =0 ((35) b ) 3 g det g αβ 1 ((35) c ) g αβ τσ 01 τ 0 σ 1 σ =0 0 ((35) d ) (35) a d Polchinski gauge τ a X + τ 32

34 b, c f f(τ,σ) g σσ g (37) τ σ τ σ σ (τ,σ) (τ,σ (τ,σ)) (38) (37) f f f f = dσ dσ f (39) σ (37) g σσ σ 2 σ 2 det root 1 cancel 1 (37) f σ b f σ σ σ τ σ = 1 σ d σf( σ, τ) (40) k(τ) 0 (40) σ (40) f (39) (39) f cancel (39) f (40) k(τ) f (τ,σ )=k(τ) (41) (41) σ Weyl (37) factor (37) f Weyl (37) g Weyl g Weyl g e 4w g (42) w Weyl g 1 c f Weyl f (37) g σσ σ b d σ (38) σ σ σ σ τ σ σ σ = σ + α(τ) (43) 33

35 σ g τσ g τσ = g τσ + g σ σ dα dτ (44) b ((44) g σ σ ) σ (44) 2 σ σ (44) g τσ 0 σ =0 (35) d 1 (44) α (44) g τσ 0 ((44) 2 ) σ g τσ σ 0 d σ =0 σ =0 σ ((44) 2 ) (35) a,b, c,d gauge (43) α α α constant det g αβ = 1 metric ( ) g αβ = g σσ g στ g στ g 1 σσ (1 g2 τσ ) (34) metric light-cone gauge (34) 2 M light-cone gauge notation M =+ X + = τ 2 X M + τ (45) α ( gg ατ ) = 0 (46) g det g = 1 g α g ατ = 0 (47) 34

36 (47) (45) α g ατ = τ g ττ + σ g στ = 0 (48) τ g σσ = σ g στ (49) g σσ σ (35) b (49) σ g στ σ σ g στ = 0 (50) g στ 2 0 constant σ g στ c 1 + c 2 σ (51) σ σ σ +2π (51) σ g τσ σ =0 0 d ((51) 1 ) g στ 0 g στ = 0 (52) (49) g σσ constant g σσ : constant (53) (49) 0 g σσ τ σ b (52,53) constant g σσ = c 1 c constant metric ( ) ( ) c 0 g αβ =, g αβ c 1 0 =, 0 c 1 0 c (54) c 1 = g σσ : constant (34) τ X M c 2 2 σx M = 0 (55) 35

37 2 1 h στ = 0 (56) h ττ = 1 2 ( c 1 h ττ + ch σσ )( c) (57) h σσ = 1 2 ( c 1 h ττ + ch σσ )c 1 (58) g 0 (56) ττ (54) metric OK (57) σσ (58) 3 (57) (58) h ττ + c 2 h σσ = 0 (59) (56) i (59) ii { i hστ =0 (60) ii h ττ + c 2 h σσ =0 h X ± h αβ = α X M β X N η MN = α X + β X α X β X + + α X i β X i (61) i 2 D 1 X + τ light-cone gauge τ 2 β τ 1 σ h h στ = σ X + σ X i τ X i h σσ =( σ X i ) 2 (62) h ττ = 2 τ X +( τ X i ) 2 i ii i, ii ± X =( ± X i ) 2 (63) ± ± 1 2 ( τ ± c σ ) (64) 36

38 X M = 0 (65) 1 2 (34) 2 (63),(65) (65) ( ) α X M (τ,σ)=x M +α p M 1/2 1 ( cτ +i α M 2 m m e im(cτ σ) + α m M e im(cτ+σ)) (66) m 0 X? X M σ τ x M constant σ τ constant term τ momentum p M momentum X M α c c metric constant linear term (66) α α α i 2 X M real parameter X M :real α m = α m, α m = α m (67) α m α m (66) i (67) 1 convention (66) m (65) + τ σ cτ ± σ (66) 2 cτ + σ cτ σ exp σ 2π 37

