Liouville Quantum Field Theory

Rémi Rhodes 1

University Paris-Est Marne La Vallée

1based on joint works with: C. Guillarmou, F. David, A. Kupiainen, H. Lacoin, V. Vargas

Outline

Motivation in 2d-quantum gravity

Liouville quantum field theory (LQFT)

Semiclassical limit

Plan of the talk

Motivation in 2d-quantum gravity

Liouville quantum field theory (LQFT)

Semiclassical limit

General principles in physics

Euclidean field theories are described by a functional (called action):

S : f ∈ Σ 7→ S(f ) ∈ R

where Σ is a functional space.

Classical physics: (principle of least action) study the minima of thisfunctional

Quantum physics: (probabilistic approach) construct the measure

e−S(f )Df

where Df is a "measure" on the functional space Σ.

Einstein-Hilbert action for gravity and matter fieldsFix a manifold M and denote Met(M) the set of Riemmannian metrics g on M.

• General relativity gives the action for the gravitational field

SEH(g) =1

2κ

∫M

Rg dVg + µ

∫M

dVg

where the functional space is Met(M).

• Coupling with matter: consider an action for matter field

ψ 7→ Sm(g, ψ)

defined on some functional space Θ made up of maps ψ : M → Rp.

The total action for Gravity+matter is

S(g, ψ) = SEH(g) + Sm(g, ψ)

acting on the functional space Σ = Met(M)×Θ.

Notations: 4g = Laplacian, ∇g = gradient , Rg=Ricci curvature, Vg=volume form

Einstein-Hilbert action for gravity and matter fieldsFix a manifold M and denote Met(M) the set of Riemmannian metrics g on M.

• General relativity gives the action for the gravitational field

SEH(g) =1

2κ

∫M

Rg dVg + µ

∫M

dVg

where the functional space is Met(M).

• Coupling with matter: consider an action for matter field

ψ 7→ Sm(g, ψ)

defined on some functional space Θ made up of maps ψ : M → Rp.

The total action for Gravity+matter is

S(g, ψ) = SEH(g) + Sm(g, ψ)

acting on the functional space Σ = Met(M)×Θ.

Notations: 4g = Laplacian, ∇g = gradient , Rg=Ricci curvature, Vg=volume form

Quantum gravity

Action for Gravity+matter is

S(g, ψ) = SEH(g) + Sm(g, ψ).

classical gravity: study the minima of this functional (yields Einstein’s fieldequation)

quantum gravity: construct the measure

e−S(g,ψ)DgDψ

on the functional space Met(M)×Θ where Dg is the volume form of theDeWitt metric.

Features:- non renormalizable in 4d .- Einstein-Hilbert functional is trivial in 2d , hence more tractable.

Polyakov’s breakthrough (1981)(works if the matter field possesses conformal symmetries in 2d).

Roughly and on the Riemann sphere:Decompose each metric g as

g = d∗(ehg)

where d is a diffeomorphism, h is a scalar function and g is the roundmetric.apply the corresponding change of variables to the DeWitt metric and get∫

F (g)e−S(g,ψ)DgDψ = C(g)

∫F (eγX g)e−SL(g,X)DX

where SL is the Liouville action functional

14π

∫M

(|∂gX |2 + QRgX + 4πµeγX)dVg

with γ ∈ (0,2), Q = 2γ + γ

2 and µ > 0.⇒ law of the metric described by Liouville QFT

Notations: 4g = Laplacian, ∂g = gradient , Rg=Ricci curvature, Vg=volume form

Sketchy state of the art1 Polyakov’s decomposition

Regularized determinants (Ray, Singer, D’Hoker, Phong, Kurzepa, Sarnak...)

Random planar maps

Ferrari, Klevtsov, Zelditch’s (2011) approach based on Kähler geometry

2 Construction of Liouville QFTAlbeverrio, Hoegh Krohn, Paycha, Scarlatti (’92) Hoegh-Krohn model

Takhtajian (’93) geometrical approachPerturbative series expansion around the minimum of the action

David, Kupiainen, Rhodes, Vargas (2014): Construction on the sphere

Guillarmou, Rhodes, Vargas (2016): Construction on hyperbolic surfacestowards a rigorous construction of string theory

3 DOZZ formula (quantitative study of Liouville QFT)formula proposed by Dorn/Otto/Zamolodchikov brothers

convincing arguments given by Teschner under a set of "axioms" supposedlysatisfied by the Liouville QFT.

