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

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. Introduction 1

I Motivation: I Motivation: How can we capture the solitonic nature of D-branes in string theory? How can we capture the solitonic nature of D-branes in string theory? • D-branes are non-perturbative object. • D-branes are non-perturbative object. ⇒ Non-perturbative formulation of string should be necessary. ⇒ Non-perturbative formulation of string should be necessary. In this work, we focus on a traditional approach: In this work, we focus on a traditional approach: String Field Theory (SFT) String Field Theory (SFT) • In the open string side, several analyses have been carried out. • In the open string side, several analyses have been carried out. (Sen’s conjecture, VSFT, analytic solutions, etc...) (Sen’s conjecture, VSFT, analytic solutions, etc...) • The question we would like to address in this talk: • The question we would like to address in this talk: “What are D-branes in closed SFT ?” “What are D-branes in closed SFT ?” 2 2

I Boundary state | B i • should play very important roles to describe D-branes in closed string sector • A single | B i is not enough. | B i ⇒ 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 [ | B i ] ??? ⇒ 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

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) µ ¶ Z ∞ 3 δ 00 ( l ) − 3 ψ † ( l ) H = dl 4 μδ ( l ) 0 Z ∞ £ ¤ ( l 1 + l 2 ) ψ † ( l 1 + l 2 ) ψ ( l 1 ) ψ ( l 2 ) + g 2 s l 1 l 2 ψ † ( l 1 ) ψ † ( l 2 ) ψ ( l 1 + l 2 ) + dl 1 dl 2 0 - Z ∞ Z ∞ dl ψ † ( l ) D ( l ) = dl ψ ( l ) ( lT ( l ) + ρ ( l )) = 0 0 • ψ † ( l ) , ψ ( l ) : creation and annihilation ops of a string with length l obeying [ ψ ( l ) , ψ † ( l 0 )] = δ ( l − l 0 ) , Z l Z ∞ ρ ( l ) = 3 δ 00 ( l ) − 3 dl 0 ψ ( l 0 ) ψ ( l − l 0 ) + g 2 dl 0 l 0 ψ † ( l 0 ) ψ ( l + l 0 ) , • T ( l ) = 4 μδ ( l ) s 0 0 4

• D ( l ) | Ψ i = 0 ⇒ loop equation: h w ( l 1 ) · · · w ( l n ) i = h 0 | ψ ( l 1 ) · · · ψ ( l n ) | Ψ i • 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) µ ¶ µ ¶ Z Z Z ∞ Z ∞ ∓ 2 dl dle − ζ l ψ † ( l ) l e ζ l ψ ( l ) d ζ V ± ( ζ ) = d ζ exp ± g s exp g s 0 0 ½ Z D-instanton d ζ V ± ( ζ ) is a creation op of ⇒ ghost D-brane (Okuda-Takayanagi) Z | Ψ i ( n + 1) -inst. = C d ζ V + ( ζ ) | Ψ i n -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

I Clues to D-branes in closed SFT (ii): boundary state in SFT for critical strings • Hashimoto-Hata introduced a BRST invariant source term h B | Φ i into the HIKKO type closed SFT. • Idempotency equation (Kishimoto-Matsuo-Watanabe) | B i ∗ | B i ∝ | B i 6

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, ¤ de fi ne 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

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

§ 2. OSp Invariant SFT 9

I Procedure for covariantizing the LC gauge SFT : OSp extension (Siegel) LC gauge SFT OSp invariant SFT ( t, α , X M ) ( t = X + , α = 2 p + , X i ) − → OSp extension X M = ( X i , X 25 , X 26 ; C, ¯ ( i = 1 , . . . , 24) C ) linear O(24) OSp( 26 | 2 ) symmetry S-matrix O(25,1) OSp( 27 , 1 | 2 ) symmetry • C ( τ , σ ) , ¯ C ( τ , σ ) : spinless Grassmann odd variables (ghost) ¡ γ n e − n ( τ − i σ ) ¢ ( C ( τ , σ ) = C 0 + 2 i π 0 τ − i P γ n e − n ( τ + i σ ) + ˜ 1 n 6 =0 n ¡ γ n e − n ( τ − i σ ) ¢ π 0 τ + i P γ n e − n ( τ + i σ ) + ˜ C ( τ , σ ) = ¯ ¯ 1 C 0 − 2 i ¯ ¯ ¯ n 6 =0 n ¯ C C ⎛ ⎞ ⎜ ⎟ δ μν • metric for ⎜ ⎟ δ ij − → η MN = ⎜ ⎟ “transverse directions”: ⎜ ⎟ ⎝ ⎠ 0 − i C ¯ i 0 C 10

Thus we may say: OSp inv. SFT is LC SFT with X M = ( X i , X 25 , X 26 ; C, ¯ C ) as transverse directions . I String fi eld action " Ã ! Z Z ∂ t − L (2) L (2) + ˜ 1 i ∂ − 2 0 0 S = dt d 1 d 2 h R (1 , 2) | Φ i 1 | Φ i 2 2 α 2 ¸ Z ­ ¯ + 2 g ¯ Φ i 1 | Φ i 2 | Φ i 3 V 0 d 1 d 2 d 3 3 (1 , 2 , 3) 3 • zero modes: wave function representation of momenta d 26 p r dr = α r d α r π ( r ) 0 d π ( r ) ⇒ momentum zero-mode integration: (2 π ) 26 id ¯ 0 2 11

