Renormalizability of Liouville Quantum Gravity at the Seiberg bound

2015/06/05 by François David, David, François, Antti Kupiainen +5
Mathematics · Physics and Astronomy · #60D05 #81T20 #81T40 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.MP #math.PR #msc:60D05 #msc:81T20 #msc:81T40

paper · pdf · doi:10.48550/arxiv.1506.01968

arxiv created 2015/06/05 · arxiv updated 2015/06/08

Abstract

Liouville Quantum Field Theory can be seen as a probabilistic theory of 2d Riemannian metrics e^ϕ(z)dz², 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 \citeDKRV 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 conical singularities in the classical setup. 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. Their construction involves methods from Gaussian Multiplicative Chaos theory at criticality.

Related

• Liouville Quantum Gravity and KPZ
• Liouville quantum gravity as a metric space and a scaling limit
• Liouville Quantum Gravity on the Riemann sphere
• Random surfaces and Liouville quantum gravity
• Gaussian curvature on random planar maps and Liouville quantum gravity