The integrable structure of Liouville theory J org Teschner DESY - PowerPoint PPT Presentation

The integrable structure of Liouville theory J¨ org Teschner DESY Hamburg Joint with A. Bytsko

What is Liouville theory? – I d 2 z � z ϕ + 4 πµe 2 ϕ ) . S [ ϕ ] = 4 π ( ∂ z ϕ∂ ¯ Σ Related to uniformization of Riemann surfaces: ds 2 = e 2 ϕ dzd ¯ z has constant negative curvature iff z ϕ = µe 2 ϕ . ϕ satisfies Liouville equation of motion ∂ z ∂ ¯ iff S [ ϕ ] is minimized. Essentially: ⇔ Teichm¨ uller theory . Liouville theory

What is Liouville theory? – II The functional S [ ϕ ] plays an important role in Teichm¨ uller theory. Let ϕ be the unique solution to ∂ ¯ ∂ϕ = 2 πµe ϕ , with the boundary conditions z ) = − 2(1 − η s ) log | z | 2 + O (1) ϕ ( z, ¯ at | z | → z s = ∞ , z ) = − 2 η i log | z − z i | 2 + O (1) ϕ ( z, ¯ at | z | → z i , i = 1 , . . . , s − 1 . Let z s − 1 = 1 , z 1 = 0 ⇒ z 2 , . . . , z s − 2 : complex analytic coordinates for M 0 ,n . Let ϕ ] = 1 � | ∂ z ϕ | 2 + µ cl e ϕ � � S cl X ǫ d 2 z � ǫ 4 π � η i � ǫ s − 1 � � � � � dx ϕ + 2 η 2 − i log ǫ + (1 − η s ) dx ϕ − 2 log ǫ , 2 πǫ 2 π ∂D i ∂D s i =1 where D i = { z ∈ C ; | z − z i | < ǫ } , D s = { z ∈ C ; | z | > 1 /ǫ } , and X ǫ = D s \ � s − 1 i =1 D i . Theorem (Takhtajan-Zograf) Let ∂ ( ¯ ∂ ) be (anti-) holomorphic components of de Rham differential. We have 2 πi∂ ¯ ∂ω = ω WP .

What is quantum Liouville theory? – III z ) = e 2 αφ ( z, ¯ z ) . Basic observables: V α ( z, ¯ Theory fully characterized by correlation functions � � V α n ( z n , ¯ z n ) . . . V α 1 ( z 1 , ¯ z 1 ) The correlation functions can be seen as giving deformations of the K¨ ahler potential in the sense that � − 1 ∼ e − 1 b 2 S [ ϕ ] � 2 (1 + O ( b 2 )) , det(∆ ϕ + 1 � � V α n ( z n , ¯ z n ) . . . V α 1 ( z 1 , ¯ z 1 ) 2 ) where S [ ϕ ] is the action functional defined above. Standard physicists rules for quantization of Lagrangian field theories ⇒ ⇒ ( Takhtajan, Teo ) formal series in b 2 . The correlation functions can be defined non-perturbatively by means of conformal bootstrap ( Belavin, Polyakov, Zamolodchikov; Al.B., A.B. Zamolodchikov; J.T.)

Quantum equivalence Liouville theory ⇔ Teichm¨ uller theory Consider T g,n : Teichm¨ uller space of Riemann surfaces with genus g and n conical singularities, deficit angles η k = bα k . Teichm¨ uller space has a K¨ ahler structure. Let m = ( m 1 , . . . , m 3 g − 3+ n ) be complex analytic coordinates for T g,n . Claim ( J.T. ): There exists a canonical K¨ ahler quantization of T g,n , characterized by the deformed Bergmann kernel B Σ (¯ n , m ) ( Karabegov ⇒ star product etc.). The Bergmann kernel B Σ is uniquely determined by • invariance under mapping class group action defined by Fock; Kashaev; Chekhov, Fock . • boundary conditions at ∂ T g,n . We then have � � B Σ ( ¯ m , m ) = V α n ( z n , ¯ z n ) . . . V α 1 ( z 1 , ¯ z 1 ) Σ m

