论文标题
外来量子差方程和积分解决方案
Exotic Quantum Difference Equations and Integral Solutions
论文作者
论文摘要
Nakajima Quiver品种的$ K $理论枚举几何形状中的基本对象之一被称为上限操作员。它是独特地确定为$ q $差方程系统的基本解决方案。这种差异方程涉及两组变量的偏移，这是在作用于多种多样的圆环和一组称为kähler参数的附加变量的圆环的变量。在[28]中，用QKZ方程鉴定了前者变量的差方程。在[30]中使用量子动力学Weyl基团的类似物在[30]中从理论上鉴定了后一个变量中的差方程。 一旦知道了这种表示理论描述，这些方程就会明显概括，我们称之为外来量子差方程。它们取决于在$ \ mathrm {pic}（x）（x）\ otimes \ mathbb {r} $中的某个超平面布置中选择的壁co，其中通常的差异方程与包含小型反动物线捆的壁co相对应。作为我们的主要结果，我们将这些方程式的基本解决方案与后代的所谓顶点相关联与Quasimap计数，并根据$ K $ - 理论稳定信封给出后代。 对于$ \ mathbb {c}^2 $中点的希尔伯特方案，我们使用量子图式代数编写了奇特的量子差方程。我们使用[6]的结果获得$ K $ - 理论稳定稳定斜率信封的公式。使用此功能，我们能够为外来差异方程的解决方案编写明确的公式。这些公式可以写为轮廓积分。作为我们结果的部分猜想的应用，我们将鞍点近似应用于这些积分，以将量子环形代数的Bethe子代数对角度化以进行任意斜率。
One of the fundamental objects in the $K$-theoretic enumerative geometry of Nakajima quiver varieties is known as the the capping operator. It is uniquely determined as the fundamental solution to a system of $q$-difference equations. Such difference equations involve shifts of two sets of variables, the variables arising as equivariant parameters for a torus that acts on the variety and an additional set of variables known as Kähler parameters. The difference equations in the former variables were identified with the qKZ equations in [28]. The difference equations in the latter variables were identified representation theoretically in [30] using an analog of the quantum dynamical Weyl group. Once this representation theoretic description is known, there is an obvious generalization of these equations, which we refer to as exotic quantum difference equations. They depend on a choice of alcove in a certain hyperplane arrangement in $\mathrm{Pic}(X)\otimes \mathbb{R}$, with the usual difference equations corresponding to the alcove containing small anti-ample line bundles. As our main result, we relate the fundamental solution of these equations back to quasimap counts using the so-called vertex with descendants, with descendants given in terms of $K$-theoretic stable envelopes. In the case of the Hilbert scheme of points in $\mathbb{C}^2$, we write our exotic quantum difference equations using the quantum toroidal algebra. We use the results of [6] to obtain formulas for the $K$-theoretic stable envelopes of arbitrary slope. Using this, we are able to write explicit formulas for the solutions of the exotic difference equations. These formulas can be written as contour integrals. As a partially conjectural application of our results, we apply the saddlepoint approximation to these integrals to diagonalize the Bethe subalgebras of the quantum toroidal algebra for arbitrary slope.
