Time-dependent backgrounds of compactified 2D string theory:

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1 Time-dependent backgrounds of compactified 2D string theory: complex curve and instantons Ivan Kostov SPhT, CEA-Saclay I.K., hep-th/007247, S. Alexandrov V. Kazakov & I.K., hep-th/ S. Alexandrov & I.K, hep-th/ Time-dependent backgrounds of compactified 2D string theory: p./25

2 Plan Non-perturbative effects (NPE) and in minimal string theories, c = 6 p(p+) Branes in minimal string theories: FZZT and ZZ String backgrounds as complex curves: ZZ brans and double points, compact and non-compact A and B cycles Comparison with the NPE in Matrix Models: ambiguity problem Non-perturbative effects in c = string theory CFT - MM dictionary Matrix Quantum Mechanics: Chiral quantization. Complex curve and NPE. Tachyon perturbations as deformations of the complex curve. Leading and subleading non-perturbative corrections. Equation for the double points. Exact bosonization of MQM in chiral quantization. Bosonization rules. Vertex operators and D-instantones. Time-dependent backgrounds of compactified 2D string theory: p.2/25

3 Sketch of the non-perturbative effects in string theory: String partition function: F F pert + F nonpert = F h gs 2h 2 + A n e D n/g s h 0 n 0 closed strings open strings F h = vacuum h-loop closed string amplitudes D n = disk amplitudes with Dirichlet b.c. along D-branes The contribution of a D-brane = _ g s _ 2 _ g s ( _ 2 g s 2 [ Polchinski 94 ] = ) + Exp _ Time-dependent backgrounds of compactified 2D string theory: p.3/25

4 Branes in minimal (c < ) string theories: Minimal string theory = Liouville + (p, q) Minimal matter CFT + Ghosts In Cardy s formalism: brane = local boundary condition (Liouville + Matter) Matter branes: (r; s) boundary conditions r p ; s q [ Cardy 89 ] Liouville branes: Extended branes or FZZT branes [ Fateev, Zamolodchikov 2, Teschner ] Labeled by the boundary cosmological constant µ B R D-branes localized at infinity or (m, n) ZZ branes [ A& Al. Zamolodchikov 00 ] Labeled by m,n N D-instanton = (m, n) Liouville brane (r, s) minimal model brane [ Martinec 03,Klebanov-Martinec-Seiberg 03 ] Time-dependent backgrounds of compactified 2D string theory: p.4/25

5 FZZT, ZZ and algebraic curve Disk partition function on FZZT brane: Φ = Φ(µ B ) x µ B = µcoshτ, ( ) y dφ dµ B = µ q/2p cosh q p τ, Geometrical interpretation of the ZZ-branes [ Seiberg&Shih 03 ]: x and y define an algebraic curve F(x, y) = 0, globally parametrized by the boundary parameter τ ZZ-branes double points of the curve: x m,n = x m, n, y m,n = y m, n. Based on a (still poorly understood) linear relation between FZZT and ZZ states Φ ZZ mn = Φ FZZT (τ m,n ) Φ FZZT (τ m, n ), τ m,n = im + i q p n Time-dependent backgrounds of compactified 2D string theory: p.5/25

6 Moduli of the complex curve Example: The theory (4, 5)... A A 2 A 3 B B 2 B singular points at x = and y = related by a non-compact B-cycles < 8 Moduli associated with the A and B cycles : H ydx = µ cosmological constant RA R ydx = µf = the -point function of the puncture operator B B mn ydx = Γ mn the instanton chemical potential R x ydx = Φ(x) the disc partition function with FZZT b.c. Time-dependent backgrounds of compactified 2D string theory: p.6/25

7 Moduli of the complex curve Example: The theory (4, 5). double points x mn, y mn -> connected by compact B mn cycles B 2 A A 2 A 3 B B 2 B Moduli associated with the A and B cycles : H ydx = µ cosmological constant RA R ydx = µf = the -point function of the puncture operator B B mn ydx = Γ mn the instanton chemical potential R x ydx = Φ(x) the disc partition function with FZZT b.c. Time-dependent backgrounds of compactified 2D string theory: p.6/25

