Abstract
Liouville Quantum Field Theory (LQFT) can be seen as a probabilistic theory of 2d Riemannian metrics e φ(z) |dz| 2 , conjecturally describing scaling limits of discrete 2d-random surfaces. The law of the random field φ in LQFT depends on weights α ∈ R that in classical Riemannian geometry parametrize power law singularities in the metric. A rigorous construction of LQFT has been carried out in [4] in the case when the weights are below the so called Seiberg bound: α < 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.
Highlights
Two dimensional statistical physics provides a large class of models of discrete random surfaces which are expected to have interesting continuous surfaces as scaling limits
In two dimensions the space of smooth metrics modulo diffeomorphisms is rather simple: its elements are eσg where σ : Σ → R and g belongs to a finite dimensional space of metrics
In this paper we will extend this theory to the case of “quantum” cusp singularities αi = Q completing the parallel with classical geometry in the setup of random surfaces
Summary
Two dimensional statistical physics provides a large class of models of discrete random surfaces (random maps) which are expected to have interesting continuous surfaces as scaling limits. The probabilistic theory has a complete parallel with the classical one with the important difference being that the parameter Qcl = 2/γ is replaced by the quantum value (1.3) It was shown in [3] that the measure (1.1) with the action (1.8) (suitably renormalized) has finite mass provided i αi > 2Q and the mass is nonzero if and only if αi < Q. In this paper we will extend this theory to the case of “quantum” cusp singularities αi = Q completing the parallel with classical geometry in the setup of random surfaces This extension requires an extra renormalization of the measure compared to the αi < Q case. The study of Q-insertions plays a prominent role in establishing convergence of the partition function of 2d-string theory where integrals over the moduli space arise (see [8])
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