Statistical physics and path integral quantization rely on a precise counting of the different states and configurations of a system. In particular, quantum gravity involves studing the possible topology and geometry of space-time. This implies deep relations between physics and topology, combinatorics and geometry, as well as with probability theory. Some seminal contributions originated from IPhT, and the Institute has a number of experts in the field, whose activity covers a large spectrum of approaches.

Some research themes

Topology and Combinatorics

Topological recursion

Since 2007, B. Eynard and his collaborators have been developing topological recursion, a method for systematically computing asymptotic expansions in matrix models, enumerative geometry, and integrable systems. Recent works support the claim that in integrable systems, topological recursion can systematically compute not only WKB-like expansions, but also non-perturbative contri- butions. This provides a geometric framework for all known integrable systems (including conformal field theories), where the systems’ properties and in particular their Tau functions can be built from the geometry of a spectral curve.

Enumerative geometry 

Enumerative geometry consists in counting the number of possible configurations of geometric objects, typically surfaces immersed in a target space. The surfaces can be either discrete, so that counting them is a problem of combinatorics, or continuous, as in string theory. Methods developped at IPhT allow to study for instance knots in the three-dimensional sphere or in Seifert manifolds .

Two-dimensional random geometry

Two-dimensional random geometry is a major field of research, with both physical (string theory, 2d quantum gravity, membrane modeling) and mathematical motivations (integrable systems, exact com-binatorics)

Random maps

Random maps (graphs embedded in a 2d surface) provide a discrete version of 2d random geometry and of 2D quantum gravity, amenable to exact enumeration. Powerful tools for map combinatorics were developed at IPhT. Recent results have been obtained on the properties of geodesics and 3 points distance correlations on random maps, of "hull perimeters", of statistics of Voronoi cells. Another open question is that of the geometry of maps decorated by random nested loops (this is retaled to statistical mechanics models). The large deviation function for the nesting properties can be computed, and compared with continuous models (Liouville theory, Conformal Loops Ensembles).

Random Delaunay triangulations

Random Delaunay triangulations generalize random maps and random circle-packing models, and may be viewed as a discretization of the 2d Polyakov string. Their relation with topological gravity and topological strings are interesting and  

Interacting Liouville theory

A continuous version of 2d random geometry is provided by the famous Liouville field theory. A first rigorous full construction of the theory by probabilistic methods has been obtained. This work eventually led to the proof of the famous DOZZ relations.

Liouville theory, SLE and CLE

Liouville Quantum gravity can also be studied by gluing a coupled pair of Continuous Random Trees (CRTs) to produce a topological sphere and to canonically embed it in the Riemann sphere. The random interface between the trees becomes a space-filling Schramm-Loewner Evolution (SLE).