几何朗兰兹对应中的戴森-施温格方程

If we want to extend the Langlands duality for the affine Lie algebra in the critical level case of k = -h∨ (h ∨ is the dual Coxeter number of Lie algebra g), to which the off-critical case, we must consider its quantum deformation. From the point of view of AdS/CFT correspondence, and more specifically the BPS/CFT correspondence, we need to consider quantum deformation of the partition functions, which they satisfy an infinite set of the Dyson-Schwinger equation, The loop equation is another name for the set. Thus, to explore the off-critical geometric Langlands correspondence, or to make the infinite set of (Dyson-Schwinger) equation solution is non-trivial, then the set of equation should follow certain quantum algebra, which is essential significance. In this case, the deformed characteristic function introduced by E. Frenkel et al., which is no longer the answer to the question. We needs to develop new characteristic function functions or understand this problem from a new perspective. This study will help to reveal the Dyson-Schwinger equation in the role, as well as its hidden quantum algebra, for the geometric Langlands correspondence.

如果要将仿射李代数上的泊松代数与经典w-代数(Lg)在临界阶情形k＝－h∨(h∨为对偶的Coxeter数)的Langlands对偶关系，推广到临界外，就必须考虑其量子形变。从AdS/CFT对应, 更具体而言是BPS/CFT对应的角度看，需要考虑形变的配分函数, 且这些配分函数满足无穷集合的戴森-施温格方程，也称为圈方程。因此，有必要探索临界外的几何Langlands对偶，以及无穷集合的(戴森-施温格)方程。要使得无穷集合的(戴森-施温格)方程得解非平凡，则该方程所遵循的量子代数具有本质意义。在这种情况下，E.Frenkel等人引入的形变特征函数，已经不可能是问题的答案，需要发展新的特征函数，或从新的角度理解此问题。 本项目研究将有利助揭示戴森-施温格方程，以及其隐藏的量子代数，在Langlands对偶中的作用。