D-branes and Closed String Field Theory Koichi Murakami (KEK) This - - PowerPoint PPT Presentation

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D-branes and Closed String Field Theory Koichi Murakami (KEK) This - - PowerPoint PPT Presentation

Apr 07, 2023 369 likes •689 views

D-branes and Closed String Field Theory Koichi Murakami (KEK) This talk is based on Yutaka Baba, Nobuyuki Ishibashi and K.M., JHEP05(2006)029 [hep-th/0603152] , and a work in preparation (talk at KEK Theory Workshop 2007, March 14, 2007) 1.

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D-branes and Closed String Field Theory

Koichi Murakami (KEK) This talk is based on Yutaka Baba, Nobuyuki Ishibashi and K.M., JHEP05(2006)029 [hep-th/0603152],

and a work in preparation (talk at KEK Theory Workshop 2007, March 14, 2007)

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§1. Introduction

1

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I Motivation: How can we capture the solitonic nature of D-branes in string theory?

  • D-branes are non-perturbative object.

⇒ Non-perturbative formulation of string should be necessary. In this work, we focus on a traditional approach: String Field Theory (SFT)

  • In the open string side, several analyses have been carried out.

(Sen’s conjecture, VSFT, analytic solutions, etc...)

  • The question we would like to address in this talk:

“What are D-branes in closed SFT ?” 2 I Motivation: How can we capture the solitonic nature of D-branes in string theory?

  • D-branes are non-perturbative object.

⇒ Non-perturbative formulation of string should be necessary. In this work, we focus on a traditional approach: String Field Theory (SFT)

  • In the open string side, several analyses have been carried out.

(Sen’s conjecture, VSFT, analytic solutions, etc...)

  • The question we would like to address in this talk:

“What are D-branes in closed SFT ?” 2

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I Boundary state |Bi

  • should play very important roles to describe D-branes in closed string sector
  • A single |Bi is not enough.

|Bi ⇒ make a single hole in a closed-string worldsheet D-brane ⇒ make an arbitrary number of holes in a closed-string worldsheet ⇒ One may guess that the D-brane state in the second-quantization looks like ∼ exp [|Bi] ??? ⇒ We will see that this is the case in a certain sense. (Of course, we will make this (meaningless) expression into a precise form later.) 3

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I Clues to D-branes in closed SFT (i): D-branes in SFT for noncritical strings

  • Hamiltonian of SFT for c = 0 (Ishibashi-Kawai, Jevicki-Rodrigues)

H = Z ∞ dl µ 3δ00(l) − 3 4μδ(l) ¶ ψ†(l) + Z ∞ dl1dl2 £ (l1 + l2)ψ†(l1 + l2)ψ(l1)ψ(l2) + g2

sl1l2ψ†(l1)ψ†(l2)ψ(l1 + l2)

¤

  • =

Z ∞ dlψ(l) (lT(l) + ρ(l)) = Z ∞ dlψ†(l)D(l)

  • ψ†(l), ψ(l): creation and annihilation ops of a string with length l obeying

[ψ(l) , ψ†(l0)] = δ(l − l0) ,

  • T(l) =

Z l dl0 ψ(l0)ψ(l − l0) + g2

s

Z ∞ dl0 l0ψ†(l0)ψ(l + l0) , ρ(l) = 3δ00(l) − 3 4μδ(l)

4

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  • D(l)|Ψi = 0

⇒ loop equation: hw(l1) · · · w(ln)i = h0|ψ(l1) · · · ψ(ln)|Ψi

  • Solitonic operators corresponding to (ghost) D-branes:

= the operators which commute with D(l), i.e. T(l) (⇐ Virasoro constraint)

(Fukuma-Yahikozawa, Hanada-Hayakawa-Ishibashi-Kawai-Kuroki-Matsuo-Tada)

Z dζV±(ζ) = Z dζ exp µ ±gs Z ∞ dle−ζlψ†(l) ¶ exp µ ∓ 2 gs Z ∞ dl l eζlψ(l) ¶ ⇒ Z dζV±(ζ) is a creation op of ½ D-instanton ghost D-brane (Okuda-Takayanagi) |Ψi(n + 1)-inst. = C Z dζV+(ζ) |Ψin-inst. (n ≥ 0)

⇓

We will show that a similar construction of solitonic operators in SFT for critical strings is possible in the OSp invariant SFT. 5

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I Clues to D-branes in closed SFT (ii): boundary state in SFT for critical strings

  • Hashimoto-Hata introduced a BRST invariant source term hB|Φi into the HIKKO

type closed SFT.

