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# Categorical Chern character and braid groups

arXiv: Geometric Topology
Nov 2018

To a braid $\beta\in Br_n$ we associate a complex of sheaves $S_\beta$ on $Hilb_n(C^2)$ such that the previously defined triply graded link homology of the closure $L(\beta)$ is isomorphic to the homology of $S_\beta$. The construction of $S_\beta$ relies on the Chern functor $CH: MF_n^{st}\to D^{per}_{C^*\times C^*}(Hilb_n(C^2))$ defined in the paper together with its adjoint functor $HC$. The properties of these functors lead us to a conjecture that $HC$ sends $D^{per}_{C^*\times C^*}(Hilb_n(C^2))$ to the Drinfeld center of $MF_n^{st}$. Modulo an explicit parity conjecture for $CH$, we prove a formula for the closure of sufficiently positive elements of the Jucys-Murphy algebra previously conjectured by Gorsky, Negut and Rasmussen.

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