Proof of the DOZZ Formula

Transcription

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

2 DOZZ formula Dorn, Otto (1994) and Zamolodchikov, Zamolodchikov (1996): C γ (α 1, α 2, α 3 ) =(π µ γ2 Γ( 4 ) Γ(1 γ2 4 ) ( γ 2 ) 4 γ2 2 ) 2Q ᾱ γ Υ (0)Υ(α 1 )Υ(α 2 )Υ(α 3 ) Υ( ᾱ 2Q 2 )Υ( ᾱ α 1 2 )Υ( ᾱ α 2 2 )Υ( ᾱ α 3 2 ) α i C, γ R, µ > 0 ᾱ = α 1 + α 2 + α 3, Q = γ γ Υ is an entire function on C with simple zeros defined by log Υ(α) = 0 (( Q 2 α)2 e t sinh2 (( Q 2 α) 2 t ) sinh( tγ 4 ) sinh( t γ ))dt t

3 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!

4 Liouville Theory Liouville field is a 2d random field φ(z) with distribution E(F(φ)) = F(φ)e S(φ) Dφ with S(φ) the Liouville action functional: S(φ) = ( z φ(z) 2 + µe γφ(z) )dz Σ φ : Σ R, Σ Riemann surface µ > 0 "cosmological constant" γ R (but γ ir also interesting) Dφ: includes also Gauge fixing, integration over moduli

5 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) with φ the γ = 3 Liouville field and α = Similar formuli for all c 1 CFTs (Potts, tricritical Ising, etc.) c = 25 6Q 2, Q = γ 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) Dφ i=1 i=1

6 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

7 Conformal Bootstrap Basic postulate of BPZ: in correlation functions operator product expansion (OPE)holds: Ψ i (z)ψ j (w) = k C k ij (z, w)ψ k(w) where Cij k (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)

8 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 γ ) DOZZ gave an explicit formula for Liouville structure constants C(α 1, α 2, α 3 ) = e α 1φ(0) e α 2φ(1) e α 3φ( ) 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"

9 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 (2Q α)φ 2. Quantum integrability (Teschner 01), Bytsko, Teschner ( 02) 3. Bootstrap Numerical checks that DOZZ solves the quadratic bootstrap equations Ψ i Ψ j }{{} Ψ kψ l }{{} = m C m ij C mkl = m Cik m C mjl = Ψ i Ψ }{{ k } Ψ jψ l }{{} 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 )!

10 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

11 Probabilistic Liouville Theory What is the meaning of e ( zφ(z) 2 +µe γφ(z) )dz Dφ? e ( zφ(z) 2 Dφ Gaussian Free Field (GFF) 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 α i φ(z i ) := i=1 R [ E e α i (c+ψ(z i )) e µ ] e γ(c+ψ(z)) dz e 2Qc dc e 2Qc is for topological reasons (work on S 2 ).

12 Renormalization GFF is not a function: Eψ(z) 2 =. Need to renormalize: e γψɛ(z) γ2 γψɛ(z) e 2 Eψɛ(z)2 and define correlations by ɛ 0 limit: Theorem (DKRV 2014) The Liouville correlations exist and are nontrivial if and only if: (A) i : α i < Q and (B) i α i > 2Q where Q := γ γ. (A), (B) are called Seiberg bounds (A), (B) = n 3: 1- and 2-point functions are.

13 0-mode [ n e α i φ(z i ) = E e α i ψ(z i ) i=1 i R e ( i α i 2Q)c e µeγc e γψ dz dc ] The c-integral converges if i α i > 2Q: [ n e α i φ(z i ) = Γ(s) µ s γ E e α i ψ(z i ) ( i=1 i e γψ dz) s ] where s := ( i α i 2Q)/γ. This explains Seiberg bound (B)

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

15 Modulus of Chaos Seiberg bound (A): The multiplicative chaos measure e γψ dz has scaling dimension γq = 2 + γ2 2 = almost surely we have 1 z z i γα e γψ(z) dz < α i i < Q. So surprisingly e αφ 0 α Q

16 Structure constants We obtain a probabilistic expression for the structure constants C(α 1, α 2, α 3 ) E ᾱ := α 1 + α 2 + α 3. ( ) 2Q ᾱ (max( z, 1)) γᾱ γ z γα M 1 1 z γα γ (dz) 2 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

17 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.

