AdS₃×S³超弦的代数及哈密顿化的研究

【作者】 王春；

【导师】 石康杰；

【作者基本信息】 西北大学 ， 理论物理， 2009， 硕士

【摘要】 探索引力和电磁、弱、强相互作用的统一是基础科学研究的宏伟目标之一,超弦理论是迄今为止最有希望的能将四种基本相互作用统一起来的理论。这一理论具有解释宇宙起源与运行以及解决近代物理学中很多难题f如夸克囚禁问题、规范场强耦合问题)的潜力,是当前国际理论物理研究的热点领域之一。论文主要作了两部分工作。首先给出了su（1,1|2）（?）su（（?）|2）李超代数的基础表示。构造8×8的矩阵,利用su（1,1|2）李超代数的γ矩阵和电荷共轭矩阵C,C’分别给出su（1,1|2）和su（（?）|2）的基础表示,然后将二者的生成元的表示做一定的组合,构造出su（1,1|2）（?）su（（?）|2）李超代数的基础表示。证明了该李超代数是自洽的。然后对该李超代数的玻色生成元和费米生成元分别作一定的组合给出了在光锥基底下该李超代数的基础表示,并给出了由Rahmfeld,Rajarama和Metsaev,Tseytlin分别给出的su（1,1|2）（?）su（（?）|2）李超代数的生成元之间的关系。这些工作对AdS₃×S³背景中Green-Schwarz IIB超弦的进一步研究具有重要意义。参数化在超弦的研究中有很重要的应用,在超弦的量子化、求解运动方程时,都必须对超弦模型进行参数化。我们对光锥规范下AdS₃×S³背景中Green-SchwarzIIB超弦进行了参数化,AdS₃部分我们用Metsaev和Tseytlin的参数化方案,S³部分我们用Kallosh,Rahmfeld,Rajaraman等给出的参数化方案。并给出了相应的Maurer-Cartan1-形式和作用量。然后对作用量中的世界面上的度规g^μv变分,给出相应的Nambu-Goto作用量。固定两个玻色变量x⁺=τ,y⁵=σ,然后对余下的玻色变量作部分勒让德变换,给出了与速度成线性关系的作用量,得到了系统的哈密顿量。证明了此场论系统是局域的,并且泊松括号可以很好的定义。利用这些结果,可进一步研究AdS₃×S³弦模型解空间的性质、解变换、流代数的结构等问题。更多还原

【Abstract】 The investigation of unified theory including gravity, electromagnetic, strong and weak interactions is one of the main tasks of the foundamental science research. String theory is the most possible candidates for such unified theory. This theory has the potential to explain the origin and evolution of our universe and can also solve many difficulties of modern physics （such as the problems of quark confinement, the couple of gauge field stength etc.）. It is one of the hot spots domain of the contemporary international theoretical physics.The paper is organized as two parts. First, the foundation representations of the Lie superalgebras su（1,1|2） and （?） were constructed by using the 7 matrix and the charge conjugation matrix C and C’ of the Lie superalgebra su（1, 1|2） with the 8×8 matrix. Then the representation of the Lie superalgebra su（1, 1|2） （?） was given by doubling the representation of the generators of the two superalgebras su（1, 1|2） and （?） correctly. We proved that the superalgebra is self-consistent. By doing some combinations of the generators of the Lie superalgebra su（1, 1|2） （?）, after some algebra, the foundation representation of the Lie superalgebra su（1, 1|2）（?） under light cone basis was obtained. We also give the relation between the generators of the Lie superalgebras su（1, 1|2） （?） given by Rahmfeld and Rajarama. and Metsaev and Tseytlin. Given the explicit representations of the Lie superalgebra su（1, 1|2） （?） under the covariant basis and the light cone basis have a great significance on the further study of the Green-SchwarzⅡB superstring in the AdS₃×S³ background.Parameterization plays an important role in the study of the superstring theory. We must parameterize the models of superstring when quantizing superstring, solving the equations of the motion, etc. We parameterize the Green-SchwarzⅡB superstring in the AdS₃×S³ background under the light cone gauge by the method of Metsaev and Tseytlin in AdS₃ and by the method of Kallosh, Rahmfeld and Rajaraman , et al in S³. After some tedious calculation, we obtain the corresponding Maurer-Cartan 1-forms and the action. Then we fix two bosonic variables x⁺ =τand y⁵ =σ, and perform the partial Legendre transformation of the remaining bosonic variables. We obtain a Lagrangian which is linear in velocity after eliminating the metric of world sheet. We also give the Hamiltonian and prove the system is local and poisson bracket of the theory can be well defined. Using these results, one can further study the properties of solution space, solution transformation and the structure of the flat currents algebra of the superstring in AdS₃×3³ background.更多还原