Spinors, strings, integrable models, and decomposed Yang-Mills theory
2014 (English)In: Physical Review D, ISSN 1550-7998, Vol. 90, no 2, 025012- p.Article in journal (Refereed) Published
This paper deals with various interrelations between strings and surfaces in three-dimensional ambient space, two-dimensional integrable models, and two-dimensional and four-dimensional decomposed SU(2) Yang-Mills theories. Initially, a spinor version of the Frenet equation is introduced in order to describe the differential geometry of static three-dimensional stringlike structures. Then its relation to the structure of the su(2) Lie algebra valued Maurer-Cartan one-form is presented, while by introducing time evolution of the string a Lax pair is obtained, as an integrability condition. In addition, it is shown how the Lax pair of the integrable nonlinear Schrodinger equation becomes embedded into the Lax pair of the time extended spinor Frenet equation, and it is described how a spinor-based projection operator formalism can be used to construct the conserved quantities, in the case of the nonlinear Schrodinger equation. Then the Lax pair structure of the time extended spinor Frenet equation is related to properties of flat connections in a two-dimensional decomposed SU(2) Yang-Mills theory. In addition, the connection between the decomposed Yang-Mills and the Gauss-Codazzi equation that describes surfaces in three-dimensional ambient space is presented. In that context the relation between isothermic surfaces and integrable models is discussed. Finally, the utility of the Cartan approach to differential geometry is considered. In particular, the similarities between the Cartan formalism and the structure of both two-dimensional and four-dimensional decomposed SU(2) Yang-Mills theories are discussed, while the description of two-dimensional integrable models as embedded structures in the four-dimensional decomposed SU(2) Yang-Mills theory are presented.
Place, publisher, year, edition, pages
2014. Vol. 90, no 2, 025012- p.
IdentifiersURN: urn:nbn:se:uu:diva-230102DOI: 10.1103/PhysRevD.90.025012ISI: 000339218100008OAI: oai:DiVA.org:uu-230102DiVA: diva2:742446