39 (63) p α m α m Xi i α i m αi m xi p i X constant term (63) x c action X M τ ẊM ẊM P M = δs δẋm = T gg τα α X M = Tc 1 τ X M (68) g =1 α τ g ττ = c 1 (68) M + M =+ X + = τ ( p + 1 ) α c P + = Tc 1 = 1 2π p+, (69) + momentum X + = τ Tc 1 p + T = 1 2πα p + 1 α c M i i 2 D 1 (68) P i (66) P i = Tc 1 τ X i = 1 1 2π pi + 2π ( α i 2α m e im(cτ σ) + α me i im(cτ+σ)) (70) m 0 momentum momentum [X i (τ,σ),p j (τ,σ )] = iδ ij δ(σ σ ) (71) (66) (70) (71) x i p j [x i,p j ]=iδ ij (72) [αm i,αj n ]=mδij δ m+n,0 (73) [ α m, i α n]=mδ j ij δ m+n,0 (74) 38

40 x c c p + c p + x p + partner (72) ± [x,p + ]=iη + = i (75) momentum 1 p p + p i p X i oscillator X i (63) (66) (63) p p = 1 2p + ( (p i ) α m 0 ) i ( α ) m α m i + α mα i m i (76) (63) (66) (63) p m 2 p M p M + m 2 = p M p M =2p + p (p i ) 2 (77) (76) 2p + (p i ) 2 m 2 = 1 i ( α α m α m i + αi m m) αi m 0 (78) physical states operator state physical state momentum k (79) [α i m,α j n]=mδ ij δ m+n,0 (80) 39

41 a a operator [a, a ] = 1 (81) a a (80) m normalization operator α 1 α m m m α i m αi m α m m α m m { αm, m > 0 α m, m > 0 (82) { αm k =0 m>0 (83) α m k =0 k α m i, αi m (m>0) state α m i, αi m m state bosonic string 1 N operator Ñ α N α mα i m, i m=1 Ñ α m i α m i (84) m=1 physical state (N Ñ) phys = 0 (85) phys pysical state level matching condition N Ñ σ constant shift generator (80) N 40

42 α Ñ α [N,αm]= mα i m i (86) [Ñ, αi m]= m α m i (87) N αm i m 1 operator X X i α α (66) N Ñ operator e iα(n Ñ) X i (τ,σ)e iα(n Ñ) = X i (τ,σ + α) (88) (86,87) σ α shift σ constant shift symmetry lightcone gauge state (85) σ state momentum operator k α m i, αi n (m, n > 0) α α 1 1 mass m 2 = 1 i ( α α n α n i + αi n n) αi (89) n 0 (84) N Ñ n α n i α n i n (α nα i n i ) m 2 = 1 i ( α α n α n i + α nαn) i i n 0 = 2 α (Ñ + N +(D 2) 41 ) n n=1 (90)

43 α i n αi n n [α i n,α j n] =nδ ij n n ζ ζ ζ(s) = n s (91) n=1 n = ζ( 1) n=1 ζ s s = n = ζ( 1) = 1 12 n=1 (92) comment string... α cancel consistent n 1 12 bosonic string D =26 Weyl anomaly symmetry light-cone gauge light-cone gauge 42

44 1: physical state states (mass) 2 k m 2 = 4 α closed tachyon massless α 1 i α j 1 k m 2 =0 φ g ij B ij trace traceless dilaton graviton tensor α 2 i α j 1 α 1 k k m 2 > 0 massive Lorentz symmetry manifest gauge gauge Lorentz symmetry Lorentz symmetry D =26 n 1 level matching 12 condition (85) N = Ñ mass squared (90) m 2 = 4 α (N 1) = 4 (Ñ 1) (93) α 5 (92) 2 ( 1 ) physical state (mass) 2 1 momentum k mass (93) N Ñ 0 state k mass squared closed string closed string tachyon k 1 mass 1 1 α 1 i α j 1 k N operator N =1 N =1 mass squared 0 massless massless α 1 i α j 1 k i j i j trace part φ trace part 1 physical state (85) level matching condition 43

45 g ij traceless B ij overview dilaton φ g ij graviton metric B ij overview tensor operator α i 2 αj 1 α k 1 k state ( 1 ) N =2 α α level matching condition (85) massive 1 α factor mass 1 tachyon closed string tachyon closed string tachyon tachyon open string superstring open string tachyon closed string tachyon over (1 ) (2 ) bosonic string light-cone gauge bosonic string 2 world-sheet action S = T d 2 σ gg αβ α X M β X M (94) 2 light-cone gauge X + = τ g αβ X i X ± X + τ X X i X i i ± 2 D 1 44

