Infinite-dimensional Lie algebras acting on the solution space of various σ models

Abstract

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 𝔤⊗R[t,t−1], where 𝔤 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 𝔥 and 𝔪 are the two eigenspaces of the involution σ defining the RSS G/H, these two graded subalgebras are 𝔥⊗R[t] and (⊕i∈N𝔥⊗t2i) ⊕(⊕i∈N𝔪⊗t2i+1).

The ADHM Construction and Non-local Symmetries of the Self-dual Yang–Mills Equations

“…The group of transformations on the one-instanton moduli space is therefore only one-dimensional. Such a collapse to a finite-dimensional action is familiar from the theory of harmonic maps (see, e.g., [1,17,20,28]), where the orbits of the group action are also, generically, of high codimension.…”

“…More specifically, in Theorem 5.1, we deduce that the only symmetries of the self-dual Yang-Mills equations that act on five-dimensional one-instanton moduli space M 1 correspond to a scaling of the instanton solutions. Such a collapse to orbits of large codimension is not unfamiliar from the theory of harmonic maps into Lie groups [1,17,20,28], where one has similar non-local symmetry algebras [10]. Date: 22 December, 2009.…”

Pluriharmonic Maps into Compact Lie Groups and Factorization into Unitons

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Grassmann manifold. The interesting problems are the removability or resolution of the singularity in the factorization for a pluriharmonic map and the explicit construction of pluriharmonic maps from a specific complex manifold into U(N).We shall introduce the notion of meromorphically pluriharmonic maps, which is the smallest class of maps invariant under the addition of meromorphic unitons, and prove a factorization theorem for meromorphically pluriharmonic maps by the energy-reduction process with Harder-Narasimhan filtration.Moreover, by the methods of [33,34,25] and [18,22,10], we also can make the action of the loop algebra and loop group on the space of pluriharmonic maps into a compact Lie group.The authors wish to thank Professor J. Eells for his kind suggestions. The firstnamed author wishes to thank Dr S. Udagawa and Dr M. Bergvelt for some useful communications.

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“…Moreover, by the methods of [33,34,25] and [18,22,10], we also can make the action of the loop algebra and loop group on the space of pluriharmonic maps into a compact Lie group.…”

Description of surfaces associated with Grassmannian sigma models on Minkowski space

“…soliton solutions, are known [16,17]. By explicit calculation one can check that the real-valued functions…”