The conformal structure of Liouville theory The general solution to to ∂ ¯ ∂ϕ = 2 πµe ϕ , can be written as | ∂ z A ( z ) | 2 �� � ϕ ( z, ¯ z ) = log 2 πµ , (1 + | A ( z ) | 2 ) 2 It is parameterized by a holomorphic function A ( z ) , which describes the uniformizing mapping. Consider instead theory on cylinder with coordinates t, σ , Minkowskian signature ∂ + ∂ − ϕ = 2 πµe ϕ , where ∂ ± = 1 2 ( ∂ t ± ∂ σ ) . Solution 2 πµ ∂ + A ( x + ) ∂ − ¯ �� � A ( x − ) ϕ ( z, ¯ z ) = log , (1 + A ( x + ) ¯ A ( x − )) 2 In other words: Liouville theory is most easily solved by means of its conformal structure : Factorization into left-movers ( A ( x + ) ) and right-movers ( ¯ A ( x − ) ).

The integrable structure of Liouville theory — 0 Why are we interested in the integrable structure of Liouville theory ? • Explain integrability of 2d gravity !!! • Integrable structure of CFT • Model for conformal field theories where conformal symmetry is not enough to solve them. A lot of work was devoted to the integrable structure of Liouville theory ( Faddeev, Volkov, Kashaev ). Missing: Link to the solution via conformal bootstrap. This is what we can now provide.

The integrable structure of Liouville theory — I Starting point: Sinh-Gordon model, ( ∂ 2 t − ∂ 2 x ) ϕ ( σ, t ) = − 8 πµb sinh(2 bϕ ( x, t )) . zero curvature condition, [ ∂ t − V ( x, t ; λ ) , ∂ σ − U ( x, t ; λ ) ] = 0 , with U , V being − im ( λe − bϕ − λ − 1 e bϕ ) b � � 2 Π U ( x, t ; λ ) = − im ( λe bϕ − λ − 1 e − bϕ ) − b 2 Π − im ( λe − bϕ + λ − 1 e bϕ ) b 2 ϕ ′ � � V ( x, t ; λ ) = − im ( λe − bϕ + λ − 1 e bϕ ) − b 2 ϕ ′ and m chosen such that m 2 = πb 2 µ . Hamiltonian formalism: { Π( σ ) , ϕ ( σ ′ ) } = 2 π δ ( σ − σ ′ ) , The time-evolution of an arbitrary observable is then given as ∂ t O ( t ) = { H , O ( t ) } , with Hamiltonian density � R dx Π 2 + ( ∂ σ ϕ ) 2 + 8 πµ cosh(2 bϕ ) � � H ShG = . 4 π 0

The integrable structure of Liouville theory — II Interesting limit: Let m → 0 , ϕ → ϕ − ξ , ξ → ∞ such that me 2 bξ → µ ⇒ � R dx H Liou = Π 2 + ( ∂ σ ϕ ) 2 + 4 πµe − 2 bϕ � � . 4 π 0 Can be performed on the level of the Lax-pair if combined with shift of spectral parameter ( Faddeev-Tirkkonen ) b − imλe − bϕ � � 2 Π U ( x, t ; λ ) = − im ( λe bϕ − λ − 1 e − bϕ ) − b 2 Π b 2 ϕ ′ − imλe − bϕ � � V ( x, t ; λ ) = − im ( λe − bϕ + λ − 1 e bϕ ) − b 2 ϕ ′ Question: Does the monodromy matrix generate enough conserved quantities?

The integrable structure of Liouville theory — III To answer this question, go to discrete versions of Sinh-Gordon / Liouville. Variables ϕ n ≡ ϕ ( n ∆) , Π n ≡ ∆Π( n ∆) , Lax matrices: µ − 1 V 2 µ − 1 V − 1 � U n + µ ¯ � n U n µ V n + ¯ n L ShG ( µ, ¯ µ ) = , µ V − 1 µ − 1 V n U − 1 µ − 1 V − 2 n U − 1 n + ¯ n + µ ¯ n b 2 Π n , V n = e − bϕ n , µ ≡ − iλm , ¯ µ ≡ − iλm − 1 . Liouville Lax matrix: where U n = e µ − 1 V 2 � U n + µ ¯ � n U n µ V n L Liou ( µ, ¯ µ ) = , µ V − 1 µ − 1 V n U − 1 n + ¯ n The corresponding monodromy matrix, T Liou ( λ ) = tr( L Liou N ( λ ) · · · L Liou ( λ )) 1 gives only L + 1 conserved quantities if N = 2 L + 1 , not enough .

