The AdS <sub> 3 </ Sub> ¡Á S <sup> 3 </ Sup> Superstring Algebra And Hamiltonian

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-Schwarzâ…¡B 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-Schwarzâ…¡B 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⁺ =Ï„and 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.