Studies on uantum integrable systems : algebraic structure and correlation functions

我们研究了以下两个问题。（1）关联函数的代数表示（i）我们得到了XXZ和XYZ自旋链关联函数的代数表示，其形式适用于物理上感兴趣的齐次链。(ii)对于XXZ链，我们发现出现在上述表示中的算符Ω可以表示为费米子湮灭算符的双线性形式。进一步构造了满足一定局部性的生成算子。我们证明了，对于由平凡算子应用生成算子得到的（准）局部算子，真空期望值可以用行列式的形式明确表示，对于一般的耦合参数值，存在一个费米子结构，这是一个值得进一步研究的新特点。（2）共形场的单式基和性质利用（2，1）-准素域的Fourier系数，得到了Virasoro极小模$M（p，p '）$（p '/p> 2 $）的一个代数单项式基.证明了酉情形下的猜想，并验证了猜想与特征的一致性。(ii)得到了sl（3）的主子空间（可积sl（3）-模的子模U（n）v $，其中v是最高向量，n是sl（3）的幂零子代数n的当前代数）的Weyl型特征公式，证明了该特征是量子户田Hamilton算子的本征函数.

We studied the following two subjects.(1)Algebraic representation of correlation functions(i) We have obtained an algebraic representation for correlation functions for the XXZ and XYZ spin chains, in a form applicable to the physically interesting homogeneous chains.(ii) For the XXZ chain, we found that the operator Omega appearing in the above representation can be expressed as a bilinear form of fermion annihilation operators. Further we constructed creation operators satisfying a certain locality property. We showed that, for the (quasi) local operators obetained from the trivial one by applying creation operators, the vacuum expectation values can be expressed explicitly in terms of determinants.The existence of a fermionic structure for generic values of coupling parameter is an unexpected new feature which deserve further investigation(2) Monomial basis and characters in conformal field thepry(i) We obtained a conjectural monomial basis for the Virasoro minimal modules $M(p,p')$ with $p'/p>2$ in terms of the Fourier coefficients of the $(2,1)$-primary field. We proved the conjecture for the unitary case, and verified the consistency of the conjecture with the character.(ii) We obtained a Weyl-type character formula for the principal subspace of sl(3) (submodule $U(\hat(n))v$ of integrable \hat{sl}(3)-modules, where v is the highest vector and \hat{n} is the current algebra for the nilpotent subalgebra n of sl(3)).As it turns out, this character is an eigenfunction of the quantum Toda Hamiltonian.