8 Algebraic curve beyond FZZ/ZZ Degeneracy problem at the (p, q) critical point: D mn = Z ZZ (m,n),() = ZZZ (,)(m,n), = ZZZ (m,),(n) = ZZZ (,n),(m,) the degeneracy is not lifted by perturbations [ V. Kazakov & I.K. 04 ] possible solution: all 4 states are physically identical [ Seiberg& Shih ] The degeneracy might by understood by studying finite perturbations off the (p, q) critical points. Then the string theory does not factorize to Matter Liouville. No-world sheet description known for such string theories. The c = string theories are more interesting: solvable models that do not reduce to Matter Liouville: sine-liouville, cigar, paperclip etc., with interesting applications in string theory. It is potentially important to study the non-perturbative effects in the c = matrix models in non-trivial (time-dependent) backgrounds. Time-dependent backgrounds of compactified 2D string theory: p.7/25

9 c = string theory: World sheet CFT Compactified Euclidean c = string theory: χ matter field Euclidean compactified time: χ + 2πR χ φ Liouville field space The action for the linear dilaton background: S c= = 4π Z d 2 σ h ( χ) 2 + ( φ) ˆRφ i + µφ e 2φ + ghosts Tachyon operators: T q d 2 σ e iqχ e (2 q )φ ; q k = k/r, k = ±, ±2,.... Perturbations by tachyon modes: δs = k (t kt k/r + t k T k/r ) E.g. sine-liouville perturbation: δs SL = λ R d 2 σ cos(χ/r) e (2 R )φ, λ = t = t Time-dependent backgrounds of compactified 2D string theory: p.8/25

10 c = string theory: Matrix model (Matrix Quantum Mechanics) Singlet sector of MQM = free fermions in upside-down gaussian potential U(x) = 2 x2. U(x) x Hamiltonian : H = 2 (p2 x 2 ) µ T q e qt Tr x q, (q > 0) + T q e qt Tr x q, (q < 0) x ± = 2 (x ± p) The ground state (fermi sea filled up to E = µ) describes the linear dilaton background. The profile of the fermi sea p 2 x 2 = µ equation of a complex curve Global parametrization: x = 2µcoshτ, p = 2µsinh τ (τ C) SL deformation of MQM [ G.Moore 92 ]; Integrability and Toda flows [ Jevicki, Moore-Plesser ] [ V. Alexandrov, V. Kazakov, I.K., D. Kutasov ] Time-dependent backgrounds of compactified 2D string theory: p.9/25

11 Chiral quantization of MQM Light-cone coordinates in the phase space: x ± = 2 (x ± p) x ± can be identified (in some sense) with the generators of the ground ring, x + x = µ the relation of the ground ring [ Witten 9 ] -particle wavefunctions in x + and x representations: Hψ E ± = Eψ E ±; H = i(x + x+ + /2) = i(x x + /2) ψ+(x E + ) = π e iφ0/2 x ie 2 +, ψ (x E ) = π e iφ0/2 x ie 2 The potential U(x) = x 2 /2 is replaced by branch point at x ± = 0 Scattering matrix Fourier operator S. The scattering phase φ 0 = φ 0 (E) is determined by the condition S = I: Z ψ E(x ) = [SψE](x ) + 2π dx + e ix + x ψ E + (x + ) Time-dependent backgrounds of compactified 2D string theory: p.0/25

12 scattering phase density of states partition function Scattering phase φ 0 (E): S ψ E + = ψe E S E = ρ 0 (E) 2π log Λ e iφ 0(E) = 2π e π 2 (E i/2) Γ(iE + /2). Small imaginary part Im φ 0 ( µ) = 2 log( + e 2πµ ) Density of states: Grand canonical partition function: ρ(e) = log Λ 2π 2π dφ 0 (E) de F (µ, β) = Z de ρ(e)log + e β(µ+e), β = 2πR 2sin µ 2R F(µ) = φ 0( µ) Time-dependent backgrounds of compactified 2D string theory: p./25

13 Non-perturbative corrections to the free energy In the integral representation of φ 0 φ 0 ( µ) = i 2 Z Λ ds s e iµs sinh s 2 the non-perturbative part comes from the poles: φ 0 = φ pert 0 + φ np 0 φ pert 0 = µlog µ + µ + O(/µ 2 ), φ np 0 = i X n 2n ( )n e 2πnµ. The non-perturbative piece of the free energy: F np (µ) {tk =0} = i X n e 2πnµ 4n( ) n sin πn R + i X n e 2πRnµ 4n( ) n sin(πrn). no pre-factors g /2 s µ /2, unlike the minimal string theories! Why? Time-dependent backgrounds of compactified 2D string theory: p.2/25

14 Flat direction > No isolated saddle points Evaluate the integral quasiclassically: E S E = ZZ Λ 0 dx + dx x+ x e ix + x +ie log x + x iφ 0(E) = 2π log Λ No saddle points but saddle contours": x + x = E e 2πin, n = 0,,2,... Dominant saddle (n = 0) Complex curve: x + x = µ x ± (τ) = µ e ±τ, τ C subdominant saddles (n =,2,...) Instantons double contours {x ± (τ ), x ± (τ ) τ R + πin, τ R πin} Time-dependent backgrounds of compactified 2D string theory: p.3/25