46 X i light-cone gauge metric fix X i ( 2 τ c 2 2 σ)x i = 0 (95) cτ τ ( 2 τ 2 σ )Xi = 0 (96) (96) (94) action X i X i (96) action S = T d 2 σ ( ( τ X i ) 2 ( σ X i ) 2) (97) 2 normalization α α operator superstring 2.2 Superstring superstring superstring world-sheet spacetime 2 world-sheet supersymmetry boson fermion (97) action scalar 2 X i scalar scalar 2 D 1 1 S = 1 d 2 ση αβ α X β X (98) 2 action boson system fermion fermion ψ ψ ψ = ( 45 ψ ψ + ) (99)

47 2 2 Majorana spinor Grassmann odd real Grassmann fermion γ S = 1 d 2 σ ( η αβ α X β X + i 2 ψγ α α ψ ) (100) γ ( ) ( ) γ 0 1 =, γ 1 1 = (101) 1 1 {γ α,γ β } =2η αβ (102) (100) action { δɛ X = ɛψ δ ɛ ψ = iγ α ɛ α X ɛ ( ) ɛ = ɛ ɛ + (103) (104) Majorana spinor X boson fermion fermion X boson (100) action δ ɛ S =0 boson fermion supersymmetry light-cone gauge (97) light-cone action η αβ S LC = T d 2 ση αβ α X i β X i (105) 2 46

48 super (100) X i partner ψ i S LC = T d 2 σ ( η αβ α X i β X i + i 2 ψ i γ α α ψ i) (106) ψ i Majorana spinor (100) i (2 D 1 ) notation ± = 1 2 ( τ ± σ ) (107) (101) γ (99) ψ S LC = T d 2 σ ( 2 + X i X i + i(ψ+ i ψ+ i + ψ i + ψ ) ) i (108) (108) ψ i + ψi ψ i ± = 0 (109) ψ ± i τ ± σ (109) superstring ψ+ i ψi ψ+ i ψi + τ σ 2 σ 2π 2 { +ψ i ψ+(τ,σ i + (τ,σ) Ramond sector (R) +2π) = ψ+(τ,σ) i Neveu Schwarz sector (NS) (110) fermion 360 superstring Ramond sector Neveu-Schwarz sector Ramond R Neveu-Schwarz NS ψ+ i ψi 2 ψ+ i ψi boundary condition R NS 47

49 R-NS NS-R 4 ψ i + R NS R NS ψi R NS NS R R-R sector NS-NS sector R-NS sector NS-R sector R R R-R sector NS-NS sector R-NS NS-R sector R R: ψ i = l s d i ne in(τ σ) n Z ψ+ i = l (111) s d i n e in(τ+σ) n Z normalization convention NS NS : ψ i = l s b i re ir(τ σ) ψ i + = l s r Z+1/2 r Z+1/2 bi r e ir(τ+σ) (112) label n r Green-Schwartz-Witten notation (111) (112) +1/2 2π fermion d i n,bi r, b i r {d i n,dj m } = δij δ n+m,0 (113) physical states Fock space creation operator state boson state state 48

50 m 2 < 0 m 2 =0 NS-NS T φ,b ij,g ij R-R C C ij C ijkl C i1...i 6 C i1...i 8 C i C ijk C ijklm C i1...i 7 NS-NS sector tachyon m 2 < 0 massless R-R sector tachyon massless massive NS-NS sector bosonic string tachyon scalar 2 tensor traceless φ massless scalar dilaton B ij B Kalb-Ramond field g ij graviton C tensor tensor R-R superstring D 10 consistent bosonic string 26 super tensor light-cone gauge i tensor C ijkl selfdual C ij C i1...i 6 C ijk C ijklm C i C i1...i 7 C C i1...i 8 dual dual 8 light-cone gauge system SO(8) symmetry manifest i SO(8) dual 8 tensor ɛ i 1...i 8 C i1...i n C i n+1...i n+8 (114) n 8 n GSO projection 49

