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Various two-dimensional σ models enjoy an infinite set of infinitesimal transformations acting on their solution space. The action of these symmetries is investigated for the Euclidean projective and Grassmannian σ models. On the (anti-) self-dual sector of the latter, the algebra of symmetries is shown to collapse to a finite-dimensional algebra isomorphic to sl(n+1,C) for the models with fields in the Grassmannians Gn+1,p. The finite action obtained by exponentiation is given in a closed form. For CPn models, this result is extended to the whole space of finite action solutions and the structure of the algebra remains sl(n+1,C). Hence the action is not transitive on the solution space.

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).

We again consider classiczl Yang-Mills equations on Minkowski space. When the gauge group is SU(2), we have shown previously that the only solutions invariant under a non-compact maximal subgroup of the conformal group are Abelian Maxwell fields. In this paper we extend the analysis to the case where the gauge group is an arbitrary semisimple compact group, with the same negative result.

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