Differential Geometry, Topology of Manifolds, Triple Systems and Physics
Differential geometry and topology of manifolds represent one of the currently most active areas in mathematics, honored by a number of Fields Medals in the recent past to mention only the names of Donaldson, Witten, Jones, Kontsevich and Perelman. The recent vitality of these areas is largely due to interactions with theoretical physics that have seen dramatic developments over the last three decades. In the 1970s, Atiyah and Faddeev, among others, pioneered the interaction between classical Yang-Mills gauge theories and integrable field theories with geometry and topology of fiber bundles. In the early 1980s Donaldson proved his celebrated theorems on classification of 4-manifolds introducing topological polynomial invariants, and these results have many unexpected consequences, most notably the existence of an uncountable number of exotic differential structures on 4-dimensional Euclidean space. The interaction between topology of manifolds and quantum field theory was provided by Witten in the late 1980s, when he constructed topological quantum field theories on 3- and 4-dimensional manifolds establishing that the Donaldson polynomials are partition functions in quantum field theory with BRST-type symmetries on 4-dimensional Riemannian manifold, and the Jones polynomial is a Wilson loop expectation value in quantum Chern-Simons theory on 3-dimensional manifold.
It is well known that a connection and its curvature are basic elements of the theory of fiber bundles, and they play an important role not only in modern differential geometry but also in theoretical physics namely in gauge field theories. Our research group of Geometry and Topology (Institute of Mathematics and Statistics) in close cooperation with the Laboratory of Theoretical Physics of University of Pierre and Marie Curie, Paris and the Laboratory of Theoretical Physics d’Orsay of University Paris-Sud develops a non-commutative geometry approach to generalization of a concept of connection and its curvature.
The idea to derive the geometric properties of space-time, and perhaps its very existence, from fundamental symmetries and interactions proper to matter’s most fundamental building blocks seems quite natural. If the space-time is to be derived from the interactions of fundamental constituents of matter, then it seems reasonable to choose the strongest interactions available, which are the interactions between quarks. At present, the most successful theoretical descriptions of fundamental interactions are based on the quark model. The only experimentally accessible states are either three-quark or three-anti-quark combinations (fermions) or the quark-anti-quark states (bosons).
Whenever one has to do with a tri-linear combination of fields (or operators), one must investigate the behavior of such states under permutations. Hence it is seems quite reasonable to develop a theory of ternary algebras and apply it to construct an algebraic model of quark interactions. A ternary algebra or triple system is an algebra, which closes under a suitable triple product. Our group together with the colleagues from the Laboratory of Theoretical Physics of University of Tartu and the mentioned above universities proposes and develops a new approach to quark model based on a concept of ternary Grassmann and Clifford algebra which provides a very natural and clear explanation for experimentally detected behavior of quarks.
Professor of Geometry and Topology, Institute of Mathematics and Statistics, University of Tartu, Estonia