Noncommutative Geometry and Vacuum String Field Theory. Y.Matsuo Univ. Tokyo July 2002, YITP WS on QFT

Transcription

1 Noncommutative Geometry and Vacuum String Field Theory Y.Matsuo Univ. Tokyo July 2002, YITP WS on QFT

2 1. Introduction and Motivation

3 A Brief History of VSFT OSFT = Noncommutative Geometry 99 Witten s OSFT Noncommutative Geometry Tachyon Condensation 85 D-brane Charge =K-theory D-brane as NC Soliton 00 Purely Cubic Theory 86 VSFT 01~

4 Motivation: What is D-brane? In effective field theory D-brane = Soliton of closed string Black hole like object In (full) string theory D-brane = Boundary condition for open string Described by (abstract) Boundary state (L n L n ) B = 0 They should be understood as Solution To the second quantized field theory

5 Why Noncommutative Geometry is relevant to understand D-brane? Open string has Chan-Paton index D-branes Φ ij : i,j : Chan-Paton Index Composition of two open strings k Φ ij Φ jk =Φ ik k i j = Multiplication of matrices

6 To pick up one D-brane, we use Projector to one specific Chan-Paton Index N-dim space Projector P P 2 = P rank(p) = k k dim linear subspace Matrix Noncommutative Geometry Projector Noncommutative Soliton

7 D-(p-2) brane out of D-p brane Idea: Use D-p brane world volume instead of Chan-Paton factor Start from D-p brane with non-zero B-field Zero mode of open string becomes noncommutative i 2 ( f g)(x) = e θ ij ( i ' j j ' i ) f (x)g(x') Moyal Product x'= x Moyal plane is the simplest example of NC geometry Projector equation f f = f f = exp 1 2θ ( x x 2 2 ) θ Blob with size is interpreted D-(p-2) brane

8 Open string as a whole as Matrix Witten s star product Ψ 1 Ψ 2 Ψ 1 Ψ 2 Ψ 1 Ψ 2 ( Ψ 1 Ψ 2 )(X) = π 2 DYDZ Π δ(x(σ) Y(σ))δ(Y(π σ) Z(σ))δ(Z(π σ) X(π σ)) σ = 0 Ψ 1(Y)Ψ 2 (Z) Witten s argument Path Integral for the overlap looks like matrix multiplication 1. *-product::noncommutative and Associative 2. Q: BRST operator : Q 2 =0 Triplet (,Q, ) 3. Integration defines Noncommutative Geometry

9 D-brane is NC soliton for Witten s star product Matrix equation Ψ Ψ=Ψ One to one correspondence between solutions? Matrix=gl( )? Virasoro Conformal Invariance (L L ) B = Open 0 string n n? Closed String

10 Progress until 2002/7 V(T) A: Witten s OSFT = D25 brane background 1 S = ( Ψ QΨ Ψ3 ) T 0 T B: Tachyon Vacuum solution by level truncation Ψ 0 S[Ψ 0 ] S[0] τ 25 = Sen, RSZ, Berkovits, Taylor B should be universal for any D-brane We want re-expand the theory from point B No analytic solution known for Ψ 0

11 Ansatz of the theory at B = VSFT RSZ Q Q VSFT : Pure ghost BRST operator NO COHOMOLOGY Splitting of variable in wave function Ψ=Ψ matter Ψ ghost Q VSFT Ψ ghost +Ψ ghost Ψ ghost = 0 Ψ matter Ψ matter =Ψ matter Exactly solvable!

12 Candidate of D-brane = Sliver state Kostelecky-Potting solution Ξ e 1 2 a+ CTa + 0 T = 1 ( 2M 0 1+ M 0 ± (1+ M 0 )(1 3M 0 )) Wedge state and Sliver state n = 0 n n m = n + m Ξ = lim( n ) n Use of square root And infinite product Is the origin of Trouble

13 2. Recent developments of VSFT

14 Topics Explicit correspondence with NC Geometry Half string Formulation Mapping Witten s star product to Moyal product Appearance of Closed string Construction of Physical State Can variation around sliver reproduces open string spectrum? Hata-Kawano state, Okawa state,

15 2.1 Explicit correspondence with NC Geometry Witten s argument uses the path integral formally. For explicit correspondence, we need to use mode expansion. 1. Split string formulation Bordes et. al., RSZ, Gross-Taylor l(σ) X(σ) r(σ) Ψ( X) Ψ( l,r) ( Ψ 1 Ψ 2 )(l,r) = dtψ 1 (l,t) Ψ 2 (t,r) l(σ) = X(σ) 0 σ π 2 π r(σ) = X(π σ) 2 σ π Except for the path integral, * product looks like matrix multiplication

