To a braid \(\beta\in \Br_n\) we associate a complex of sheaves \(S_\beta\) on \(\Hilb_n(\CC^2)\) 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
\(\CH\colon \MF_n^{\mathrm{st}}\to \mathrm{D}^{\mathrm{per}}_{\CC^*\times\CC^*}(\Hilb_n(\CC^2))\)  defined in the paper together with its adjoint functor
\(\HC\).  The properties of these functors lead us to a conjecture that \(\HC\) sends
\(\mathrm{D}^{\mathrm{per}}_{\CC^*\times\CC^*}(\Hilb_n(\CC^2))\)
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.

§ INTRODUCTION

#1 Ad_#1

#1 _#1

#1 ^#1

In <cit.> we constructed a homomorphism \(\Phi\) from the braid group \(\Br_n \) to the convolution algebra of \(\Tqt=\CCsq\times\CCst\)-equivariant
matrix factorizations:

\begin{equation}
\label{eq:stfrmf}
\MF^{\st}=\MFGLT\left(\big(\gln\times \CCn\times \TsFl\times\TsFl\big)^{\rst},W\right), \quad
\end{equation}

where \(\TsFl=(\frn\times\GLn)/B\) is the affine cotangent space to the flag variety, $\amu\colon\TsFl\to \glns
%\times (\CC^n)^*
$ is the corresponding generalized moment map and $(\cdots)^{\rst}$ denotes an open  stability condition defined later. This categorical representation of the braid group was used to construct a triply-graded link invariant (homology):

\[\HHH(\beta)=\mathbb{H}(\mathcal{E}\mathrm{xt}(\Phi(\beta),\Phi(1))\otimes \Lambda^\bullet \calB),\]

where \(\calB\) is the tautological vector bundle and $\mathbb{H}(\dots)$ is the hypercohomology. The $q$- and $t$-gradings are weights of $\Tqt$, while the a-grading is the exterior power in $\Lambda^\bullet \calB$.

In this paper we construct a pair of functors which we call a Chern functor and a co-Chern functor:

\begin{equation}
\label{eq:mnchcchf}
\begin{tikzcd}
\MF^{\st}\arrow[rr,bend left,"\CH^{\st}_{\loc}"]&&\DperT(\Hilb)\arrow[ll,bend left,"\HC^{\st}_{\loc}",pos=0.435]
\end{tikzcd},
\end{equation}

where \(\Hilb\) is the Hilbert scheme of \(n\) points on \(\CC^2\), while \(\DperT(\Hilb)\) is the derived category of
two-periodic \(\Tqt\)-equivariant complexes on the Hilbert scheme.
In a subsequent paper <cit.> we will put these functors in context of a 2-category of a 3D TQFT (equivariant B-model).

Here we explore the properties of the functors and show that

* the functors \(\CHfl\) and \(\HCfl\) are adjoint,

* the functor \(\HCfl\) is monoidal,

* the image of \(\HCfl\) commutes with the elements \(\Phi(\beta)\), \(\beta\in \Br_n\).

Finally as a manifestation of the categorified Riemann-Roch formula, we obtain a new interpretation for the triply-graded homology:

For any \(\beta\in \Br_n\) we have:

\[\HHH(\beta)=\Hom(\calO,\CHfl(\Phi(\beta))\otimes \Lambda^\bullet\calB).
\]

Moreover, for any \(k\) we have

\[
\CHfl(\Phi(\beta\cdot FT^k))=
\CHfl(\Phi(\beta))\otimes \det(\calB)^k\]

where \(FT\) is the full-twist braid.

The advantage of this new interpretation is that the Hilbert scheme is smooth, unlike the flag Hilbert scheme which
is a homological support of \(\mathcal{E}\mathrm{xt}(\Phi(\beta),\Phi(1))\). So the complexes
on \(\Hilb\) are  more manageable than their flag counter-part. As an example, we apply the Chern functor to the sufficiently positive Jusys-Murphy braids in $\Br_n$ and use a combination of the `parity property' of the resulting complexes of sheaves together with localization formula in order to establish an explicit formula for their link homology.

