Batalin-Vilkoviski ( ) 2016 2 22 at SFT16 based on arxiv:1511.04187 BV Analysis of Tachyon Fluctuation around Multi-brane Solutions in Cubic String Field Theory 1 / 43
Introduction 2 / 43
in Cubic open SFT (Schnabl 05) KBc (Okawa 06, Erler+Schnabl 09) KBc (Murata+Schnabl 11, 12, Hata+Kojita 11, 12, ) Boundary Condition Changing Op. (Erler+Maccaferri 14 ) 3 / 43
Cubic SFT D25-brane N -brane N 2 2 brane ( ) Batalin-Vilkovisky 2 brane? 2 2 = 4 4 / 43
: 5 / 43
CSFT : S[Ψ] = EOM: Q B Ψ + Ψ Ψ = 0 ( 1 2 Ψ Q BΨ + 1 ) 3 Ψ3 KBc [B, K] = 0, {B, c} = 1, B 2 = c 2 = 0, Q B B = K, Q B K = 0, Q B c = ckc. N gh (K) = 0, N gh (B) = 1, N gh (c) = 1. i = 1 = 1/2 = 2 = 1 convention 6 / 43
: Pure-gauge type Ψ sol = UQ B U 1 = c K Bc (1 G(K)) G(K) with U = 1 Bc (1 G(K)), U 1 = 1+ 1 Bc (1 G) G Okawa 06, Erler+Schnabl 09 K = 0 ( K = ) 7 / 43
K = 0 K ε -regularization: K K ε K + ε Ψ sol ε = ( UQ B U 1) K K ε = c K ε G(K ε ) Bc (1 G(K ε)) pure-gauge EOM O(ε) : Q B Ψ sol ε + Ψ sol ε Ψ sol ε = ε c K ε G ε c(1 G ε ) ε 0 EOM? (EOM?) N D25-brane? 8 / 43
K = & (Hata+Kojita 12, 13) K n 0 (K 0) G(K) (1/K) n (K ) ( Re(K) > 0 ) n 0 = 0, ±1 n = 0, ±1 Energy density = 1 2π ( n 2 0 n ) N = ( n 0 n + 1) brane EOM: Ψ sol ε (n 0, n ) ( Q B Ψ sol ε N = & Ψ sol ε EOM 0 + (Ψ sol ε ) 2) = 0 9 / 43
K = 0 K = N = 0 : G(K) = K 1 + K, 1 (tachyon ) 1 + K N = 2 : G(K) = 1 + K K, 1 + K N = 3 : G(K) = (1 + K) 2 K = 0 tachyon 2 brane K 10 / 43
K = 0 K = : KBc K K = 1 K, B B = B K 2, c c = ck 2 Bc Masuda-Noumi-Takahashi 12, Erler 12, Hata+Kojita 13 K = 0 K = K K εη = K ε 1 + ηk ε, B B εη = c c εη = c(1 + ηk ε ) 2 Bc B (1 + ηk ε ) 2 11 / 43
BV BV-basis 12 / 43
BRST-charge Ψ = Ψ sol ε + Φ CSFT S[Ψ] = S[Ψ sol ε ] + Φ ( Q B Ψ sol ε ( 1 + 2 Φ QΦ + 1 ) 3 Φ3 with + (Ψ sol ε ) 2) } {{ } S[Φ] : QA Q B A + Ψ sol ε A ( 1) A A Ψ sol ε Q 2 = 0 EOM Q 2 A = [ Q B Ψ sol ε + (Ψ sol ε ) 2, ] A 13 / 43
BV CSFT ( )BV S[Φ] BV eq: ( ) δs[ψ] 2 = 0 δψ ( ) δs[φ] 2 = 0 δφ nilpotency Q 2 = 0 ( EOM) BV S[Φ] BRST BRST (on-shell) nilpotency unphysical sector 14 / 43
BV { u i (p) } Φ (p µ : ): Φ = u i (p) φ i (p) }{{} p i component field ( p ) d 26 p (2π) 26 CSFT ω ij (p) u i (p ) u j (p) = ω ij (p) (2π) 26 δ 26 (p + p ) : ω ij (p) = ω ji ( p) ω ij (p) 0 only when N gh (u i ) + N gh (u j ) = 3 15 / 43
BV component field BV ω ij δs δs (p) δφ i (p) δφ j ( p) = 0 p i,j ω ij (p) ω ij (p) : j ω ij (p)ω jk (p) = δ i k (det ω ij 0 ) 16 / 43
