Using a geometric argument building on our new theory of graded sheaves, we
compute the categorical trace and Drinfel'd center of the (graded) finite Hecke
category $H_W^gr = Ch^b(SBim_W)$ in terms
of the category of (graded) unipotent character sheaves, upgrading results of
Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type $A$, we relate the
categorical trace to the category of $2$-periodic coherent sheaves on the
Hilbert schemes $Hilb_n(C^2)$ of points on $C^2$
(equivariant with respect to the natural $C^* C^*$
action), yielding a proof of a conjecture of Gorsky-Negut-Rasmussen which
relates HOMFLY-PT link homology and the spaces of global sections of certain
coherent sheaves on $Hilb_n(C^2)$. As an important
computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich
on the formality of the Hochschild homology of $H_W^gr$.
Description
Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$
%0 Generic
%1 ho2023graded
%A Ho, Quoc P.
%A Li, Penghui
%D 2023
%K Character homflypt sheaves
%T Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of
points on $C^2$
%U http://arxiv.org/abs/2305.01306
%X Using a geometric argument building on our new theory of graded sheaves, we
compute the categorical trace and Drinfel'd center of the (graded) finite Hecke
category $H_W^gr = Ch^b(SBim_W)$ in terms
of the category of (graded) unipotent character sheaves, upgrading results of
Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type $A$, we relate the
categorical trace to the category of $2$-periodic coherent sheaves on the
Hilbert schemes $Hilb_n(C^2)$ of points on $C^2$
(equivariant with respect to the natural $C^* C^*$
action), yielding a proof of a conjecture of Gorsky-Negut-Rasmussen which
relates HOMFLY-PT link homology and the spaces of global sections of certain
coherent sheaves on $Hilb_n(C^2)$. As an important
computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich
on the formality of the Hochschild homology of $H_W^gr$.
@misc{ho2023graded,
abstract = {Using a geometric argument building on our new theory of graded sheaves, we
compute the categorical trace and Drinfel'd center of the (graded) finite Hecke
category $\mathsf{H}_W^\mathsf{gr} = \mathsf{Ch}^b(\mathsf{SBim}_W)$ in terms
of the category of (graded) unipotent character sheaves, upgrading results of
Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type $A$, we relate the
categorical trace to the category of $2$-periodic coherent sheaves on the
Hilbert schemes $\mathsf{Hilb}_n(\mathbb{C}^2)$ of points on $\mathbb{C}^2$
(equivariant with respect to the natural $\mathbb{C}^* \times \mathbb{C}^*$
action), yielding a proof of a conjecture of Gorsky-Negut-Rasmussen which
relates HOMFLY-PT link homology and the spaces of global sections of certain
coherent sheaves on $\mathsf{Hilb}_n(\mathbb{C}^2)$. As an important
computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich
on the formality of the Hochschild homology of $\mathsf{H}_W^\mathsf{gr}$.},
added-at = {2023-05-03T13:02:46.000+0200},
author = {Ho, Quoc P. and Li, Penghui},
biburl = {https://www.bibsonomy.org/bibtex/2d30867748f19fccb7b4b47715a4e7679/dragosf},
description = {Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$},
interhash = {125db38ae10877d6f82ae1b3de6ada50},
intrahash = {d30867748f19fccb7b4b47715a4e7679},
keywords = {Character homflypt sheaves},
note = {cite arxiv:2305.01306},
timestamp = {2023-05-03T13:02:46.000+0200},
title = {Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of
points on $\mathbb{C}^2$},
url = {http://arxiv.org/abs/2305.01306},
year = 2023
}