Mathematical physics, structures and models

Mathematical physics, in its wide anglo-saxon meaning, spans a wide range of topics in theoretical physics, from almost pure mathematics to some studies which can be related to experiments. Despite this variety, many of these topics are connected by the fact that they use common mathematical and theoretical tools (quantum field theory, integrable systems, conformal field theory, string theories, random matrices, combinatorics, probability and random processes), and that mathematical rigor and consistency, and the obtention of precise and often explicit results, are very important. Many of these tools have been developed over the years at IPhT. On several subjects, contacts and collaborations between IPhT and mathematicians develop quickly. This does not reflect a move towards more abstraction, but a genuine interest of mathematicians for problems from physics, and of physicsts for the ideas and the new mathematical tools developped by mathematicians. Mathematical physics at IPhT also has close contacts with statistical physics and high energy physics. For instance, important progress are made on the problem of evaluating scattering amplitudes, which is now finding its way in LHC phenomenology, in cosmology and in statistical physics.

Michel BAUER, Iosif BENA, Michel BERGERE°, Jérémie BOUTTIER, John-Joseph CARRASCO, François DAVID, Bertrand DUPLANTIER, Bertrand EYNARD, Mariana GRANA, Monica GUICA, Riccardo GUIDA, Emmanuel GUITTER, Ivan KOSTOV, Ruben MINASIAN, Stéphane NONNENMACHER*, Vincent PASQUIER, Eric PERLMUTTER, Sylvain RIBAULT, Didina SERBAN, Pierre VANHOVE, André VOROS°

The physics and mathematics of the theory of quantum systems, of their manipulations, and of their applications for instance for the technology of information is a fast evolving and important field. Researcher at IPhT are involved in many directions of research, for instance: quantum non-demolition measurements and the properties of "quantum jumps" and "quantum spikes" ; quantum chaos, escape rates and hyperbolicity; number theory, the Riemann hypothesis, and its relationship with quantum chaos. These themes overlap with other themes pursued at IPhT, in mathematical physics (from integrable systems to string theory and black holes) and in condensed matter and statistical mechanics.

Quantum field theory is the pillar of modern quantum theory and of high energy physics, and of part of statistical mechanics. Conformal field theories (CFT's , i.e. theories invariant under Virasoro symmetry - local angle-preserving transformations - and of possibly higher extended symmetries) are of special interest. After more than 40 years, they still undergo tremendous developments, e.g. the conformal bootstrap program and Liouville theory, non-unitary CFT's, lattice regularisationsof CFT's, with applications to condensed matter and statistical physics. A vast and partly overlapping field is the study of integrable systems, classical and quantum systems with an infinite number of conserved quantities (like energy and momentum), but non-local, associated with infinite dimensional symmetry algebras. They allow to study non-perturbatively physical systems with strong statistical and quantum fluctuations. Born in the 1930's with Bethe's ansatz, integrable systems are studied at IPhT since the 1960's until now, in many different contexts: mathematics and combinatorics (cluster algebras, plane tilings), quantum spin chains, integrable gauge theories, out-of-equilibrium physics and integrable probabilities, etc. Again, these themes overlap and interact with many others studied at IPhT.

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.

Topological recursion is a method for systematically computing asymptotic expansions in matrix models, enumerative geometry, and integrable systems, where it should allows to systematically compute not only WKB-like expansions, but also nonperturbative contri- butions.

Random maps (graphs embedded in a two dimensional surface) provide a discrete version of two dimensional random geometry. Various methods (random matrix models, recursion relations, tree bijections) from theoretical physics and mathematics, allow to study many difficult problems: statistics of distances and of Voronoi cells on random maps, nesting of loops and statistical models on random maps, random Delaunay triangulations and 2d gravity, etc.

A continuous version of 2d random geometry is provided by the famous Liouville field theory (or Liouville Quantum Gravity). It is studied by rigorous probabilistic methods, through a general construction of the measure and of all the correlations functions in various topology, as well as by using coupled Continuous Random Trees (CRT), Schramm-Loewner Evolution processes (SLE) and Conformal Loop Ensembles (CLE).

String Theory is not only the most viable quantum theory of gravity but is has also led to remarkable new insights into strongly coupled quantum field theory, black-holes, cosmology and even perturbative amplitudes in field theory. Developing the essential physics of string theory has also spun-off vibrant new areas of research in mathematics.

The String-Theory Group at the IPhT uses string theory to study gravity at its extremes: black holes and cosmology. This research is also deeply linked with a wide-ranging research program in holographic field theory. While string theory is expected to become manifest at such extreme “boundaries” of our universe, the string-theory group is also developing new methods, like generalized geometry and topological techniques, to explore how low-energy physics and small, positive cosmological constants can emerge from a description of the universe based in string theory.

Much of this research has a healthy symbiosis with other activities at the IPhT. In particular, conformal field theory, integrable systems, large-N matrix models, QCD amplitudes, gravitational waves, cosmology, and particle physics beyond Standard Model provide an invaluable background that informs and inspires the research into string theory.

#854 - Last update : 12/09 2020