An introduction to 2d Liouville conformal field theory and random surfaces
Mathematical Physics and Probability Seminar
This is an expository talk on 2d Liouville theory, a scalar quantum field theory with an exponential self-interaction (first introduced by Hoegh-Krohn 50+ years ago). It is a building block for Polyakov's theory of quantum gravity (i.e. summing over random surfaces), which is related to a head-spinning number of developments over the past 40 years. Guillarmou, Kupiainen, Rhodes, Vargas and others have used probabilistic methods to show that when parameters are correctly tuned, Liouville theory is conformally invariant, satisfies Segal's axioms, and can be explicitly solved (confirming formulas originally found by physicists). Independently Sheffield, Miller and others have related LQG to a number of other points of view on random surfaces and curves. My aim in this introduction is to convey how ideas from probability (which are new for me) have enlivened a venerable subject.
For a magisterial overview google Scott Sheffield's ICM lecture What is a random surface? I will be talking about a few details which are (rather modestly) omitted from the last part of his lecture on LQG.