Proof of the DOZZ Formula Antti Kupiainen joint work with R. - PowerPoint PPT Presentation

Proof of the DOZZ Formula Antti Kupiainen joint work with R. Rhodes, V. Vargas Diablerets February 12 2018

DOZZ formula Dorn, Otto (1994) and Zamolodchikov, Zamolodchikov (1996): Γ( γ 2 4 − γ 2 4 ) 2 Q − ¯ α 4 ) ( γ C γ ( α 1 , α 2 , α 3 ) =( π µ 2 ) ) 2 γ Γ( 1 − γ 2 Υ ′ ( 0 )Υ( α 1 )Υ( α 2 )Υ( α 3 ) × α − α 1 ¯ α − α 2 ¯ α − α 3 ¯ Υ( ¯ α − 2 Q )Υ( )Υ( )Υ( ) 2 2 2 2 ◮ α i ∈ C , γ ∈ R , µ > 0 α = α 1 + α 2 + α 3 , Q = γ 2 + 2 ◮ ¯ γ Υ is an entire function on C with simple zeros defined by � ∞ 2 − α ) 2 e − t − sinh 2 (( Q 2 − α ) t 2 ) ) dt (( Q log Υ( α ) = sinh ( t γ 4 ) sinh ( t t γ ) 0

Structure Constant C γ ( α 1 , α 2 , α 3 ) is the structure constant of Liouville Conformal Field Theory Physics: ◮ String theory, 2d quantum gravity, ◮ 4d Yang-Mills; AGT correspondence Mathematics: ◮ Fractal random surfaces ◮ Integrable systems, Quantum cohomology The DOZZ formula is an exact expression for a fundamental object in a non-trivial quantum field theory, a rare thing!

Liouville Theory Liouville field is a 2d random field φ ( z ) with distribution � F ( φ ) e − S ( φ ) D φ E ( F ( φ )) = with S ( φ ) the Liouville action functional : � ( | ∂ z φ ( z ) | 2 + µ e γφ ( z ) ) dz S ( φ ) = Σ ◮ φ : Σ → R , Σ Riemann surface ◮ µ > 0 "cosmological constant" ◮ γ ∈ R (but γ ∈ i R also interesting) ◮ D φ : includes also Gauge fixing, integration over moduli

Gravitational Dressing Liouville theory enters study of spin systems on planar maps Example: Ising model. Let ◮ σ be scaling limit of Ising spin ◮ ˜ σ be scaling limit of Ising spin on a planar map Then σ ( z ) = e αφ ( z ) σ ( z ) ˜ √ 5 with φ the γ = 3 Liouville field and α = 3 . √ 2 Similar formuli for all c ≤ 1 CFTs (Potts, tricritical Ising, etc.) ◮ c = 25 − 6 Q 2 , Q = γ 2 + 2 γ ◮ α given by the KPZ relation Hence we need to understand correlation functions of vertex operators e αφ ( z ) in Liouville theory: n n � � � e α i φ ( z i ) � = e α i φ ( z i ) e − S ( φ ) D φ � i = 1 i = 1

Conformal Field Theory Liouville theory is a Conformal Field Theory . Belavin, Polyakov, Zamolodchikov ’84: Conformal Field Theory is determined by ◮ Central charge c of Virasoro algebra ◮ Spectrum : the set of primary fields Ψ i , i ∈ I ◮ Transform like tensors under conformal transformations ◮ E.g. in Ising model spin and energy are primary fields ◮ Three point functions � Ψ i ( z 1 )Ψ j ( z 2 )Ψ k ( z 3 ) � By Möbius invariance suffices to find structure constants C ( i , j , k ) = � Ψ i ( 0 )Ψ j ( 1 )Ψ k ( ∞ ) � BPZ: spectrum and structure constants determine all correlation functions by Conformal Bootstrap

Conformal Bootstrap Basic postulate of BPZ : in correlation functions operator product expansion (OPE)holds: � C k Ψ i ( z )Ψ j ( w ) = ij ( z , w )Ψ k ( w ) k where C k ij ( z , w ) are given in terms of structure constants C ( i , j , k ) . Iterating this n -point function is given in terms of structure constants. BPZ found C ( i , j , k ) for minimal models (e.g. Ising). Liouville model should be a CFT so can one solve it? BPZ failed to find structure constants for Liouville Conformal Field Theory is an "unsuccesful attempt to solve the Liouville model" (Polyakov)

DOZZ Conjecture The spectrum of Liouville was conjectured to be (Braaten, Curtright, Thorn, Gervais, Neveu, 1982): Ψ α = e αφ , α = Q + iP , P > 0 ( Q = γ 2 + 2 γ ) ’94-96 DOZZ gave an explicit formula for Liouville structure constants C ( α 1 , α 2 , α 3 ) = � e α 1 φ ( 0 ) e α 2 φ ( 1 ) e α 3 φ ( ∞ ) � Its original derivation was somewhat mysterious: "It should be stressed that the arguments of this section have nothing to do with a derivation. These are rather some motivations and we consider the expression proposed as a guess which we try to support in the subsequent sections"

Evidence for the DOZZ formula 1. Assume the full machinery of CFT (Teschner ’95) ◮ Fusion rules of degenerate fields ◮ Bootstrap of 4-point functions to 3-point functions ◮ A mysterious reflection relation e αφ = R ( α ) e ( 2 Q − α ) φ 2. Quantum integrability (Teschner ’01), Bytsko, Teschner (’02) 3. Bootstrap ◮ Numerical checks that DOZZ solves the quadratic bootstrap equations � � C m C m � Ψ i Ψ j ���� Ψ k Ψ l � �� � � = ij C mkl = ik C mjl = � Ψ i Ψ k � �� � Ψ j Ψ l ���� � m m ◮ Also DOZZ seems to be the only solution for c > 1 i.e. Liouville is the unique (unitary) CFT with c > 1 (Collier et al arxiv 1702.00423)!

