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 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 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 Conformal Field Theory: the legacy of Polyakov Polyakov introduces LCFT: Polyakov (1981): Quantum geometry of bosonic strings. Conformal Field Theory to solve LCFT: Belavin, Polyakov, Zamolodchikov (1984): A. Polyakov Infinite Conformal Symmetry in Two- Dimensional Quantum Field Theory. 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 jrφj +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 2 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 1 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 2 S: < eα1φ(z)eα2φ(0)eα3φ(1)eα4φ(1) > Z 1 DOZZ DOZZ 2 = Cγ (α1; α2; Q − iP)Cγ (Q + iP; α3; α4)jFP (z)j 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 2 n i 2 C R The Υ γ function defined as analytic continuation of 2 ! Z 1 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 2 C n iR. For all µ > 0 and γ 2 C n iR and α1; α2; α3 2 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 action: Z 1 2 γX (z) 2 SL(X ) := jrg X (z)j + 2QX (z) + 4πµe g(z) d z 4π S2 4 2 γ with g(z) = (1+jzj2)2 round metric, γ 2]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 8i; 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 Free Field 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: 0 1 n 1 Y αi φ(zi ) Y −s −s h e iγ,µ = A @ α α A µ Γ(s)E[Z1 ] jzj − zk j j k i=1 1≤j<k≤n 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 jz − zi j i C i=1 with Xg GFF with vanishing mean on the sphere. The KPZ formula Theorem (DKRV, 2014) Let (αi )i satisfy (∗). If is a M¨obiustransform, we have n n n Y αi φ( (zi )) Y 0 −2∆α Y αi φ(zi ) h e iγ,µ = j (zi )j 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(1) = jz1 − z2j jz2 − z3j jz1 − z3j he e e iγ,µ where: • ∆12 = ∆α3 − ∆α1 − ∆α2 , etc... α φ(0) α φ(1) α φ(1) • he 1 e 2 e 3 iγ,µ is the 3 point structure constant α φ(0) α φ(1) α φ(1) 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(1) = jz1 − z2j jz2 − z3j jz1 − z3j he e e iγ,µ where: • ∆12 = ∆α3 − ∆α1 − ∆α2 , etc... α φ(0) α φ(1) α φ(1) • he 1 e 2 e 3 iγ,µ is the 3 point structure constant α φ(0) α φ(1) α φ(1) 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(1)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 jzj jz − 1j with Xg GFF with vanishing mean on the sphere (and 4 g(z) = (1+jzj2)2 ). The DOZZ formula Recall the (∗) condition P3 i=1 αi 2 8i; αi < Q and Q − < ^ inf (Q − αi )(∗) 2 γ 1≤i≤3 Theorem (Kupiainen, Rhodes, V., 2017) For all γ 2 (0; 2) and (αi ) satisfying (∗) the following identity holds α1φ(0) α2φ(1) α3φ(1) DOZZ he e e iγ,µ = Cγ,µ (α1; α2; α3) Some comments on the previous theorem • Previous result provides analytic continuation for γ 2 C n 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 {− 2 ; − γ g 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(1) he 2 e 2 e 3 iγ,µ γ (α1− )φ(0) α φ(1) α φ(1) 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 .