  • Idempotency equation (Kishimoto-Matsuo-Watanabe)

|Bi ∗ |Bi ∝ |Bi 6

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I What we did

  • We construct BRST invariant solitonic states in the OSp invariant SFT.

⇐ states in which D-branes (or ghost D-branes) are excited

  • We perturbatively calculate the overlap between these states

⇒ the vacuum cylinder amplitude (×4) for the D-branes is reproduced.

Our solitonic ops create two D-branes (or ghost D-branes) and not a single. (← We do not know why...)

  • Using the above solitonic states, we calculate the disk amplitude of two closed

string tachyons. For this purpose, ¤ define BRST invariant observables in the OSp invariant SFT ¤ Green’s functions, S-matrix As a bonus, determine the sign ambiguity between D-brane and ghost D-brane 7

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Plan of the talk §1. Introduction §2. OSp invariant SFT §3. Boundary state and Solitonic operators §4. Observables §5. Green’s functions and S-matrix elements §6. Disk amplitude §7. Summary and Discussions 8

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§2. OSp Invariant SFT

9

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I Procedure for covariantizing the LC gauge SFT : OSp extension (Siegel) LC gauge SFT OSp invariant SFT (t = X+, α = 2p+, Xi)

(i = 1, . . . , 24)

− → OSp extension

(t, α, XM) XM = (Xi, X25, X26; C, ¯ C) linear symmetry O(24) OSp(26|2) S-matrix symmetry O(25,1) OSp(27, 1|2)

  • C(τ, σ), ¯

C(τ, σ): spinless Grassmann odd variables (ghost) ( C(τ, σ) = C0 + 2iπ0τ − i P

n6=0 1 n

¡ γne−n(τ+iσ) + ˜ γne−n(τ−iσ)¢ ¯ C(τ, σ) = ¯ C0 − 2i¯ π0τ + i P

n6=0 1 n

¡ ¯ γne−n(τ+iσ) + ˜ ¯ γne−n(τ−iσ)¢

  • metric for

“transverse directions”: δij − → ηMN =

C ¯ C C ¯ C

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ δμν −i i ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ 10

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Thus we may say: OSp inv. SFT is LC SFT with XM = (Xi, X25, X26; C, ¯ C) as transverse directions . I String field action S = Z dt " 1 2 Z d1d2 hR(1, 2) |Φi1 Ã i ∂ ∂t − L(2) + ˜ L(2) − 2 α2 ! |Φi2 + 2g 3 Z d1d2d3 ­ V 0

3 (1, 2, 3)

¯ ¯ Φi1|Φi2|Φi3 ¸

  • zero modes: wave function representation of momenta

⇒ momentum zero-mode integration: dr = αrdαr 2 d26pr (2π)26 id¯ π(r)

0 dπ(r)

11

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I Relationship between S-matrix in LC SFT and S-matrix in OSp invariant SFT

  • S-matrix in LC SFT

SLC ¡ pLC, ²LC¢ = δLC(pLC) F ¡ pLC

r

· pLC

s

, pLC

r

· ²LC

s

, ²LC

r

· ²LC

s

¢ pLC

r

= (kr,0, kr,1, pr,i) (k+ = 2α) ²LC

r : polarizations

  • S-matrix in OSp inv SFT

SOSp ¡ pOSp, ²OSp¢ = δOSp(pOSp) F ¡ pOSp

r

· pOSp

s

, pOSp

r

· ²OSp

s

, ²OSp

r

· ²OSp

s

¢ pOSp

r

= (pLC

r

; pr,25, pr,26 ; π(r)

0 , ¯

π(r)

0 ) , ²OSp r

= (²LC

r

; ²r,25, ²r,26 ; ²r,C, ²r, ¯

C)

Because of the OSp(27, 1|2) invariance, the function F is common to the two theories.