The integrable structure of Liouville theory — IV But consider also: µ − 1 V 2 µ − 1 V − 1 � U n + µ ¯ � n U n µ V n + ¯ ¯ n L Liou ( µ, ¯ µ ) = . µ − 1 V n U − 1 ¯ n The corresponding monodromy matrix, T Liou ( λ ) = tr(¯ ¯ N ( λ ) · · · ¯ L Liou L Liou ( λ )) 1 Poisson-commutes with T Liou ( λ ) and gives another L + 1 conserved quantities. One of these coincides with a quantity from T Liou ( λ ) , ”zero mode”. Together: Number of conserved quantities = Number of degrees of freedom .

The integrable structure of Liouville theory — V Recall 2 πµ ∂ + A ( x + ) ∂ − ¯ �� A ( x − ) � ϕ ( z, ¯ z ) = log (1 + A ( x + ) ¯ A ( x − )) 2 Claim : T Liou ( λ ) : Conserved quantitites constructed from A ( x + ) , T Liou ( λ ) : Conserved quantitites constructed from ¯ ¯ A ( x − ) .

Quantum integrable structure of Liouville theory — I Quantization: U n , V n → operators, algebra U n V n = q V n U n , q = e πib 2 . Realize on L 2 ( R N ) as U n = e πb (2 x n + p n ) , V n = e πb p n , [ p n , x m ] = (2 πi ) − 1 δ nm . Quantum Lax-matrices: µ − 1 V n U n V n µ − 1 V − 1 � U n + µ ¯ � µ V n + ¯ n L ( λ ) = , µ V − 1 µ − 1 V n U − 1 µ − 1 V − 1 n U − 1 n V − 1 n + ¯ n + µ ¯ n Transfer matrices: T ( λ ) = tr(¯ ¯ L N ( λ ) · · · ¯ T ( λ ) = tr( L N ( λ ) · · · L 1 ( λ )) , L 1 ( λ )) . Mutual commutativity: [ T ( λ ) , ¯ [¯ T ( λ ) , ¯ [ T ( λ ) , T ( λ ′ )] = 0 , T ( λ ′ )] = 0 , T ( λ ′ )] = 0 .

Quantum integrable structure of Liouville theory — II Key tool for study of the spectrum of T ( λ ) , ¯ T ( λ ) : The Baxter Q-operators. Solution to conditions T ( λ ) Q ( λ ) = a ( λ ) Q ( qλ ) + d ( λ ) Q ( q − 1 λ ) � � (i) T ( λ )¯ ¯ a ( λ )¯ Q ( qλ ) + ¯ d ( λ )¯ Q ( q − 1 λ ) Q ( λ ) = ¯ T, ¯ T, Q, ¯ (ii) Q commute for arbitrary values of the spectral parameter. This means that T, ¯ T, Q, and ¯ Q can be simultaenously diagonalized. The eigenvalues T ( λ ) , ¯ T ( λ ) and Q ( λ ) , ¯ Q ( λ ) must satisfy the Baxter equations T ( λ )¯ ¯ a ( λ ) ¯ Q ( qλ ) + ¯ d ( λ ) ¯ T ( λ ) Q ( λ ) = a ( λ ) Q ( qλ ) + d ( λ ) Q ( q − 1 λ ) , Q ( q − 1 λ ) Q ( λ ) = ¯ If we know the Q-operators explicitly ⇒ Analytic and asymptotic properties of Q ( λ ) , ¯ Q ( λ ) ⇒ Definition of the space of admissible solutions of Baxter’s equations ⇒ Quantization conditions.