Recall that the JM subgroup $\fJMn\subset\Br_n$ is generated by elementary JM braids

\[
\delta_i=\sigma_i\sigma_{i+1}\dots\sigma_{n-1}^2\dots\sigma_{i+1}\sigma_i,
\]

and a JM braid can be presented as their product:

\[
\mJMa:= \prod_{i=2}^{n}\delta^{a_i}_i,\qquad \text{where}\quad
\vec{a} = (a_1,\ldots,a_{n-1})\in\ZZ^{n-1}.
\]

Denote two special elements of $\ZZ^{n-1}$ as
\(\vec{1}=(1,\dots,1)\) and \(\vec{\rho}=(1,2,\dots,n-1)\).

We propose the following geometric parity  conjecture which we plan to revisit in the later paper:

For any $\vec{b}\in \ZZ^{n-1}$ there is  an integers $K$ such that for any \(k>K\), if \(\vec{a}=\vec{b}+m\vec{\rho}\),
then \(\CH^{\rfs}_{\rloc}(\CH_{\rloc}^{\rfs}(\delta^{\vec{a}}))\) is a sheaf (not just a two-periodic complex) supported in the even
homological degree.

Modulo this conjecture we have

For any $\vec{b}\in \ZZ^{n-1}$ there are  integers $K,M$ such that for any \(k>K,m>M\), if \(\vec{a}=\vec{b}+k \vec{1}+m\vec{\rho}\),

$(q,t,a)$-character of the homology of $\mJMa$  is given by the formula

\[
\dim_{a,Q,T}\HHH(\mJMa) =
\sum_{\ytb} \prod_i\frac{z_i^{b_i}(1+az_i^{-1})}{1-z^{-1}}\prod_{1\le i<j\le n}\zeta(\frac{z_i}{z_j}),\]

\[\zeta(x)=\frac{(1-x)(1-QTx)}{(1-Qx)(1-Tx)},\qquad Q=q^2,\qquad T=t^2/q^2,\]

while $\ytb$ is the set of all standard Young tableaux, we denote
\(z_i=Q^{a'(i)}T^{l'(i)}\) and \(a'(i),l'(i)\) are co-arm and co-leg of the $i$-labeled  square in the standard tableau.

Altogether we construct the Chern and co-Chern functors for three braid related categories of matrix factorizations: the unframed, the unstable framed and the stable framed category (<ref>).
We showed in <cit.> that the first two categories
provide  representations of the affine braid group, while the third one provides a representation of the ordinary braid group.

In section <ref> we recall the basic theory of
matrix factorizations and define our `unframed' categoriy together with its Chern and co-Chern functors \(\HC\)
and \(\CH\), proving their properties. In section <ref> we introduce the
framed and stable versions of our category and its Chern and co-Chern functors. In this section
we also discuss the linear Koszul duality relating the category of matrix factorizations to the category of coherent sheaves and
we show that we can combine the linear Koszul duality functor with the (co-) Chern functor to obtain the functor (<ref>).
In the section <ref> we prove theorem <ref>.

In section <ref> we use Chern functor theory to prove theorem <ref>. Finally, section <ref> we consider a version
of the Chern functor to the category of coherent sheaves on the space which is `between the flag Hilbert scheme and the ordinary Hilbert scheme'.
We expect that this functor could be related to the results of Bezrukavnikov <cit.>. We also discuss a conjectural
description of the Drinfeld centers of our main monoidal categories.

We would like to thank Dmitry Arinkin, Eugene Gorsky, Ivan Losev, Roman Bezrukavnikov and Andrei Neguţ for useful discussions.
The work of A.O. was supported in part by  the NSF CAREER grant DMS-1352398, NSF FRG grant DMS-1760373 and Simons Fellowship.
The work of L.R. was supported in part by  the NSF grant DMS-1108727.

