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Elliptic Integrable Systems: a Comprehensive Geometric Interpretation
About this Title
Idrisse Khemar
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 219, Number 1031
ISBNs: 978-0-8218-6925-3 (print); 978-0-8218-9114-8 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00651-4
Published electronically: February 22, 2012
MSC: Primary 53B20, 53B35; Secondary 53B50
Table of Contents
Chapters
- Introduction
- Notation, conventions and general definitions
- 1. Invariant connections on reductive homogeneous spaces
- 2. $m$-th elliptic integrable system associated to a $k’$-symmetric space
- 3. Finite order isometries and twistor spaces
- 4. Vertically harmonic maps and harmonic sections of submersions
- 5. Generalized harmonic maps
- 6. Generalized harmonic maps into $f$-manifolds
- 7. Generalized harmonic maps into reductive homogeneous spaces
- 8. Appendix
- List of symbols
Abstract
In this paper, we study all the elliptic integrable systems, in the sense of C. L. Terng [66], that is to say, the family of all the $m$-th elliptic integrable systems associated to a $k’$-symmetric space $N=G/G_0$. Here $m\in \mathbb N$ and $k’\in \mathbb N^*$ are integers. For example, it is known that the first elliptic integrable system associated to a symmetric space (resp. to a Lie group) is the equation for harmonic maps into this symmetric space (resp. this Lie group). Indeed it is well known that this harmonic maps equation can be written as a zero curvature equation: $d\alpha _{\lambda } + \frac {1}{2}[\alpha _{\lambda }\wedge \alpha _{\lambda }]=0, \quad \forall \lambda \in \mathbb {C}^*,$ where $\alpha _\lambda = \lambda ^{-1}\alpha _1’ + \alpha _0 + \lambda \alpha _1''$ is a 1-form on a Riemann surface $L$ taking values in the Lie algebra $\mathfrak {g}$. This 1-form $\alpha _\lambda$ is obtained as follows. Let $f\colon L\to N=G/G_0$ be a map from the Riemann surface $L$ into the symmetric space $G/G_0$. Then let $F\colon L\to G$ be a lift of $f$, and consider $\alpha =F^{-1}.dF$ its Maurer-Cartan form. Then decompose $\alpha$ according to the symmetric decomposition $\mathfrak {g}=\mathfrak {g}_0\oplus \mathfrak {g}_1$ of $\mathfrak {g}$ : $\alpha =\alpha _0 + \alpha _1$. Finally, we define $\alpha _\lambda := \lambda ^{-1}\alpha _1’ + \alpha _0 + \lambda \alpha _1''$, $\forall \lambda \in \mathbb {C}^*$, where $\alpha _1’,\alpha _1''$ are the resp. $(1,0)$ and $(0,1)$ parts of $\alpha _1$. Then the zero curvature equation for this $\alpha _\lambda$, for all $\lambda \in \mathbb {C}^*$, is equivalent to the harmonic maps equation for $f\colon L\to N=G/G_0$, and is by definition the first elliptic integrable system associated to the symmetric space $G/G_0$. Thus the methods of integrable system theory apply to give generalised Weierstrass representations, algebro-geometric solutions, spectral deformations, and so on. In particular, we can apply the DPW method [23] to obtain a generalised Weierstrass representation. More precisely, we have a Maurer-Cartan equation in some loop Lie algebra $\Lambda \mathfrak {g}_\tau =\{\xi \colon S^1\to \mathfrak {g} |\xi (-\lambda )=\tau (\xi (\lambda ))\}$. Then we can integrate it in the corresponding loop group and finally apply some factorization theorems in loop groups to obtain a generalised Weierstrass representation: this is the DPW method. Moreover, these methods of integrable system theory hold for all the systems written in the forms of a zero curvature equation for some $\alpha _\lambda =\lambda ^{-m}\hat {\alpha }_{-m} + \cdots +\hat {\alpha }_0 + \cdots + \lambda ^m\hat {\alpha }_m$. Namely, these methods apply to construct the solutions of all the $m$-th elliptic integrable systems. So it is natural to ask what is the geometric interpretation of these systems. Do they correspond to some generalisations of harmonic maps? This is the problem that we solve in this paper: to describe the geometry behind this family of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction below gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.- Ilka Agricola, Connections on naturally reductive spaces, their Dirac operator and homogeneous models in string theory, Comm. Math. Phys. 232 (2003), no. 3, 535–563. MR 1952476, DOI 10.1007/s00220-002-0743-y
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