Darboux basis {u i (p)} {v i (p), v i (p)} vi (p) v j (p ) = a i (p) δ ij (2π) 26 δ 26 (p + p ) vi v j = v i v j = 0 { } ϕ i (p), ϕ (p) i field anti-field S[ϕ, ϕ ] BV eq: a i (p) 1 δs δs p δϕ i i ( p) δϕ i (p) = 0. 17 / 43
: Darboux basis Darboux basis anti-field ϕ L : ϕ i = δυ[ϕ] δϕ i ( Υ[ϕ] = gauge-fermion (gauge ) field ϕ gauge-fixed action BRST-transf. : Ŝ[ϕ] = S[ϕ, ϕ ] L δ B ϕ i (p) = a i (p) 1 δs[ϕ, ϕ ] δϕ i ( p) L ) δ B Ŝ[ϕ] = 0 & ( δ B ) 2 ϕ EOM 18 / 43
BV basis : I photon u 1A 6 u i BV-basis (photon sector A µ µ χ): N gh = 0 : u 0 = p = e (π/4)k e ip X e (π/4)k N gh = 1 : u 1A = p µ α µ 1 c 1 p, u 1B = c 0 p N gh = 2 : u 2A = p µ α µ 1 c 0c 1 p, u 2B = c 1 c 1 p N gh = 3 : u 3 = c 1 c 0 c 1 p Q B : ( ) ( ) iq B u 0 = u 1A + k 2 u1a k 2 u 1B, Q B = (u u 1B 1 2A + u 2B ) ( ) ( u2a 1 iq B = k u 2B 1) 2 u 3, Q B u 3 = 0 19 / 43
{u i } Darboux basis ( ) ( ) ω1a,2a ω 1A,2B p = 2 0, ω 0 1 0,3 = 1 ω 1B,2A ω 1B,2B (v i, v i ) = (u 1A, u 2A ), (u 2B, u 1B ), (u 0, u 3 ) {u i } : } Ψ = {u 0 C+u 1A χ+u 1B C +u 2A χ +u 2B C+u 3 C p ( ) ( ) field C =, ant-field C ( χ χ ), C C N gh (field) + N gh (anti-field) = 1 20 / 43
kinetic term S 0 = 1 Ψ Q B Ψ 2 { = 1 ( p 2 χ( p) + C ( p) ) ( p 2 χ(p) + C (p) ) p 2 + ( ip 2 C( p) χ ( p) ) } C(p) Siegel gauge: anti-field = C = χ = C = 0 Ŝ 0 = p Totally Unphysical System { 1 } 2 p2 χ( p) p 2 χ(p) + ip 2 C( p)c(p) δ B χ(p) = C(p), δ B C(p) = ip 2 χ(p), δ B C(p) = 0 21 / 43
BV basis : II BV basis N gh = 1 : u 1 (p) = e π 4 K c e ip X e π 4 K N gh = 2 : u 2 (p) = e π 4 K ckc e ip X e π 4 K Q B u 1 (p) = ( p 2 1 ) u 2 (p), Q B u 2 (p) = 0 ω 1,2 = 1 Ψ = u1 (p)ϕ(p) + u 2 (p)ϕ (p) p( ), S 0 = 1 ϕ( p) ( p 2 ) 1 ϕ(p) 2 p Ŝ 0 [ϕ] = S 0 = S 0, δ B ϕ = ϕ δs 0 =0 δϕ ϕ =0 = 0 22 / 43
brane BV 23 / 43
K = 0 2 brane with G(K) = 1 + K K tachyon with G(K) = K 1 + K? BV BRST-charge Q QA Q B A + Ψ sol ε A ( 1) A A Ψ sol ε???? 24 / 43
Ψ sol ε = (UQ B U 1 ) K Kε U ε U K Kε P ε : ( ) P ε = Uε 1 Ψ sol ε + Q B Uε [ ] = Uε 1 (UQB U 1 ) ε U ε Q B Uε 1 Uε = ε 1 G ε cg ε Bc(1 G ε ) U ε P ε = O(ε) Q = Q B + O(ε): QA Q B A + P ε A ( 1) A A P ε = Q B A + O(ε) 25 / 43
1. P ε 2. photon 6 states unphysical BV-basis 3. brane P ε BV-basis(6 ) 4. BV-basis ω ij ω ij (det ω ij 0) ω ij 26 / 43
BV-basis : û 0 = B e π 4 K c e ip X e π 4 }{{} K } {{ } K ε homotopy op. û 1A, û 1B, û 2A Q B û 0 = û 1A û 1B, Q B û 1A =Q B û 1B = ( 1 p 2) û 2A û 1A = e π 4 K c e ip X e π 4 K û 1B =(1 p 2 ) B K ε e π 4 K ckc e ip X e π 4 K + ε K ε e π 4 K c e ip X e π 4 K û 2A = e π 4 K ckc e ip X e π 4 K 27 / 43
