将运用量子逆散射方法，结合李（超）代数表示论和Drinfeld twisting 理论等方法，研究一般边界可积自旋链的能谱以及相关模型关联函数的解析表示。主要将构造一般边界可积自旋链的F-basis，利用该基底计算关联函数并构造它们的行列式表示；通过半经典展开的方法，研究相对应边界Gaudin 模型的关联函数。同时将发展完善推广的边界代数Bethe ansatz 方法，利用该方法精确求解与Bn 、Cn 、Dn 李代数相关的一般边界可积自旋链的能谱问题；利用可积模型与N=2超对称规范理论的对应关系，构造与边界可积自旋链相对应的超对称规范理论并研究关联函数行列式表示在该规范理论中对应量。

Correlation functions are central to mathematical physics, particularly since they encode essential information about quantum many-body systems. Based on Quantun Inverse Scattering Method, Representation Theory of Lie super algebras and Drinfeld Twisting Theory, this proposal will develop innovative techniques to make further fundamental advances in the exact solutions to quantum integrable systems and the analytic computation and applications of correlation functions of the systems. Through the duality bewteen the quantum integrable spin chains and N=2 supersymmetric gauge theories proposed by N. Nekrasov et al, the corresponding gauge theories to quantum boundary integrable systems will be investigated. Moreover, the counterparts corresponding to Determinant Representations of correlation functions will be studied. Success of the project will provide an enhanced capacity for the understanding of the associated quantum supersymmetric gauge theories.