⇓

Using Pairisi-Sourlas formula, for the S-matrix elements of the external states with ²OSp

∓

= ²OSp

C

= ²OSp

¯ C

= 0, we have (after k0 → k2 = −ik0, p26 → p0 = ip26), Z Y

r

∙dkr,1dk2 2π id¯ π(r)

0 dπ(r)

¸ SOSp(pOSp, ²OSp) µ = SOSp(pOSp, ²OSp) ¯ ¯ ¯ ¯ ¶ = SLC(pμ, ²μ) . 12

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I BRST symmetry = J−C of the OSp(27, 1|2),

(Siegel-Zwiebach, Bengtsson-Linden)

δBΦ = QBΦ + gΦ ∗ Φ

  • BRST charge

QB = C0 2α(L0 + ˜ L0 − 2) − iπ0 ∂ ∂α + i α

∞

X

n=1

à γ−nLn − L−nγn n + ˜ γ−n ˜ Ln − ˜ L−n˜ γn n !

Ln, ˜ Ln: Virasoro generators Ln ≡ 1 2 X

m

  • αM

n+mαN −mηMN

  • ,

˜ Ln ≡ 1 2 X

m

  • ˜

αM

n+m ˜

αN

−mηMN

  • In particular, L0 = α0

4 pμpμ +

∞

X

m=1

αμ

−mαmμ + iπ0¯

π0 + i

∞

X

m=1

(γ−m¯ γm − ¯ γ−mγm) , ˜ L0 = . . .

13

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§3. Boundary state and Solitonic operators

14

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I Boundary state in OSp invariant theory

  • matter sector: P μ(0, σ)|B0i = Xi(0, σ)|B0i = 0

(as usual)

  • ghost sector: C(0, σ)|B0i = ¯

C(0, σ)|B0i = 0 I def |B0(l)i = |B0iδ(α − l) ⇒ α|B0(l)i = l|B0(l)i I Regularization |B0(l)i → |B0(l)i² = e− L0+ ˜

L0−2 |α|

|B0(l)i

  • This regularization is BRST invariant:

² α ³ L0 + ˜ L0 − 2 ´ = {QB , 2²¯ π0}

  • Z

²hB0(l)|B0(l0)i² = π e

π2|l| ²

l0δ(l + l0)(1 + O(e− 1

² )

  • propagation of open string tachyon

I def |n(l)i = |B0(l)i² e− π2

2² |l|

I Remark: Naively

R hB0(l)|B0(l0)i = 0, due to the lack of π0 and ¯

π0. However, for |B0(l)i² these zero modes are supplied by e−

² |α| (L0+˜

L0−2).

15

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I An expansion of the string field |Φi = |ψi + | ¯ ψi = Z ∞ dl £ |n(l)iφ(l) + |n(−l)i¯ φ(l) + (orthogonal complement) ¤

  • canonical quantization:

h |ψir , | ¯ ψis i = |R(r, s)i ⇒ |ψi: α > 0 mode → annihilation ; | ¯ ψi: α < 0 mode → creation of a string ⇒ This yields h φ(l) , ¯ φ(l0) i = 1 πlδ(l − l0)

  • vacuum in the second quantization: |ψi|0i

i = h h0|h ¯ ψ| = 0, h h0|0i i = 1 I similarity of the above part and SFT for c = 0 noncritical strings critical string noncritical string φ(l) ∼ r 2 π 1 gslψ(l) ¯ φ(l) ∼ gs √ 2π ψ†(l) BRST invariance ∼ Virasoro constraints This similar structure suggests ⇒ we may be able to construct solitonic

  • perators in the OSp inv SFT in an anal-
  • gous way to those in SFT for c = 0.

