A warm-up for solving noncompact sigma models: The Sinh-Gordon model

Transcription

1 A warm-up for solving noncompact sigma models: The Sinh-Gordon model Jörg Teschner Based on A. Bytsko, J.T., hep-th/ , J.T., hep-th/

2 Goal: Understand string theory on AdS-spaces Building blocks: nonlin. sigma-models on super-groups. P SL(1, 1 2) AdS3 S3 P SL(2, 2 4) AdS5 S5 To write out worldsheet-actions, use Poincare coordinates: ds 2 = 1 z 2(dz2 + d x 2 ). Writing z = e ϕ world-sheet Hamiltonian of the form H =... + e 2ϕ (... ) + e 2ϕ (... ) +... Exponential interactions! On the other hand: Hope for integrability! (Bena, Polchinski, Roiban)

3 Nice recent story concerning limit (Hofman, Maldacena) J, E J = fixed, y R 1 AdS = fixed Gauge-fixing: J R = J/ y, R: world-sheet radius. Hg.f. = 2πR dσ (... + e 2ϕ (... ) + e 2ϕ (... ) +... ) 0 Conjecture: Hg.f. is integrable. Proposal for factorized scattering theory in infinite volume R ( J ) (Beisert, Staudacher, Hofman, Maldacena...) : Elementary excitations ( magnons ) 8B 8F. Proposal for 2 2 scattering matrix (for all y!!!) Proposal has passed highly nontrivial tests!

4 Important problem: Spektrum for finite R finite J? Asymptotic Bethe ansatz (Arutyunov, Frolov, Staudacher) Pretend magnons are free except for magnons crossing (δ-like interactions), Ansatz for coord. space wave-fct.: Plane-waves + crossings (S-matrix) Periodicity of wave-fct. e ip ar = b a S(pa, pb). Problem: Interactions are not δ-like (vacuum polarization) Expect corrections to As. Bethe ansatz!

5 Classes of integrable models: (A) Compact integrable models. Models associated to compact groups: WZNW, σ-models, XXX, XXZ, Ising, Interactions e ibφ : Sine-Gordon... (B) Noncompact integrable models - Real type Models associated to non-compact groups: WZNW, σ-models, XXX, XXZ... Interactions e bφ : Liouville theory, Sinh-Gordon, (Affine) Toda... Class (B) is important: String theory: Strings on noncompact target spaces (black holes, cosmology), Gauge theory - via AdS-CFT correspondence, Condensed matter: Integer quantum Hall, electron systems with disorder. But: Class (B) is very different from (A): Prediction: Bethe ansatz fails in class (B)!

6 Consider prototype: Sinh-Gordon model (on circle with circumference R) HShG = R dx { 4π Π π ( σϕ) 2 + 2µ cosh(bϕ) }. 0 Infinite volume R : Spectrum: One massive particle, E = m cosh ϑ, p = m sinh ϑ. Scattering: S-matrix factorizes into two-particle scattering S(ϑ1 ϑ2) S(ϑ) = sinh ϑ i sin ϑ 0. sinh ϑ + i sin ϑ0 Fields, correlation functions:...?! Finite volume R: Spectrum: Conserved quantities, classification of eigenvectors - today! First example for QFT from class (B) where spectrum was understood for finite volume!

7 First step: Construct integrable lattice regularization of the Sinh-Gordon model: Hilbert space H Hamiltonian H, set Q = {T0, T1,... } of commuting conserved charges 1. Definition of H: Discretize Sinh-Gordon variables as Πn Π(x), Φn Φ(x), x = n. Quantize: [ Πn, Φn ] = 1 i δ n,m. Hilbert space H (L 2 (R)) N.

8 2. Construction of Q. Let M(u) ( A(u) B(u) C(u) D(u) ) LN(u)... L2(u) L1(u), where L(u) ( L11(u) L12(u) L21(u) L22(u) ), ( ) L11(u) = e +β 8 (Π n+2s) 1 + e β(φ n+s) e +β 8 (Π n+2s) L12(u) = sinh ( πbu + β 2 Φ n ) L21(u) = sinh ( πbu β 2 Φ ) β = b 8π n ( ) L22(u) = e β 8 (Π n 2s) 1 + e +β(φ n s) e β 8 (Π n 2s) Consider the one parameter family of operators: T(u) = tr ( M(u) ) = A(u) + D(u). The operators Tm which are defined by T(u) = e πbnu N m=0 ( e 2πbu ) m Tm, are positive self adjoint and commuting, [Tm, Tn] = 0.