§ CHERN CHARACTER

§.§ General facts about equivariant matrix factorizations

In this section we recall conventions of matrix factorization theory also define the unframed version of our main category.
For an algebraic variety \(X\) and a polynomial \(F\in \CC[X]\) Orlov <cit.> defines a DG category \(\MF(X,F)\) whose objects
are `curved' homologically $\ZZ_2$-graded finite rank differential modules

\begin{equation}
\label{eq:mfdf}
(M,D) = \bigl(\begin{tikzcd}M_0 \arrow[r,shift left=0.5ex,"D_{01}"] & M_1  \arrow[l,shift left=0.5ex,"D_{10}"]\end{tikzcd}\bigr)
\end{equation}

`Curved' means that the square of the total differential $D = D_{01} + D_{10}$ is equal to $F$: $D^2 = F\xId_M$. The morphisms $\Hom_{\MF}(M_1,M_2)$ between two matrix factorizations are defined as homomorphisms $\Hom_{\CC[X]}(M_1,M_2)$ equipped with the differential $d = [D,\cdot]$, where $D = D_1 + D_2$. Note that $d^2 = 0$.

If $F$ has a presentation $F = f_1 g_1 +\cdots + f_m g_m$, $f_i,g_i\in\CC[X]$, then one defines a Koszul matrix factorization

\[
\begin{bmatrix}f_1&g_1\\f_2&g_2\\ \vdots&\vdots\\f_m&g_m\end{bmatrix}
= \bigotimes_{i=1}^m \begin{tikzcd}(\CC[X] \arrow[r,shift left=0.5ex,"f_i"] & \CC[X] \arrow[l,shift left=0.5ex,"g_i"])\end{tikzcd}
\]

Equivalently, the Koszul matrix factorization can be presented with the help of odd variables $\theta_1,\ldots,\theta_m$: its module is $M = \CC[X]\otimes \CC[\theta_1,\ldots,\theta_m]$, while its differential is $D = \sum_{i=1}^m f_i \theta_i + g_i \frac{\partial}{\partial \theta_i}$. If $f_1,\ldots,f_m$ form a regular sequence, then the Koszul matrix factorization is independent of the choice of $g_1,\ldots, g_m$ up to an isomorphism, so we use an abbreviated notation \(\mathrm{K}^F(f_1,\dots,f_m)\in \MF(X,F)\).

If an algebraic group $G$ acts on \(X\)  and \(F\in \CC[X]^G\), then we define a category of
\(G\)-equivariant matrix factorizations \(\MF_G(X,F)\), see <cit.> for details.
If \(G\) is not reductive, then the definition of the equivariant matrhix factorization includes the  complex \(\CE_\frg\), \(\frg=\mathrm{Lie}(G)\).
is included in the definition of the equivariant
matrix factorization,
see sections 2 and 3 of <cit.> for details.

We define \(\DperG(X)\) as a derived category of two-periodic \(G\)-equivariant complexes of
\(\CC[X]\)-modules.

For a matrix factorization \(\calF=(M,D)\in \MF_G(Z,F)\) of (<ref>) we define a conjugate matrix factorization \(\calF^*\in\MFG(X,-F)\)
by inverting the sign of the component $D_{01}$ of  the differential:
$ \calF^* = \bigl(\begin{tikzcd}M_0 \arrow[r,shift left=0.5ex,"-D_{01}"] & M_1  \arrow[l,shift left=0.5ex,"D_{10}"]\end{tikzcd}\bigr)$.

Since the potentials of matrix factorizations
add under a tensor product, we define:

\[\mathcal{E}\mathrm{xt}(\calG,\calF):=\CE_\frg(\calG\otimes \calF^*)\in \Dper(X^G),\]

where \(X^G=\Spec(\CC[X]^G).\)

For two \(G\)-varieties \(X,Y\) with \(G\)-invariant  potentials \(F,H\) and  an equivariant map \(f\colon X\to Y\) such
that \(f^*(H)=F\), there is an obvious pull-back functor \(f^*\colon \MF_G(Y,H)\to \MF_G(X,F)\).
Moreover, if \(f\) is a smooth projection or a regular embedding then there is a well-defined
push-forward functor \(f_*\colon \MF_G(X,F)\to \MF_G(Y,G)\) which is adjoint to $f_*$, see <cit.>.
The pull-back and push-forward functors satisfy the base relations for commuting squares of maps.

