Integrability of boundary Liouville conformal field theory

Rémy, Guillaume; Zhu, Tunan

doi:10.48550/arxiv.2002.05625

Cited by 3 publications

(10 citation statements)

References 28 publications

(86 reference statements)

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“…Building upon this work, the same authors proved in [KRV19a] the DOZZ formula for the 3-point function of LCFT on the sphere, first proposed in physics in [DO94,ZZ96]. Similar methods were used in the recent works [Rem20,RZ18,RZ20] to study LCFT on simply connected domain with boundary and solve several open problems about the distribution of one-dimensional GMC measures. The next step in this program is to prove a bootstrap statement such as (1.1) for the torus.…”

Section: Introductionmentioning

confidence: 77%

“…For (c), the analyticity in α of moments of Gaussian multiplicative chaos has already been shown to hold in several works such as [KRV19a,RZ20]. To reduce our GMC to the one studied in [RZ20], one can map the unit disk D to the upper-half plane H by the map z → −i z−1 z+1 . The circle parametrized by x ∈ [0, 1] becomes the real line R and the point x goes to y = −i e 2πix −1 e 2πix +1 .…”

Section: 2mentioning

confidence: 98%

“…The function R(α, 1, e −iπ γχ 2 +πγP ) is the reflection coefficient for boundary Liouville CFT, also known as the boundary two-point function. It was introduced in its most general form and computed in [RZ20]. An analogous function first appeared in the case of the Riemann sphere in [KRV19a] and a special case of R was computed in [RZ18].…”

Section: Operator Product Expansions For Conformal Blocksmentioning

confidence: 99%

“…These shift equations are inhomogenous first order difference equations with difference 2χ for χ ∈ { γ 2 , 2 γ }. Similar homogeneous versions were proposed for the DOZZ formula in [Tes95] and used in its proof in [KRV19a], while other versions have played a role in the recent works [Rem20,RZ18,RZ20]. To find the desired result from the shift equations, we use the fact that the equality holds when N := − α γ is an integer and that solutions to the shift equations are unique up to a constant factor.…”

Section: Introductionmentioning

confidence: 99%

“…This argument is a generalization of the one used in [KRV19a] to prove the DOZZ formula, although that case only involved a single homogeneous hypergeometric ODE. We mention also that the OPE for χ = 2 γ requires an intricate reflection argument making use of the results and the techniques of [RZ20].…”

Section: Introductionmentioning

confidence: 99%

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Probabilistic conformal blocks for Liouville CFT on the torus

Ghosal,

Remy,

Sun

et al. 2020

Preprint

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Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by Polyakov in the context of string theory. Conformal blocks are objects underlying the integrable structure of CFT via the conformal bootstrap equation. The present work provides a probabilistic construction of the 1-point toric conformal block of Liouville theory in terms of a Gaussian multiplicative chaos measure corresponding to a one-dimensional log-correlated field. We prove that our probabilistic conformal block satisfies Zamolodchikov's recursion, and we relate it to the instanton part of Nekrasov's partition function by the Alday-Gaiotto-Tachikawa correspondence. Our proof rests upon an analysis of Belavin-Polyakov-Zamolodchikov differential equations, operator product expansions, and Dotsenko-Fateev type integrals. 2.1. Definition of Gaussian multiplicative chaos 2.2. A GMC construction for the 1-point toric conformal block 2.3. 1-point toric conformal block and Nekrasov partition function 2.4. Statement of the main results 3. BPZ equation for deformed conformal blocks 3.1. Proof of Proposition 3.2 3.2. Proof of Theorem 3.4 3.3. Analyticity in α 4. From BPZ to Gauss hypergeometric equations 4.1. Separation of variables for the BPZ equation 4.2. Construction of a particular solution 4.3. Analyticity of solutions in α 5. Operator product expansions for conformal blocks 6. Equivalence of the probabilistic conformal block and Nekrasov partition function 6.1. Shift equations for series coefficients of the conformal block 6.2. Shift equations for Z q γ,P (α) 6.3. Proof of Theorem 2.11 6.4. Proof of Proposition 6.2 6.5. Preliminaries for Zamolodchikov's recursion 6.6.

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Section: Introductionmentioning

confidence: 77%

Section: 2mentioning

confidence: 98%

Section: Operator Product Expansions For Conformal Blocksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probabilistic conformal blocks for Liouville CFT on the torus

Ghosal,

Remy,

Sun

et al. 2020

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A Family of Probability Distributions Consistent with the DOZZ Formula: Towards a Conjecture for the Law of 2D GMC

Ostrovsky

2020

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A three parameter family of probability distributions is constructed such that its Mellin transform is defined over the same domain as the 2D GMC on the Riemann sphere with three insertion points (α 1 , α 2 , α 3 ) and satisfies the DOZZ formula in the sense of Kupiainen et al. (Ann. Math. 191 (2020) 81 -166). The probability distributions in the family are defined as products of independent Fyodorov-Bouchaud and powers of Barnes beta distributions of types (2, 1) and (2, 2). It is conjectured that these distributions give the law of the GMC for some background metric.

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The moduli of annuli in random conformal geometry

Ang¹,

Rémy²,

Sun³

2022

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We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The first is for the law of the modulus of the Brownian annulus describing the scaling limit of uniformly sampled planar maps with annular topology, which is as predicted from the ghost partition function in bosonic string theory. The second is for the law of the modulus of the annulus bounded by a loop of a simple conformal loop ensemble (CLE) on a disk and the disk boundary. The formula is as conjectured from the partition function of the O(n) loop model on the annulus derived by Cardy (2006). The third is for the annulus partition function of the SLE 8/3 loop introduced by Werner (2008). It again confirms a prediction of Cardy (2006). The physics principle underlying our proofs is that 2D quantum gravity coupled with conformal matters can be decomposed into three conformal field theories (CFT): the matter CFT, the Liouville CFT, and the ghost CFT. At the technical level, we rely on two types of integrability in Liouville quantum gravity, one from the scaling limit of random planar maps, the other from the Liouville CFT. We expect our method to be applicable to a variety of questions related to the random moduli of non-simply-connected random surfaces.

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Integrability of boundary Liouville conformal field theory

Cited by 3 publications

References 28 publications

Probabilistic conformal blocks for Liouville CFT on the torus

Probabilistic conformal blocks for Liouville CFT on the torus

A Family of Probability Distributions Consistent with the DOZZ Formula: Towards a Conjecture for the Law of 2D GMC

The moduli of annuli in random conformal geometry

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