I Relationship between S-matrix in LC SFT and S-matrix in OSp invariant SFT • S-matrix in LC SFT ¡ p LC , ² LC ¢ ¡ ¢ = δ LC ( p LC ) F p LC · p LC , p LC · ² LC , ² LC · ² LC S LC r s r s r s = ( k r, 0 , k r, 1 , p r,i ) ( k + = 2 α ) p LC ² LC r : polarizations r • S-matrix in OSp inv SFT ¡ p OSp , ² OSp ¢ ¡ ¢ = δ OSp ( p OSp ) F p OSp · p OSp , p OSp · ² OSp , ² OSp · ² OSp S OSp r s r s r s ; p r, 25 , p r, 26 ; π ( r ) π ( r ) p OSp = ( p LC 0 ) , ² OSp = ( ² LC 0 , ¯ ; ² r, 25 , ² r, 26 ; ² r,C , ² r, ¯ C ) r r r r 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 ² OSp = ² OSp = ² OSp = 0 , we have (after k 0 → k 2 = − ik 0 , p 26 → p 0 = ip 26 ), ¯ ∓ C C ¯ ∙ dk r, 1 dk 2 ¸ µ ¶ Z Y ¯ π ( r ) 0 d π ( r ) ¯ S OSp ( p OSp , ² OSp ) = S OSp ( p OSp , ² OSp ) id ¯ = S LC ( p μ , ² μ ) . ¯ 0 2 π 0 r 12

I BRST symmetry = J − C of the OSp( 27 , 1 | 2 ), (Siegel-Zwiebach, Bengtsson-Linden) δ B Φ = Q B Φ + g Φ ∗ Φ • BRST charge à ! X ∞ γ − n ˜ L n − ˜ Q B = C 0 ∂α + i ∂ γ − n L n − L − n γ n + ˜ L − n ˜ γ n 2 α ( L 0 + ˜ L 0 − 2) − i π 0 α n n n =1 X X L n ≡ 1 L n ≡ 1 L n , ˜ ◦ α M n + m α N ˜ α M α N ◦ ◦ ◦ ◦ L n : Virasoro generators − m η MN ◦ , ◦ ˜ n + m ˜ − m η MN ◦ 2 2 m m X ∞ X ∞ In particular, L 0 = α 0 4 p μ p μ + α μ ˜ − m α m μ + i π 0 ¯ π 0 + i ( γ − m ¯ γ m − ¯ γ − m γ m ) , L 0 = . . . m =1 m =1 13

§ 3. Boundary state and Solitonic operators 14

I Boundary state in OSp invariant theory • matter sector: P μ (0 , σ ) | B 0 i = X i (0 , σ ) | B 0 i = 0 (as usual) • ghost sector: C (0 , σ ) | B 0 i = ¯ C (0 , σ ) | B 0 i = 0 I def | B 0 ( l ) i = | B 0 i δ ( α − l ) ⇒ α | B 0 ( l ) i = l | B 0 ( l ) i I Regularization | B 0 ( l ) i → | B 0 ( l ) i ² = e − L 0+ ˜ L 0 − 2 | B 0 ( l ) i | α | • This regularization is BRST invariant: ³ ´ ² L 0 + ˜ L 0 − 2 = { Q B , 2 ² ¯ π 0 } α Z π 2 | l | ² h B 0 ( l ) | B 0 ( l 0 ) i ² = π e l 0 δ ( l + l 0 )(1 + O ( e − 1 ² ) • ² - propagation of open string tachyon | n ( l ) i = | B 0 ( l ) i ² e − π 2 I def 2 ² | l | R I Remark: Naively h B 0 ( l ) | B 0 ( l 0 ) i = 0 , due to the lack of π 0 and ¯ π 0 . | α | ( L 0 +˜ ² However, for | B 0 ( l ) i ² these zero modes are supplied by e − L 0 − 2) . 15

I An expansion of the string fi eld Z ∞ £ ¤ | Φ i = | ψ i + | ¯ | n ( l ) i φ ( l ) + | n ( − l ) i ¯ φ ( l ) + ( orthogonal complement ) ψ i = dl 0 h i | ψ i r , | ¯ • canonical quantization: ψ i s = | R ( r, s ) i | ¯ ⇒ | ψ i : α > 0 mode → annihilation ; ψ i : α < 0 mode → creation of a string h i = 1 φ ( l ) , ¯ φ ( l 0 ) π l δ ( l − l 0 ) ⇒ This yields h 0 |h ¯ • vacuum in the second quantization: | ψ i| 0 i ψ | = 0 , i = h h h 0 | 0 i i = 1 I similarity of the above part and SFT for c = 0 noncritical strings critical string noncritical string r This similar structure suggests 2 1 φ ( l ) ∼ g s l ψ ( l ) ⇒ we may be able to construct solitonic π operators in the OSp inv SFT in an anal- g s ¯ 2 π ψ † ( l ) φ ( l ) ∼ √ ogous way to those in SFT for c = 0 . BRST invariance ∼ Virasoro constraints 16