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arXiv 0901.1646hep-thgr-qcmath-phmath.MPmath.RT
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谱生成共形和拟共形U-对偶群、超引力和球矢量

Spectrum Generating Conformal and Quasiconformal U-Duality Groups, Supergravity and Spherical Vectors

Murat Gunaydin, Oleksandr Pavlyk

备注：
Section 4 is split with the addition of a subsection on quadratic Casimir operators; references added; typos corrected. Latex file; 53 pages
发表信息：
JHEP 1004:070,2010
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摘要：

在回顾了N=2的Maxwell-Einstein超引力理论（MESGT）在具有对称标量流形的5维和4维空间的几何结构的基础上，给出了它们的三维U-对偶群作为谱生成拟共形群的统一实现.它们是F_{4(4)}，E_{6(2)}，E_{7(-5)}，E_{8(-24)}和SO（n+2，4）.我们的公式对于相应的五维超引力理论SL（3，R），SL（3，C），SU*（6），E_{6(6)}和SO（n-1，1）× SO（1，1）的U-对偶对称群是协变的.我们确定所有这些群的拟共形实现的球向量扭曲的一个单一的字符。给出了它们的二次Casimir算子并确定了它们的值。我们的工作奠定了代数基础，这些群的几何拟共形作用，其中包括四元数离散级数诱导的酉表示的构建。对于等级2的情况，SU（2，1）和G_{2(2)}，对应于四维和五维中简单的N=2超重力，该程序在arXiv：0707.1669中进行。我们还讨论了N=4和N>4超引力理论中物质对称性的代数结构。它们导致U-对偶群的分裂实形式的拟共形实现作为四元数实形式的直接扩展。

英文摘要：

After reviewing the algebraic structures that underlie the geometries of N=2 Maxwell-Einstein supergravity theories (MESGT) in five and four dimensions with symmetric scalar manifolds, we give a unified realization of their three dimensional U-duality groups as spectrum generating quasiconformal groups. They are F_{4(4)}, E_{6(2)}, E_{7(-5)}, E_{8(-24)} and SO(n+2,4). Our formulation is covariant with respect to U-duality symmetry groups of corresponding five dimensional supergravity theories, which are SL(3,R), SL(3,C), SU*(6), E_{6(6)} and SO(n-1,1)X SO(1,1), respectively. We determine the spherical vectors of quasiconformal realizations of all these groups twisted by a unitary character. We also give their quadratic Casimir operators and determine their values. Our work lays the algebraic groundwork for constructing the unitary representations of these groups induced by their geometric quasiconformal actions, which include the quaternionic discrete series. For rank 2 cases, SU(2,1) and G_{2(2)}, corresponding to simple N=2 supergravity in four and five dimensions, this program was carried out in arXiv:0707.1669. We also discuss the corresponding algebraic structures underlying symmetries of matter coupled N=4 and N>4 supergravity theories. They lead to quasiconformal realizations of split real forms of U-duality groups as a straightforward extension of the quaternionic real forms.

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