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

The integrable structure of Liouville theory

J¨

• rg Teschner

DESY Hamburg Joint with A. Bytsko

What is Liouville theory? – I

S[ϕ] =

• Σ

d2z 4π (∂zϕ∂¯

zϕ + 4πµe2ϕ) .

Related to uniformization of Riemann surfaces: ds2 = e2ϕdzd¯ z has constant negative curvature iff ϕ satisfies Liouville equation of motion ∂z∂¯

zϕ = µe2ϕ.

iff S[ϕ] is minimized. Essentially: Liouville theory ⇔ Teichm¨

uller 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 = 2πµeϕ, with the boundary conditions ϕ(z, ¯ z) = −2(1 − ηs) log |z|2 + O(1) ϕ(z, ¯ z) = −2ηi log |z − zi|2 + O(1) at |z| → zs = ∞, at |z| → zi, i = 1, . . . , s − 1. Let zs−1 = 1, z1 = 0 ⇒ z2, . . . , zs−2: complex analytic coordinates for M0,n. Let Scl

ǫ

• ϕ] = 1

4π

• Xǫ d2z
• |∂zϕ|2 + µcleϕ

−

s−1

• i=1

ηi 2πǫ

• ∂Di

dx ϕ + 2η2

i log ǫ

• + (1 − ηs)

ǫ 2π

• ∂Ds

dx ϕ − 2 log ǫ

• ,

where Di = {z ∈ C; |z − zi| < ǫ}, Ds = {z ∈ C; |z| > 1/ǫ}, and Xǫ = Ds \ s−1

i=1 Di.

Theorem (Takhtajan-Zograf) Let ∂ (¯ ∂) be (anti-) holomorphic components of de Rham differential. We have 2πi∂ ¯ ∂ω = ωWP .

What is quantum Liouville theory? – III

Basic observables: Vα(z, ¯ z) = e2αφ(z,¯

z).

Theory fully characterized by correlation functions

• Vαn(zn, ¯

zn) . . . Vα1(z1, ¯ z1)

• The correlation functions can be seen as giving deformations of the K¨

ahler potential in the sense that

• Vαn(zn, ¯

zn) . . . Vα1(z1, ¯ z1)

• ∼ e− 1

b2S[ϕ]

det(∆ϕ + 1

2)

−1

2(1 + O(b2)) ,

where S[ϕ] is the action functional defined above. Standard physicists rules for quantization of Lagrangian field theories ⇒ ⇒ (Takhtajan, Teo) formal series in b2. 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 Tg,n: Teichm¨ uller space of Riemann surfaces with genus g and n conical singularities, deficit angles ηk = bαk. Teichm¨ uller space has a K¨ ahler structure. Let m = (m1, . . . , m3g−3+n) be complex analytic coordinates for Tg,n. Claim (J.T.): There exists a canonical K¨ ahler quantization of Tg,n, characterized by the deformed Bergmann kernel BΣ(¯ n, m) (Karabegov ⇒ star product etc.). The Bergmann kernel BΣ is uniquely determined by

• invariance under mapping class group action defined by Fock; Kashaev;

Chekhov, Fock.

• boundary conditions at ∂Tg,n.

We then have BΣ( ¯ m, m) =

• Vαn(zn, ¯

zn) . . . Vα1(z1, ¯ z1)

• Σm

The conformal structure of Liouville theory

The general solution to to ∂ ¯ ∂ϕ = 2πµeϕ, can be written as ϕ(z, ¯ z) = log

• 2πµ

|∂zA(z)|2 (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 = 2πµeϕ, where ∂± = 1

2(∂t ± ∂σ). Solution

ϕ(z, ¯ z) = log

• 2πµ ∂+A(x+)∂− ¯

A(x−) (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) = −8πµb sinh(2bϕ(x, t)) .

zero curvature condition, [ ∂t − V (x, t; λ) , ∂σ − U(x, t; λ) ] = 0 , with U, V being U(x, t; λ) =

• b

2Π

−im(λe−bϕ − λ−1ebϕ) −im(λebϕ − λ−1e−bϕ) −b

2Π

• V (x, t; λ) =
• b

2ϕ′

−im(λe−bϕ + λ−1ebϕ) −im(λe−bϕ + λ−1ebϕ) −b

2ϕ′

• and m chosen such that m2 = πb2µ.

