The boundary state from open string fields. Yuji Okawa University of Tokyo, Komaba. March 9, 2009 at Nagoya

Transcription

1 The boundary state from open string fields Yuji Okawa University of Tokyo, Komaba March 9, 2009 at Nagoya Based on arxiv: in collaboration with Kiermaier and Zwiebach (MIT) 1

2 1. Introduction Quantum field theory S = d 4 x [ 1 4 F µνf µν + i Ψ DΨ + Perturbation theory ] Feynman diagrams 2

3 String theory: perturbation theory for on-shell scattering amplitudes? String field theory: Action String perturbation theory 3

4 String theory: perturbation theory for on-shell scattering amplitudes S String field theory: Action String perturbation theory 3-a

5 Background independence A universal set of degrees of freedom? Open string field theory boundary CFT s classical solutions 4

6 Construction of analytic solutions Tachyon condensation Marginal deformations in the bosonic string Schnabl, hep-th/ ; Schnabl, hep-th/ Kiermaier, Okawa, Rastelli and Zwiebach, hep-th/ ; Fuchs, Kroyter and Potting, arxiv: ; Kiermaier and Okawa, arxiv: Marginal deformations in the superstring Erler, arxiv: ; Okawa, arxiv: and arxiv: ; Fuchs and Kroyter, arxiv: ; Kiermaier and Okawa, arxiv:

7 However, it has been difficult to extract information on the boundary CFT from the solution Ψ. Recent progress Ellwood, arxiv: Gauge-invariant observables W (V, Ψ) V: on-shell closed string vertex operator (open string field theory) The one-point function of V on a disk (boundary CFT) Our motivation: construction of the boundary state from Ψ. (on-shell V arbitrary off-shell closed string states) 6

8 We construct a class of closed string states B (Ψ) for any open string field theory solution Ψ. (1) The state B (Ψ) is BRST invariant, Q B (Ψ) = 0, (2) and changes by a BRST-exact term under a gauge transformation of Ψ: B (Ψ + δ χ Ψ) = B (Ψ) + ( Q exact ). 7

9 The contraction with an on-shell closed string state V provides a gauge-invariant observable: V (c 0 c 0 ) B (Ψ + δ χ Ψ) = V (c 0 c 0 ) B (Ψ). New feature The contraction with off-shell closed string states in the Fock space is well defined and regular. Our claim The state B (Ψ) coincides with the boundary state up to a possible BRST-exact term. 8

10 The construction of the state B (Ψ) depends on a choice of a propagator strip. If we choose the Schnabl propagator strip, the state B (Ψ) becomes explicitly calculable for wedge-based solutions. (Details later.) [Based on the results in Kiermaier and Zwiebach, One-loop Riemann Surfaces in Schnabl Gauge, arxiv: ] We precisely reproduce the boundary state without any BRST-exact term for all the analytic solutions we consider. 9

11 Plan 1. Introduction 2. Open string field theory 3. Construction of the closed string state B (Ψ) 4. The boundary state from analytic solutions 5. Discussion 10

12 2. Open string field theory The open bosonic string Ground state tachyonic scalar T (p) in 26 dimensions First-excited state massless vector field A µ (p) Degrees of freedom of string field theory. { T (p), A µ (p), } S [ T (p), A µ (p), ] 11

13 Free case S = d 26 p (2π) 26 [ 1 (p 2 T ( p) 2 1α ) T (p) A µ( p)(p 2 η µν p µ p n )A ν (p) ] or S = d 26 p (2π) 26 [ 1 (p 2 T ( p) 2 1α ) T (p) A µ( p)p 2 A ν (p) + B( p)p µ A µ (p) 1 2 B( p)b(p) ] 12

14 Equations of motion: (p 2 1α ) T (p) = 0, p 2 A ν (p) p µ B(p) = 0, B(p) p µ A µ (p) = 0. Gauge transformations: δ χ A µ (p) = p µ χ(p), δ χ B(p) = p 2 χ(p). 13

15 SU(2) gauge fields A a µ (x), a = 1, 2, 3 a single 2 2 matrix field A µ (x) A µ (x) = a=1 A a µ (x) σa. { T (p), A µ (p), } string field Ψ = a state in a two-dimensional conformal field theory Ψ = d 26 p (2π) 26 [ 1 α T (p) c 1 0; p + 1 α A µ(p) α µ 1 c 1 0; p + 1 ] B(p) c 0 0; p

