The Faddeev Knots and the Skyrme Solitons

Polytechnic University Of New York, Brooklyn NY

Investigators

Abstract

Abstract Award: DMS-0406446 Principal Investigator: Yisong Yang This project is to develop a general existence theory for the knotted solitons arising in Faddeev's quantum field theory model. The difficult structure of the problem requires innovative techniques and ideas beyond the well developed tools such as the method of concentration-compactness. Recently, the PI has collaborated with F. H. Lin on the existence of these Faddeev knots and a series of important results have been obtained. A notable feature in their study is that, although compactness may fail, existence may be achieved as a consequence of two key inequalities: the first inequality, called by them the Substantial Inequality, ensures the existence of at least one topological minimizer; the second inequality, giving an optimal sublinear upper growth estimate for the Faddeev energy in terms of the associated topological charge, implies the existence of an infinite family of topological minimizers, hence the existence of the Faddeev knots. This progress opens new directions for achieving a higher level of understanding of various unsettled aspects of the Faddeev knots, including the existence of unit-charge unknots (ring-like solitons), existence of high-charge knots and their stability, and transition pictures of knots in view of their topological and energetical characteristics. The similarity between the Faddeev energy and the Skyrme energy may be exploited to tackle the difficult existence problem of the Skyrme solitons. Indeed, a by-product of the work of F. H. Lin and the PI will be a proof of the existence of unit-charge Skyrme solitons which corrects the flawed (but well-known) proof of M. Esteban published in 1986. This study may also be extended to the Skyrme model in two spatial dimensions which has applications in condensed matter physics and cosmology. This project also suggests new directions for the study of the existence problems of global energy minimizers with topological characteristics in other important related areas. The concept of knots is of foundational importance in science. For example, Lord Kelvin first explored the idea of using knots to model atoms and molecules. More recently, the concept of knots has inspired various fundamental areas, including particle and condensed-matter physics, statistical mechanics, cosmology, molecular biology, and synthetic chemistry. Up until recently, mathematical work had been focused on topological and combinatorial classifications of knots, but not on existence. In 1995, Faddeev and Niemi started a seminal study on the existence of knots as the soliton solutions of a quantum field theory model known as the Faddeev model using high-power computer simulations. The proposed work develops a mathematical existence theory of such knots. This work will give new insight into the mathematical structure of knots and their computer realization.