An introduction to Liouville Quantum Field theory

Transcription

1 An introduction to Liouville Quantum Field theory Vincent Vargas ENS Paris

2 Outline 1 Quantum Mechanics and Feynman s path integral 2 Liouville Quantum Field theory (LQFT) The Liouville action The Gaussian Free Field Kahane s Gaussian multiplicative chaos Relation with random planar maps

3 Plan of the talk 1 Quantum Mechanics and Feynman s path integral 2 Liouville Quantum Field theory (LQFT) The Liouville action The Gaussian Free Field Kahane s Gaussian multiplicative chaos Relation with random planar maps

4 Quantum mechanics: the Schrödinger equation Particle of mass m in potential V described by Schrödinger equation: i ψ 2 2 ψ (x, t) = (x, t) + V (x)ψ(x, t), t 2m x 2 ψ(x, t = 0) = ψ 0 (x) Figure : Erwin Schrodinger

5 The Feynman path integral Feynman path integral (1948): ( ) x(t )=B ψ(b, T ) = e i S(x) Dx ψ 0 (A)dA R d x(0)=a where Dx Lebesgue measure on functions and S is the action: S(x) = T 0 m 2 (ẋ(t) 2 V (x(t)) ) dt

6 Euclidean formalism: Wick rotation Problem: Feynman path integral ill defined. Wick rotation: t iτ. This leads to the Euclidean integral: x(it )=B x(0)=a e i S(x) Dx = x(t )=B e 1 x(0)=a T 0 m 2 (ẋ(τ) 2 +V (x(τ)))dτ Dx Idea: go back to Feynman path integral by analytical continuation. Problem: Euclidean integral still ill defined...

7 Euclidean formalism Idea: write the partition function x(t )=B e 1 x(0)=a x(t )=B = x(0)=a T 0 ( e 1 T m 2 (ẋ(τ)2 +V (x(τ))dτ Dx 0 V (x(τ))dτ ) e m 2 T 0 ẋ(τ)2 dτ Dx }{{} Can one make sense of this measure?

8 Brownian motion τ = T N and x N (i τ) = x i. Brownian motion (B(τ)) τ T (Bachelier, Einstein, Wiener): RN N F ((x N (τ)) τ T ) e i=1 (x i x i 1 ) 2 /(2 τ) (2π τ) N/2 N dx i i=1 E[F ((B τ ) τ T )] N

9 Feynman-Kac formula Formally, we have defined E[F ((B τ ) τ T )] =: hence we set x(0)=0 x(t )=B e 1 T 0 x(0)=a T 0 V (A+ := E[e 1 F ((x(τ)) τ T )e 1 2 m 2 (ẋ(τ)2 +V (x(τ))dτ Dx m Bτ )dτ 1 A+ m B T =B ] T 0 ẋ(τ)2 Dx

10 Brownian motion: a few facts Since formally E[F ((B τ ) τ T )] =: x(0)=0 F ((x(τ)) τ T )e 1 2 T 0 d 2 x dτ 2 (τ)x(τ) Dx, Brownian motion is a Gaussian random function with covariance E[B s B t ] = ( d 2 dτ 2 ) 1 (s, t) = s t. Brownian motion is the canonical random path: by Donsker s theorem (1952), limit of discrete random functions with independent increments.

11 Plan of the talk 1 Quantum Mechanics and Feynman s path integral 2 Liouville Quantum Field theory (LQFT) The Liouville action The Gaussian Free Field Kahane s Gaussian multiplicative chaos Relation with random planar maps

12 LQFT on the Riemann sphere: the 2d analog with exponential potential Consider the following partition function on the sphere (Polyakov, 1981) Z = e SL(φ,g) Dφ, where S L is the Liouville action: S L (φ, g) := 1 ( g φ 2 (x) + QR g (x)φ(x) + 4πµe γφ(x)) g(x)dx 4π S 2 and g some metric on the sphere, γ ]0, 2[, Q = 2 γ + γ 2 and µ > 0. Goal: construct a Conformal Field theory with action given by S L. This action only recently defined (see David, Kupiainen, Rhodes, Vargas, 2015).

13 LQFT on the Riemann sphere: the 2d analog with exponential potential Consider the following partition function on the sphere (Polyakov, 1981) n Z = e α i φ(x i ) e SL(φ,g) Dφ, n 3 i=1 where S L is the Liouville action: S L (φ, g) := 1 ( g φ 2 (x) + QR g (x)φ(x) + 4πµe γφ(x)) g(x)dx 4π S 2 and g some metric on the sphere, γ ]0, 2[, Q = 2 γ + γ 2 and µ > 0. Goal: construct a Conformal Field theory with action given by S L. This action only recently defined (see David, Kupiainen, Rhodes, Vargas, 2015).

