Integrability of SLE and Liouville CFT through Conformal Welding Xin Sun University of Pennsylvania THU-PKU-BNU Probability Webinar 2020 1 Ang-Holden-S.[2020] Conformal welding of quantum disks. 2 Remy-S.-Zhu, in progress. 3 Ang-Holden-Remy-S., in progress. 4 Ang-S., in progress. 0 / 30 Two Riemannian manifolds ( , g ) and ( , g ) are S1 1 S2 2 conformally equivalent if diffeomorphism : 1 2 and a 9 ' S !S function ' on 2 s.t. g1 = e g2.( : conformal embedding.) S ⇤ A random geometry on coming from quantum gravity, S conditioned on being conformally equivalent to ( , g), S can be written as ( , e'g) for some random function ' on . S S Polyakov (1981), Quantum geometry of bosonic strings. The random function ' is governed by the Liouville field theory, the 2D quantum field theory defined by Liouville action. Liouville field theory is a conformal field theory, and is a locally trivial but globally nontrivial perturbation of Gaussian free field. 2D quantum gravity and Liouville field theory : a topological surface. S 2D quantum gravity on = random geometry on . S S 1 / 30 Polyakov (1981), Quantum geometry of bosonic strings. The random function ' is governed by the Liouville field theory, the 2D quantum field theory defined by Liouville action. Liouville field theory is a conformal field theory, and is a locally trivial but globally nontrivial perturbation of Gaussian free field. 2D quantum gravity and Liouville field theory : a topological surface. S 2D quantum gravity on = random geometry on . S S Two Riemannian manifolds ( , g ) and ( , g ) are S1 1 S2 2 conformally equivalent if diffeomorphism : 1 2 and a 9 ' S !S function ' on 2 s.t. g1 = e g2.( : conformal embedding.) S ⇤ A random geometry on coming from quantum gravity, S conditioned on being conformally equivalent to ( , g), S can be written as ( , e'g) for some random function ' on . S S 1 / 30 2D quantum gravity and Liouville field theory : a topological surface. S 2D quantum gravity on = random geometry on . S S Two Riemannian manifolds ( , g ) and ( , g ) are S1 1 S2 2 conformally equivalent if diffeomorphism : 1 2 and a 9 ' S !S function ' on 2 s.t. g1 = e g2.( : conformal embedding.) S ⇤ A random geometry on coming from quantum gravity, S conditioned on being conformally equivalent to ( , g), S can be written as ( , e'g) for some random function ' on . S S Polyakov (1981), Quantum geometry of bosonic strings. The random function ' is governed by the Liouville field theory, the 2D quantum field theory defined by Liouville action. Liouville field theory is a conformal field theory, and is a locally trivial but globally nontrivial perturbation of Gaussian free field. 1 / 30 Today’s talk Explain an ongoing effort to mix the two directions. Present a few results that this effort has achieved. Two directions inspired by Polyakov’s work Direction 1: random planar geometry Liouville quantum gravity (LQG): random geometry for exp(Gaussian free field + perturbations). LQG = scaling limit of random planar maps; coupling with Schramm-Loewner evolution. Direction 2: Liouville conformal field theory (CFT) Study Liouville field theory as a primary example of 2D CFT. constructive quantum field theory, path integral; rich integrable structure as a CFT = exact formulae. ) 2 / 30 Two directions inspired by Polyakov’s work Direction 1: random planar geometry Liouville quantum gravity (LQG): random geometry for exp(Gaussian free field + perturbations). LQG = scaling limit of random planar maps; coupling with Schramm-Loewner evolution. Direction 2: Liouville conformal field theory (CFT) Study Liouville field theory as a primary example of 2D CFT. constructive quantum field theory, path integral; rich integrable structure as a CFT = exact formulae. ) Today’s talk Explain an ongoing effort to mix the two directions. Present a few results that this effort has achieved. 2 / 30 1 Convergence holds almost surely in H− (D). 1 h is not pointwise defined but h H− (D). 2 E[h(x)h(y)]⇢1(x)⇢2(y) dxdy := E[ h⇢1dx h⇢2dy]. · E[h(x)h(y)]= log x y + a smooth function. R − | − | R R Gaussian free field on the disk 'n n 1: non-constant eigenfunctions { } ≥ of the Laplace operator ∆ on the unit disk D, normalized by ' 2 = 2⇡ and ' = 0. D |r n| @D n R R Gaussian free field (GFF) on D h := n1=1 ↵n'n, where ↵ are i.i.d. standardP Gaussian random variables. { n} 3 / 30 Gaussian free field on the disk 'n n 1: non-constant eigenfunctions { } ≥ of the Laplace operator ∆ on the unit disk D, normalized by ' 2 = 2⇡ and ' = 0. D |r n| @D n R R Gaussian free field (GFF) on D h := n1=1 ↵n'n, where ↵ are i.i.d. standardP Gaussian random variables. { n} 1 Convergence holds almost surely in H− (D). 