论文标题
Instanton计数，量子几何和代数
Instanton Counting, Quantum Geometry and Algebra
论文作者
论文摘要
这本回忆录的“ HabilitationàDirigerdes Recherches”的目的是呈现超对称仪表理论的量子几何和代数方面，该理论来自Instantons引起的真空结构的非扰动性质。我们从简要摘要instanton模量空间的模棱两可的定位，并展示如何以三种方式获得激体顿分区函数及其对颤动仪表理论和超级组理论的概括：等激索引公式，轮廓集成式，式集成式公式和组合公式。然后，我们探索基于Seiberg-Witten几何形状的$ \ Mathcal {n} = 2 $量规理论的几何描述以及其字符串/M理论的透视图。通过与集成系统的关系，我们通过$ω$ - 定义量规理论来展示如何量化这种几何结构。我们还讨论了由超对称真空引起的潜在量子代数结构。我们介绍了通过对塞伯格（Seiberg）的几何形状进行双重量化构建的箭量W- algebra的概念，并显示了其特定特征：仿射箭量W-Algebras，分数W-Algebras及其椭圆形变形。
The aim of this memoir for "Habilitation à Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometric description of $\mathcal{N} = 2$ gauge theory based on Seiberg-Witten geometry together with its string/M-theory perspective. Through its relation to integrable systems, we show how to quantize such a geometric structure via the $Ω$-deformation of gauge theory. We also discuss the underlying quantum algebraic structure arising from the supersymmetric vacua. We introduce the notion of quiver W-algebra constructed through double quantization of Seiberg-Witten geometry, and show its specific features: affine quiver W-algebras, fractional quiver W-algebras, and their elliptic deformations.