16 Subtlety in split string Boundary condition at the midpoint? Neumann at M Labeled by Even integers Neumann or Neumann Dirichlet Original Variable X(σ) = x 0 + 2Σ n 0 x n cos(nσ) l(σ) = l 0 + r(σ) = r 0 + Dirichlet at M l(σ) = r(σ) = 2Σ e l e cos(eσ) (e even, positive) 2Σ e r e cos(eσ) 2Σ o l o cos(oσ) (o even, positive) 2Σ o r o cos(oσ) Labeled by Even and Odd integers Labeled by Odd integers

17 Translation between even and odd mode T eo = π 4 π 2 dσ cos(eσ)cos(oσ) = 2( 1)(e +o 1)/ 2 1 π o + e + 1 o e 0 R oe = () T 1 oe ( ) e /2 T 0e X in Gross-Jevicki, Gross-Taylor H odd T H even TR = RT =1 Zero mode part R v o = 1 2 T 0,o H odd, w e = 2( 1) e /2+1 H even with Tv = 0, v = Tw, TT =1, TT =1 vv

18 These relation breaks associativity (RT)v = v but R(Tv) = 0 (TT)w = w but T(Tw) = Tv = 0 It is not very clear that this anomaly produces the associativity anomaly of * product itself. As we see later, any string amplitude can be written in terms of only one matrix written in terms of T and vector by w. In the following discussion, we will use the finite dimensional regularization and use ordinary multiplication rule of matrix everywhere.

19 Associativity anomaly in purely cubic theory Purely cubic theory (Yoneya, Friedan, Witten) S cubic = 1 3 Ψ 3 e.o.m Ψ 2 = 0 Solution (Horowitz,Lykken, Rohm, Strominger) Ψ 0 = Q L I I : Identity operator Q L : half BRST operator Q L = π /2 0 j BRS σ ()dσ Expansion around Ψ 0 Reproduces Witten s action S cubic [Ψ 0 +Ψ 1 ] = S Witten [Ψ 1 ] It reproduce correct Open string spectrum!

20 Closed string sector in (old) VSFT How to write space-time reparametrization by open string degree of freedom? Space-Time translation (Horowitz, Strominger) Λ=P L I, [Λ,Ψ[X]] = Ψ[X + ε] ε It breaks associativity explicitly. (P 1L + P 2L ) V 4 = 0, (x1 x 3) V 4 = 0 but [ P 1L + P 2L,x1 x 3]= i 2 Closed string sector breaks associativity? In terms of split string variables, Anomaly of T, R, v, w P L =Σ o v o lo,p R =Σ o v o ro Anomaly from closed string

21 Moyal Formulation (Bars, Bars-Matsuo) Split string Moyal ( Ψ 1 Ψ 2 )(l,r) = Ψ 1 (l,t)ψ 2 (t,r) dt Fourier Transformation A(x, p) = Ψ x+y, x y 2 2 ( ) ( )e ipy dy F Ψ (x, p) F( Ψ 1 ) F( Ψ 2 )= F( Ψ 1 Ψ 2 ) A 1 A 2 ( )(x, p) = e i 2 ( x p ' p x ') A 1 (x, p)a 2 (x', p') x= x' p= p'

22 Extension to OSFT Ax ( even,x odd )= Π o dx o e 2iΣ e,o p e T eo x o Ψ( x 0,x e,x ) o 1. Matrix T is needed to translate p odd to p even 2. On LHS, we do not need split string wave function but original wave function 3. Witten s star product is now realized infinite direct product of Moyal planes with same for all the planes

23 Note Associativity breaking mode (Moore-Taylor, Bars-Matsuo) Kink at the midpoint = zero mode of K 1 (RSZ) = generator of space-time translation Closed string vertex (Hashimoto-Izhaki, GRSZ) δs = V( π 2 )Ψ V : closed string vertex Gauge invariant form

24 Another formulation of MSFT [ x(κ), y(κ') ] = iθ()δ κ ( κ κ' ) Liu, Douglas, Moore, Zwiebach θκ ()= 2tanh( πκ 4 ), κ 0, Continuous parameter v x(κ) = 2Σ e= 2 v e (κ) ex e, y(κ) = 2Σ o (κ) o>0 o In terms of discrete variable x, p, [ ] = iθ e,o, n,m 1 x e, p o p o Θ e,o = 2T e,o Comparison with Bars : Fourier transformation without T

25 Explicit computation in MSFT Bars, Matsuo Any SFT computation is drastically simplified in MSFT Operator Formalism MSFT Identity: Projector: Nontrivial Building block e Σ na n + ( 1) n a n + ψ = e a + CTa + 0 MT 2 (1+ M)T + M = 0 [rr] M = CV 3 Neumann Coefficients 0 1 A = e ξ Mξ, ξ = x e p e m 2 =1, (m = Mσ) σ = 0 i i 0 Perturbative vacuum