16 Equations of motion: Q Ψ = 0. Gauge transformations: δ χ Ψ = Q χ with χ = 1 2α Action : S = 1 2 Ψ, Q Ψ. (Q : BRST operator) d 26 p (2π) 26 χ(p) 0; p. Analogy BPZ inner product A, B trace trab Just like A a µ (x) = tr σa A µ (x), Ψ can be specified by ϕ, Ψ for all ϕ in the Fock space. 15

17 Interacting case U(N) gauge transformation δa µ = µ χ δa µ = µ χ + i ( χa µ A µ χ ) Witten s string field theory Nucl. Phys. B268 (1986) 253 S = 1 [ 1 α 3 gt 2 2 Ψ, Q Ψ + 1 ] 3 Ψ, Ψ Ψ. g T : open string coupling constant. A, B : BPZ inner product A B : Witten s star product (noncommutative A B B A) 16

18 Q Ψ + Ψ Ψ = 0. δ χ Ψ = Q χ + Ψ χ χ Ψ. δ χ S = 0. (Chern-Simons-like) Analogy string field A matrix A ij A, B tr AB = A ij B ji A B (AB) ij = A ik B kj 17

19 CFT description string field = a state in the 2D CFT The state-operator correspondence 18

20 Sliver frame Rastelli and Zwiebach, hep-th/ z = f(ξ) = 2 π arctan ξ. [ f c(ξ) = ( df(ξ) dξ ) ] 1 c(f(ξ)), f c(0) = π 2 c(0) 19

21 BPZ inner product and star product 20

22 Examples T = c 1 0 = ( n π ) 3 sin π(z 1 z 2 ) n sin π(z 1 z 3 ) n sin π(z 2 z 3 ) n. 21

23 22

24 Star product of wedge states W α W β = W α+β (Star symbols will be implicit in the rest of the talk.) 23

25 3. Construction of the closed string state B (Ψ) The boundary state B (c 0 c 0 ) ϕ c (0) disk = B (c 0 c 0 ) ϕ c. ϕ c : closed string state Cut the unit disk along a circle of radius e π2 /s : B (c 0 c 0 ) ϕ c = B e π2 s (L 0+ L 0 ) e π2 s (L 0+ L 0 ) (c 0 c 0 ) ϕ c. 24

26 The boundary state from a half-propagator strip 25

27 Repeat the construction using the propagator for the background associated with the solution. In the particle case, In the open string case, 26

28 The closed string state B (Ψ) : Riemann surface after inserting open string Fock-space states 27

29 We can replace e sl 0 with the propagator strip in any regular linear b-gauge. Kiermaier, Sen and Zwiebach, arxiv: The gauge condition: B Ψ = 0. B: the zero mode of the b ghost in a frame z = f(ξ). dz B = 2πi z b(z) = dξ f(ξ) 2πi f (ξ) b(ξ). The propagator strip e sl with L = {Q, B }: 28

30 The propagator strip e sl in the w frame related to the z frame via z = 1 2 ew. The boundary state B can be constructed as and the state B (Ψ) can also be defined. 29

31 An annulus constructed from a half-propagator strip and an annulus with slits in the ζ frame defined by ( ) ( ) 2πi 2πi ζ = exp ln 2z = exp s s w. The classical solution Ψ is glued to each slit. 30

32 The Schnabl gauge with B = dξ 2πi f(ξ) f (ξ) b(ξ), f(ξ) = 2 π arctan ξ can be obtained as a limit of regular linear b gauges. For example, the limit λ 0 of the λ-regulated Schnabl gauge with f λ (ξ) = 1 2 arctan(e λ ξ) arctan(e λ ) λ > 0. Kiermaier, Sen and Zwiebach, arxiv: Explicit calculations are possible in the limit λ 0. 31

33 The propagator strip e sl with λ = 10 4 and s = 1: in the λ-regulated Schnabl gauge 32

34 The main ingredient for the construction of B (Ψ) is thus a half-propagator strip P(s a, s b ). Line integrals L R (t) of the energy-momentum tensor L R (t) γ( π 2 )+t t [ dw 2πi T (w) + d w 2πi T ( w) γ( π ): the open string mid point (ξ = i) in the w frame. 2 ]. 33

35 Note the t dependence of L R (t). The surface P(s a, s b ) can be expressed as the path-ordered exponential: P(s a, s b ) = Pexp [ sb s a dt L R (t) ]. We also define the corresponding b-ghost line integrals B R (t): γ( π 2 )+t [ dw d w B R (t) b(w) + b( ] w). t 2πi 2πi [ L R (t) is the BRST transformation of B R (t). ] 34

