The Ads <sub> 2 </ Sub> ¡Á S 1 The Context Of Su (1,1 | 1) Spinor Form Of The Category Iib Superstring Model And Several Lie Algebra Basis

The theory of superstring and condensed matter physics has many-sided internal relation.On the one hand, methods in the theory of field are widely used in many domain of condensed matter physics and they have a very important significance in the study ofcondensed matter physics;on the orther hand, some important mechanism of the theory of condensed matter physics are closely related to some the theory of fields , such as soliton, scattering amplitude and so on. AdS/CFT correspondence sets up the relation between the theory of superstring and supersymmetric gauge theory ,which implys some integrable systems existing in the model of superstring. Some integrable curve of spectrums can be got from the theory of superstring's near low energy. So with the thoroughly researching on the theory of superstring, it must can provide some useful help to condensed matter physics from aspects of backgrounds and methods and that some method in the model of condensed matter physics such as method of Bethe-ansaz can also resolve some important issues of the theory of superstring. This paper study the Green-Schwarz integrable type IIB superstring on AdS⊕ S background. It is a kind of the theory of integrable field thereby its research result has a significance on the corresponding system of condensed matter physics simultaneous having Boson and Fermi.The coset model of PSU(2,2| 4) supergroup's superstring(The model of MT) is the one extending on AdS5 ⊕ S~5 background from the famous theory of Green-Schwarz IIB superstring on flat space. psu(2,2| 4) Lie superalgebras's self-consistency need satisfysome self-consistent conditions, they are foundation of the model's founding. These consistency conditions and their effect on the model's derivation are never given before.We deduce detailedly these self-consistent conditions by making use of psu(2,2| 4) Liesuperalgebra's Jacobi identities.It is so difficulty to validate Jacobi-identities that above Lie superalgebra shoud satisfy. By constructing a nonsingular representation of the Lie superalgebra , we can prove that the Lie superalgebra is self-consistent. This paper constructs detailedlysu(2,2\4) , 5?(l,l|2) psu(2,2\2) and st/(l,l|l) Lie superalgebra's foundational spinorrepresentation and proves that these Jacobi-identities are legitimacy. We still obtain some rules from above construction, which present the conditions that/matrices in these Lie superalgebra's Jacobi identities should satisfy. These are never given definitely before.su(\, l| 1) Lie superalgebra's coset model of IIB superstring on AdS2 <8> Sl background is the simplest one in the model of the type MT, but it has some primary character of the model of the type MT. The model can be regarded as the model of Toy to make research on the model of the type MT. This paper study the model of Toy : the action is presented and the variational form of corresponding action is worked out from 5w(l,l|l) Liesuperalgebra. That the model has k -symmetry that the theory of GS should satify is proved. Euler-Lagrange equations of movement are worked out. From equations of movement flat connections with spectrums are worked out and thereout it can be proved that the model is likely to be a model of integrable field. The coset model belongs to the model of the noncritical type MT , witch is never carefully studied before in documents. The k -symmetry with form of Nambo-Goto, flat connections with spectrums and infinite conserved current are given by us for the first time.