Plan of the talk

Motivation in 2d-quantum gravity

Liouville quantum field theory (LQFT)

Semiclassical limit

Classical Liouville theory (uniformization)

(a) E. Picard (b) H.Poincaré

Goal: Given a 2d Riemann manifold (M,g0), find a metric g conformallyequivalent to g0, i.e. of the form

g = eγX g0

for some γ ∈ R, with constant Ricci scalar curvature −2πγ2µ

Rg = −2πγ2µ (?)

Notations: 4g = Laplacian, ∂g = gradient , Rg=Ricci curvature, Vg=volume form

Classical Liouville theory (uniformization)Goal: Given a 2d Riemann manifold (M,g0), find a metric g conformallyequivalent to g0, i.e. of the form

g = eγX g0

for some γ ∈ R, with constant Ricci scalar curvature −2πγ2µ

Rg = −2πγ2µ (?)

Liouville equation: If X : M → R solves the equation

γ4g0X − Rg0 = 2πµγ2eγX

then the metric g = eX g0 satisfies (?).

Liouville action: Find such a X by minimizing the functional

X 7→ SL(g0,X ) =1

4π

∫M

(|∂g0X |2 + QclRg0X + 4πµeγX

)dVg0

with Qcl = 2γ

Notations: 4g = Laplacian, ∂g = gradient , Rg=Ricci curvature, Vg=volume form

LQFT on the Riemann sphere S2 = C ∪ {∞}We see the Riemann sphere as the complex plane C equipped with the roundmetric

g(z) =4

(1 + |z|2)2 ,

hence Rg = 2.

Fix γ ∈]0,2], Q = 2γ + γ

2 and µ > 0.

Theorem (DKRV, 2014)

Pick a g in the conformal class of g. One can make sense of

e−SL(X ,g)DX

where DX is the "uniform measure" on maps X : S2 → R and SL Liouvilleaction:

SL(g,X ) :=1

4π

∫S2

(|∂gX |2 + QRgX + 4πµeγX)dVg

as a measure on H−s(S2,g) for s > 0.

Notations: 4g = Laplacian, ∂g = gradient , Rg=Ricci curvature, Vg=volume form

Idea of the construction

νL(DX ) = e−Q

4π

∫S2 RgX dVg−µ

∫S2 eγX dVg × e−

14π

∫S2 |∂gX |2 dVg DX︸ ︷︷ ︸

Gaussian measure (GFF) on H−s(S2,g)

Figure: J.-P. Kahane

The exponential is ill definedon H−s(S2,g) with s > 0⇒ make sense of∫

S2eγX dVg

by means of renormalization theory:

Gaussian multiplicative chaos, Kahane 1985

Definition of the correlation functions

Important feature of Quantum Field Theories: the correlation functions

they stand for observables of the systemthey are in some cases exactly computable

In Liouville QFT, they are defined for distinct zi ∈ S2 and αi ∈ R by

〈n∏

i=1

eαiφ(zi )〉g :=

∫ ( n∏i=1

eαiφ(zi ))

e−SL(X ,g)DX

where φ is the Liouville field

φ(z) = X (z) +Q2

ln g(z).

Link with classical Liouville theory

Formally

〈n∏

i=1

eαiφ(zi )〉g := C∫

e−S(zi ,αi )iL (X ,g)DX

where

S(zi ,αi )iL (X ,g) =

14π

∫Σ

(|∂gX |2 + QRgX + 4πµeγX)dVg −

∑i

αiX (zi )

Classically, the minimum X of this functional solves the Liouville equation withsources

γ4gX − Rg = 2πµγ2eγX − 2πγ∑

i

αiδzi .

The metric eγX g has uniformized curvature on S2 \ {z1, . . . , zn} and conicalsingularities at the points (zi )i .

Existence of the correlation functions

Theorem (DKRV, 2014)

The correlation 〈∏n

i=1 eαiφ(zi )〉g exists and is non trivial if and only if:

∀i , αi < Q andn∑

i=1

αi > 2Q (Seiberg bounds)

In particular, existence implies n ≥ 3!