26 Wedge state and sliver in MSFT Wedge state 0 A 0 = N 0 exp ξm 0 ξ n ( A 0 ) Sliver state ( ), M 0 = κ e 0, Z = Tκ 1 0 Z o T = N n exp( ξm n ξ), M n σ = (1 + m 0 )n (1 m 0 ) n (1 + m 0 ) n + (1 m 0 ), m = M n 0 0σ m s = M s σ = lim n M n σ = m 0 ( m 0 v κ ) = tanh π κ ( ( 4 )v κ ) < κ <, at κ = 0 indefinite m, m 2 2 s =1 0 ( m s v κ) ( = εκ ()v κ ) Singularity at =0!

27 Relation between OSFT and MSFT Every Neumann coeffs are expressed in terms of M 0 and w ( ) V n Ψ 1 L Ψ n = Tr A 1 L A n A i = F( Ψ ) i For example, 3-string vertices are expressed as M 0 = m m , M + = 2 m 0 +1 m , M = 2 1 m 0 m V 0 = 4m 0 2 ( ) W, V + = 3 m V 00 = W 4m 2 0 m W Which satisfies all Gross-Jevicki s nonlinear identities.

28 Spectroscopy of Neumann coefficients RSZ M 0, M +/- are simultaneously diagonalized K 1 = L 1 + L 1, K 1 v (κ ) = κv (κ ), κ 0 n=1 z n n (κ v ) n = 1 ( κ 1 exp [ κ tan 1 z] ) (κ M 0 v ) 1 n = 1+ 2cosh πκ /2 ( ) v n (κ ), M ± v n (κ ) = In Moyal language, this is automatic ±πκ /2 1+ e 1+ 2cosh( πκ /2) v (κ ) n Every Neumann coefficients are written by single matrix m 0 m 0 = tanh( π K ) 4 1

29 2.3 Physical States Expansion around Sliver state should reproduce open string living on corresponding D-brane (up to gauge transformation) Variation around Ψ 0 (Ψ 0 2 =Ψ 0 ) Ψ'=Ψ 0 + Ψ 1, Ψ' 2 =Ψ' Ψ 1 =Ψ 0 Ψ 1 +Ψ 1 Ψ 0 Very simple!

30 The Issue For finite dim noncommutative geometry (=finite matrix), any such variation becomes pure gauge Naively, there is no matter Virasoro in E.O.M. How can it reproduce every physical state correctly?

31 Possible Solutions Midpoint subtlety Infinite dimensionality Infinite conformal transformation associated with sliver state Hata-Kawano tachyon state Okawa state

32 Hata-Kawano state Hata-Kawano Ansatz T = e Σ nt n a n + a 0 e ipx 0 Ξ By tuning t n, tachyon state satisfies e.o.m ( ) If we expand, x 0 = Roughly x +Σ speaking. e>0 w ex e, x We = Xneed π delicate deviation 2 Parameters {t} are from tuned that in to such reproduce way to correct cancel mass-shell x e dependence condition. of ipx 0 T = e ip x Ξ With this form, e.o.m follows directly.

33 Pathology from infinite product φ e.o.m = 0 for φ in Fock space but φ e.o.m = 0 for φ in sliver state = We have to be very Careful to define The definition Of Hilbert space Where e.o.m. is imposed

34 Okawa s state BCFT consideration (Abstract argument) D-brane Boundary state (L n L n) B = 0 Physical open string states on D-brane δ B = dσ j σ () B (L n L n)δ B = 0 Solution in closed string sector

35 Mapping from boundary state to sliver RSZ, YM Closed string B B + δ B Open string Hilbert space 0 BB H BB Sliver Ξ BB = ( 0 ) BB Okawa s state

36 Some features of Okawa state It correctly reproduces mass-shell condition for any vertex operator Conformal invariance requires the vertex operator to have dimension one The brane tension computed from three tachyon coupling gives correct value.

37 Remaining questions Both HK and Okawa states solve e.o.m. It seems that there are too many solutions. We need to reexamine the definition of Hilbert space more carefully. So far only (infinite) conformal transformation associated with sliver gives the right mass-shell conditions. Only conformal dimension gives onshell condition. Does it also describe gauge degree of freedom correctly?

38 Conclusion Noncommutative geometry MSFT gives handy description of OSFT Now we do not need Neumann coefficients! Correct description of physical state on D-brane seems to be given. Many problems remain Associativity anomaly Extra (unphysical) solutions Closed string sector Supersymmetric extension