Hall algebras and gauge theory on a surface

我的研究成果包括构造和刻画了一条固定的光滑射影复曲线X的二维上同调霍尔代数。在论文ARXIV：1801.03482中，我和O·希夫曼一起定义了与X的DolBeaut模堆相关的上同调霍尔代数(即X上的Higgs层参数模叠)。我们刻画了这样一个代数，例如它的一组生成元.回想到De Rham模堆栈是X上具有平坦连通的堆栈参数化向量丛，而Betti模堆栈是X的基本群的有限维表示的堆栈参数化表示.在文章Arxiv：1903.07253中，我与M.Porta一起构造了X的de Rham模堆栈和Betti模堆栈的上同调Hall代数.这一结果是在DolBeaut、de Rham和Betti模堆上凝聚层的有界导出范畴上卷积代数结构的更一般构造的结果(这产生了范畴化的Hall代数)。在杜美特的例子中，所得到的范畴化的霍尔代数确实对用希夫曼构造的代数进行范畴化。此外，我还建立了这三个范畴Hall代数之间的一些关系，它们可以解释为Riemann-Hilbert对应的Hall代数版本(在de Rham&Betti情形下)和非阿贝尔Hodge对应(在DolBeaut&de Rham情形下)。

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.