15 Time-dependent backgrounds Tachyon perturbations can be introduced by chanhing the asymptotics of the fermionic wave functions: Ψ E ± (x ± ) = x±ie 2 ± exp " i kmax 2 φ(e) + X k= t ±k x k/r ± +...!# x ± The saddle point equations for the integral E S E = 2π conditions for the complex curve at x ± : log Λ gives asymptotic x + x = µ + R = µ + R X k X k kt k x k/r (x + ) kt k x k/r +... (x ) E.g. fore Sine-Liouville perturbation, t = t = λ, 2 µf = 2Rlog u: x ± = ue ±τ + λ R u 2R e ±τ( /R), u 2 + R R 3 λ 2 u 2 R 2 = µ Time-dependent backgrounds of compactified 2D string theory: p.4/25

16 Sub-dominant saddles = double points of the deformed curve For the deformed curve the degeneracy is lifted: Double points: x + (τ ) = x + (τ ) and x (τ ) = x (τ ) τ = iθ n, τ = iθ n, sin (θ n ) = λ R u 2R sin ` R R θ n, Quasiclassical wave functions: Ψ E ± (x ± ) p ±2π τ x ± E S E = 2π log Λ eiφ pert e iφ/2 exp» log Λ + P n= Z x± i p π 2 n e Γ n x dx ± «= e iφ( µ) log Λ φ pert = R x (τ) dx (τ), + n = x+ τ x τ iθ n iθ n Γ n = i R iθ n iθ n x (τ) dx + (τ) dx dx + dx + dx iθ n iθ n «/2 Time-dependent backgrounds of compactified 2D string theory: p.5/25

17 Subleading NPE for the free energy F F pert + i π 2 2log Λ X n>0 n sin µs n 2R e S n. A n n sin µs n 2R = sin 2 θ R» x+ τ x τ iθ n iθ n x+ τ x τ iθ n iθ n «/2. For sine-liouville t = t = λ: A(ξ, y) = q e R χ sin 2 θ sin 2 θ R C `` cot ` R θ cot θ R R πr C = i 4 2log Λ The pre-exponents scale as g /2 s and are non-analytic in the limit λ 0 Time-dependent backgrounds of compactified 2D string theory: p.6/25

18 Bosonization in the perturbed theory Tachyon modes = collective excitations of fermions (left and right moving waves on the Fermi surface). After second quantization these modes form a bosonic field In (x +,x ) representation the fermions can be exactly bosonized. Scnd quantized fermion fields: Ψ ˆ ± (x ±, t) = R de e Et/2 Ψ `e t E x ± ± b(e), Ψ ˆ ± (x±, t) = R de e Et/2 Ψ `e t E ± x ± b (E), {b (E),b(E )} = δ(e E ) Thermal vacuum: µ b (E) b(e ) µ = δ(e E ) +e β(µ+e). Fermion phase space density operator: Û(x +, x,t) = 2π e ix + x Ψ ˆ (x, t) Ψ ˆ + (x +, t) Time-dependent backgrounds of compactified 2D string theory: p.7/25

19 Bosonization rules Phase Bosonic field: Φ± ˆ (x ± ) = V ± (x ± ) + ˆφ E log x 2 ± + D ˆ ± (x ± ), V ± (x ± ) = P n>0 t ±nx n/r ±, ˆφ = R E ˆ D ± (x ± ) = P n n x n/r ± t±n. Fermion wave function Bekher-Akhieser function D E Ψ µ (x ) = (2π x ± ± ± ) /2 µ : e i Φ ˆ ± (x ) ± : µ, normal product sign (: :) all derivatives moved to the right expectation value: Ô = Z Ô Z. Operator formula for the fermion bilinears: Ψˆ (x ) Ψ ˆ + (x + ) = (x x R+ 2πR + ) 2R : e iˆφ(x +,x ) :, where ˆΦ(x+,x ) = ˆ Φ + (x + ) + ˆ Φ (x ). Time-dependent backgrounds of compactified 2D string theory: p.8/25