51 boson R-NS NS-R fermion boson consistent consistent state 2 consistent 2 Type IIA Type IIB Type IIA NS-NS R-R Type IIB Type IIA Type IIB NS-NS R-R C i C ijk... φ, B ij,g ij CC ij C ijkl... NS-NS sector Type IIA Type IIB tachyon projection dilaton B μν,b graviton R-R Type IIA Type IIB Type IIA Type IIB scalar 2 tensor 4 Type 0 supersymmetry supersymmetry tachyon closed string closed string tachyon superstring superstring supersymmetry Type IIA Type IIB boson fermion fermion Type IIA Type IIB fermion chirality 1 boson part fermion part partner spacetime 10 supersymmetry Lagrangean (108) 2 2 massive mode fermion state 50

52 2 X i ψ i boson fermion graviton Type II 10 3 D-brane D-brane D-brane R-R charge R-R C i1...i n tensor charge 4 interaction S int e dx μ A μ dt (115) dt x ( 22) t 1 dim x 22: gauge interaction gauge interact notation A μ dx μ A (115) e A μ dx μ dt dt = e 1dim dt 51 A (116)

53 1 light-cone gauge 2 9 light-cone gauge covariant 0 9 dx M 1 dx Mn C n C M1...M n dx M 1 dx Mn = C n (117) n tensor n tensor charge (116) S int = e C n (118) n dim (116) world-line ( 22) 1 C n n dx M n 22 n ( 23) n t n dim x 23: D-brane n tensor interaction D-brane p +1 p +1 tensor C p+1 couple C p+1 charge Dp-brane Dp-brane p D-brane D Polchinski D- brane D-brane D Dp-brane string Dp-brane open string p +1 52

54 p+1 dim 24: D-brane p +1 open string ( 24) open string D-brane R-R n tensor p +1 tensor charge Polchinski string second revolution D-brane Type IIA Type IIB R-R Type IIA Type IIB Dp-brane D-brane R-R couple D-brane non BPS D-brane R-R charge D-brane BPS D-brane Type IIA Type IIB ( 2) Type IIA tensor p IIA D0 D2 D4 D6 D8 IIB D( 1) D1 D3 D5 D7 D9 2: BPS D-brane 1 Type IIB 1 1 p +1 p = 1 p 1 0 D-instanton D-brane R-R charge open string non BPS D-brane 53

55 tachyon tachyon non BPS tachyon D-brane D-brane tachyon 3.2 open string Dp-brane physical state open string open string closed string open string string open string string ( 25) t open string 25: open string closed string X M ( ) α X M (τ,σ)=x M + α p M 1/2 1 ( τ + i α M 2 m m e im(τ σ) + α m M e im(τ+σ)) (119) m 0 α m M τ σ part α m M τ + σ part 2 part τ σ τ + σ left mover right mover ( 26) left right closed string left mover closed string open string ( 25) left right 54

56 left mover right mover 26: closed string open string σ σ 0 σ π σ =0,π left right (119) αm M αm m fermion oscillator d M n d M n b M r b M r αm m, d M n, b M r closed string closed string brane open string brane open string 10 brane open string momentum brane brane momentum mass bosonic closed string m 2 = 4 (N 1) (120) α open superstring m 2 = 1 (N + a) (121) α superstring N fermion N α m M m operator fermion N a 1 superstring R-sector NS-sector a = { 0 (R) 1/2 (NS) (122) 55

57 state boson fermion NS-sector R-sector NS-sector closed string R-R R-NS open string left right NS R 2 NS-sector states m 2 0; k m 2 = 1 2α T :tachyon b i 1/2 0; k m2 =0 A i :gauge b I 1/2 0; k m2 =0 Φ I :scalar state momentum state m 2 (121) N 0 m 2 = 1 tachyon 2α state b i 1/2 bi 1/2 i state massless N operator operator N 1/2 (122) 1/2 cancel massless i brane 2 p I p+1 9 b I 1/2 massless creation operator state NS-sector massive 0; k tachyon b i 1/2 0; k Dp-brane x0 x p i Dp-brane p +1 vector 3 massless gauge b I 1/2 0; k D-barane Lorentz scalar NS-sector R-sector R-sector fermion closed string GSO projection GSO projection projection 3 0, 1 light-cone gauge 56