Hamiltonian formalism: { Π(σ) , ϕ(σ′) } = 2π δ(σ − σ′), The time-evolution of an arbitrary observable is then given as ∂tO(t) = { H , O(t) } , with Hamiltonian density HShG = R dx 4π

• Π2 + (∂σϕ)2 + 8πµ cosh(2bϕ)
• .

The integrable structure of Liouville theory — II

Interesting limit: Let m → 0, ϕ → ϕ − ξ, ξ → ∞ such that me2bξ → µ ⇒ HLiou = R dx 4π

• Π2 + (∂σϕ)2 + 4πµe−2bϕ

. Can be performed on the level of the Lax-pair if combined with shift of spectral parameter (Faddeev-Tirkkonen) U(x, t; λ) =

• b

2Π

−imλe−bϕ −im(λebϕ − λ−1e−bϕ) −b

2Π

• V (x, t; λ) =
• b

2ϕ′

−imλe−bϕ −im(λe−bϕ + λ−1ebϕ) −b

2ϕ′

• 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 ϕn ≡ ϕ(n∆) , Πn ≡ ∆Π(n∆) , Lax matrices: LShG(µ, ¯ µ) = Un + µ¯ µ−1V2

nUn

µVn + ¯ µ−1V−1

n

µV−1

n + ¯

µ−1Vn U−1

n + µ¯

µ−1V−2

n U−1 n

• ,

where Un = e

b 2Πn, Vn = e−bϕn, µ ≡ −iλm, ¯

µ ≡ −iλm−1. Liouville Lax matrix: LLiou(µ, ¯ µ) = Un + µ¯ µ−1V2

nUn

µVn µV−1

n + ¯

µ−1Vn U−1

n

• ,

The corresponding monodromy matrix, TLiou(λ) = tr(LLiou

N (λ) · · · LLiou 1

(λ)) gives only L + 1 conserved quantities if N = 2L + 1, not enough.

The integrable structure of Liouville theory — IV

But consider also: ¯ LLiou(µ, ¯ µ) = Un + µ¯ µ−1V2

nUn

µVn + ¯ µ−1V−1

n

¯ µ−1Vn U−1

n

• .

The corresponding monodromy matrix, ¯ TLiou(λ) = tr(¯ LLiou

N (λ) · · · ¯

LLiou

1

(λ)) Poisson-commutes with TLiou(λ) and gives another L + 1 conserved quantities. One

• f these coincides with a quantity from TLiou(λ), ”zero mode”. Together:

Number of conserved quantities = Number of degrees of freedom .

The integrable structure of Liouville theory — V

Recall ϕ(z, ¯ z) = log

• 2πµ ∂+A(x+)∂− ¯

A(x−) (1 + A(x+) ¯ A(x−))2

• Claim:

TLiou(λ): Conserved quantitites constructed from A(x+), ¯ TLiou(λ): Conserved quantitites constructed from ¯ A(x−).