Suppose \(X,Y,F,H,f \) are as above, and \(f\) is either a smooth projection or a regular embedding. Then
for any \(\calF\in \MF_G(X,F)\), \(\calG\in \MF_G(Y,H)\) there is a relation

\[f^*\mathcal{E}\mathrm{xt}(\calF,f_*(\calG))= \mathcal{E}\mathrm{xt}(f^*\calF,\calG).\]

If $f$ is a smooth projection, the statement is immediate. Suppose that \(f\) is a regular embedding and
the image is defined by the ideal with generators \(g_1,\dots,g_m\) which form a regular sequence.
To simplify notations in our proof, we assume \(m=1\).
The construction of the push-forward <cit.> produces  a matrix factorization
\(f_*(\calG)\otimes \calF^*\) which is a two-periodic curved complex

\begin{equation}\label{eq:2per}
\dots\xrightarrow{} M_1\otimes \calF^*\xrightarrow{g_1} M_0\otimes\calF^* \xrightarrow{D'} M_1\otimes \calF^*\xrightarrow{g_1} M\otimes\calF^*\xrightarrow{g_1}\dots
\end{equation}

where \(M_i\) are curved complexes such \(M_0\oplus M_1\) modulo ideal \((g_1)\) is equal to \(\calG\).
Hence in complex (<ref>) we can contract the even arrows. Since \(g_1\) is a non-divisor of zero,
only even terms of the complex (<ref>) survive after the contraction, thus the correction differentials
as in lemma 9.3 <cit.> vanish,
and we obtain the complex
\(\calG\otimes f^*(\calF)\) after the contraction.

Denote $G = \GLn$, $B\subset G$ being its Borel subgroup (of upper-triangular matrices).
Recall that the flag variety $\Fl$ is a quotient: $\Fl = G/B$ and similarly its cotangent bundle $\TsFl$ is a quotient: $\TsFl = (G \times\frn)/B$, where $\frn\subset \frg$ is the nilpotent radical (of strictly upper-triangular matrices) in $\frg = \gln$. The moment map $\TsFl\rightarrow \frg$ corresponding to the action of $G$ has an explicit form $\mu(g,Y) = \Adv{g} Y$, where $(g,Y)\in G\times\frn$.

§.§ Monoidal category and Drinfeld center as categories of matrix factorizations

The main (unframed) monoidal category in <cit.> is the category of equivariant matrix factorizations

\begin{equation}
\label{eq:stfrmf1}
\MF=\MFGsLT\left(\frg\times  \TsFl\times\TsFl,\mu_1-\mu_2\right)
\end{equation}

\[\scX:=\frg\times (G  \times \frn)\times (G \times\frn),\quad W(X,g_1,Y_1,g_2,Y_2)=\Tr(X(\Ad_{g_1}Y_1-\Ad_{g_2}Y_2))\]

and the action of an element $(h,\grbo,\grbt)\in \GBt$ on $\scX$ is

\[
(h;\grbo,\grbt)\cdot(X;g_1,Y_1;g_2,Y_2) = (\Adv{h}X;hg_1\grbo^{-1},\Adv{\grbo}Y_1;
\]

For the precise definition of \(B\)-equivariant matrix factorizations see <cit.>.

The (candidate) Drinfeld center category is also a category of equivariant matrix factorizations:

\[
\MF_{\dr}:=\MF_{G}(\scc,W_{\dr}),\quad
\scc=\frg\times G\times \frg,\quad W_{\dr}(Z,g,X)=\Tr(X(Z-\Ad_gZ)),
\]

the group $G$ acting on components of $\scc$ by conjugation.

§.§ Construction of the Chern functor: motivation

The discussion in this
section meant to provide a geometric intuition behind our construction and the technical
details are postponed till the next subsection.
The Chern character functor \(\CH\colon \MF\rightarrow \MF_{\Dr}\) is the Fourier-Mukai transform going through the category of $(\GBd)$-equivariant matrix factorizations $\MFGBdT(\scZCHd; \WCHxd)$, where

\[
\scZCHd = \frg\times (G\times\frnD) \times G,\qquad
\WCHxd(X;g,Y;h) = \Tr (
X (\Ad_{gh}Y - \Ad_{g}Y)),
\]