58 b M r ( 1)F operator { ( 1) F 0; k = 0; k {( 1) F,b M r } =0 (123) projection operator P = 1 2 (1 + ( 1)F ) (124) P phys = phys (125) state P (123) ( 1) F P 0; k = 0 (126) tachyon gauge b i 1/2 0; k P ( 1) F operator ( 1) F b M r ( 1) F (123) 1 ( 1) F b i 1/2 0; k = b i 1/2( 1) F 0; k = b i 1/2 0; k (127) gauge P projection operator Pb i 1/2 0; k = b i 1/2 0; k (128) scalar b I 1/2 0; k I A i, Φ I physical state consistent 3.3 D-brane D-brane 1 N D-brane 57

59 1 2 3 N 27: D-brane ( 27 ) ( 27 ) D-brane N D-brane open string open string D-brane D-brane open string D-brane D-brane a, b 1 N 1 open string D-brane label open string gauge scalar D-brane N 2 open string 4 open string 1 D-brane gauge scalar gauge scalar a b (A i ) a b, (ΦI ) a b N N gauge N N non-abelian U(N) gauge Dp-brane p +1 U(N) gauge non-abelian gauge string 4 string (a b ) 58

60 3.4 D-brane D-brane 2 analogy QED photon Feynman graph ( 28) Coulomb 28: photon exchange D-brane ( 29) D-brane 2 D-brane 29: closed string exchange Dp-brane Dp-brane closed string Feynman graph string closed string closed string exchange closed string closed string 59

61 dilaton φ graviton g R-R graph ( 29) R-R charge closed string spectrum R-R R-R charge open string string Feynman graph world-sheet Feynman graph closed string graph string closed string exchange ( 29) 30 open string 30: open string 1-loop open string 1 open string 1-loop diagram open-closed duality closed string tree level graph open string 1-loop open string closed string open string 1-loop 1-loop vacuum graph analogy free part interaction free scalar D path integral e F = Dφe d D xφ( 2 m 2 )φ = ( det( 2 m 2 ) ) 1/2 (129) 60

62 path integral exp F F = 1 2 log det( 2 m 2 ) (130) log det Tr log operator trace momentum = 1 2 Tr log( 2 m 2 ) = 1 d D k 2 (2π) k D log(k2 + m 2 ) k (131) momentum k k k k V D = 1 2 d D k (2π) D V D log(k 2 + m 2 ) (132) normalization log d D k dt = V D (2π) D 0 2t e (k2 +m2 )t (133) k Gauss 1 dt ( π ) D/2 = V D e m 2 t (2π) D 0 2t t (134) open string gauge scalar massive mode e m2t f(t) e m2 i t e m2 i t (135) i:boson i:fermion mass m scalar 1 string massive mode fermion fermion 1-loop 61

63 fermion (134) e m2t factor (135) f(t) f(t) 1 m 2 brane brane D-brane y 31: D-brane 1 D-brane y ( 31) y open string open string open string tension 1 energy mass 2πα m 2 = 1 ( y ) 2 α (N + a)+ (136) 2πα D-brane N operator operator a R-sector R-sector fermion NS-sector 2 boson { 0 (R) fermion a = (137) 1/2 (NS) boson 62

64 f(t) =e ( y 2πα )2 t }{{} e 1 2α t }{{} n=1 ( n=1 1 1 e n 1 α t }{{} α n ( 1 e (n 1 2 ) 1 α t }{{} b (n 1/2) ) 8 } {{ } P ( 1) F ) 8 1 [ 2 16 e 1 2α t }{{} n=1 ( 1+ e (n 1 2 ) 1 α t }{{} b (n 1/2) ) 8 } {{ } P 1 ( ) 8 ] 1+ e n 1 α t }{{} n=1 d n (138) α n NS-sector b (n 1 ) GSO projection 2 2 P 1 ( 1) F P GSO projection projection operator R-sector d n (137) a (136) (135) state mass state NS-sector state 0; k α b α n1...α nk b r1...b rl 0; k (139) state mass e m2 i t mass (136) N operator N = n n k + r r l (140) m 2 1 α NS-sector a = 1 2 ( y 2πα ) 2 mass factor b b fermionic operator 1 b (n 1 2 ) mass (n 1 2 ) 1 α 1 n 1 1 e (n 1 2 ) 1 α t state 8 i i 8 63

