ISSN:
1089-7658
Quelle:
AIP Digital Archive
Thema:
Mathematik
,
Physik
Notizen:
Infinite-dimensional Lie algebras of infinitesimal transformations acting on the solution space of various two-dimensional σ models are investigated. The main tools are (i) Takasaki's interpretation [Commun. Math. Phys. 94, 35 (1984)] of the solutions of the associated linear system in terms of points in an infinite-dimensional Grassmann manifold and (ii) Mikhaïlov's reduction procedure [Physica D 3, 73 (1981)] for linear systems. Takasaki's approach leads, for the σ models with values in a Lie group G, to a set of transformations that has the structure of the loop algebra @Fg⊗R[t,t−1], where @Fg is the Lie algebra of G. (This algebra has already been encountered by Dolan [Phys. Rev. Lett. 47, 1371 (1981)] and by Wu [Nucl. Phys. B 211, 160 (1983)] among others.) The σ models with a Wess–Zumino term are also considered; the algebraic structure is found to be the same. Finally, Mikhaïlov's procedure is used to study the σ models with values in a Riemannian symmetric space (RSS) G/H which is not a Lie group. The algebra in these cases is a subalgebra of the loop algebra found for the principal models but it does not seem to be graded. However, it contains two graded infinite-dimensional subalgebras with the following structure: if @Fh and @Fm are the two eigenspaces of the involution σ defining the RSS G/H, these two graded subalgebras are @Fh⊗R[t] and (⊕i∈N@Fh⊗t2i) ⊕(⊕i∈N@Fm⊗t2i+1).
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1063/1.527736
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