$\frnD=\frn$ is (identified with) the diagonal in the factor $\frn\times\frn$ of the space $\scX$, the group $\Bd=B$ is (identified with) the diagonal  in $B^2$ acting on that space and the action of $\GBd$ on $\scX_\Delta$ is

\[
(h,\grb)\cdot(X;g_1,Y;g) = (\Adv{h}X;hg_1\grb^{-1},\Adv{\grb}Y; \Adv{h}g)
\]

We define two equivariant maps

\[
\begin{tikzcd}
& \scZCHd
\ar[dl,"\hat{f}_{\dr}"']
\ar[dr,"\hat{\pi}_{\dr}"]
\\
\scX && \scc
\end{tikzcd}
\]

explicitly as

\begin{equation}
\label{eq:pdr}
\hat{f}(X; g,Y;h) = (X; gh,Y;h,Y),
\qquad
\hat{\pi}_{\dr}(X;g,Y;h) = (X,g,\Adv{h}Y).
\end{equation}

Both maps $\hat{f}_\Delta$ and $\hat{\pi}_{\dr}$ have a clear geometric meaning.
The map $\hat{f}_\Delta$ identifies $\scZCHd$ with a subvariety of $\scX$ corresponding to the diagonal $\frnD\subset\frn\times \frn$, which is invariant with respect to the action of the diagonal subgroup $\Bd\subset B^2$. As for $\hat{\pi}_{\dr}$, note that $\scZCHd/\Bd = \frg\times\TsFl\times G$ and the $\Bd$-equivariant map $\hat{\pi}_{\dr}$ descends on the quotient to $\frg\times\TsFl\times G\rightarrow \frg\times \frg\times G$ acting as the Springer resolution (composed with the natural embedding $\frn\hookrightarrow\frg$) on the second factor.

Now the Chern functor and its adjoint co-Chern functor are compositions of a pull-back and a push-forward:

\[
\CH = \hat{\pi}_{\dr*}\circ\hat{f}_\Delta^*,\qquad\HC = \hat{f}_{\Delta*}\circ\hat{\pi}^*_{\dr}.
\]

From the computational perspective it is convenient to split $\pDr$ into a composition of two maps

\[
\begin{tikzcd}
\scZCHd
\ar[r,hook,"\emi"']
\ar[rr,bend left,"\hat{\pi}_{\dr}"]
\scZ_{\CH}\ar[r,"\pDr"']
\scc
\end{tikzcd}
\]

where $\scZ_{\CH} = \frg\times G\times\frg\times G\times \frb$ and the embedding $\emi$ is generated by the natural inclusion $\frnD\hookrightarrow \frb$ combined with the adjoint action \(\emi(X,g,Y,h)=(\Ad_h Y,g,X,h,Y)\). The map \(\pi_{\dr}\)
is a projection on the first three factors.

Let us also point out that a formula for co-Chern functor in this section requires many clarifications and correction:
it is not immediately clear what is the adjoint functor to the restriction of \(B^2\)-equivariant structure to
the \(B_\Delta\)-equivariant structure. We spell out the omitted technical in the next subsection.

§.§ The Chern functor: details

As we explained in the previous subsection, we need two auxiliary spaces in order to define the Chern functor:

\[\scz_{\CH}=\frg\times G\times\frg\times G\times\frn,\quad \scZ_{\CH}=\frg\times G\times\frg\times G\times\frb\]

The action of \(G\times B\) on these spaces is

\[(k,b)\cdot (Z,g,X,h,Y)=

and the invariant potential is

\[W_{\CH}(Z,g,X,h,Y)=\Tr(X(\Ad_{gh}(Y)-\Ad_{h}(Y))).\]

The spaces \(\scC\) and \(\scX\) are endowed with the standard \(G\times B^2\)-equivariant structure, the action of  \(B^2\) on \(\scC\) is trivial.
The following maps

\[\pi_{\Dr}:\mathscr{Z}_{\CH}\rightarrow \mathscr{C},\quad f_\Delta:\scZ_{\CH}^0\rightarrow \mathscr{X}\quad j^0:\scZ^0_{\CH}\to \scZ_{\CH}.\]

\[\pi_{\Dr}(Z,g,X,h,Y)=(Z,g,X),\quad f_\Delta(Z,g,X,h,Y)=(X,gh,Y,h,Y)\]

are  fully equivariant if we restrict the \(B^2\)-equivariant structure on
\(\scX\) to the \(B\)-equivariant structure via the diagonal embedding
\(\Delta:B\rightarrow B^2\). Note that $j^0$ is the inclusion map.