65 2 projection operator ( 1) F ( 1) F b operator 1 operator e (n 1 2 ) 1 α t ( 1) F operator α n e n 1 α t Taylor 1+e n 1 α t + e 2n 1 α t +... α n bosonic operator operator 1 Taylor 1 e n 1 α t 8 n 3 R-sector R-sector d n operator 1 1 n 1 m 2 α (138) boson fermion notation f(t) = 1 2 e ( y 2πα )2t f 1 (q) 8 ( f 3 (q) 8 f 4 (q) 8 f 2 (q) 8) (141) q e 1 2α t f 1,f 2,f 3,f 4 f 1 (q) =q 1 12 (1 q 2n ) (142) n=1 f 2 (q) = 2q 1 12 f 3 (q) =q 1 24 f 4 (q) =q 1 24 (1 + q 2n ) (143) n=1 (1 + q 2n 1 ) (144) n=1 (1 q 2n 1 ) (145) n=1 f 3 (q) 8 f 4 (q) 8 f 2 (q) 8 =0 64

66 (135) (boson ) (fermion ) 0 boson fermion cancel supersymmetry open string D-brane supersymmetric boson fermion boson fermion boson fermion cancel open string 1-loop closed string exchange graviton R-R R-R cancel miss leading closed string cancel (130) F string overall F 0 dt p+1 t t 2 e ( y 2πα )2t f 1 (q) 8 ( NS NS exchange {}}{ f 3 (q) 8 }{{} NS 1 R R exchange {}}{ f 4 (q) 8 }{{} NS ( 1) F NS NS exchange {}}{ ) f 2 (q) 8 }{{} R (146) f(t) (141) (134) e m2t D D Dp-brane p +1 D p +1 (141) (134) (146) 1 (146) 1 NS-sector 1 P operator 1 2 NS-sector ( 1) F R-sector f 2 closed string 1 3 NS-NS closed string exchange 2 R-R sector R-R exchange 3 D-brane tachyon ( ) 65

67 4 tachyon Type II closed string open string open string D-brane D-brane brane open string D9-brane 10 D9-brane 1+9 brane Open string D-brane 1 10 D-brane 10 open string Type IIA 53 2 Type IIA D9-brane R-R charge D-brane D9-brane R-R charge D-brane open string R-R charge open stirng open string 1-loop vacuum amplitude (146) F 0 dt t t 5 f 1 (q) 8 ( NS NS exchange {}}{ f 3 (q) 8 }{{} NS 1 R R exchange {}}{ f 4 (q) 8 }{{} NS ( 1) F NS NS exchange {}}{ ) f 2 (q) 8 }{{} R (147) 5 D9-brane R-R charge (147) 2 dt F t t 5 f 1 (q) ( 8 f 3 (q) 8 f 2 (q) 8) (148) 0 open string 1 NS-sector GSO projection projection operator (124) 1 2 NS-sector projection operator (124) ( 1) F part 3 R-sector R-R charge 2 open string projection 5 (146) e ( y 2πα )2t D9-brane brane y zero D9-brane p =9 66

68 ( 1) F open string GSO projection open string open string GSO projection state open string spectrum physical state tachyon gauge scalar fermion massive mode (fermion, massive mode) GSO projection tachyon GSO projection tachyon tachyon D9-brane A i Φ I i 2 p I p +1 9 p 9 I scalar Φ I p tachyon gauge A i 10 gauge spectrum T A i Φ I fermion ( ) massive ( ) 10 D9-brane D-brane N D9-brane R-R charge D-brane non BPS D-brane N non BPS D9-brane tachyon gauge N N U(N) gauge U(N) adjoint U(N) adjoint tachyon gauge N D9-brane tachyonic Type IIA N N N non BPS D9-brane tachyonic Type IIA T,A i : N N U(N) adjoint T = T,A i = A i Type IIB Type IIB R-R charge D9-brane 53 2 D9-brane 67

