A probabilistic approach to Liouville field theory

Vincent Vargas 1

ENS Paris

1based on a series of works with: David, Kupiainen, Rhodes Outline

1 LCFT in the

2 Path integral formulation of LCFT and probabilistic statement of the DOZZ formula

3 Ideas of the probabilistic proof of the DOZZ formula Plan of the talk

1 LCFT in the conformal bootstrap

2 Path integral formulation of LCFT and probabilistic statement of the DOZZ formula

3 Ideas of the probabilistic proof of the DOZZ formula : the legacy of Polyakov

Polyakov introduces LCFT: Polyakov (1981): of bosonic strings. Conformal Field Theory to solve LCFT: Belavin, Polyakov, Zamolodchikov (1984): A. Polyakov Infinite in Two- Dimensional .

The DOZZ proposal for LCFT in the bootstrap: Dorn-Otto (1994), Zamolodchikov-Zamolodchikov (1996). Unifying LCFT

Conformal Scaling limit Bootstrap of planar maps

Our program in LCFT ...... MIT/Cambridge ... program

Path integral: R 2 γφ R Q αk φ(zk ) − M |∇φ| +µe φ k e e Dφ

γ → 0 µγ2 → Λ

φ∗ γφ → φ∗, ∆φ∗ = Λe Motivation from discrete planar maps: courtesy of F. David

Figure: The scaling limit of large circle packed triangulations should be described by LCFT Motivation from discrete planar maps

Figure: Circles of the circle packed triangulation Motivation from discrete planar maps

The KPZ relation reads:

n Qn γφ(x1) γφ(x2) γφ(x3) Qn αφ(xi ) Y h i=4 σ(xi )ihe e e i=4 e iγ,µ h Φ(xi )i = heγφ(x1)eγφ(x2)eγφ(x3)i i=4 γ,µ where: 2 • x1, x2, x3 ∈ S fixed points in the embedding of the map in the sphere • σ scaling limit of primary field on regular lattice with conformal weight ∆σ • Φ scaling limit of same primary field on planar map

• Conformal weight condition: ∆σ + ∆α = 1 Introduction to LCFT in the conformal bootstrap

Main ingredients of LCFT in the conformal bootstrap:

• Shift equations to determine 3 point structure constant DOZZ Cγ (α1, α2, α3)

αφ(z) • Base of vertex operators e where α in S = Q + iR γ 2 (Reminder: Q = 2 + γ ).

• Recursive procedure called the Operator Product Expansion (OPE): Z ∞ α1φ(z1) α2φ(z2) X Q+iP ¯ (Q+iP)φ(z2) e e = Cα1,α2 (z1, z2)LL(e )dP −∞ L,L¯

where L and L¯ are differential operators acting on z2 andz ¯2. The 4 point correlation function

Using the OPE, one can then get an expression for the 4 point correlation function for (αi )i ∈ S:

< eα1φ(z)eα2φ(0)eα3φ(1)eα4φ(∞) > Z ∞ DOZZ DOZZ 2 = Cγ (α1, α2, Q − iP)Cγ (Q + iP, α3, α4)|FP (z)| dP −∞

where FP := F are the (universal) conformal blocks of (∆αi )i ,P CFT. Similarly for general n point correlations, etc... Zamolodchikov’s Υ γ function for γ ∈ \ i 2 C R

The Υ γ function defined as analytic continuation of 2

! Z ∞ 2 Q t 2 Q  −t (sinh(( 2 − z) 2 )) dt ln Υ γ (z) = − z e − 2 2 tγ t t 0 sinh( 4 ) sinh( γ )

for 0 < <(z) < <(Q). Remarkable functional relation

γ γz γ 1−γz 2 2z γ 4z −1 Υ γ (z+ ) = `( )( ) Υ γ (z), Υ γ (z+ ) = `( )( ) γ Υ γ (z) 2 2 2 2 2 2 γ γ 2 2

with `(x) = Γ(x)/Γ(1 − x). The DOZZ formula for γ ∈ C \ iR.