1 h is not pointwise defined but h H− (D). 2 E[h(x)h(y)]⇢1(x)⇢2(y) dxdy := E[ h⇢1dx h⇢2dy]. · E[h(x)h(y)]= log x y + a smooth function. R − | − | R R 3 / 30 1 Qc Let (h, c) H− (D) R be sampled from e− dP dc. 2 ⇥ D Liouville field on D γ 1( ) = + LFD is the measure on H− D describing the law of X h c. The random function X is called the Liouville field on D. Take-home: Liouville field is a simple modification of GFF. Liouville field on the disk Global parameters: γ (0, 2) and Q = γ + 2 . 2 2 γ 1 PD: probability measure on H− (D) for GFF on D. 1 dP dc: product measure on H− (D) R;(non-probability). D ⇥ Remark: Most measures in this talk are non-probability! But we will still use vocabulary from probability such as “sample”, “law”, “random” etc. 4 / 30 Liouville field on the disk Global parameters: γ (0, 2) and Q = γ + 2 . 2 2 γ 1 PD: probability measure on H− (D) for GFF on D. 1 dP dc: product measure on H− (D) R;(non-probability). D ⇥ Remark: Most measures in this talk are non-probability! But we will still use vocabulary from probability such as “sample”, “law”, “random” etc. 1 Qc Let (h, c) H− (D) R be sampled from e− dP dc. 2 ⇥ D Liouville field on D γ 1( ) = + LFD is the measure on H− D describing the law of X h c. The random function X is called the Liouville field on D. Take-home: Liouville field is a simple modification of GFF. 4 / 30 Liouville field with insertions LFγ (↵ , ,↵ ; β , ,β ):= m V (w ) n V (z ) LFγ . m,n 1 ··· m 1 ··· n i=1 ↵i i j=1 βj j · D Vertex insertion operator Q Q 1 ↵X(w) ↵X"(w) Var[↵X"(w)] V↵(w)=e := lim" 0 e − 2 β ! β 1 β X(z) X"(z) Var[ X"(z)] Vβ(z)=e 2 := lim" 0 e 2 − 2 2 . ! ↵,β R and w D, z @D. X": regularization of X. 2 2 2 γ γ LFm,n is obtained from LFD via a Cameron-Martin shift. Locally amounts to add ↵ log w and β log z − i |·− i | − i |·− j | γ If X is a sample from LFm,n, X is called a Liouville field with insertions at wi of weight ↵i , and zj of weight βj . wi : bulk insertion; zj : boundary insertion. •izj 5 / 30 2 γ /2 γh" 2 := lim" 0 " e d z in probability, γ (0, 2). ! 2 h"(z): average over the circle of radius " around z. eγhdx = Gaussian multiplicative chaos of h. Kahane (1985). γ-LQG boundary length for GFF Dup.-She. (2008) γ h Lh := e 2 dx; also via Gaussian multiplicative chaos. γ-LQG metric Dh for GFF (not needed for this talk) Dubedat-Ding-Dunlap-Falconet(2019) + Gwynne-Miller (2019) (Ah, Lh, Dh): the γ-LQG random geometry induced by h. If X is a Liouville field, we can similarly define (AX , LX , DX ). Liouville quantum gravity γ-LQG area for GFF Duplantier-Sheffield (2008) γh 2 Ah := e d z 6 / 30 eγhdx = Gaussian multiplicative chaos of h. Kahane (1985). γ-LQG boundary length for GFF Dup.-She. (2008) γ h Lh := e 2 dx; also via Gaussian multiplicative chaos. γ-LQG metric Dh for GFF (not needed for this talk) Dubedat-Ding-Dunlap-Falconet(2019) + Gwynne-Miller (2019) (Ah, Lh, Dh): the γ-LQG random geometry induced by h. If X is a Liouville field, we can similarly define (AX , LX , DX ). Liouville quantum gravity γ-LQG area for GFF Duplantier-Sheffield (2008) 2 γh 2 γ /2 γh" 2 Ah := e d z := lim" 0 " e d z in probability, γ (0, 2). ! 2 h"(z): average over the circle of radius " around z. 6 / 30 γ-LQG boundary length for GFF Dup.-She. (2008) γ h Lh := e 2 dx; also via Gaussian multiplicative chaos. γ-LQG metric Dh for GFF (not needed for this talk) Dubedat-Ding-Dunlap-Falconet(2019) + Gwynne-Miller (2019) (Ah, Lh, Dh): the γ-LQG random geometry induced by h. If X is a Liouville field, we can similarly define (AX , LX , DX ). Liouville quantum gravity γ-LQG area for GFF Duplantier-Sheffield (2008) 2 γh 2 γ /2 γh" 2 Ah := e d z := lim" 0 " e d z in probability, γ (0, 2). ! 2 h"(z): average over the circle of radius " around z. eγhdx = Gaussian multiplicative chaos of h. Kahane (1985). 6 / 30 If X is a Liouville field, we can similarly define (AX , LX , DX ). Liouville quantum gravity γ-LQG area for GFF Duplantier-Sheffield (2008) 2 γh 2 γ /2 γh" 2 Ah := e d z := lim" 0 " e d z in probability, γ (0, 2).