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): Cγ(α1, α2, α3) =(π µ

Γ( γ2

4 )

Γ(1− γ2

4 ) ( γ

2 )

4−γ2 2

)

2Q− ¯ α γ

× Υ′(0)Υ(α1)Υ(α2)Υ(α3) Υ( ¯

α−2Q 2

)Υ(

¯ α−α1 2

)Υ(

¯ α−α2 2

)Υ(

¯ α−α3 2

)

◮ αi ∈ C, γ ∈ R, µ > 0 ◮ ¯

α = α1 + α2 + α3, Q = γ

2 + 2 γ

Υ is an entire function on C with simple zeros defined by log Υ(α) = ∞ (( Q

2 − α)2e−t − sinh2(( Q

2 − α) t 2 )

sinh( tγ

4 ) sinh( t γ )

)dt t

Structure Constant

Cγ(α1, α2, α3) 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

• bject in a non-trivial quantum field theory, a rare thing!

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) =

• Σ

(|∂zφ(z)|2 + µeγφ(z))dz

◮ 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

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) = eαφ(z)σ(z) with φ the γ = √ 3 Liouville field and α =

5 2 √ 3.

Similar formuli for all c ≤ 1 CFTs (Potts, tricritical Ising, etc.)

◮ c = 25 − 6Q2, Q = γ 2 + 2 γ ◮ α given by the KPZ relation

Hence we need to understand correlation functions of vertex

• perators eαφ(z)in Liouville theory:
• n
• i=1

eαiφ(zi) =

• n
• i=1

eαiφ(zi)e−S(φ)Dφ

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 Ψi, i ∈ I

◮ Transform like tensors under conformal transformations ◮ E.g. in Ising model spin and energy are primary fields

◮ Three point functions Ψi(z1)Ψj(z2)Ψk(z3)

By Möbius invariance suffices to find structure constants C(i, j, k) = Ψi(0)Ψj(1)Ψk(∞) 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: Ψi(z)Ψj(w) =

• k

Ck

ij (z, w)Ψk(w)

where Ck

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): Ψα = eαφ, α = Q + iP, P > 0 (Q = γ

2 + 2 γ )

’94-96 DOZZ gave an explicit formula for Liouville structure constants C(α1, α2, α3) = eα1φ(0)eα2φ(1)eα3φ(∞) 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 eαφ = 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 ΨiΨj ΨkΨl =

• m

Cm

ij Cmkl =

• m

Cm

ik Cmjl = ΨiΨk

ΨjΨl

• ◮ 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 e−

• (|∂zφ(z)|2+µeγφ(z))dzDφ ?

◮ e−

• (|∂zφ(z)|2Dφ → Gaussian Free Field (GFF)

◮ Work on sphere S2 = C ∪ {∞}. ◮ On S2 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
• i=1

eαiφ(zi) :=

• R

E

• eαi(c+ψ(zi))e−µ
• eγ(c+ψ(z))dz

e−2Qcdc e−2Qc is for topological reasons (work on S2).

Renormalization

GFF is not a function: Eψ(z)2 = ∞. Need to renormalize: eγψǫ(z) → eγψǫ(z)− γ2

2 Eψǫ(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 : αi < Q and (B)

• i

αi > 2Q 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=1

eαiφ(zi) = E

• i

eαiψ(zi)

• R

e(

i αi−2Q)ce−µeγc

eγψdzdc

• The c-integral converges if

i αi > 2Q:

• n
• i=1

eαiφ(zi) = Γ(s)

µsγ E

• i

eαiψ(zi)(

• eγψdz)−s
• where s := (

i αi − 2Q)/γ.

This explains Seiberg bound (B)

Reduction to Multiplicative Chaos

Shift (Cameron-Martin theorem) ψ(z) → ψ(z) +

• i

αi Eψ(z)ψ(zi)

• =− log |z−zi|

Result: Liouville correlations are given by

• n
• i=1

eαiφ(zi) =

Γ(s) µs

i<j |zi−zj|αi αj E

i

1 |z − zi|γαi eγψ(z)dz −s

◮ eγψ(z)dz random metric (volume) ◮ Conical curvature singularities at insertions αi

Modulus of Chaos

Seiberg bound (A): The multiplicative chaos measure eγψdz has scaling dimension γQ = 2 + γ2

2

= ⇒ almost surely we have

• 1

|z − zi|γαi eγψ(z)dz < ∞ ⇔ αi < Q. So surprisingly ”eαφ ≡ 0” α ≥ Q

Structure constants

We obtain a probabilistic expression for the structure constants C(α1, α2, α3) ∝ E (max(|z|, 1))γ ¯

α

|z|γα1|1 − z|γα2 Mγ(dz) 2Q− ¯

α γ

¯ α := α1 + α2 + α3. 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 αi satisfying

Seiberg bounds

◮ Real αi are relevant for random surfaces

Dilemma

The DOZZ proposal CDOZZ(α1, α2, α3) is a meromorphic function of αi ∈ C. In particular for real α’s CDOZZ(α1, α2, α3) = 0 if αi > Q The probabilistic C(α1, α2, α3) is identically zero in this region: C(α1, α2, α3) ≡ 0 αi ≥ Q What is going on? DOZZ is too beautiful to be wrong!

