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  • Kupiainen, Antti; Rhodes, Remi; Vargas, Vincent (2020)
    Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the three point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e., to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first nontrivial rigorous integrability result on Gaussian Multiplicative Chaos measures.
  • David, Francois; Kupiainen, Antti; Rhodes, Remi; Vargas, Vincent (2017)
    Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics e(phi(z)) |dz|(2), conjecturally describing scaling limits of discrete 2d-random surfaces. The law of the random field phi in LQFT depends on weights alpha is an element of R that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [3] in the case when the weights are below the so called Seiberg bound: alpha <Q where Q parametrizes the random surface model in question. These correspond to studying uniformized surfaces with conical singularities in the classical geometrical setup. An interesting limiting case in classical geometry are the cusp singularities. In the random setup this corresponds to the case when the Seiberg bound is saturated. In this paper, we construct LQFT in the case when the Seiberg bound is saturated which can be seen as the probabilistic version of Riemann surfaces with cusp singularities. The construction involves methods from Gaussian Multiplicative Chaos theory at criticality.
  • Kupiainen, Antti; Oikarinen, Joona (2020)
    We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere S-2 by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on S-2. In particular we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In Kupiainen et al. (Commun Math Phys 371:1005-1069, 2019) the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in Oikarinen (Ann Henri Poincare 20(7):2377-2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.