量子Toroidal代数的表示、应用及推广

Quantum toroidal algebra is a two-parameter analogue of W_(1+∞). It has abundant structures and has attracted a lot of attention of many mathematicians and physicists in the world. Recently this algebra has been applied to several areas in mathematical physics, such as the refined topological vertex and the Alday-Gaiotto-Tachikawa correspondence. . The vector representation, Fock-module, Elliptic quantum Toroidal algebra has been established. Based on the Boson-Fermion correspodence, we construct a fermion analogue of Fock representation of this algebra. We give the fermion realization of quantum toroidal algebra on the framed moduli space of torsion free sheaves on P^2, and we will consider the blow up case. We develop the operator product formulae in the algebraic approach of the refined topological vertex. we introduce the generalized vertex operators and discuss its application in the refined topological vertex. We will consider the gluing cases. And we want to construct the representation of Elliptic quantum toroidal algebra, and consider its application.

量子Toroidal 代数是一个两参数推广的W_(1+∞)代数，它具有丰富的数学结构。近些年来许多国内外著名数学物理学家发现它在数学物理许多方面都有重要应用，比如：精细拓扑顶点，Alday-Gaiotto-Tachikawa（AGT）对应等。 . 现在这个代数的矢量表示，Fock 模，以及它的推广代数（椭圆量子Toroidal代数）等已被实现。我们利用玻色费米对应研究了量子Toroidal代数的费米实现，并给出了它在二维复射影空间的高阶无挠层的构架模空间上的费米实现，我们希望进一步考虑这个代数的2维费米实现以及它与双参数loop群的gerbal表示间的联系；我们还从这个代数出发引入了两个推广的顶点算子，准确得到了精细拓扑顶点中两个边界为空的顶点算子实现，我们想进一步考虑精细拓扑顶点的粘合情形；另外我们也想构造椭圆量子Toroidal代数的表示，并考虑它的一些应用。