QuantGMC SIGNED

Project "QuantGMC" data sheet

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 Project objective

The proposed goal for our research program is to attack some mathematical problems arising in constructive two dimensional Quantum Field Theory (QFT) and two dimensional Quantum Gravity (QG) using probabilistic methods.

The physical theory of Quantum Gravity has the aim of providing a unified framework which encompasses the two descriptions of nature provided by quantum mechanics and general relativity. The two dimensional version of the theory is more tractable than the one corresponding to the four dimensional space-time and thus is used as a testing workbench to understand higher dimensional physics.

In order to reinforce the rigourous mathematical understanding of this theory, we wish to explore two particular aspects of QFT which are based on a probabilistic construction called Gaussian Multiplicative chaos. The objectives of QuantGMC are:

A- To obtain an explicit construction of canonical random surfaces equipped with a structure of Kähler manifold. In technical terms this corresponds to the construction of a path integral corresponding to the coupling of Liouville functional and the Mabuchi K-energy on 2D manifold of arbitrary genus.

B- To enhance the current understanding of the Quantum Sine-Gordon model, which can be interpreted as a random version of the Sine-Gordon equation. This model is conjectured to undergo an infinite sequence of collapse transitions when the inverse temperature increases. However up to now, rigorous renormalization theory of the model can only allow to witness the three first of these transitions. We plan to use Gaussian Multiplicative Chaos to provide a more efficient renormalization scheme which would allow to account for all the transitions.