69 D-brane D-brane R-R interaction S int C 10 (149) D9-brane couple R-R 10-form 10 S int C M1...M 10 dx M 1 dx M 10 (150) 10 dim = C d 10 x (151) 10 dim light-cone gauge C 10 kinetic term 0 field strength light cone gauge Dp-brane p couple 10 action M 1 M (150) C 0 9 (151) interaction term C field strength kinetic term kineteic term interaction term C δs δc 10 =0 1 = 0 (152) consistency R-R R-R tadpole cancelation R-R 10-form 1 interaction open string closed string R-R 10-form coupling 68

70 open string open string anomaly cancel open string fermion 1loop anomaly D9-brane D9 charge D9 anti D9- brane charge interaction (149) tadpole cancel D9-brane D9 D9 _ D9 1 2 > < 3 32: D9 D9 system 10 open string D9-brane 2 open string D9 2 open string D9 D9 open string 32 1, 2, BPS D-brane gauge gauge gauge 2 (A i,ãi) Dp-brane spectrum scalar Type IIA D9-brane gauge 9 scalar scalar boson mass (fermion massive mode ) 10 gauge 2 D9 D9 3 open string 1 (146) 3 open string 1 loop vacuum amplitude (146) 1 R-R charge D9 D9 69

71 closed string 2 R-R charge R-R charge F 0 dt t t 5 f 1 (q) 8 ( NS NS exchange {}}{ f 3 (q) 8 + }{{} NS 1 R R exchange {}}{ f 4 (q) 8 }{{} NS ( 1) F NS NS exchange {}}{ ) f 2 (q) 8 }{{} R (153) open string 1 NS-sector projection operator 1 2 NS-sector projection operator ( 1) F (146) 2 projection operator (124) 2 P = 1 ( ) 1 ( 1) F (154) 2 open string GSO projection projection operator 1+ R-R charge 2 ( 1) F operator 1 state GSO projection tachyon gauge scalar GSO projection tachyon GSO projection gauge scalar P 2 tachyon Type IIB Type IIA D9 D9 2 open string 32 2 tachyon complex D9 D9 N gauge U(N) U(N) gauge 1 open string A i gauge 2 Ãi gauge 3 tachyon A i gauge 1 U(N) adjoint Ãi gauge 2 U(N) adjoint tachyon tachyon N N 2 T a b open string a b D9-brane D9-brane bifundamental tachyon 70

72 D9-D9 N = U(N) U(N) A i adj 1 Ã i 1 adj T a b tachyon N gauge symmetry string 4.1 Tachyon potential tachyon potential IIA IIB ( IIA V (t) tr e T 2) (T = T ) (155) ( ) IIB V (t) tr e T T (T : complex) (156) overall normalization T 2 T T IIA tachyon IIB tachyon complex tachyon potential 33 Gauss potential tachyon V(T) T 33: tachyon potential 33 tachyon potential minimum T T/ 1+T 2 T T tachyon T 1 1 potential minimum 34 minimum 71

73 ~ V(T) 1 1 ~ T 34: tachyon potential kinetic term potential potential kinetic term tachyon m 2 tachyon m 2 higgs m 2 potential minimum tachyon potential minimum potential minimum Sen 35 potential minimum V(T) T 35: potential minimum 72

74 potential minimum open string closed string closed string Sen conjecture open string non BPS D9-brane D9-D9 IIA IIB R-R charge brane cancel brane brane charge brane charge tachyon potential minimum tachyon open string open string open string closed string Type IIA Type IIB Type II string potential minimum potential potential tachyon Type II string 73

75 V(T) T 36: potential tachyon 5 D-brane section 4 tachyon D-brane soliton ( D-brane Dp-brane ) 5.1 1:Type IIA Type IIA non BPS D9-brane 1 N 1 tachyon condense tachyon potential minimum non BPS D9-brane 1 D9-brane 1 tachyon real potential (155) ( 33) T = ± T = ± potential minimum tachyon x 9 T (x 0,x 1,...,x 9 )=ux 9 (157) parameter u real parameter x 9 x 0 x tachyon (157) configuration x 9 x 9 tachyon x 9 74

76 0~8 energy x T~+ x 9 T~ non BPS D9 37: tachyon x 9 x 0~8 D8-brane x 9 38: kink tachyon Sen conjecture tachyon potential minimum non BPS D9-brane conjecture (x 9 ± ) non BPS D9 x 9 0 energy u parameter 38 x 9 =0 energy 1+8 string 1+8 D-brane D8-brane formulation non BPS D-brane open string D8-brane tachyon (157) configuration 1+8 D8-brane u 75

