A Rigorous Perspective on Liouville Quantum Gravity & KPZ

Опубликовано 8 сентября 2016, 17:07

Polyakov first understood in 1981 that the summation over random Riemannian metrics involved in transition amplitudes in gauge theory or string theory could be represented mathematically by the now celebrated Liouville theory of quantum gravity. The quantum gravity measure is formally defined by d mu_gamma = exp[gamma h(z)]dz, where dz is the 2D Euclidean measure; exp[gamma h(z)] is the random conformal factor of the Riemannian metric, with h an instance of the Gaussian free field (GFF) on a bounded domain; and gamma is a constant, 0 <= gamma < 2. In 1988, Knizhnik, Polyakov and Zamolodchikov predicted that corresponding critical exponents (x) of a conformally invariant statistical model in the Euclidean plane and in Liouville quantum gravity (Delta) would obey the the universal KPZ