16

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I Solitonic state |D±i i ≡ Z dζV±(ζ)|0i i V±(ζ) = λ exp ∙ ± √ 2π Z ∞ dl e−ζl ¯ φ(l) ¸ exp ∙ ∓ √ 2π Z ∞ dl0 eζl0φ(l0) ¸ e

±

C²2 √ 2π g

³ ζ+ π2

2²

´2

  • |D±i

i is BRST invariant: δB|D±i i = 0

  • δB

∙µZ dζV(ζ) ¶n |0i i ¸ = 0 (n = 1, 2, . . . )

  • perturbatively (g << 1)

In the ζ integration, contribution at ζ ∼ − π2

2² is dominant. ⇒ saddle pt approx.

|D±i i ' λ0 exp ∙ ± √ 2π Z ∞ dl e

π2 2² l ¯

φ(l) ¸ |0i i = λ0 exp " ± Z ∞ dl l r 2 π Z h ¯ ψ|B0(l)i² # |0i i ⇐ what we guessed!!

  • Remark:

( |D+i i for g > 0 |D−i i for g < 0 ⇐ “unstable” 17

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I Vacuum amplitude h hD±|e−iT ˆ

H|D±i

i ' exp h 22 × (cylinder amplitude for a single D-brane) i

  • correct integration measure for the moduli parameter of the cylinder is derived
  • independent of T ⇐ ˆ

H is BRST exact

  • factor 22 ⇒ |D±i

i should be identified with the state in which two D-branes or ghost D-branes are excited 18

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§4. Observables

19

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I on-shell physical states

  • n-shell:

à i ∂ ∂t − L0 + ˜ L0 − 2 α ! | i = 0 , physical: QB| i = 0 Hamiltonian L0+˜

L0−2 α

is QB-exact ⇒ We may set t = 0 in the QB-cohomology I Relationship between QB in OSp invariant theory and Kato-Ogawa’s QKO

B

  • identification

no counter part in (b.c) system C0 = 2αc+ ¯ π0 =

1 2αb+

¯ C0 , π0 , α γn = inαcn ¯ γn = 1

αbn

  • contained in QB

˜ γn = inα˜ cn ˜ ¯ γn = 1

α˜

bn = ⇒ QB = QKO

B

− iπ0 µ ∂ ∂α + 1 α ¶ I QB cohomology: Solve QB| i = 0 ⇒ | i = 1 α|physi + ¯ π0|physi + g1(α)π0|physi + g2(α)π0¯ π0|physi |physi: QKO

B -physical ,

g1(α), g2(α): ∀ function of α 20

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I boundary condition for α direction (C, ¯ C-representation) For the bra or ket states, 0 < |α| < ∞ ⇒ introduce ω s.t. α = eω, then −∞ < ω < ∞ ⇒ integration measure: Z ∞ αdα = Z ∞

−∞

dωe2ω ⇒ The wave functions should be expand with respect to exp (−ω + iωxω) ⇒ • the wave functions are delta-function normalizable

  • QB is hermitian
  • the QB-exact states are null

etc... We should focus on the state | i = 1 α|physi + 1 απ0¯ π0|physi %

  • auxiliary fields

physical external states I Observable associated with |primaryiX = |primaryiX(2π)26δ26(pμ − kμ) O(t, k) = i 2 Z ∞

−∞

dα Z d¯ π0dπ0 C ¯

Ch0| ⊗ Xhprimary|Φ(t, α, π0, ¯

π0, k)i 21

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§5. Green’s functions and S-matrix elements

22

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I Two-point Green’s function for Or(tr) (r = 1, 2): |primary1iX = |primary2iX with

Xhprimary|primaryiX = 1 ;

mass M D D ˜ O1(E1) ˜ O2(E2) E E ≡ Z dt1dt2eiE1t1+iE2t2h h0|TO1(t1)O2(t2)|0i i =

2

Y

r=1

µ i 2 Z dαrd¯ π(r)

0 dπ(r)

¶ iδ(1, 2)2πδ(E1 + E2) α1E1 − p2

1 − M 2 − 2iπ(1) 0 ¯

π(1) + i² . . .

