Hall algebras and gauge theory on a surface

我的研究成果包括构造和刻画了固定光滑射影复曲线X的二维上同调Hall代数。Schiffmann，我已经定义了与X的Dolbeaut模栈（即，X上希格斯层参数化的模栈）相关的上同调霍尔代数。我们已经描述了这样一个代数的特征，例如，它的一组生成元。回想一下，德拉姆模栈是参数化X上具有平坦连接的向量丛的栈，而贝蒂模栈是参数化X的基本群的有限维表示的栈。在arXiv：1903.07253的论文中，M. Porta，我分别构造了X的de Rham和Betti模栈的上同调Hall代数。这个结果是一个更一般的卷积代数结构的建设上的有界衍生类别的相干层的Dolbeaut，德拉姆和贝蒂模栈（这引起了分类霍尔代数）的后果。在Dolbeaut的情况下，所得到的分类化的Hall代数确实分类了用Schiffmann构造的代数。此外，我还建立了这3类Hall代数之间的关系，它们在de Rham & Betti情形下可以被解释为Riemann-Hilbert对应的Hall代数版本，在Dolbeaut & de Rham情形下可以被解释为非交换Hodge对应的Hall代数版本。

My research achivevements consisted of the construction and characterization of the two-dimensional cohomological Hall algebras of a fixed smooth projective complex curve X.In the paper arXiv:1801.03482, together with O. Schiffmann, I have defined the cohomological Hall algebra associated with the Dolbeaut moduli stack of X (that is, the moduli stack parameterizing Higgs sheaves on X). We have characterized such an algebra describing, for example, a set of generators of it.Recall that the de Rham moduli stack is the stack parameterizing vector bundles with flat connections on X, while the Betti moduli stack is the stack parameterizing finite-dimensional representations of the fundamental group of X. In the paper arXiv:1903.07253, together with M. Porta, I constructed cohomological Hall algebras for the de Rham and Betti moduli stacks of X, respectively. This result is a consequence of a more general construction of convolution algebra structures on the bounded derived category of coherent sheaves on the Dolbeaut, de Rham and Betti moduli stacks (this gives rise to categorified Hall algebras). In the Dolbeaut case, the resulting categorified Hall algebra indeed categorifies the algebra constructed with Schiffmann. In addition, I have established some relations between these 3 categorified Hall algebras, which can be interpreted as Hall algebra versions of the Riemann-Hilbert correspondence, in the de Rham & Betti case, and of the non-abelian Hodge correspondence, in the Dolbeaut & de Rham case.