18 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 )

19 Belavin-Polyakov-Zamolodchicov equation Consider a 4-point function F(u) := e χφ(u) e α 1φ(0) e α 2φ(1) e α 3φ( ) Theorem (KRV2016) For χ = γ 2 or χ = 2 γ, F satisfies a hypergeometric equation 2 uf + a u(1 u) b uf 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.

20 Solution We have the multiplicative chaos representation: ( F(u) E (max( z, 1)) γᾱ ) 2Q ᾱ γ z u γχ z γα M 1 1 z γα γ (dz) 2 Asymptotics as u 0: if α 1 + χ < Q then F(u) = C(α 1 χ, α 2, α 3 )+A(α 1 )C(α 1 +χ, α 2, α 3 ) u γ 2 (Q α 1) +... with A explicit ratio of 12 Γ-functions! This can be stated as a fusion rule: e χφ e αφ = e (α χ)φ + A(α)e (α+χ)φ which is postulated in the bootstrap approach.

21 Periodicity The BPZ equation implies a relation between the coefficients C(α 1 + χ, α 2, α 3 ) = D(χ, α 1, α 2, α 3 )C(α 1 χ, α 2, α 3 ) with χ = γ 2 or 2 γ. and D(χ, α 1, α 2, α 3 ) explicit. D(α 0, α 1, α 2, α 3 ) = 1 Γ( χ 2 )Γ( α 0 α 1 )Γ(χα 1 χ 2 )Γ( χ 2 (ᾱ 2α 1)) πµ Γ( χ 2 (ᾱ 2Q))Γ( χ 2 (ᾱ 2α 3))Γ( χ 2 (ᾱ 2α 2)) Γ(1 + χ 2 (2Q ᾱ))γ(1 + χ 2 (2α 3 ᾱ))γ(1 + χ 2 (2α 2 ᾱ)) Γ(1 + χ 2 )Γ(1 + χα 1 )Γ(1 χα 1 + χ 2 )Γ(1 + χ 2 (2α 1 ᾱ)) Teschner 95: DOZZ is the unique analytic solution of these equations if we could extend C(α 1, α 2, α 3 ) beyond the region α i < Q.

22 Reflection What happens if if α 1 + χ > Q? F(u) has same asymptotics with the replacement: C(α 1 + χ, α 2, α 3 ) R(α + χ)c(2q (α 1 + χ)), α 2, α 3 ) This can be stated informally as the reflection relation e αφ = R(α)e (2Q α)φ, α > Q. This is postulated in bootstrap since DOZZ satisfies it. We prove it with a probabilistic expression for R.

23 Reflection coefficient R(α) is given in terms of tail behavior of multiplicative chaos: Let α < Q, D any neighborhood of origin and 1 Z D := M(dz) z γα D Then P(Z > x) = R(α) x 2(Q α) γ (1 + o(z)) R(α) enters since asymptotics of the four point function F(u) = e χφ(u) e α 1φ(0) e α 2φ(1) e α 3φ( ) as u 0 is controlled by the singularity at α 1. R(α) has an explicit expression in terms of multiplicative chaos (2-point Quantum Sphere of Duplantier and Sheffield)

24 Integrability We use R to obtain analytic continuation of C(α 1, α 2, α 3 ) to α 1 > Q: C(α 1, α 2, α 3 ) = R(α 1 )C(2Q α 1, α 2, α 3 ) DOZZ formula satifies this with R DOZZ (α) = (( γ 2 ) γ2 2(Q α) Γ( 2 2 µ) γ 2 (α Q))Γ( 2 γ (α Q)) γ Γ( γ 2 (Q α))γ( 2 γ (Q α)). To conclude the proof of DOZZ formula we need to show the probabilistic R equals R DOZZ. We prove R can be expressed in terms of the structure constants = periodicity relations for R that determine it.

25 Outlook Our proof settles some physics puzzles: Duality: DOZZ invariant under γ 4 γ with: µ µ = (µπl( γ2 4 )) 4 γ 2 (πl( 4 γ 2 )) 1 (l(x) = Γ(x) Γ(1 x) ). but Liouville action functional is not! Reflection: where does the relation e αφ = R(α)e (2Q α)φ come from? Should one e.g. modify the theory by adding a µe 4 γ φ term in the action? The answer is no! Challenges: Find the spectrum and prove general bootstrap Analyticity in γ: Liouville theory is believed to exist for all c R. In paricular for c < 1 (γ ir) it is conjectured to describe random cluster Potts models (Ribault, Santa-Chiara)