9 3. Construction of H There exists an operator H which has the following properties. (a) The operator H is local: N H = n=1 Hn,n+1. (b) H commutes with the conserved charges Tk: [ H, Tk ] = 0, for k = 0,..., N. (c) Classical continuum limit (Πn Π(x), Φn Φ(x), x = n.) 0 N with { } R = N /2π fixed. m = 4 e πbs 1 H 0 HG,cl n n,n+1 2πR 0 dx ( 1 2 Π ( xφ) 2 + m2 β 2 cosh βφ ) + const

10 Good news: We do have integrable lattice regularization Strategy: Diagonalize T(u) Diagonalization of H Bad news: The Bethe ansatz fails: one would need pseudo-vacuum Ω annihilated by B(u) Since interactions involve real exponentials e bϕ : B(u)Ω = 0 leads to equations like e φ Ω = 0 no normalizable solution Ω H!!! Reasons: exponential interactions e φ, non-compactness of target space! However, there is something better than Bethe ansatz: Q-operators and separation of variables

11 Assume we have an operator Q(u) related to T(u) via the Baxter equation Q(u)T(u) = ( a(u) ) N Q(u ib) + ( d(u) ) NQ(u + ib), which furthermore satisfies (a) Q(u) is normal, Q(u)Q (v) = Q (v)q(u), (b) Q(u) Q(v) = Q(v) Q(u), (c) Q(u) T(v) = T(v) Q(u), Q(u) is repackaging of conserved quantities Diagonalization of Q(u) diagonalization of T(u). Eigenvalues q(u) of Q(u) must satisfy the Baxter equation. Explicit construction of Q(u) (Bytsko, J.T.) analytic and asymptotic properties of q(u): Quantization conditions!

12 Bytsko,J.T. Necessary conditions on the spectrum: A function t(u) can be an eigenvalue of the transfer matrix T(u) only if there exists a function qt(u) which satisfies (ii) qt(u) is meromorphic in C, with poles in ± Υ s, e +iπn σ u iπ 2 Nu2 for u, arg(u) < π 2 (iii) qt(u) e iπn σ u iπ 2 Nu2 for u, arg(u) > π 2. where d(u) = a( u) = 1 + ( m 4 )2 e πb(2u+ib), (i) t(u) qt(u) = ( a(u) ) N q t(u ib) + ( d(u) ) N q t(u + ib),

13 Separation of Variables Main idea (Sklyanin): Diagonalize B(u), parametrize eigenvalues b(u) as b(u) sinh 2πb(u yk). wave-functions Ψ(y1... yn). Key observations: (Sklyanin) T(u)Ψt = t(u)ψt(u) t(yk)ψ(... yk... ) = (a(yk)) N Ψ(... yk ib... ) + (d(yk)) N Ψ(... yk + ib... ) (Bytsko, J.T.) Properties (i)-(iii) Ansatz N Ψt = k=1 qt(yk) yields normalizable eigenstates of T(u).

14 Bytsko,J.T. Full characterization of the spectrum: A function t(u) is eigenvalue of the transfer matrix T(u) if and only if there exists a function qt(u) which satisfies (i) t(u) qt(u) = ( a(u) ) N q t(u ib) + ( d(u) ) N q t(u + ib), where d(u) = a( u) = 1 + ( m 4 )2 e πb(2u+ib), (ii) qt(u) is meromorphic in C, with poles in ± Υ s, e +iπn σ u iπ 2 Nu2 for u, arg(u) < π 2 (iii) qt(u), e iπn σ u iπ 2 Nu2 for u, arg(u) > π 2. Task: Classify set Q of solutions to the Baxter equation (i)

15 Let YM be the set of all functions Y (ϑ) which satisfy the integral equation (I)L log Y (ϑ) = dϑ C 4π σ(ϑ ϑ ) log(w (ϑ ) + Y (ϑ )) ( ) ( ) cosh(ϑ + iτ) cosh(ϑ iτ) N arctan N arctan sinh σ sinh σ M + log S(ϑ ϑa i π 2 ) + 1 M log S(ϑ ϑa i π 2 2 ). a=1 a=m +1 and which have the properties (Y1)L log Y (ϑ) iδn((ϑ σ i π 2 )2 τ 2 ) for u, arg(±u) < π 2, (Y2)L Y (ϑ) is meromorphic with poles of maximal order N in ± Υ s±iτ, (Y3)L There exist complex numbers ϑa S, a = 1,..., M, such that W (ϑ) + Y (ϑ) = 0 if ϑ = ϑa ± i π 2. ( σ(ϑ) 4 sin ϑ 0 cosh ϑ, ϑ0 π b cosh 2ϑ cos 2ϑ0 2 δ, σ π s 2 δ, τ π δ ) 2 δ.