77 u limit x 9 0 localize 1+8 (157) kink 5.2 2:Type IIB Type IIB Type IIB D9-D9 1 tachyon complex scalar T 2 potential (156) e potential minimum kink vortex vortex u parameter T (x) =u(x 8 + ix 9 ) (158) complex scalar i x 8 x 9 x 8 x 9 ( 39) x 8 (7 + 1) D7-brane x 0~7 x 9 x 8 39: vortex x 9 0 D-brane x 8 x 9 nonzero u D-brane Sen conjecture D-brane x 8 = x 9 =0 energy 7+1 D7-brane 76

78 5.3 analogy t Hooft-Polyakov monopole 4 4 SU(2) gauge Higgs tachyon Higgs T adjoint Lagrangian gauge kinetic term tachyon kinetic term potential L = 1 2 TrF 2 μν +Tr(D μt ) 2 V (T ) (159) D μ T adjoint gauge couple D μ T = μ T +[A μ,t] (160) potential V (T )=λ(trt 2 2v 2 ) 2 (161) 40 double well potential potential min- V(T) 1 2 Tr 2 T =v 2 T 40: 4 SU(2) gauge theory adjoint Higgs potential imum 1trT 2 = v 2 2 adjoint Higgs base ( ) v T = g g 1, g SU(2) (162) v 77

79 adjoint SU(2) traceless v, v SU(2) symmetry SU(2) g SU(2) symmetry U(1) x 1 x 3 3 energy x 3 x 2 x 1 41: tachyon Higgs ( ) v T g g 1, ( x ) (163) v potential energy energy density energy energy 0 configuration 3 x S 2 S 2 g SU(2) (S 2 ) g S 2 SU(2) map S 2 g tachyon g SU(2) effective g ) (e iα g g (164) e iα g tachyon factor U(1) part 78

80 x 3 x 2 x 1 S 2 42: x S 2 U(1) part tachyon tachyon S 2 SU(2) map U(1) S 2 SU(2)/U(1) (165) map map S 2 map Π 2 2 homotopy Z Π 2 (SU(2)/U(1)) Z (166) Z label monopole charge t Hooft-Polyakov monopole Lagrangian topological Type IIB Type IIB D9-D9 N U(N) U(N) gauge tachyon bifundamental (T :(, )) tachyon nonzero U(N) U(N) 79

81 diagonal U(N) tachyon 1 1 T u..., u (167) 1 tachyon bifundamental U(N) U(N) (g 1,g 2 ) tachyon T g 1 Tg 1 2 tachyon (167) g 1 g 2 2 U(N) 1 U(N) U(N) U(N) x 0 x p x p+1 x 9 9 p 2 ( 43) x p+1 x 9 x p+1 ~ x 9 S 8 p 43: 9 p x p+1 x 9 0 energy energy 0 D-brane x p+1 x 9 8 p S 2 S 8 p SU(2)/U(1) SU(2) gauge gauge U(N) U(N) U(1) U(N) tachyon S 8 p U(N) U(N)/U(N) map S 8 p U(N) U(N)/U(N) (168) 80

82 Π 2 2 S 2 2 Π 8 p U(N) U(N)/U(N) Π 8 p (U(N)) ( ) U(N) U(N) Π 8 p =Π 8 p (U(N)). (169) U(N) homotopy N 0 Z N homotopy p even 0 p odd { 0 (p :even) Π 8 p (U(N)) Z (p : odd) (N : ) (170) Type IIB Dp-brane p odd section Type IIB Type IIA even odd section 3 supersymmetry D-brane topological D-brane section 3 string Dp-brane Dp-brane D-brane boundary state D-brane 3 6 D-brane string 1 N K 6 Tsuguhiko Asakawa, Shigeki Sugimoto and Seiji Terashima, Exact Description of D-branes via Tachyon Condensation, arxiv:hep-th/

83 Π 8 p (U(N)) = K(R 9 p ) (171) K K D-brane K Witten 98 K D-brane K tachyon K K tachyon 6 section 6 82