Hall algebras and gauge theory on a surface

在arXiv:1801.03482论文中，我与O. Schiffmann共同定义了与X的Dolbeaut模堆栈（即参数化X上的希格斯束的模堆栈）相关的上同调霍尔代数。我们已经描述了这样一个代数，例如，描述了它的一组生成器。回想一下，de Rham模堆栈是X上具有平连接的堆栈参数化向量束，而Betti模堆栈是X的基本群的堆栈参数化有限维表示。在论文arXiv:1903.07253中，我与M. Porta分别构造了X的de Rham和Betti模堆栈的上同调Hall代数。这一结果是在Dolbeaut， de Rham和Betti模栈上相干束的有界派生范畴上更一般地构造卷积代数结构的结果（这产生了分类霍尔代数）。在Dolbeaut的情况下，所得到的分类霍尔代数确实对希夫曼构造的代数进行了分类。此外，我还建立了这三种分类霍尔代数之间的一些关系，它们可以被解释为黎曼-希尔伯特对应的霍尔代数版本，在de Rham和Betti情况下，以及非阿贝尔霍奇对应的霍尔代数版本，在Dolbeaut和de Rham情况下。

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.