Conformal bootstrap in Liouville Theory

Guillarmou, Colin; Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent

doi:10.48550/arxiv.2005.11530

Cited by 13 publications

(39 citation statements)

References 47 publications

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“…Finally, let us emphasize that the massless sine-Gordon model is an essential example of a two-dimensional non-conformal perturbation of a CFT. For conformal field theories, a lot of recent progress has been made, in particular for the Ising model (see [10,11,33] and references therein) and for the Liouville CFT (see [31,37] and references therein). Moreover, we mention that models related to the massless Thirring model have also been studied in detail, in particular recently in the form of interacting dimers [28,29].…”

Section: Heuristics and Previous Resultsmentioning

confidence: 99%

The Coleman correspondence at the free fermion point

Bauerschmidt¹,

Webb²

2020

Preprint

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We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on R 2 with β = 4π exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence in the non-perturative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with β = 4π decay exponentially and that the corresponding probabilistic field is localized.

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Section: Heuristics and Previous Resultsmentioning

confidence: 99%

The Coleman correspondence at the free fermion point

Bauerschmidt¹,

Webb²

2020

Preprint

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“…for some constant C > 0 independent of 0 < ε 1. By Lemma 2.10 and the embedding f L p f B δ p,p (εZ 2 ) , inequality (32) implies that…”

Section: Difference Spaces and Besov Spaces On Latticementioning

confidence: 93%

“…The Sinh-Gordon model (in the limit case of zero mass m → 0) is an Integrable Quantum Field Theory see, e.g, [51,62] and has received considerable attention in the physics literature [58,68,69]. We remark also that a probabilistic approach developed to the Lioville model (V (ϕ) = exp(βϕ), m = 0) developed by Kupianen, Rhodes, Vargas and their coauthors has found remarkable success [32,42,43,44]. As of this moment we are not aware of a probabilistic construction of the Sinh-Gordon in the case m = 0, and giving such a construction appears to be an interesting and challenging problem.…”

Section: Introductionmentioning

confidence: 99%

Elliptic stochastic quantization of Sinh-Gordon QFT

Barashkov¹,

Vecchi²

2021

Preprint

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The (elliptic) stochastic quantization equation for the (massive) cosh(βϕ)2 model, for the charged parameter in the L 2 regime (i.e. β 2 < 4π), is studied. We prove the existence, uniqueness and the properties of the invariant measure of the solution to this equation. The proof is obtained through a priori estimates and a lattice approximation of the equation. For implementing this strategy we generalize some properties of Besov space in the continuum to analogous results for Besov spaces on the lattice. As a final result we show as how to use the stochastic quantization equation to verify the Osterwalder-Schrader axioms for the cosh(βϕ)2 quantum field theory, including the exponential decay of correlation functions.

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“…Structure constants are model dependent. A highly non-trivial consequence of our work is a proof of the convergence of the series in q defining the Liouville CFT conformal blocks for almost all p (this fact was unknown except for the torus 1-point case by [GRSS20] and the sphere 4-point case in our previous work [GKRV20]).…”

Section: Introductionmentioning

confidence: 96%

“…Concerning Step 3, the importance of understanding the spectral analysis of the Hamiltonian of LCFT was stressed in physics by Teschner in [Te01]. This was recently mathematically settled in our previous work [GKRV20] using tools from scattering theory, semigroups and probability, allowing us to prove the conformal bootstrap for the sphere.…”

Section: Introductionmentioning

confidence: 99%

Segal's axioms and bootstrap for Liouville Theory

Guillarmou,

Kupiainen,

Rhodes

et al. 2021

Preprint

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In 1987 Graeme Segal gave a functorial definition of Conformal Field Theory (CFT) that was designed to capture the mathematical essence of the Conformal Bootstrap formalism pioneered in physics by A.A. Belavin, A.M Polyakov and A.B. Zamolodchikov. In Segal's formulation the basic objects of CFT, the correlation functions of conformal primary fields, are viewed as functions (in fact sections of a line bundle) on the moduli space of Riemann surfaces with marked points which behave naturally under gluing of surfaces. In this paper we give a probabilistic realization of Segal's axioms in Liouville Conformal Field Theory (LCFT) which is a CFT that plays a fundamental role in the theory of random surfaces and two dimensional quantum gravity. Namely, to a Riemann surface Σ with marked points and boundary given by a union of parametrized circles, we assign a Hilbert-Schmidt operator A Σ , called the amplitude of Σ, acting on a tensor product of Hilbert spaces assigned to the boundary circles. We show that this correspondence is functorial: gluing of surfaces along boundary circles maps to a composition of the corresponding operators. Correlation functions of LCFT, constructed probabilistically in earlier works by the authors and F. David, can then be expressed as compositions of the amplitudes of simple building blocks where Σ is a sphere with b ∈ {1, 2, 3} disks removed (hence b boundary circles) and 3 − b marked points. These amplitudes in turn are shown to be determined by basic objects of LCFT: its spectrum and its structure constants determined in earlier works by the authors. As a consequence, we obtain a formula for the correlation functions as multiple integrals over the spectrum of LCFT, the structure of these integrals being associated to a pant decomposition of the surface. The integrand is the modulus squared of a function called conformal block: its structure is encoded by the commutation relations of an algebra of operators called the Virasoro algebra and it depends holomorphically on the moduli of the surface with marked points. The integration measure involves a product of structure constants, which have an explicit expression, the so called DOZZ formula. Such holomorphic factorization of correlation functions has been conjectured in physics since the 80's and we give here its first rigorous derivation for a nontrivial CFT.

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Conformal bootstrap in Liouville Theory

Cited by 13 publications

References 47 publications

The Coleman correspondence at the free fermion point

The Coleman correspondence at the free fermion point

Elliptic stochastic quantization of Sinh-Gordon QFT

Segal's axioms and bootstrap for Liouville Theory

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