• Remark. One can renormalize eαφ for α ≥ Q so that

C(α1, α2, α3) = 0. However the result does not satisfy DOZZ.

Analyticity

Theorem (KRV 2017) The DOZZ formula holds for the probabilistic C(α1, α2, α3). 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(α1, α2, α3) outside the Seiberg bound region

◮ Use the periodicity relations to determine C(α1, α2, α3)

Belavin-Polyakov-Zamolodchicov equation

Consider a 4-point function F(u) := e−χφ(u)eα1φ(0)eα2φ(1)eα3φ(∞) Theorem (KRV2016) For χ = γ

2 or χ = 2 γ , F satisfies a

hypergeometric equation ∂2

uF +

a u(1 − u)∂uF − b 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.

Solution

We have the multiplicative chaos representation: F(u) ∝ E

• (max(|z|, 1))γ ¯

α

|z − u|−γχ|z|γα1|1 − z|γα2 Mγ(dz) 2Q− ¯

α γ

Asymptotics as u → 0: if α1 + χ < Q then F(u) = C(α1−χ, α2, α3)+A(α1)C(α1+χ, α2, α3)|u|

γ 2 (Q−α1)+. . .

with A explicit ratio of 12 Γ-functions! This can be stated as a fusion rule: e−χφ × eαφ = e(α−χ)φ + A(α)e(α+χ)φ which is postulated in the bootstrap approach.

Periodicity

The BPZ equation implies a relation between the coefficients C(α1 + χ, α2, α3) = D(χ, α1, α2, α3)C(α1 − χ, α2, α3) with χ = γ

2 or 2 γ . and D(χ, α1, α2, α3) explicit.

D(α0, α1, α2, α3) = − 1 πµ Γ(−χ2)Γ(−α0α1)Γ(χα1 − χ2)Γ( χ

2 ( ¯

α − 2α1)) Γ( χ

2 ( ¯

α − 2Q))Γ( χ

2 ( ¯

α − 2α3))Γ( χ

2 ( ¯

α − 2α2)) × Γ(1 + χ

2 (2Q − ¯

α))Γ(1 + χ

2 (2α3 − ¯

α))Γ(1 + χ

2 (2α2 − ¯

α)) Γ(1 + χ2)Γ(1 + −χα1)Γ(1 − χα1 + χ2)Γ(1 + χ

2 (2α1 − ¯

α))

Teschner ’95: DOZZ is the unique analytic solution of these equations if we could extend C(α1, α2, α3) beyond the region αi < Q.

Reflection

What happens if if α1 + χ > Q? F(u) has same asymptotics with the replacement: C(α1 + χ, α2, α3) → R(α + χ)C(2Q − (α1 + χ)), α2, α3) This can be stated informally as the reflection relation eαφ = 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.

Reflection coefficient

R(α) is given in terms of tail behavior of multiplicative chaos: Let α < Q, D any neighborhood of origin and ZD :=

• D

1 |z|γα M(dz) 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)eα1φ(0)eα2φ(1)eα3φ(∞) as u → 0 is controlled by the singularity at α1. R(α) has an explicit expression in terms of multiplicative chaos (2-point Quantum Sphere of Duplantier and Sheffield)

Integrability

We use R to obtain analytic continuation of C(α1, α2, α3) to α1 > Q: C(α1, α2, α3) = R(α1)C(2Q − α1, α2, α3) DOZZ formula satifies this with RDOZZ(α) = −(( γ

2 )

γ2 2 −2˜

µ)

2(Q−α) γ

Γ( γ

2(α − Q))Γ( 2 γ (α − Q))

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 RDOZZ. We prove R can be expressed in terms of the structure constants = ⇒ periodicity relations for R that determine it.

Outlook

Our proof settles some physics puzzles:

◮ Duality: DOZZ invariant under γ → 4 γ with:

µ → ˜ µ = (µπℓ( γ2

4 ))

4 γ2 (πℓ( 4

γ2 ))−1

(ℓ(x) =

Γ(x) Γ(1−x)).

but Liouville action functional is not!

◮ Reflection: where does the relation eαφ = 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)