Fall 2012 – Spring 2013 Program Details
visit the program web page
The Simons Center will host a program on `Conformal Geometry’ for the Spring
semester of 2013. This will cover subjects representing some of the most successful
examples of the cross-fertilization between mathematics and physics in this
century. It has led to the award of three Fields Medals in mathematics and a
resurgence of interest in conformal field theory (CFT) and other areas of physics
and mathematics where conformal symmetry emerges. Prior to these major
developments conformal symmetry had been in explicitly observed only in a few
`one-dimensional’ random processes: Brownian motion, random trees, and the like.
Inspired by work of physicists in the 70s and 80s on conformal invariance and field
theories in two dimensions, a number of leading probabilists and combinatorialists
began thinking about spatial process in two dimensions: percolation, polymers,
dimer models, Ising models. Major breakthroughs by Kenyon, Schramm, Lawler,
Werner, Smirnov, Sheffield and others led to a rigorous underpinning of conformal
invariance in two-dimensional systems and paved the way for a new era of two-
dimensional probability theory. In recent years probability theory (and here we
mean probability theory in the largest sense, comprising combinatorics, statistical
mechanics, algorithms, simulation) has made immense progress in understanding
the basic two-dimensional models of statistical mechanics and random surfaces. At
the same time deep links have emerged with various manifestations of integrable
Conformal Field Theory, as originally developed to describe critical phenomena,
focuses on the operator algebra of local operators and representations of infinite-
dimensional algebras. Conversely the probability theory of conformal processes
focuses on stochastic processes and random objects which have a geometric
interpretation. Together, these problems constitute a newly emerging field of
conformal stochastic geometry, or simply `Conformal Geometry.’
This emergent field of Conformal Geometry establishes new interesting links
and synergy between probability theory, geometry, representation theory and
analysis. The methods emerging within this field have proved to be efficient in many
important applications in physics. These range across many different physics sub-
disciplines: from traditional models of statistical mechanics, disordered systems,
quantum gravity, and random matrices, to quantum and classical non-equilibrium
phenomena and fluid mechanics.
The subject of Topological Phases of Matter has been building over a number of years, and is currently very active. The field includes many diverse experimental aspects, but this program will focus on theory (with some visits by experimentalists). The theoretical side ranges from experimental predictions and interpretation to deep theoretical concepts and use of sophisticated tools. The emphasis in the program should include the deepest theoretical aspects, as described below.
A topological phase of matter can be broadly defined as a (quantum-mechanical) phase of matter in which the ground state has a gap for all local excitations. This seemingly uninspiring definition gives rise to interesting phenomena for a number of reasons. Ground state correlation functions of local operators decay exponentially fast with distance, so the leading asymptotically low-energy (below the scale of the gap) and long-distance observable properties will be independent of the metric or of the length scale in the correlation functions they are topological invariants (however, such observables might all be zero, giving the trivial topological phase). Moreover, the observable properties will remain unchanged under a small change in Hamiltonian, because the Hamiltonian is the integral over space of a local operator, which cannot affect the topological observables.
Examples of such topological observable properties are (i) the Hall conductivity, which is quantized (but possibly zero) in these systems; (ii) ground state degeneracy when the space on which the phase lives is topologically non-trivial, for example a 2-torus as opposed to a 2-sphere; (iii) existence of “topologically non-trivial” quasiparticle excitations above the true ground state, and the quantum numbers and non-trivial statistics (behavior under adiabatic braiding) of these are topological properties (the braiding is equivalent to expectation values of knots or links formed by Wilson lines in spacetime). Related topologically “protected” effects include (iv) gapless excitations on the boundary of a region filled by the topological phase, which may be described by a conformal field theory, and cannot be rendered gapped (massive) by any perturbation in the Hamiltonian. Most importantly, topological phases exist in nature, and more are being discovered. Many are two-dimensional, but examples in three dimensions are now being uncovered. Theory is leading experiment: for example, three-dimensional topological insulators were predicted, then confirmed.