The kernel of the Fourier-Mukai transform is the Koszul matrix factorization

\[\mathrm{K}_{\CH}:=[X-\Ad_{g^{-1}}X,\Ad_hY-Z]\in \MF(\scZ_{\CH},\pi_{\Dr}^*(W_{\Dr})-f^*_\Delta(W)).\]

and we define the Chern functor:

\begin{equation}\label{eq:CH}
\CH(\calC):=\pi_{\Dr*}(\CE_{\frn}(\mathrm{K}_{\CH}\otimes (j^0_*\circ f^*_\Delta(\calC)))^{T}).
\end{equation}

We also define the co-Chern functor $\HC$ as the adjoint functor that goes in the opposite direction:
\(\HC\colon\MF_{\Dr}\rightarrow \MF.\)
Thus, the functor $\HC$ is the composition of adjoints of all the functors that appear in the formula (<ref>).

The product \(\scZ_{\CH}\times B\) has a \(B\times B\)-equivariant structure: for $(p,g)\in \scZ_{\CH}\times $ we define

\[
(h_1,h_2)\cdot (p, g) = (h_1\cdot p,h_1 g h_2^{-1})
\]

Then the following map is \(B^2\)-equivariant:

\[\tilde{f}_{\Delta}:\scZ_{\CH}^0\times B\rightarrow \scX\times B,\]

\begin{equation*}\tilde{f}_\Delta(Z,g,X,h,Y,b)=(X,gh,Y,hb,\Ad_bY,b).\end{equation*}

The map \(\tilde{f}_\Delta\) is a composition of the projection along the first factor of \(\scZ_{\CH}\) and
the embedding inside \(\scX\times B\). The embedding is defined by the formula

\[\Ad_bY_1=Y_2,\]

so it is a regular embedding. Thus since

\[j^{0*}(\mathrm{K}_{\CH}\otimes \tilde{\pi}_{\Dr}^*(\calD))\in\MF_{G\times B^2}(\scZ_{\CH}\times B,\tilde{f}_\Delta^*(W)),\]

where \(\tilde{\pi}_{\Dr}:\scZ_{\HC}\times B\rightarrow\scC\) is a natural extension of map
\(\pi_{\Dr}\) by the projection along \(B\),
we have a well-defined matrix factorization \(\tilde{f}_{\Delta *}\circ j^{0*}(\mathrm{K}_{\CH}\otimes \pi_{\Dr}^*(\calD))\in \MF_{G\times B^2}(\scX\times B,\pi_B^*(W)),\) where \(\pi_B\) is the projection along the last factor. Now we can define:

\begin{equation}\label{eq:HC}\HC(\calD):=\pi_{B*}(\tilde{f}_{\Delta*}\circ j^{0*}(\mathrm{K}_{\CH}\otimes \pi_{\Dr}^*(\calD))).\end{equation}

The functors \(\HC\) and \(\CH\) are adjoint, that is, for
any \(\calC\in\MF\) and \(\calD\in\MF_{\Dr}\) we have
an isomorphism of doubly-graded two-periodic complexes
of \(\CC[\frg]^{G}\)-modules:

\[\mathcal{E}\mathrm{xt}(\calD,\CH(\calC))=\mathcal{E}\mathrm{xt}(\HC(\calD),\calC),
\quad \mathcal{E}\mathrm{xt}(\CH(\calC),\calD)=\mathcal{E}\mathrm{xt}(\calC,\HC(\calD)).\]

We have to compare two objects of \(\DperG(\frg)\):

\[\calD\otimes \pi_{\Dr*}(\CE_{\frn}(\mathrm{K}^*_{\HC}\otimes j^{0}_*\circ f_{\Delta}^*(\calC^*))^T) \quad\text{and}\quad
\CE_{\frn^2}(\pi_{B*}\circ\tilde{f}_{\Delta*}\circ j^{0*}
(\mathrm{K}_{\HC}\otimes\tilde{\pi}_{\Dr}^*(\calD))\otimes \calC^*)^{T^2}.\]

