**Abstract:** |
The main technical tool in our proof of $P=W$ conjecture was the construction of an action of a certain algebra, which we called deformed $W_{1+\infty}$-algebra, on the cohomology of the moduli space of stable Higgs bundles on the curve $C$. It turns out that this algebra is nothing else than the cohomological Hall algebra of finite length sheaves on $T^*C$ (see my talk at Academia Sinica in 2019). This isomorphism can be extended to any (cohomologically pure) surface $S$, which opens the way for algebraic study of cohomology of various moduli spaces of sheaves. I will explain the proof of this isomorphism, and speculate about applications to $P=C$ phenomena for $K3$ surfaces and computation of ring structure of $H^*(Hilb S)$ and its cousins. Based on joint work with A. Mellit, O. Schiffmann, E. Vasserot. |
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