Quantum integrable structure of Liouville theory — I

Quantization: Un, Vn → operators, algebra UnVn = qVnUn, q = eπib2. Realize on L2(RN) as Un = eπb(2xn+pn), Vn = eπbpn, [pn, xm] = (2πi)−1δnm. Quantum Lax-matrices: L(λ) = Un + µ¯ µ−1VnUnVn µVn + ¯ µ−1V−1

n

µV−1

n + ¯

µ−1Vn U−1

n + µ¯

µ−1V−1

n U−1 n V−1 n

• ,

Transfer matrices: T(λ) = tr(LN(λ) · · · L1(λ)), ¯ T(λ) = tr(¯ LN(λ) · · · ¯ L1(λ)). Mutual commutativity: [T(λ), T(λ′)] = 0, [T(λ), ¯ T(λ′)] = 0, [¯ T(λ), ¯ 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 (i)

• T(λ)Q(λ) = a(λ)Q(qλ) + d(λ)Q(q−1λ)

¯ T(λ)¯ Q(λ) = ¯ a(λ)¯ Q(qλ) + ¯ d(λ)¯ Q(q−1λ)

• (ii)

T, ¯ T, Q, ¯ 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)Q(λ) = a(λ)Q(qλ) + d(λ)Q(q−1λ), ¯ T(λ)¯ Q(λ) = ¯ a(λ) ¯ Q(qλ) + ¯ d(λ) ¯ Q(q−1λ)

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.

Quantum integrable structure of Liouville theory — III

Explicit construction by kernels in x-representation: xn|x = xn |x . x′ | QLiou

s

(w) | x =

N

• r=1

W −

w−s(x′ r−1 + xr) ¯

Ww+s(x′

r − xr)W + +2s(xr−1 + xr) ,

x′ | ¯ QLiou

s

( ¯ w) | x =

N

• r=1

W −

−2s(x′ r + x′ r+1) ¯

W ¯

w−s(x′ r − xr)W + ¯ w+s(x′ r−1 + xr) ,

where the functions ¯ Wa(x), W +

a (x) and W − a (x) are defined respectively as

W +

a (x) = ζ−1 e−iπ

2(x+a 2)2 wb

a

2 − x

• ,

W −

a (x) = ζ eiπ

2(x−a 2)2wb

a

2 + x

• ,

¯ W2a(x) =

• R

dt e2πixtwb(x + a) wb(x − a), where ξ = e

πi 24(b2+b−2), wb(x) is the non-compact q-dilogarithm (Faddeev, Kashaev),

wb(x) = ξ exp πi 2 x2 − dt 4t e−2πitx sinh bt sinh b−1t

• .

Quantum integrable structure of Liouville theory — IV

Now we are in business:

• Extract analytic properties of Q-operators ⇒ quantization conditions.
• Use separation of variables (Sklyanin) to show that all admissible solutions of

the Baxter equation indeed give quantum states.

• Reformulate in terms of nonlinear integral equations.
• Take continuum limit ⇒ quantization conditions of the continuum QFT.
• Use ODE-IM-correspondence for more efficient calculation of q-functions.

Quantum integrable structure of Liouville theory — V

Main results:

• The spectrum of Liouville theory can be described as direct integral over Fock

spaces H ≃ ⊕

R+

dP FP ⊗ ¯ FP , where FP, ¯ FP: Fock spaces.

• Eigenstates of the Hamiltonian H are uniquely labeled by P and tuples k, ¯

k of positive integers. The eigenvalues of the Hamiltonian are H = 2π R

• 2P 2 − 1

12 +

• a∈K

ka +

• a∈K

¯ ka

• .

Quantum integrable structure of Liouville theory — V

Main results (ctd.):

• The states vP with H =

2π R (2P 2 − 1 12) (”Fock-vacua”) have an analytic

continuation to negative values of P which satisfy the reflection relations vP = R(P)v−P, R(P) = −ρ−8iδP Γ(1 + 2ibP)Γ(1 + 2ib−1P) Γ(1 − 2ibP)Γ(1 − 2ib−1P) , in which we have used the abbreviation ρ ≡ R 2π m 4√π Γ

• 1

2 + 2b2

• Γ
• 1 +

b2 2 + 2b2

• .

We observe nontrivial quantitative agreement with the bootstrap approach:

• VQ

2 +iP(1)VQ 2 +iP ′(0)

• = δ(P − P ′)R(P) .