For all µ > 0 and γ ∈ C \ iR and α1, α2, α3 ∈ C,

2 γ γ 2 2Q−α¯ C DOZZ (α , α , α ) = (π µ `( )( )2−γ /2) γ γ,µ 1 2 3 4 2 0 Υ γ (0)Υ γ (α1)Υ γ (α2)Υ γ (α3) 2 2 2 2 × α¯−2Q α¯−2α α¯−2α α¯−2α Υ γ ( )Υ γ ( 1 )Υ γ ( 2 )Υ γ ( 3 ) 2 2 2 2 2 2 2 2

withα ¯ = α1 + α2 + α3.

Reminder: `(x) = Γ(x)/Γ(1 − x). Plan of the talk

1 LCFT in the conformal bootstrap

2 Path integral formulation of LCFT and probabilistic statement of the DOZZ formula

3 Ideas of the probabilistic proof of the DOZZ formula Path integral of LCFT on the Riemann sphere S2

Formal definition of correlations: n Z n ! Y αi φ(zi ) Y αi φ(zi ) −SL(X ) h e iγ,µ := e e DX , i=1 i=1 where • DX ”Lebesgue measure” on functional space

• SL Liouville : Z 1 2 γX (z) 2 SL(X ) := |∇g X (z)| + 2QX (z) + 4πµe g(z) d z 4π S2 4 2 γ with g(z) = (1+|z|2)2 round metric, γ ∈]0, 2], Q = γ + 2 and µ > 0. Q • Liouville field: φ(z) = X (z) + 2 ln g(z). Existence of the correlation functions

Theorem (DKRV, 2014)

Qn αi φ(zi ) One can define the correlations h i=1 e iγ,µ by a regularization procedure. The correlations are non trivial if and only if: Pn i=1 αi 2 ∀i, αi < Q and Q − < ∧ inf (Q − αi )(∗) 2 γ 1≤i≤n In particular, existence implies n ≥ 3!

Remark: see region I and II in Harlow, Maltz, Witten (2011).

Idea of proof: interpret the gradient term in Liouville action as Gaussian with average distributed as Lebesgue (zero mode). An explicit expression for the correlation functions

The existence is in fact based on the following explicit expression:

  n 1 Y αi φ(zi ) Y −s −s h e iγ,µ = A  α α  µ Γ(s)E[Z1 ] |zj − zk | j k i=1 1≤j

Pn i=1 αi −2Q where s = γ , A some constant (depending on the αi and γ) and

n ! Z 2 γ 2 Y 1 γ Pn γXg (z)− 2 E[Xg (z) ] 1− 4 i=1 αi 2 Z1 = e γα g(z) d z |z − zi | i C i=1

with Xg GFF with vanishing mean on the sphere. The KPZ formula

Theorem (DKRV, 2014)

Let (αi )i satisfy (∗). If ψ is a M¨obiustransform, we have

n n n Y αi φ(ψ(zi )) Y 0 −2∆α Y αi φ(zi ) h e iγ,µ = |ψ (zi )| i h e iγ,µ i=1 i=1 i=1

αi αi αi φ(z) where ∆αi = 2 (Q − 2 ) is the conformal weight of e .

γ 2 Reminder: Q = 2 + γ . 2 Central charge: cL = 1 + 6Q ≥ 25. DOZZ The DOZZ formula Cγ,µ (α1, α2, α3)...

The 3 point correlation function

By conformal covariance:

3 Y αi φ(zi ) h e iγ,µ i=1 2∆12 2∆23 2∆13 α1φ(0) α2φ(1) α3φ(∞) = |z1 − z2| |z2 − z3| |z1 − z3| he e e iγ,µ

where:

• ∆12 = ∆α3 − ∆α1 − ∆α2 , etc...

α φ(0) α φ(1) α φ(∞) • he 1 e 2 e 3 iγ,µ is the 3 point structure constant

α φ(0) α φ(1) α φ(∞) Exact expression for he 1 e 2 e 3 iγ,µ? The 3 point correlation function

By conformal covariance:

3 Y αi φ(zi ) h e iγ,µ i=1 2∆12 2∆23 2∆13 α1φ(0) α2φ(1) α3φ(∞) = |z1 − z2| |z2 − z3| |z1 − z3| he e e iγ,µ

where:

• ∆12 = ∆α3 − ∆α1 − ∆α2 , etc...

