New developments of vertex algebra theory

(1)我们引进了手征微分算子的渐近代数的概念。然后，我们已经通过手征哈密顿约化，构建了一个这样的代数上的决议的交集的Slodowy片与幂零锥。我们计算了这个代数的整体截面的空间，从而证明了仿射W-代数在临界水平上的一个局部化定理。这是一个与T。Kuwabara和F.马利科夫(2)我们引入了半无限限制函子的概念，并证明了它与仿射Kac-Moody代数表示的容许性是相容的，从而可以归纳研究Kac-Wakimoto容许表示。

(1) We have introduced the notion of an asymptotic algebra of chiral differential operators. We then have constructed, via a chiral Hamiltonian reduction, one such algebra over a resolution of the intersection of the Slodowy slice with the nilpotent cone. We compute the space of global sections of this algebra thereby proving a localization theorem for affine W-algebras at the critical level. This is a joint work with T. Kuwabara and F. Malikov.(2) We introduced a notion of semi-infinite restriction functor, and shows that it is compatible with admissibility of representations of affine Kac-Moody algebras, enabling an inductive study of Kac-Wakimoto admissible representations.