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University of Cambridge > Talks.cam > Isaac Newton Institute Seminar Series > Towards quantum Kähler geometry

## Towards quantum Kähler geometryAdd to your list(s) Download to your calendar using vCal - Remi Rhodes (Université Paris-Est)
- Friday 20 July 2018, 11:15-12:15
- Seminar Room 1, Newton Institute.
If you have a question about this talk, please contact info@newton.ac.uk. RGMW06 - RGM follow up We propose a natural framework for probabilistic Kähler geometry on a one-dimensional complex manifold based on a path integral involving the Liouville action and the Mabuchi K-energy. Both functionals play an important role respectively in Riemannian geometry (in the case of surfaces) and Kähler geometry. The Weyl anomaly of this path integral, which encodes the way it reacts to changes of background geometry, displays the standard Liouville anomaly plus an additional K-energy term. Motivations come from theoretical physics where these type of path integrals arise as a model for fluctuating metrics on surfaces when coupling (small) massive perturbations of conformal field theories to quantum gravity as advocated by A. Bilal, F. Ferrari, S. Klevtsov and S. Zelditch in a series of physics papers. Interestingly, our computations show that quantum corrections perturb the classical Mabuchi K-energy and produce a quantum Mabuchi K-energy: this type of correction is reminiscent of the quantum Liouville theory. Our construction is probabilistic and relies on a variant of Gaussian multiplicative chaos (GMC), the Derivative GMC (DGMC for short). The technical backbone of our construction consists in two estimates on (derivative and standard) GMC which are of independent interest in probability theory. Firstly, we show that these DGMC random variables possess negative exponential moments and secondly we derive optimal small deviations estimates for the GMC associated with a recentered Gaussian Free Field. This talk is part of the Isaac Newton Institute Seminar Series series. ## This talk is included in these lists:- All CMS events
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