The Superstring Integrable Study

The classical/quantum integrabilities in string theories are expected to play an important role in establishing the AdS/CFT correspondence which have attracted much attention. In the present thesis, we will investigate the integrabilities of two different formalisms (the Green-Schwarz formalism and the Hybrid formalism) for superstring in various AdS backgrounds.In the second chapter, we study the Green-Schwarz IIB superstring propagating in the AdS₅×S⁵ background. Firstly, we recall the necessary facts about the Z₄ graded Lie superalgebra psu(2,2|4). Secondly, the action of superstring is constructed, then it can be written in the light-cone coordinates, equivalently. Finally, we derive the Lax connection with the spectral parameter which ensures the integrability of the system.In the third chapter, the integrability in the Hybrid superstring is investigated in the AdS₂×S2 background. Firstly, according to the Z₄ graded Lie superalgebra psu(1, 1|2), we construct the action for the hybrid superstring in an AdS₂×S² background with R-R flux. Furthermore, we give the equations of motion (EOM) and the Maurer-Cartan equations (MCE), at the same time, a new linear combination of MCE and EOM is also made. By the defination of the Hodge star about the bosonic and fermionic currents, respectively, it is shown that there exists a twisted duality symmetry between MCE and EOM. Finally, from the twisted duality transformation, we construct the Lax connection with the spectral parameter and the monodromy matrix.In the four chapter, the manifold AdS₃×S³ can be described as the group manifold SU(1,1)×SU(2). By TsT transformations on S³ and AdS₃ respectively, we derive the string action inγ-deformed AdS₃×S³ backgrounds. According to TsT transformations, we can derive the local Lax connections and the monodromy matrices in 7-deformed backgrounds generated TsT transformations on AdS₃ and S³ which ensure the classical integrabilities of the string theories.In the five chapter, the group manifold SL(2, R) is locally isomorphic to SU(1,1), the manifold AdS₃ can also be understood as the group manifold SL(2,R), then the process of computations in the last chapter is repeated, we also derive the Lax connection in 7-deformed backgrounds generated TsT transformation on AdS₃.In the six chapter, we review the construction of the infinite set of non-local conserved currents for Sigma model by the BIZZ procedure and Inverse BIZZ procedure. Furthermore, we generalize the BIZZ procedure and Inverse BIZZ procedure, and the infinite set of non-local conserved currents for WZW model is constructed.