Introduction SFT Tachyon condensation in SFT SFT ( ) at 1 / 38

( ) 2011 5 14 at 1 / 38

Introduction? = String Field Theory = SFT 2 / 38

String Field : ϕ(x, t) x ϕ x / ( ) X ( σ) (string field): Φ[X(σ), t] X(σ) Φ (Φ X(σ) ) X(σ) & / 3 / 38

SFT with Lorentz & Gauge Invariance SFT Lorentz ( Yang-Mills ) Φ[X(σ), t] = Φ[X µ (σ), b(σ), c(σ) ] t X 0 (σ) } {{ } ghost t X 0 (σ) 4 / 38

Φ S(Φ): SFT S(Φ) = 1 2 ΦQ BΦ + 1 3 Φ3 +... ( ) : δ Λ Φ = Q B Λ + Φ Λ Λ Φ +... δ Λ S(Φ) = 0 Λ = Λ[X µ (σ), b(σ), c(σ)] 5 / 38

( / ) : 6 / 38

SFT Covariant & Gauge-invariant SFT ( 85 ) Yang-Mills ( ) (Effective potential, Instanton, Large-N,...)! 7 / 38

SFT ( ) Tachyon condensation (1999 )/ (2005) SFT 8 / 38

SFT Light-cone SFT by Kaku-Kikkawa ( 74) BRST 1st quantization (Kato-Ogawa 83) Batalin-Vilkovisky formalism ( 83) Covariant & Gauge Invariant SFT Cubic SFT (Witten) HIKKO (Hata-Itoh-Kugo-Kunitomo-Ogawa) Non-Polynomial closed SFT (S-Z,K-K-S) ( 85 ) Boundary SFT (Witten, 92) Tachyon condensation in SFT ( 99 ) for tachyon vacuum (Schnabl, 2005) 9 / 38

Plan of this talk 1. Introduction 2. SFT 3. Tachyon condensation 4. SFT Super-SFT ( ) Up to sign (or Up to factor) 10 / 38

SFT SFT Batalin-Vilkoviski(BV) systematic ( SFT BV ) 11 / 38

BV SFT : index I : : : String field: : ( X µ (σ), b(σ), c(σ) ) = I DX(σ)Db(σ)Dc(σ) = Φ[X µ (σ), b(σ), c(σ)] = Φ I δ δφ[x(σ), b(σ), c(σ)] = Φ I I 12 / 38

( )BV SFT action S(Φ) BV : ( )BV ( ) 2 S = 0 I Φ I BV : ( = ) Gauge invariance of SFT action S(Φ) Procedure of Gauge-Fixing and BRST invariance 13 / 38

Λ[X µ (σ), b(σ), c(σ)] = Λ I 2 S ( ) : δ Λ Φ I = Λ J Φ I Φ J ( I ): δ Λ S(Φ) = S δ Λ Φ I = S Φ I Φ I ( S = 1 2 Λ J Φ J ) 2 J 2 S Λ J Φ I Φ J = 0 Φ } {{ I } = 0 (BV eq) 14 / 38

Gauge BRST BV gauge S(Φ) 1. Gauge Ŝ(φ) 2. BRST δ B with BRST : δ B Ŝ(φ) = 0 (On-shell) Nilpotency: (δ B ) 2 = 0 up to EOM ( ) 15 / 38

BV S(Φ): S(Φ) = 1 2 Q IJΦ I Φ J + V (3) IJK Φ IΦ J Φ K + V (4) Φ IJKL IΦ J Φ K Φ L +... BV : ( S/ Φ I ) 2 = 0 Q IJ Q JK = 0 Q = Q Kato-Ogawa B (3) Q II V + Q I JJ V (3) + Q JK IJ KK V (3) = 0 K IJK I J,K,L Q II V (4) I JKL + QV (N) + N 1 M=3 V (3) V (3) IJM (I,J,K,L) V (N M+2) V (M) = 0 MKL = 0 16 / 38

1 2 Φ IQ IJ Φ J 1 2 Φ IQ IJ Φ J with Q = Q Kato-Ogwa Q B = π δ dσ 0 δb [ 1 2 ( ( δ δx ) 2 ) ( + (X ) 2 + i c b + δ δb B δ δc )] ( +ic X δ δx + δ ( δ c δc + δb Fock space : Φ(x, b 0 ) = b φ(x) 0 + ψ(x) } {{ } 0 (Siegel-gauge) b 0 : b(σ) x µ : X µ (σ) = string φ(x) (Open SFT): φ(x) = 0 t(x) + α µ 0 1 A µ(x) + α µ 0 2 W µ(x) + α µ 1 αν 1 0 v µν(x) + c 1 b 1 0 u(x) +... ) ) c 17 / 38

Φ Q B Φ = = { d 26 x d 26 x φ(x) L0 φ(x) 2 + (mass) 2 t( 2 1)t + A µ ( 2 )A µ + W µ ( 2 + 1)W µ } + v µν ( 2 + 1)v µν u( 2 + 1)u +... t : Tachyon (m 2 = 1) A µ : Photon (m 2 = 0) W µ, v µν, u,... : Massive (m 2 1) 18 / 38