Axiomatic of Conformal Field Theories

Let (αi )i satisfy the Seiberg bounds then

Theorem (DKRV, 2014)

1) Conformal covariance: If ψ is a Mobius transform, we have

〈n∏

i=1

eαiφ(ψ(zi ))〉g =n∏

i=1

|ψ′(zi )|−2∆αi 〈n∏

i=1

eαiφ(zi )〉g

where ∆αi = αi2 (Q − αi

2 ) is called conformal weight of eαi X(z).

2) Conformal anomaly: If g′ = eϕg then

ln〈∏n

i=1 eαiφ(zi )〉g′〈∏n

i=1 eαiφ(zi )〉g= −1 + 6Q2

96π

∫C

(|∂gϕ|2 + 2Rgϕ

)dVg

Plan of the talk

Motivation in 2d-quantum gravity

Liouville quantum field theory (LQFT)

Semiclassical limit

Context: Quantify how the quantum theory is dominated by the classical one

Setup: consider the following probability law P on H−s(S2,g) with expectation

E [F (φ)] := C∫

F (φ)( n∏

i=1

eαiφ(zi ))

e−SL(X ,g)DX

whereφ = X + Q

2 ln g is the Liouville field

F is a continuous functional on H−s(S2,g)

C is a renormalization constant such that E [1] = 1

Goal: Find the limit in law of the field γφ under the probability law P when

γ → 0 and µ =Λ

γ2 , and ∀i , αi =χi

γ

for fixed χi < 2.

Semiclassical limit (LRV, 2014)

As γ → 0, the field γφ converges in probability under P towards the solutionφ? to the equation

4φ? = 2πΛeφ? − 2π∑

i

χiδzi

Fluctuations (LRV, 2014)

As γ → 0, the field φ− 1γφ? converges in law under P towards a Massive Free

Field in the metric eφ?(z)dz2

One can also prove a large deviation principle.

Remark: This result combined with the BPZ equations provides a proof ofPolyakov’s conjecture for the accessory parameters (see V. Vargas’ talk)

Make the change of variables γX− > X ′ in the correlation functions∫ ( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX

to get for small γ∫ ( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼∫

e−1γ2 S(X ′)DX ′

withS(h) :=

14π

∫ (|∂gh|2 + 2Rgh + 4πΛeh

)dVg −

∑i

χih(zi ).

Hence (Laplace’s method)∫ ( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼ e−1γ2 S(h?)

whereS(h?) = min

hS(h)

Take a linear functional L on H−s. Similarly (γX− > X ′)∫eL(X/γ)

( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼∫

e−1γ2

(S(X ′)−L(X ′)

)DX ′

Hence (Laplace’s method)∫eL(X/γ)

( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼ e−1γ2 minh

(S(h)−L(h)

).

Otherwise stated

γ2 ln E [eγ−2L(γX)] ∼ −min

h

(S(h)− S(h?)− L(h)

).

By standard large deviation results

P[γX − h? ∈ A] ∼ e−1γ2 minh∈A(S(h+h?)−S(h?))

for A ⊂ H−s.

Take a linear functional L on H−s. Similarly (γX− > X ′)∫eL(X/γ)

( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼∫

e−1γ2

(S(X ′)−L(X ′)

)DX ′

Hence (Laplace’s method)∫eL(X/γ)

( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼ e−1γ2 minh

(S(h)−L(h)

).

Otherwise stated

γ2 ln E [eγ−2L(γX)] ∼ −min

h

(S(h)− S(h?)− L(h)

).

By standard large deviation results

P[γX − h? ∈ A] ∼ e−1γ2 minh∈A(S(h+h?)−S(h?))

for A ⊂ H−s.

Take a linear functional L on H−s. Similarly (γX− > X ′)∫eL(X/γ)

( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼∫

e−1γ2

(S(X ′)−L(X ′)

)DX ′

Hence (Laplace’s method)∫eL(X/γ)

( n∏i=1

eαi X(zi ))

e−SL(X ,g)DX ∼ e−1γ2 minh

(S(h)−L(h)

).

Otherwise stated

γ2 ln E [eγ−2L(γX)] ∼ −min

h

(S(h)− S(h?)− L(h)

).

By standard large deviation results

P[γX − h? ∈ A] ∼ e−1γ2 minh∈A(S(h+h?)−S(h?))

for A ⊂ H−s.

Thanks!