14 LQFT on the Riemann sphere For the sake of simplicity, we will consider the toy problem of defining Z = e S L(X ) DX, X D =0 where S L is the (simplified) Liouville action: S L (X ) := 1 ( X 2 (x) + 4πµe γx (x)) dx 4π D where D bounded domain of R 2, γ ]0, 2[ and µ > 0.

15 LQFT on the Riemann sphere For the sake of simplicity, we will consider the toy problem of defining Z = e S L(X ) DX = (e µ ) D eγx (x) dx e 1 4π D X 2 (x) }{{ dx DX} X D =0 X D =0 where S L is the (simplified) Liouville action: S L (X ) := 1 ( X 2 (x) + 4πµe γx (x)) dx 4π D where D bounded domain of R 2, γ ]0, 2[ and µ > 0. Gaussian Free Field

16 The Gaussian Free Field First define F (X )e 1 4π D X 2 (x) dx DX = X D =0 X D =0 F (X )e 1 4π D X (x)x (x) dx DX, The Gaussian Free Field (GFF) X is the random Gaussian (distribution) with covariance E[X (x)x (y)] = ( D ) 1 (x, y) = G D (x, y)

17 The Gaussian Free Field Green function satisfies G D (x, y) = ln where ϕ smooth function. 1 + ϕ(x, y) x y Problem: G D (x, x) = hence X is not a function but a distribution! How to make sense of e γx (x) dx? Answer: consider mollifying sequence X ε (x) and look for sequence c ε such that c ε e γxε(x) dx converges to something non trivial. Renormalization theory!

18 Gaussian multiplicative chaos: Kahane s approach Let X ε be a smooth sequence of Gaussian fields converging to X with 1 E[X ε (x)x ε (y)] ln x y + ε Theorem (Kahane, 1985) There exists a random measure M γ such that the following limit exists almost surely in the space of Radon measures: γ2 γxε(x) e 2 E[Xε(x)2] f (x)dx ε 0 M γ (dx). The law of M γ is independent of the cut-off procedure. M γ is called Gaussian multiplicative chaos associated to G D.

19 Gaussian multiplicative chaos: Kahane s approach Theorem (Kahane, 1985) The measure M γ is different from 0 if and only if γ < 2. Theorem (Kahane, 1985) For γ < 2, the above measure M γ lives almost surely on a set of Hausdorff dimension 2 γ2 2.

20 Density of Gaussian multiplicative chaos for γ = 0.2: courtesy of R. Rhodes

21 Density of Gaussian multiplicative chaos on a line as a function of γ γ2 γxε(x) Figure : x e 2 E[Xε(x)2] on a line

22 Summary We have made sense (in fact on a toy model!) of F (e γφ(x) dx)e S L(φ,g) Dφ, where S L is the Liouville action: S L (φ, g) := 1 ( g φ 2 (x) + QR g (x)φ(x) + 4πµe γφ(x)) g(x)dx 4π S 2 Brownian motion is universal (Donsker); what about e γφ(x) dx?

23 Circle packing of a regular triangular lattice: converges to usual Euclidean geometry

24 Uniform circle packed random triangulation: limit? Figure : Circle packed random triangulation

25 Uniform circle packed triangulation: courtesy of F. David Figure : Circle packed triangulation

26 Uniform Circle packed triangulation: courtesy of F. David

27 Uniform Circle packed triangulation Conjecture: measure on uniform circle packed triangulation converges to e γφ(x) 8 dx with γ = 3. (Curien: made progress for a related model). Other problem: does the distance of uniform circle packed triangulation converge to something? Yes, to the so-called Brownian map... but in the Gromov Hausdorff sense (equivalence class on isometric metric spaces). Contributors: Bettinelli, Bouttier, Chassaing, Di Francesco, Guitter, Le Gall, Marckert, Miermont, Mokkadem, Schaeffer... Related topic: construct an isometry between the Brownian map (m, D, V ) and a metric space (S 2, d, V d ) on the sphere S 2 (Miller, Sheffield: construction for a related model).

28 Perspectives and open problems Planar map side (no embedding in the sphere, i.e. in the Gromov Hausdorff sense): construction of the distance? What is the diameter (dimension) of space? Planar map side (with embedding in the sphere): convergence of triangulations weighted by a statistical physics model (Ising, spanning trees, etc...). Measure expected to converge to e γφ(x) dx with γ depending on the model. Liouville Quantum field theory: LQFT is integrable and physicists have derived many formulas for correlation functions, etc... Can we derive these formulas?

29 Remercions Jérémie Bouttier! Bouttier and Guitter (2008) found an exact expression for the distribution of (D (x 1, x 2 ), D (x 2, x 3 ), D (x 3, x 1 )) where x 1, x 2, x 3 are three points uniformly chosen on (m, V ). Merci à Jérémie pour ses théorèmes et l organisation du séminaire!