We  simplify the second expression.  Observe that the  functor commutes with the
push-forward. Since the first factor of \(B\) in \(B^2\) acts freely on \(\scZ_{\CH}\times B\), the functor
\(\CE_{\frn^2}(\dots)^{T^2}\) is equivalent to the composition of the restriction to the
unity inside \(B\) and the functor \(\CE_{\frn}(\dots)^T\) with respect to the diagonal action of \(B\). In other words:

\[\CE_{\frn^2}(\mathrm{K}_{\HC}\otimes\tilde{\pi}_{\Dr}^*(\calD))^{T^2}=\CE_{\frn}(\mathrm{K}_{\CH}\otimes\pi_{\Dr}^*(\calD))^T.\]

When restricted to the unit inside \(B\), the map \(\tilde{f}_{\Delta}\) becomes \(f_{\Delta}\),
so now we have to compare

\[\calD\otimes \pi_{\Dr*}(\CE_{\frn}(\mathrm{K}^*_{\HC}\otimes j^{0}_*\circ f_{\Delta}^*(\calC^*))^T)\quad\text{and}\quad f_{\Delta*}(\CE_{\frn}(
\mathrm{K}_{\HC}\otimes j^{0*}\circ {\pi}_{\Dr}^*(\calD))\otimes \calC^*)^{T}).\]

The pull-back functors are adjoint to the corresponding push-forward functors, hence the statement follows.

§.§ Properties

Recall that the category \(\MF\) has a convolution algebra structure defined with the auxiliary convolution space

\[\scX_{cnv}:=\frg\times (G\times\frn)\times (G\times\frn)\times (G \times\frn).\]

This space has a unique \(B^3\)-equivariant structure such that the  projection maps

\[\pi_{ij}(X,g_1,Y_1,g_2,Y_2,g_3,Y_3)=(X,g_i,Y_i,g_j,Y_j),\]

commute with the action of each $B$. The convolution is defined in the usual way as a push-forward of the tensor product of two pull-backs, the push-forward includes taking derived invariants with respect to the action of the middle Borel subgroup $B$:

\begin{equation}\label{eq:conv}\calF\star\calG:=\pi_{13*}(\CE_{\frn^{(2)}}(\pi_{12}^*(\calF)\otimes\pi_{23}^*(\calG))^{T^{(2)}}).
\end{equation}

The convolution provides a monoidal structure on \(\MF\), hence there is a well-defined Drinfeld center \(\mathfrak{Z}(\MF)\) of this category. We expect that the functor \(\HC\)  could be enhanced to the functor
\(\HC\colon\MF_{\Dr}\rightarrow \mathfrak{Z}(\MF).\) In this paper we prove a weaker statement which is
just about the convolution algebra structure, see proposition <ref>.

The Koszul matrix factorization
\[\calO:=[g-1,[X,Z]]\in \MF_{\dr}\]
is a unit (see proposition <ref>) in the natural convolution algebra \(\MF_{\Dr}, \star\) with the convolution defined by:

\begin{equation}\label{eq:convDr}
\calF\star\calG:=\pi_{3*}(\pi_1^*(\calF)\otimes\pi_2^*(\calG))
\end{equation}

where the maps \(\pi_i\) are the maps \(\frg^2\times G^2\rightarrow\scC\) are

\[\pi_1(Z,X,g_1,g_2)=(Z,X,g_1),\quad \pi_2(Z,X,g_1,g_2)=(\Ad_{g_1}Z,X,g_2),\]

\[\pi_3(X,Y,g_1,g_2)=(Z,X,g_1g_2).\]

Moreover, the functor \(\HC\) respects the convolution product.

The functor \(\HC\) is monoidal.

