From Quantum Cohomology to Algebraic Combinatorics | the Example of Flag Manifolds

The computation and understanding of quantum cohomology is a very hard problem in mathematical physics (string theory). We review in non-technical terms how in case of the ag manifolds this problem turns out to be at its core a non-trivial problem in algebraic combinatorics. The basic quest of physics in the 20th century is that for a uniied theory of the physical world. During the 80's string theory emerged as the most promissing candidate of such a \Theory of Everything". In string theory the classical point particle is replaced by a closed string (a copy of the \circle" S 1). This has at least two advantages: 1. the singularities (innnities of potentials) of the classical theory are avoided, and 2. strings may have vibrational eigenstates (just as mechanical strings) thereby representing different kinds of particles. In 1988 Witten Wt1] has established the notion of the-model as a fundamental ingredient of a string theory and subsequently quantum cohomology as the core of the technical machinerie of-models Wt2]. To give a rst idea what quantum cohomology is about consider the cup product of ordinary cohomology classes: its structure constants represent the intersection properties of submanifolds or subvarieties of a given manifold or variety, where intersection means intersection in points. Now with the deformation of classical point particles to strings the new quantum version of intersection has the meaning of intersection with a common rational curve (without changing the manifold or varitety itself and without requiering that the subobjects meet at all in a common point). The main diiculty with this new and \axiomatically" introduced quantum cohomology (cf. KM]) was to show the associativity of the quantum cup product. Thus it came as a major breakthrough, when in 1995 Ruan and Tian rst established this associativity and therefore the existence of quantum cohomology for a reasonably broad class of manifolds (semi-positive symplectic manifolds and Ricci-at KK ahler = Calabi-Yau manifolds RT]).