α φ(0) α φ(1) α φ(∞) • he 1 e 2 e 3 iγ,µ is the 3 point structure constant

α φ(0) α φ(1) α φ(∞) Exact expression for he 1 e 2 e 3 iγ,µ? DOZZ The DOZZ formula Cγ,µ (α1, α2, α3)... Exact expression the 3 point structure constants

We have the following expression withα ¯ = α1 + α2 + α3:

2Q−α¯ α¯ − 2Q  2Q−α¯ heα1φ(0)eα2φ(1)eα3φ(∞)i = Aµ γ Γ [Z γ ] γ,µ γ E 1

where A is some constant (depending on the αi and γ) and

γ 2 1− α¯ Z γ 2 g(z) 4 γXg (z)− E[Xg (z) ] 2 Z1 = e 2 d z γα1 γα2 C |z| |z − 1|

with Xg GFF with vanishing mean on the sphere (and 4 g(z) = (1+|z|2)2 ). The DOZZ formula

Recall the (∗) condition

P3 i=1 αi 2 ∀i, αi < Q and Q − < ∧ inf (Q − αi )(∗) 2 γ 1≤i≤3

Theorem (Kupiainen, Rhodes, V., 2017)

For all γ ∈ (0, 2) and (αi ) satisfying (∗) the following identity holds

α1φ(0) α2φ(1) α3φ(∞) DOZZ he e e iγ,µ = Cγ,µ (α1, α2, α3) Some comments on the previous theorem

• Previous result provides analytic continuation for γ ∈ C \ iR of the path integral approach

DOZZ • Observation: Cγ,µ (α1, α2, α3) invariant under duality:

4 γ2 γ 2 (µπ`( )) γ2 ↔ , µ ↔ µ˜ = 4 2 γ 4 π`( γ2 ) Path integral interpretation? Plan of the talk

1 LCFT in the conformal bootstrap

2 Path integral formulation of LCFT and probabilistic statement of the DOZZ formula

3 Ideas of the probabilistic proof of the DOZZ formula The quantum Fuchsian equation: the BPZ differential equation of order 2

− γ φ − 2 φ The fields e 2 and e γ satisfy BPZ of order 2: Theorem (Kupiainen, Rhodes, V., 2015) γ 2 For real (αi )i staisfying (∗), one has for α ∈ {− 2 , − γ }

1 3 3 ∆ 3 2 αφ(z) Y αi φ(zi ) X αk αφ(z) Y αi φ(zi ) 2 ∂zz he e iγ,µ + 2 he e iγ,µ α (z − zk ) i=1 k=1 i=1 3 1 3 X αφ(z) Y αi φ(zi ) + ∂zk he e iγ,µ = 0, z − zk k=1 i=1 Consequences of the BPZ equation

γ − φ(z) Q3 αi φ(zi ) By studying he 2 i=1 e iγ,µ around z = 0, 1, one gets by monodromy argument

γ (α1+ )φ(0) α φ(1) α φ(∞) he 2 e 2 e 3 iγ,µ γ (α1− )φ(0) α φ(1) α φ(∞) he 2 e 2 e 3 iγ,µ 2 2 γ γα1 α1γ γ γ γ 1 `(− 4 )`( 2 )`( 2 − 4 )`( 4 (¯α − 2α1 − 2 )) = − γ γ γ γ γ γ . πµ `( 4 (¯α − 2 − 2Q))`( 4 (¯α − 2α3 − 2 ))`( 4 (¯α − 2α2 − 2 ))

γ 2 and a dual equation where 2 ↔ γ .

Reminder: `(x) = Γ(x)/Γ(1 − x) andα ¯ = α1 + α2 + α3. Perspectives and open problems

Perspectives:

• Construction of the of LCFT based on the two Ward identities: probabilistic construction of eαφ(z) where α in S = Q + iR?(Kupiainen, Rhodes, V.). • DOZZ type formulas for LCFT with a boundary: Teschner, Zamolodchikov brothers (probabilistic statement: Remy).

• Can one make sense of the path integral for γ ∈ iR using the path integral? Can one relate this path integral to the imaginary DOZZ formula? Related to critical FK model, CLE, etc...