BV-basis {u 0, u 1A, u 1B, u 2A, u 2B, u 3 } N gh =0 : u 0 = Lû 0 R 1, N gh =1 : u 1A =Lû 1A R 1, u 1B =Lû 1B R 1 [ P ε, Lû 0 R 1], N gh =2 : u 2A =Lû 2A R 1, u 2B = { P ε, Lû 1A R 1} N gh =3 : u 3 =[P ε, u 2A ] = [ P ε, Lû 2A R 1] with ( Qu 0 = u 1A u 1B EOM = QB P ε + P 2 ε = O(ε) ) ( ) ( ) u1a 1 [(1 Q = ) ] ( ) p 2 0 u u 1B 1 2A + u 2B + [EOM, u 1 0 ], ( ) ( ) ( ) u2a 1 0 Q = u 2B p 2 u 1 3 + [EOM, u 1 1A ], Qu 3 = i [EOM, u 2A ] 28 / 43
L = L(K ε ) R = R(K ε ) ε 0 EOM ui EOM = 0 ui [EOM, u j ] = 0 u i Q 2 u j = 0 SFT u EOM K = 0 singular u u 1A u 1B u 2A u 2B (photon BV basis...) 29 / 43
ω ij ui Qu j = ( 1) u i (Qui ) u j, ui [EOM, u j ] = 0 ω ij ω ij (p) (i, j = 0, 1A, 1B, 2A, 2B, 3) ω 0,3 (p) ω 1A,2A (p) ω 1A,2B (p) 2 ω 1B,2A = ω 1A,2A +ω 0,3, ω 1B,2B = ω 1A,2B +(p 2 1)ω 0,3 u 1A/B Qu 1A/B = (1 p 2 ) ω 1A,2A + ω 1A,2B u 0 Q (u 2A, u 2B) =(1, p 2 1) ω 0,3 30 / 43
: tachyon 31 / 43
tachyon with G = K 1 + K ( : Ellwood+Schnabl 06) 1. EOM : u 1A/B EOM = 0, u 1A/B [EOM, u 0 ] = 0 L(K ε ) R(K ε ) 2. ω ij (p) (3 ) 3. kinetic term S 0 = (1/2) Φ QΦ 32 / 43
1. EOM u1a/b EOM = 0 L = 1 + O(K ε ), R = 1 + O(K ε ) OK (L, R) u1a/b (p) [EOM, u 0 ( p)] = ( O ε ) min(2p2 1,1) space-like p µ (p 2 > 1 ) OK 2 p µ p 2 > 1 2 2. ω ij (p) ω 0,3 = 1 + O(p 2 1) ( ) ( ) ω1a,2a ω 1A,2B 1 0 [1 = + O(p 2 0 1 p 2 1) ] ω 1B,2A ω 1B,2B 33 / 43
det ω ij = (p 2 1) 2 + O ( (p 2 1) 3) 0 totall unphysical system (photon tachyonic ) 34 / 43
: 2 brane 35 / 43
2 brane with G = 1 + K K ( ) 1. EOM L = 1 + O(K ε ), R = 1 + O(K ε ) O(K ε ) u 1A/B [EOM, u 0 ] = 0 L = O(K ε ), R = O(K ε ) 1 R = 1 {1 + π K ε 4 K ε + π ( π ) } 8 4 + 2 Kε 2 + O(Kε 3 ) EOM 2 (L = O(K ε ) ) 36 / 43
2. ω ij (p) ω 0,3 (p) = O ( ε min(2p2,1) ) ω ij ω 0,3 = 0 ( ) ω1a,2a ω 1A,2B = ω 1B,2A ω 1B,2B ( ) 1 0 [1 + O(p 2 1) ] 1 0 } {{ } ω 1A,2A (p) 6 u i u 0 0, u 2B 0, u 3 0, u 1A u 1B 37 / 43
u 1A u 2A ( ) Φ = u 1A (p) χ(p) + u 2A (p) χ (p) p field anti-field 1 Φ QΦ = 1 ω 1A,2A (p) χ( p)(p 2 1)χ(p) 2 2 p ω 1A,2A = 1 + O(p 2 1) 38 / 43
tachyon : K = 0 Q u i O(ε) 2 brane : K = 0 39 / 43
1. 2 2 2 = 4 3? 2. 6 BV-basis{u i }? u 0 u 1A/B u 2A/B ( ) 3. (L, R) for u 0 = Lû 0 R 1 O(K 0 ε ) O(K 1 ε )? (2 brane EOM L R O(K ε ) ) 40 / 43
4. u i hermiticity P ε hermiticity : P ε = W P ε W 1, W G ε (1 G ε ) Ψ = P ε + u i (p)φ i (p) p i ( φ i (p) = φ i( p) ) u i (p) hermiticity u i (p) = W u i( p) W 1 u i (p) 41 / 43
hermiticity basis U i (p) = 1 [ ui (p) + W 1 u i ( p) ] W 2 EOM 42 / 43
43 / 43