BV V (n) Cubic Open SFT (Witten, 86) V (3) = HIKKO Open SFT ( 86): Φ[X µ (σ), b(σ), c(σ), α] V (3) =, V (4) = dα string-length HIKKO Closed SFT ( 86) V (3) = 19 / 38

BV V (n) Non-Polynomial Closed SFT (Saadi-Zwiebach, Kugo-Kunitomo-Suehiro, 89) V (3) =, V (N=4,, ) = d 2N 6 l... Boundary SFT (Open SFT, 92) [ ] SFT action is given implicitly as a solution to 2π ds = 1 2 0 dθdθ do(θ) { Q B, O(θ ) } λ 20 / 38

V (3) in Cubic SFT V (3) = V (3) V (3) (3) +V V (3) =0 IJM MKL LIM MJK I V V + M =0 J M L K I J V V L K? I J L K = I J L K M M (I, J, K, L) 21 / 38

BV S(Φ) String Feynman (=world sheet) Propagator + Vertex : Light cone, HIKKO Witten 22 / 38

Tachyon condensation in SFT SFT (?) (?) 23 / 38

Asoke Sen in bosonic (1998) T 25 U D25-brane D25 : T 25 = Φ C Φ D25-brane U(Φ C ) = T 25 1 2π 2 α 13 24 / 38

Off-shell ( ) SFT! Level truncation in CSFT ( 88, 99 ) (mass) 2 = L 1 (L = Level) φ = }{{} 0 t + c 1 b 1 0 u + α µ 1 αµ 0 1 v +... } {{ } level-0 level-2 Lorentz : (t, u, v,...) 25 / 38

Level truncation L U(Φ C )/T 25 0 0.684616 2 0.959377 4 0.987822 6 0.995177 8 0.997930 10 0.999182 12 0.999822 14 1.000174 16 1.000375 18 1.000494 20 1.000563 L = 20 Takahashi- Kishimoto(2009) 26 / 38

for : - (2002): U(Φ C ) Schnabl (2005) EOM Chern-Simons EOM: Q B Φ C + Φ C Φ C = 0 ( 1 S(Φ C ) = 2 Φ CQ B Φ C + 1 ) 3 Φ3 C = 1 6 Φ 3 C Φ C = UQ B U 1 (pure-gauge) U U(Φ C ) = T 25 No open-string modes around Φ = Φ C 27 / 38

SFT SFT : 1. SFT( Closed SFT) 2. SFT 28 / 38

BV BV Closed SFT action (HIKKO, Non-polynomial) Loop! S-matrix Unitarity BV SFT action : BV ( ) 2 S = i 2 S I Φ I I Φ I Φ I 29 / 38

BV S(Φ) BV SFT Ŝ(φ) BRST δ B SFT : Dφ exp ( ) i Ŝ(φ) measure Dφ BRST δ B { i Ŝ(φ) + ln Dφ } = 0 30 / 38

BV S(Φ) = S (0) (Φ) + S } {{ } (1) (Φ) + 2 S (2) (Φ) +... BVeq!!!... ( ) 31 / 38

: SFT= BV S (0) (Φ) S-matrix unitariy? vertex V (N) x µ = 2π dσx µ (σ) : 0 ( ) 2 ( ) 2 V (N) exp = x µ exp + 2 x 0 [ : QB ( / x µ ) 2 + ] m 2 SFT 32 / 38

Cubic SFT : : Cubic SFT S CSFT = 1 2 ΦQ BΦ + 1 3! V (3) IJK }{{} Φ IΦ J Φ K t =X 0 ( π 2 ) t (?) 2 S CSFT Φ I Φ I = 0 33 / 38

SFT? SFT ( closed SFT) ( ) : S(Φ) = 1 2 ΦQ BΦ + V (3) IJK Φ IΦ J Φ K +... Q B η µν graviton : Φ (x) = α µ 1 αν 1 0 h µν (x) +... g µν (x) η µν + h µν (x) 34 / 38

Einstein-Hilbert action S EH = d D x g R SFT! (Pregeometrical SFT) V (N) ( ) V (N 4) = 0 SFT Q = 0 BV eq V (3) S Pregeom. = 1 3! V (3) IJK Ψ IΨ J Ψ K V (3) HIKKO = 35 / 38

Q = Q B SFT Pregeom. SFT Ψ : Ψ = Ψ + Φ S Preg. = 1 2 Φ I V (3) IJK Ψ K } {{ } (Q B ) IJ Φ J + 1 3! V (3) IJK Φ IΦ J Φ K EOM: V (3) IJK Ψ J Ψ K = 0 Ψ 36 / 38

... Pregoemetrical SFT! 37 / 38

... Pregoemetrical SFT! 37 / 38

... Pregoemetrical SFT! /? 37 / 38

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