36 The half-propagator strip has four boundaries. Bottom: with the open string boundary conditions (e.g. Neumann boundary conditions) Left: the left half of the open string Right: the right half of the open string Top: the boundary for a segment of the closed string (closed string boundary) 35

37 The star multiplication glues together the left and right boundaries. P(s a, s b ) P(s b, s c ) = P(s a, s c ). Σ(s a, s b ) = P(s a, s 1 ) A 1 P(s 1, s 2 ) A 2 P(s k, s b ). A 1, A 2,..., A k : open string states 36

38 We can construct a closed string state by identifying the left and right boundaries. We denote the resulting closed string state from Σ(s b, s a ) by s b s a Σ(s b, s a ). Examples: s b s a P(s a, s 1 ) A 1 P(s 1, s 2 ) A 2 P(s k, s b ), s P(0, s 1) [ L R (s 1 ), A 1 ] P(s 1, s 2 ) {B R (s 2 ), A 2 } P(s 2, s). 37

39 The state B (Ψ) from B in terms of P(0, s) We can write e sl = e s {Q, B}. The BRST operator Q Q associated with the new background given by Q A QA + ΨA ( ) A A Ψ. Therefore, the propagator strip e sl e s{q, B}. The action of {Q, B} decomposes into right and left pieces: {Q, B}A = [ L R A + ( ) A (B R A)Ψ ( ) A B R (AΨ) ] + [ L L A + Ψ(B L A) + B L (ΨA) ] with L = L R + L L and B = B R + B L. 38

40 Replacing L R (t) L R (t) + {B R (t), Ψ} in P(s a, s b ), we define the modified half-propagator strip by P (s a, s b ) Pexp [ The expansion in Ψ: P (s a, s b ) = P(s a, s b ) sb sb s a dt [ L R (t) + {B R (t), Ψ} ] s a ds 1 P(s a, s 1 ) {B R (s 1 ), Ψ} P(s 1, s b ) ]. + sb s a ds 1 sb s 1 ds 2 P(s a, s 1 ) {B R (s 1 ), Ψ} P(s 1, s 2 ) {B R (s 2 ), Ψ} P(s 2, s b ) + 39

41 We now define B (Ψ) e π2 s (L 0+ L 0 ) s P (0, s). The expansion in Ψ: with B (0) (Ψ) = B, B (Ψ) = k=0 B (k) (Ψ) B (1) (Ψ) = e π2 (L s 0+ L 0 ) s s 0 ds 1 P(0, s 1 ) {B R (s 1 ), Ψ} P(s 1, s),. 40

42 The BRST transformation: Q P (s a, s b ) = sb s a dt P (s a, t) (Q {B R (t), Ψ}) P (t, s b ). Q: integral of the BRST current along the right half of the open string closed string boundary the left half of the open string. Since Q {B R (t), Ψ} = [ L R (t) + {B R (t), Ψ}, Ψ ] when Q Ψ + Ψ 2 = 0, we have Q P (s a, s b ) = sb s a dt P (s a, t) [ L R (t) + {B R (t), Ψ}, Ψ ] P (t, s b ) = sb s a dt t [ P (s a, t) Ψ P (t, s b ) ] = [ Ψ, P (s a, s b ) ]. 41

43 It thus follows that Q B (Ψ) = 0. The gauge transformation: δ χ Ψ = Q χ + [ Ψ, χ ]. δ χ P (s a, s b ) = Q sb sb s a dt P (s a, t) [ B R (t), χ ] P (t, s b ) + { Ψ, dt P (s a, t) [ B R (t), χ ] P (t, s b ) } [ χ, P (s a, s b ) ]. s a We thus conclude that δ χ B (Ψ) = Q exact. 42

44 The state B (Ψ) depends on B and s. Variation of B: B R (t) B R (t) + δb R (t). δ B (Ψ) = Q exact. Variation of s: s B (Ψ) = Q exact. The s 0 limit For regular solutions, lim V (c 0 c 0 ) B (Ψ) V (c 0 c 0 ) B = 4πi W (V, Ψ). s 0 V: an on-shell closed string state W (V, Ψ): gauge-invariant observables Hashimoto and Itzhaki, hep-th/ ; Gaiotto, Rastelli, Sen and Zwiebach, hep-th/ (No precise regularity conditions identified.) 43