δ(1, 2) ≡ 2δ(α1 + α2)(2π)26δ26(p1 + p2)i(¯ π(1) + ¯ π(2)

0 )(π(1)

+ π(2)

0 )

= 2πδ(E1) 2πδ(E2) p2

1 + M 2

(2π)26δ26(p1 + p2) I def Ofr

r ≡

Z dtrfr(tr)Or(tr) with Z dtrfr(tr) = 1 , then D D Of1

1 Of2 2

E E = 1 p2

1 + M 2 (2π)26δ26(p1 + p2)

⇐ Euclidean propagator for a particle of mass M with correct normalization 23

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I N-point Green’s functions ¿ ¿ N Y

r=1

˜ Or(Er) À À ≡

N

Y

r=1

µZ dtreiErtr ¶ h h0|T

N

Y

r=1

Or(tr)|0i i ≡

N

Y

r=1

à i 2 Z dαrd¯ π(r)

0 dπ(r)

i αrEr − p2

r − M 2 r − 2iπ(r) 0 ¯

π(r) ! ×δOSp µ N X

s=1

pOSp

s

¶ Gtranc.(pOSp

1

, . . . , pOSp

N

) (For p2

r + M 2 r ∼ 0)

∼ −i à N Y

r=1

2πδ(Er) p2

r + M 2 r

! (2π)26δ26 µ N X

r=1

pr ¶ Gtranc.(pOSp

r

) ¯ ¯ ¯ ¯ µ ¯ ¯ ¯ ⇔ set p2

r + M 2 r = Er = αr = π(r)

= ¯ π(r) = 0 ¶ Hence ¿ ¿ N Y

r=1

˜ Ofr

r

À À ∼ −i à N Y

r=1

1 p2

r + M 2 r

! (2π)26δ26 µ N X

r=1

pr ¶ Gtranc.(pOSp

r

) ¯ ¯ ¯ ¯ ⇐ 26 dim. Euclidean Green’s function = ⇒ LC SFT’s S-matrix elements 24

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§6. Disk amplitude

25

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I Green’s function for two closed string tachyons in the presence of the soliton ¿ ¿ ˜ OT

1 (E1) ˜

OT

2 (E2)

À À

D±

≡ h hD±| ˜ OT

1 (E1) ˜

OT

2 (E2)|D±i

i h hD±|D±i i where OT : tachyon ⇔ |primaryiX = |0iX

  • O(g) term of the above ⇒ disk Green’s function
  • S-matrix element for this process

ST T D± = ±2ig r 2 π s (2π)26 (8π2)p+1 Y

μ∈N

(2π)δ(p1,μ + p2,μ) B µ t 2 − 1, 2s − 1 ¶ s = X

μ∈N

p1,μpμ

1 =

X

μ∈N

p2,μpμ

2 ,

t =

26

X

μ=1

(p1 + p2)μ(p1 + p2)μ 26

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  • t-channel pole

ST T D± ∼ ∓2iDμν √ 2π s (2π)26 (8π2)p+1 −i (p1 + p2)2 ig(p1 − p2)μ(p1 − p2)ν

%

  • coupling between (ghost) D-brane and graviton

tachyon-tachyon-graviton Dμν = ( ημν for μ, ν ∈ N −δμν for μ, ν ∈ D In our convention, Gμν(x) ∼ ημν + 4ghμνeip·x. The above result means that ( |D+i i for g > 0 |D−i i for g < 0 ⇐ should be identified with the ordinary D-brane. Otherwise ghost D-brane. 27

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§7. Summary and Discussions

28

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¥ D-brane ⇔ BRST invariant state in OSp invariant SFT looks like bosonization ¥ D-brane or ghost D-brane D-brane state is unstable ⇐ consistent with the instability of D-brane in bosonic string theory ¥ How to include open string degrees of freedom associated with D-brane? ¥ similar construction in superstrings? ¤ single D-brane state? why two D-brane state?? etc..... 29