The Legendre transform and its generalizations, originally found in supersymmetric σ-models, are techniques that can be used to give local constructions of hyperkähler metrics. We give a twistor space interpretation to the generalizations of the Legendre transform construction. The Atiyah-Hitchin metric on the moduli space of two monopoles is used as a detailed example.
We find the complex structure on the dual of a complex target space. For N = (2, 2) systems, we prove that the space orthogonal to the kernel of the commutator of the left and right complex structures is always integrable, and hence the kernel is parametrized by chiral and twisted chiral superfield coordinates. We then analyze the particular case of SU (2) × SU (2), and are led to a new N = 2 superspace formulation of the SU (2) × U (1) WZW-model.
We define an sl(N ) analog of Onsager's Algebra through a finite set of relations that generalize the Dolan Grady defining relations for the original Onsager's Algebra. This infinite-dimensional Lie Algebra is shown to be isomorphic to a fixed point subalgebra of sl(N ) Loop Algebra with respect to a certain involution. As the consequence of the generalized Dolan Grady relations a Hamiltonian linear in the generators of sl(N ) Onsager's Algebra is shown to posses an infinite number of mutually commuting integrals of motion.
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