Gradient representations and affine structures in AEn
Gradient representations and affine structures in AEn
复制标题
AEn 中的梯度表示和仿射结构
DOI：
10.1088/0264-9381/22/21/004
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发表时间：
2005
影响因子：
3.5
通讯作者：
H. Nicolai
中科院分区：
文献类型：
--
作者：
A. Kleinschmidt;H. Nicolai
我们研究不定型卡茨 - 穆迪代数\(A\!E_n\)，它出现在爱因斯坦理论从\((n + 1)\)维时空到一维（时间）的约化过程中，以及它们特殊的极大正则子代数和\(A^{(1)}_{n - 2}\)。对于\(n = 3\)，利用这两个子代数之间的相互作用来确定\(A\!E_3\)内“梯度生成元”的对易关系。测地线\(\sigma\)模型在陪集空间\(A\!E_n / K(A\!E_n)\)上的低阶截断被证明映射到\(SL(n)/SO(n)\)非线性\(\sigma\)模型的一个适当截断版本，该\(SL(n)/SO(n)\)非线性\(\sigma\)模型是由\((n + 1)\)维爱因斯坦方程约化到\((1 + 1)\)维得到的。可以进一步截断到对角解，以定义此类解与无限维陪集空间上的类光测地轨迹之间的一一对应，其中\(H\)是（扩展的）海森堡群及其极大紧子群。我们阐明\(H\)与杰罗赫群的相应子群之间的关系。
We study the indefinite Kac–Moody algebras AEn, arising in the reduction of Einstein's theory from (n + 1) spacetime dimensions to one (time) dimension, and their distinguished maximal regular subalgebras and A(1)n−2. The interplay between these two subalgebras is used, for n = 3, to determine the commutation relations of the ‘gradient generators’ within AE3. The low-level truncation of the geodesic σ-model over the coset space AEn/K(AEn) is shown to map to a suitably truncated version of the SL(n)/SO(n) nonlinear σ-model resulting from the reduction Einstein's equations in (n + 1) dimensions to (1 + 1) dimensions. A further truncation to diagonal solutions can be exploited to define a one-to-one correspondence between such solutions, and null geodesic trajectories on the infinite-dimensional coset space , where is the (extended) Heisenberg group, and its maximal compact subgroup. We clarify the relation between and the corresponding subgroup of the Geroch group.