New solutions for any open string background

Transcription

1 New solutions for any open string background Carlo Maccaferri Torino University and INFN to appear, with Ted Erler

2 OSFT CONJECTURE Given OSFT defined on a given D-brane system Classical solutions describing any other D-brane system Background independence by expanding around the solution plus field redefinition

3 EM solution ( Erler, CM 2014) Ψ = Ψ 0 tv Σ Ψ* tv Σ BCFT 0 tv TV tv Equation of motion Q tv Σ = QΣ + Ψ 0 tv Σ ΣΨ* tv = 0 Q tv Σ = QΣ + Ψ* tv Σ ΣΨ 0 tv = 0 ΣΣ = 1

4 In 2014 we realised this structure as Σ = Q tv ( Σ = Q tv ( K Bσ K ) K Bσ K ) using identity-like insertions of weight zero matter bcc operators σ = U* 1 σ(0) 0 σ = U* 1 σ(0) 0

5 Fundamental property to satisfy eom σσ = 1 achieved thanks to time trick (no collision, weight=0) = e ip hx 0 (c=25) = e ip hx 0 (c=25) Only static background (lumps, multiple branes, but no time dependence) Broken Lorentz invariance by pure gauge excitations in time CFT Potential associativity anomalies (real problem for superstring!) σσ = g * = 1 BCFT* g 0 1 (σσ)σ σ(σσ)

6 Therefore we search for a new realisation of the EM basic ingredients First note that we can write Σ = Q tv (Aτ) Σ = Q tv (τa) Q tv A = 1 A 2 = 0 ΣΣ = Q tv ( τa) Q tv (Aτ) = Q tv ( τaq tv (Aτ)) = Q tv ( τaτ) = 1 Provided τaτ = A

7 Simplest (but too singular) realization Ψ tv = 1 K c 1 K A = B σbσ = B More generally Ψ tv = Fc KB 1 F c F A = B 1 F K τ = 1 F K σ K 1 F τ = K 1 F σ 1 F K τaτ = A

8 So we are after σbσ = B Can (σ, σ) be something else than identity-like? The idea of the Flags σ = σ = L R L R BCFT * BCFT * f σ(0) f σ(0) f f σ(0) σ(0)

9 Multiplying in the order σ σ : degenerating surface ϵ 0 L R BCFT * L R BCFT * = BCFT * BCFT * f σ(0) f σ(0) = 1 g * I σ(0) σ(0) 1 (*) (normalization) σσ = 1

10 Multiplying in the order σ σ : new kind of surface state L R IDENTIFICATION L R = L R It looks like the identity string field towards the midpoint but it has a non degenerate boundary and it is left/right factorized towards the endpoints! σ σ = projector

11 Therefore we have a new kind of Partial Isometry σσ = 1 σ σ = P Now associativity is OK (no multiple collisions) σ σ σ = σ σ σ σ = σ

12 But to have a solution we need σ B σ = B σ B σ = B σσ [B, σ] σ = B σ b σ B = dz 2π b(z) σ b [B, σ] = BCFT * B + f σ(0) BCFT * f σ(0) B But (see later) we have proven that = BCFT * B + f σ(0) TO ANALYZE CAREFULLY! lim ϵ 0 BCFT * σ b Ω ϵ σ = 0 σ B σ = B f σ(0) B

13 So we have a solution of the EM form (correct energy and boundary state) Ψ = Ψ 0 tv Σ Ψ* tv Σ Q Ψ,0 Σ = Q tv Σ = 0 Q 0,Ψ Σ = Q tv Σ = 0 Q Ψ (ΣΣ) = Q tv (ΣΣ) = 0 Fluctuations and background independence S (0) (Ψ + Σϕ Σ) = 1 2π 2 (g 0 g * ) + S (*) (ϕ) Field redefinition (see also Kishimoto, Masuda, Takahashi) ψ = ΣϕΣ, (... )

14 MULTIBRANES Easily realized by orthogonal flags Σ i Σ j = δ ij Obtained by diagonalizing the matrix of inner products δ ij = I σ i (0)σ j (0) Universal multibranes: take (σ i, σ j ) in the matter Verma module of the identity and diagonalize the Gram matrix.

15 EXPLICIT REALIZATION OF THE FLAGS Take the flags to be horizontal semi-infinite strips (needed for having simple Schwartz-Christoffel map) β/2 α β/2 z u Tr[Ω β σ Ω α σ] = B σ(a) σ( A) B 5 f σ(0) f σ(0) -5 5

16 Basic properties of the strip flags lim ϵ 0 σ Ω ϵ σ = 1 lim ϵ 0 σ BΩ ϵ σ = B This is all very good, but one has to be careful with the c ghost lim ϵ 0 cω ϵ σ = 0 lim ϵ 0 σω ϵ c = 0 These anomalies can be consistently avoided by uplifting the c ghost in the bulk c iϵ σσ = σσ c iϵ = c iϵ, c iϵ e iϵk ce iϵk, ϵ > 0

17 FOCK SPACE COEFFICIENTS The solution is concretely defined if we can expand it in Fock space Ψ = i ψ i cϕ i (0) 0 + (... ) From the very structure of the solution we will have that ψ i = K(h i, h σ ) C iσ σ K is a universal function which depends on the choice of tachyon vacuum and on the details of the Schwarz-Christoffel map (quite complicated) We started analysing the following ghost number zero toy model Ψ toy = K σ K σ K K σ 1 * 1 + K σ * K

18 TOY MODEL MARGINAL DEFORMATIONS Coefficients of the marginal field and the tachyon Ψ (marg) toy (λ) = t(λ) 0 + λ SFT (λ) j (... ) 0.25 λ SFT t(λ) λ λ Same qualitative behaviour as in (Schnabl-CM 15) for the solution (CM) Ψ = K Qσ K σ + Q(... ) This behaviour seems generic

19 TOY MODEL LUMPS Choose the bcc to be the well known ND twist fields (h=1/16) at radius R Ψ (lump) toy = n t n e i n R X (0) 0 + (... ) t(x) = t n>0 t n cos n R x R = R = R = R = R = Same qualitative behaviour as in Siegel gauge and previous EM (POSITIVE ENERGY) Zero momentum coefficient not always negative for positive energy solutions

20 FINAL COMMENTS Our previous construction is now realized in a fully associative way. The old realization can be obtained in a limit where flags degenerate (changing the flags is a gauge transformation) Solutions can be consistently composed using multi-flags. Partial isometries play a structural role (cfr GMS, Schnabl, 2000): solution generating technique from the tachyon vacuum. This is the first concrete analytic solution outside the wedge (or related) algebra (generalizations?) Do we necessarily have to pass through the TV? More general (and mysterious) solutions are available (see Ted s talk at SFT@HRI Feb 2018) This is a better starting point for generalization to the superstring (ULTIMATE GOAL) THANK YOU!

21 Mar, May, Simons-GGI Scientists Paolo Di Vecchia, Hirosi Ooguri, Ashoke Sen March 2019: Review Lectures on worldsheet aspects of String Theory April 2019: Focus Week 6-10 May 2019Int. Conference on String Field Theory and String Perturbation Theory APPLY AT