Collapse and exponentiation of infinite symmetry algebras of Euclidean projective and Grassmannian σ models

Abstract

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.

The ADHM construction and non-local symmetries of the self-dual Yang-Mills equations

“…3 From a CP 1 point of view, we are viewing U 1 ∪U 2 = CP 3 \L∞ as the total space of the normal bundle O(1)⊕O (1) of the rational curve L 0 ⊂ CP 3 \ L∞, where 0 denotes the origin in R 4 . X and Y are then linearly independent sections of this normal bundle.…”

“…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].…”

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.…”

“…Since our considerations are aimed at making a connection between the local aspects of the self-dual Yang-Mills equations and the global aspects, and the literature in these fields generally have completely different notation, it was deemed necessary to give an integrated, relatively detailed description of both approaches in a consistent notation. This accounts for the length of this section 1 . In Section 3, we show how the ADHM construction may be used to yield explicit patching matrices for holomorphic bundles over subsets of CP 3 , to which we may apply the results of [9] concerning the action of the symmetry algebra of the self-dual Yang-Mills equations.…”