Harmonic maps into Lie groups: classical solutions of the chiral model

Abstract

Table of ContentsPART 1 2 1. Basic formulas and characteristic notation 2 2. The canonical associated linear problem or Lax pair 5 3. The Dolan representation 8 4. The variation formulas for the extended solution 12 5. The representation of $/ (S 2 , G) on holomorphic maps C* -• G 14 6. The representation of J#k(S 2 , G) on extended harmonic maps and Backlund transformations 20 7. The additional S ι action 23 8. Harmonic maps into Grassmannians 24 PART II 27 9. The single uniton 27 10. The fixed points of the S ι action 31 11. Global conservation laws and finiteness 35 12. Adding a uniton by singular Backlund transformation 37 13. The minimal uniton number 40 14. The unique factorization theorem 42 15. Complex Grassmannian manifolds again 45 16. Additional questions and problems 47

Гармонические Отображения

“…Указанная связь была установлена Уленбек в известной работе [22], результатом которой явилось проникновение методов теории интегрируемых систем в теорию гармонических отображе-ний. На их основе было получено полное описание гармонических отображений в унитарную группу ( [22], [24]). Это направление в теории гармонических отображений, представленное в главе 5, было в значительной степени мотивировано работами физиков по изучению классических решений т.н.…”

Further remarks on the chiral model approach to self-dual gravity

“…The aim of the present section is to show how the considerations concerning the harmonics maps into the Lie group U(N) given in an excellent paper by Uhlenbeck [5] ( See also Pohlmeyer [13], Ward [6] and Anand [7]) can be carry over to sel f-dual gravity. Our results are very far from that to be complete; nevertheless, we intend to indicate the direction of further investigations.…”

“…Section 4 is devoted to indicating how Uhlenbeck's considerations on harmonic maps into finite Lie groups [5] can be carried over to the case of the groups defined by the Moyal * -product. Finally, in Section 5 we consider the Wess-ZuminoWitten-lik e (WZW-like) action [9,10] within the Moyal formalism and it is shown that the action for Husain's heavenly equation is theh → 0 limit of the WZW-like action in the Moyal algebra.…”

“…results of [3] should be very closed related to the considerations of Fairlie, Fletcher and Zachos [4] on the su(N) algebra as embedded in the Moyal bracket algebra and also to some results on the harmonic maps, especially to ones given by Uhlenbeck [5] (see also [6,7,8]). …”

Isothermic submanifolds of symmetric R-spaces

“…In general, the Birkhoff factorisation is non-local and so difficult to compute but the key observation of Terng and Uhlenbeck [55,58] is that, for certain basic elements of G − , the simple factors, the factorisation can be carried out explicitly (see [13, §4.3] for a conceptual discussion of this). For our setting, we define the simple factors as follows: let r o ≤ g C be a parabolic subalgebra of height one and w ∈ C × \ {±1} such that (r 0 , τ o r o ) are a complementary pair and r = r 0 , for w ∈ R and r = τ o r 0 otherwise.…”