45 4. The boundary state from analytic solutions The propagator strip for the λ-regulated Schnabl gauge in the z frame The left and right boundaries of the half-propagator strip becomes vertical in the limit λ 0. 44

46 Consider a calculation of s s ds 1 s 0 s 1 ds 2 P(0, s 1 ) {B R (s 1 ), A α1 } P(s 1, s 2 ) {B R (s 2 ), A α2 } P(s 2, s). A α1, A α2 : wedge-based open string states. The Riemann surface before inserting the states A α1 and A α2 : 45

47 The Riemann surface after inserting the states A α1 and A α2 : The conformal transformation from the frame for the solution to the z frame consists of only scaling and translation! 46

48 What happened to the closed string boundary? It is hidden at z = i but cannot be neglected. A remarkable result from arxiv: by Kiermaier and Zwiebach: all Riemann surfaces associated with closed string states of the form P(0, s 1) A 1 P(s 1, s 2 ) A 2 P(s k, s) s coincide in the Schnabl limit λ 0 when A i are wedge-based states. We can explicitly map the surface to an annulus whose modulus only depends on s. 47

49 Results of explicit calculations (1) Schnabl s solution Ψ S for tachyon condensation The result: Ψ S = lim N B (Ψ S ) = for any finite s. [ [ N ] ψ n ψ N n=0 1 + k=1, ψ n d dn ψ n. ] s k e s k! ( 1)k B = 0 e s 1 Consistent with Sen s conjecture that the D-brane disappears at the tachyon vacuum. This can be thought of as an off-shell and finite s generalization of the calculations in Ellwood, arxiv: and Kawano, Kishimoto and Takahashi, arxiv:

50 (2) Regular marginal deformations in Schnabl gauge Schnabl, hep-th/ ; Kiermaier, Okawa, Rastelli and Zwiebach, hep-th/ Ψ = λ n Ψ (n) λ : deformation parameter 0 n=1 Contributions of O(λ 2 ) from B (2) (Ψ (1) ) and B (1) (Ψ (2) ). The insertions of the marginal operator V in the z frame From B (2) (Ψ (1) ) : e s s s ( ds e s 1 ds 2 e s 1 e s 1 V e s 1 + ) ( es 2 e s 2 e s e s 1 V e s 1 + ) es e s 2. e s 1 s 1 From B (1) (Ψ (2) ) : e s e s 1 s 0 ds dt 1 e s 1 V ( e s e s 1 + t 1 e s 1 e s 1 ) e s 1 V ( e s e s 1 + es t 1 e s 1 e s 1 49 ).

51 The two contributions are combined to give Γ (2) du 1du 2 V (u 1 ) V (u 2 ) with Γ (2) given by 1 e s u 1 u 2 e s, 0 u 2 u 1 1. We can show that e s e s du 1du 2 V (u 1 ) V (u 2 ) = du Γ (2) 1 du 2 V (u 1 ) V (u 2 ). 1 u 1 The calculation can be generalized to all orders in λ and we obtain B (Ψ) = exp [ λ 2π 0 dθ V (θ) ] B. 50

52 5. Discussion We constructed a class of BRST-invariant closed string states B (Ψ) for any open string field solution Ψ. We calculated B (Ψ) for various known analytic solutions choosing the Schnabl propagator strip and found that B (Ψ) precisely coincides with the boundary state. The first construction of the full boundary state from solutions of open string field theory. Our results in particular imply that the wildly oscillatory rolling tachyon solution of open string field theory actually describes the regular closed string physics studied by Sen using the boundary state. Our interpretation: the wild oscillatory behavior is due to the description of the regular physics in the closed string channel in terms of the open string degrees of freedom. 51

53 In all the examples we considered, the state B (Ψ) factorized as B (Ψ) = B (matter) (Ψ) B (bc), with B (bc) being the boundary state of the bc CFT. The state B (Ψ) factorized this way can be a consistent boundary state without any BRST-exact term. We presented a sufficient condition on the solution. The state B (Ψ) can be regarded as a generalized energymomentum tensor in string theory. Its form in terms of the path-ordered exponential is reminiscent of the energy-momentum tensor of noncommutative gauge theory in terms of open Wilson lines. Okawa and Ooguri, hep-th/ Relation to open-closed string field theory. We hope that exciting developments on the closed string physics in open string field theory await us in the near future! 52