16 Theorem 1. There is a one-to-one correspondence between the solutions Y (ϑ) YM of the integral equations (I)L and the elements Q QM. Given Q QM, get Y (ϑ): W (ϑ) + Y (ϑ) = Q(ϑ + i π 2 )Q(ϑ iπ 2 ). Z = {ϑ1,..., ϑm}: set of zeros of Q(ϑ) within S. Given Y (ϑ) YM (zeros: ϑa S \ S, ϑ b S) get Q Q M as (Q)L log Q(ϑ) = + C M a=1 dϑ 4π ϑ Ca log(w (ϑ ) + Y (ϑ )) cosh(ϑ ϑ ) dϑ 1 sinh(ϑ ϑa) ( ) cosh ϑ N arctan sinh σ M M b=1 ϑ Ca dϑ 1 sinh(ϑ ϑ b ).

17 Continuum limit: Easy! Continuum limit: N, s such that mr 2 sin π b 2δ 2N exp ( π ) 2δ s is kept constant. Main claim:

18 Main claim: The Hilbert space of the Sinh-Gordon model contains (is equal to!?) HTBA = M=0 H M. The spaces HM have ONB ek labelled by k = (k1,..., km) Z M,, k1 > > km, ek: eigenvect. to the conserved quantities of the model. The corresponding function Qk(ϑ) can be represented as (Q) log Qk(ϑ) = R dϑ 2π log(w (ϑ ) + YT k (ϑ )) mr cosh ϑ M + cosh(ϑ ϑ ) 2 sin ϑ0 a=1 ϑ Ca dϑ 1 sinh(ϑ ϑa). where Tk = [ϑ1,..., ϑm] is the unique solution of (B) 2πka+mR sinh ϑa+ M arg S(ϑa ϑb)+i b=1 b a R dϑ 2π σ(ϑ a ϑ+i π 2 ) log(1+y T k (ϑ)) = 0, with YT(u) being defined as the unique solution to dϑ (A) log YT(ϑ) R 2π σ(ϑ ϑ ) log(1+yt(ϑ ))+mr cosh ϑ+ M log S(ϑ ϑa i π 2 ) = 0. a=1

19 The energies are calculated from YT(ϑ) as follows: Ek = M m cosh ϑa m a=1 R dϑ 2π cosh ϑ log(1 + Y T(ϑ)) Excited state TBA! (Generalizes work of Al.B. Zamolodchikov, S. Lukyanov for the ground state)

20 The IR limit R Considering IR limit R, notice that YT(ϑ) = O(e mr ). Particle picture: M: number of particles, ϑa: Rapidity of particle a, Ea = m cosh ϑa, pa = m sinh ϑa, ϑa quantized by asymptotic Bethe ansatz equations. e irp a = M b=1 b a S(ϑa ϑb)

21 For finite R: where Φa R e ir(p a+φa) = M b=1 b a S(ϑa ϑb), dϑ 2πR σ(ϑ a ϑ + i π 2 ) log(1 + Y T(ϑ) = O(e mr ). Φa: Effects of vacuum polarization! The UV limit R 0: Connection with Liouville theory!

22 Compare with String Bethe Ansatz : Valid for J, J: angular momentum on S5 Radius of world-sheet cylinder in gauge-fixed action. For finite J: corrections O ( e 2πJ/ λ ) to String Bethe Ansatz (Schäfer-Nameki, Zamaklar, Zarembo) counterparts of e mr -corrections in Sinh-Gordon Gauge theory side: Wrapping interactions : (see Kotikov, Lipatov, Rej, Staudacher, Velizhanin) Prediction: Bethe ansatz will fail for AdS5 sigma model, finite J. But don t forget: There is sthg. better than Bethe ansatz!