Proof of the DOZZ conjecture Our proof of DOZZ: ◮ Rigorous construction of Liouville functional integral in terms of multiplicative chaos DKRV2014 ◮ Proof of the CFT machinery (Ward identities, BPZ equations) KRV2016 ◮ Probabilistic derivation of reflection relation KRV2017

Probabilistic Liouville Theory What is the meaning of ( | ∂ z φ ( z ) | 2 + µ e γφ ( z ) ) dz D φ ? � e − ( | ∂ z φ ( z ) | 2 D φ → Gaussian Free Field (GFF) ◮ e − � ◮ Work on sphere S 2 = C ∪ {∞} . ◮ On S 2 Ker (∆) = { constants } ◮ φ = c + ψ , c constant and ψ ⊥ Ker (∆) ◮ Inclusion of c is necessary for conformal invariance Then define the Liouville functional integral as n � � e γ ( c + ψ ( z )) dz � � � e α i φ ( z i ) � := e α i ( c + ψ ( z i )) e − µ e − 2 Qc dc � E R i = 1 e − 2 Qc is for topological reasons (work on S 2 ).

Renormalization GFF is not a function: E ψ ( z ) 2 = ∞ . Need to renormalize: e γψ ǫ ( z ) → e γψ ǫ ( z ) − γ 2 2 E ψ ǫ ( z ) 2 and define correlations by ǫ → 0 limit: Theorem (DKRV 2014) The Liouville correlations exist and are nontrivial if and only if: � ( A ) ∀ i : α i < Q and ( B ) α i > 2 Q i where Q := γ 2 + 2 γ . ◮ (A), (B) are called Seiberg bounds ◮ (A), (B) = ⇒ n ≥ 3: 1- and 2-point functions are ∞ .

0-mode �� � n � � i α i − 2 Q ) c e − µ e γ c � e γψ dz dc e α i φ ( z i ) � = E e α i ψ ( z i ) e ( � � R i = 1 i The c -integral converges if � i α i > 2 Q : �� � n � � e α i φ ( z i ) � = Γ( s ) e α i ψ ( z i ) ( e γψ dz ) − s � µ s γ E i = 1 i where s := ( � i α i − 2 Q ) /γ . This explains Seiberg bound (B)

Reduction to Multiplicative Chaos Shift (Cameron-Martin theorem) � ψ ( z ) → ψ ( z ) + α i E ψ ( z ) ψ ( z i ) � �� � i = − log | z − z i | Result: Liouville correlations are given by n � � � 1 � � − s Γ( s ) e α i φ ( z i ) � = | z − z i | γα i e γψ ( z ) dz � i < j | z i − z j | α i α j E µ s � i = 1 i ◮ e γψ ( z ) dz random metric (volume) ◮ Conical curvature singularities at insertions α i

Modulus of Chaos Seiberg bound (A): The multiplicative chaos measure e γψ dz has scaling dimension γ Q = 2 + γ 2 = ⇒ almost surely we have 2 � 1 | z − z i | γα i e γψ ( z ) dz < ∞ ⇔ α i < Q . So surprisingly ” e αφ ≡ 0 ” α ≥ Q

Structure constants We obtain a probabilistic expression for the structure constants �� ( max ( | z | , 1 )) γ ¯ � 2 Q − ¯ α α γ C ( α 1 , α 2 , α 3 ) ∝ E | z | γα 1 | 1 − z | γα 2 M γ ( dz ) α := α 1 + α 2 + α 3 . ¯ The DOZZ formula is an explicit conjecture for this expectation. Note! ◮ DOZZ formula is defined for α in spectrum i.e. α = Q + iP ◮ Probabilistic formula is defined for real α i satisfying Seiberg bounds ◮ Real α i are relevant for random surfaces

Dilemma The DOZZ proposal C DOZZ ( α 1 , α 2 , α 3 ) is a meromorphic function of α i ∈ C . In particular for real α ’s C DOZZ ( α 1 , α 2 , α 3 ) � = 0 if α i > Q The probabilistic C ( α 1 , α 2 , α 3 ) is identically zero in this region: C ( α 1 , α 2 , α 3 ) ≡ 0 α i ≥ Q What is going on? DOZZ is too beautiful to be wrong! Remark. One can renormalize e αφ for α ≥ Q so that C ( α 1 , α 2 , α 3 ) � = 0 . However the result does not satisfy DOZZ.

Analyticity Theorem (KRV 2017) The DOZZ formula holds for the probabilistic C ( α 1 , α 2 , α 3 ) . Ideas of proof ◮ Prove ODE’s for certain 4-point functions ◮ Combine the ODE’s with chaos methods to derive periodicity relations for structure constants ◮ Identify reflection coefficient R ( α ) as a tail exponent of multiplicative chaos ◮ Use R ( α ) to construct analytic continuation of C ( α 1 , α 2 , α 3 ) outside the Seiberg bound region ◮ Use the periodicity relations to determine C ( α 1 , α 2 , α 3 )

Belavin-Polyakov-Zamolodchicov equation Consider a 4-point function F ( u ) := � e − χφ ( u ) e α 1 φ ( 0 ) e α 2 φ ( 1 ) e α 3 φ ( ∞ ) � Theorem (KRV2016) For χ = γ 2 or χ = 2 γ , F satisfies a hypergeometric equation a b ∂ 2 u F + u ( 1 − u ) ∂ u F − u ( 1 − u ) F = 0 Proof: Gaussian integration by parts & regularity estimates. Remark : In CFT jargon e − χφ ( u ) are level 2 degenerate fields . In bootstrap approach this is postulated, we prove it.