20 Pre-exponential factor as annulus amplitude D : ˆΦ(x E +, x )ˆΦ(y +, y ) : c = log sinh τ(x ) τ(y + ) sinh τ(y ) τ(x + ) 2R 2R sinh τ(x + ) τ(y + ) sinh τ(x ) τ(y ) 2R 2R + log(x /R + y /R + ) + log(x /R D µ Ψ ˆ (x ) Ψ ˆ E + (x + ) µ = 2πR (x + x ) R+ 2R exp =» iφ(x +, x ) 2 y /R ), D : ˆΦ(x E +, x )ˆΦ(x +, x ) : c = e iφ(x +,x ) 2π p τ x + τ x 2R sinh τ(x ) τ(x + ) 2R. Time-dependent backgrounds of compactified 2D string theory: p.9/25

21 NPC from bosonization The denominator diverges on the surface of the Fermi sea, τ(x + ) = τ(x ), where the quasiclassics breaks down D 2R sin( 2R µ) µ Ψ ˆ (x ) Ψ ˆ E + (x + ) µ = e iφ(x +,x ) 2π p = Ψ µ τ x + τ x (x )Ψ µ + (x + ) N µ Tr I µ = Z dx + dx µ Û(x +, x ) µ = µ 2π log Λ 2πR µf, sin ` 2R µ N = 4πR (log Λ + µφ( µ)). Z 2π dx + dx e ix + x iφ(x +,x ) p τ x + τ x = log Λ. The evaluation of this integral by the saddle point method gives the non-perturbative corrections to the zero mode φ of the bosonic field Time-dependent backgrounds of compactified 2D string theory: p.20/25

22 Several fermion bilinears = ny k= 2π µ ny j= e iφ(xk +,xk ) q τ x k + τx k ˆ Ψ (x j ) ˆ Ψ + (x j + ) µ = det i,j 2R sinh τ(xi ) τ(xj + ) 2R A. exact (?) after ω ± = e τ e i µ The parameter τ plays the same role as the boundary parameter parametrizing the boundary cosmological constant µ B = µ coshτ in Liouville theory. Here there are two boundary cosmological constants": x + (τ) and x (τ). Two disk amplitudes" Φ + (x + ) = R x by Legendre transformation + x dx + and Φ (x ) = R x x + dx related Time-dependent backgrounds of compactified 2D string theory: p.2/25

23 Back to (x,p) representation ) Stationary background x + (τ), x (τ) x(τ), w(τ) = dφ/dx Φ(x) - the disk partition function with Dirichlet matter FZZT µb =x. The curve x + (τ) + x (τ) = 2x(τ), x + (τ) x (τ) = w(τ + iπ) w(τ iπ) x(τ) = p 2µ coshτ, 2µ w(t) = π τ sinh τ is a Z 2 orbifold of the universal cover of the hyperboloid x + x = µ, obtained by identifying the points τ and τ. The branch points of y = w(x) are images of infinitely many points τ = ±iπn, the (degenerate) limit of the double points of the minimal string curves τ mn = i(m/b + nb), τ m, n = i(m/b nb) Time-dependent backgrounds of compactified 2D string theory: p.22/25

24 Back to (x,p) representation 2) Time-dependent background x(τ) = = 2 e 2R χ hcoshτ + P k max k= a k cosh `( k R )τ i, p(τ) = = 2 e 2R χ hsinh τ + P k max k= a k sinh `( k R )τ i. p(τ) = w(τ + iπ (τ)) w(τ iπ (τ)), 2i where the function π (t) is obtained as the solution of the transcendental equation sin π + kx max k= a k sinh `( k R )t sinh t sin `( k R )π = 0, t R. As a consequence, the parameter τ does not uniformize the curve y = w(x). In general, w(τ) will have infinitely many branch points. Time-dependent backgrounds of compactified 2D string theory: p.23/25

25 Sheets of W(x) horizontal: Im τ vertical : t = Re τ The lines give the solutions of the transcendental eqn for π k (t) Time-dependent backgrounds of compactified 2D string theory: p.24/25

26 Conclusion The time-dependent backgrounds of the c = string theory are described by complex curves with two singular points. All NPC are associated with double points (=vanishing cycles) of the complex curve. The leading exponents and the subleading factors of the non-perturbative corrections are expressed in terms of the one- and two-point functions of a bosonic field defined on the complex curve. The subleading correction is non-analytic in the perturbation couplings Questions:? Is there a D-brane interpretation of the correlation functions of the fermion bilinears??? If so, identify the CFT boundary conditions for these chiral branes. For φ large and negative, these branes are extended in χ + φ direction and localized in χ φ direction, or vice versa. D P h ie Φ(x+,x ) = n T n/r x n/r + T + n/r x n/r =??? disk sphere Time-dependent backgrounds of compactified 2D string theory: p.25/25