Consider the embedding of the nilpotent cone $j^0\colon\caln\rightarrow\frg$.

$$\MF_{\Dr}^0:=\MF_{G\times B^2 }(\scC^0,W_{\dr}), \quad \scC^0=\frg\times G\times\caln

By restricting the maps $\pi_i$ in (<ref>) to the nilpotent locus we obtain a definition of the
convolution structure on the category $\MF_{\dr}^0$ such that the pull-back functor:

is a homomorphism of convolution algebras.
Define an analog of the functor $\HC$ for the nilpotent version of our category:

$$\HC^0:\MF_{\dr}^0\to \MF,\quad \HC^0(\calD):=\pi_{B*}(\tilde{f}_{\Delta*}(\mathrm{K}_{\CH}\otimes \pi_{\Dr}^*(\calD))).$$

Since the maps $j^0$ and $\pi_{\dr}$ commute,

the previous functor is the composition of the new one with the pull-back:

$$\HC=\HC^0\circ j^0.$$

By the previous remark, it is enough to show that the functor $\HC^0$ is monoidal.
Our proof relies on the base change along the rectangle of maps:

\[
\begin{tikzcd}
\scX& \scX_{conv}\arrow[l,"\pi_{13}"]\arrow[r,"i_{cnv}"]&\scX\times\scX\\
\scZ^0\times B\arrow[u,dotted,"\tilde{f}_{\Delta}\circ\pi_{B}"]\arrow[d,dotted,"\pi_{\Dr}\times 1"]& \mathscr{Y}\arrow[r,dotted,"\hat{i}_{cnv}"]\arrow[u,dotted,"\hat{f}"]\arrow[l,dotted,"\pi_Y"]&\scZ^0\times\scZ^0\times B\arrow[u,"f_{\Delta}\times\tilde{f}_{\Delta}\circ \pi_B"]\arrow[d,"\pi_{\Dr}\times\pi_{\Dr}\times \pi_B"]\\
\mathscr{C}^0\times  B&\scC^0_{cnv}\times G\times B\arrow[l,dotted,"\pi_3\times \pi_G"]\arrow[r,dotted,"j_{cnv}"]\arrow[u,"i_C",dotted]&\scC^0\times\scC^0
\end{tikzcd}
\]

\[\mathscr{Y}=\frg^3\times G^3\times\frn\times B\]
and the dotted maps will be explained below.
First of all, we explain why pushing along the solid arrows results in \(\HC^0(\calD')\star\HC^0(\calD'')\). Indeed,  expand
the expression for the convolution:

\[\pi_{13*}(\CE_{\frn^{(2)}}(i^*_{cnv}(\tilde{f}_{\Delta*}\circ\pi_{B*}(\mathrm{K}\otimes \pi^*_{\Dr}(\calD'))\boxtimes \tilde{f}_{\Delta*}\circ\pi_{B*}(\mathrm{K}\otimes \pi^*_{\Dr}(\calD'')) ))^{T^{(2)}}),\]

where \(i_{cnv}\colon \scX_{cnv}\rightarrow \scX\times\scX\) is the natural inclusion:

\[i_{cnv}(X,g_1,Y_1,g_2,Y_2,g_3,Y_3)=(X,g_1,Y_1,g_2,Y_2)\times (X,g_2,Y_2,g_3,Y_3).\]

Next we notice that \(\frn^{(2)}\) and \(T^{(2)}\) act freely on \(\scZ^0\times B\times \scZ^0\times B\) and since
this space is a domain for \(\mathrm{K}\otimes\pi_{\Dr}^*(\calD')\boxtimes\mathrm{K}\otimes \pi_{\Dr}^*(\calD'')\),
we replace the functor \(\CE_{\frn^{(2)}}(\dots)^{T^{(2)}}\) with the restriction functor \((\dots)_{b'=1}\) where
\(b'\) is an element of the first copy of \(B\) in \(\scZ^0\times B\times \scZ^0\times B\). Thus

\[\HC^0(\calD')\star\HC^0(\calD'')=\pi_{13*}(i^*_{cnv}(\tilde{f}_{\Delta*}(\mathrm{K}\otimes\pi_{\Dr}^*(\calD'))\boxtimes
\tilde{f}_{\Delta*}\circ\pi_{B*}(\mathrm{K}\otimes \pi^*_{\Dr}(\calD''))))).\]

Define the maps \(\hat{i}_{cnv}\) and \(\hat{f}\):

\[\hat{i}_{cnv}(Z_1,Z_2,X,g_1,g_2,g_3,Y,b)=(Z_1,g_1g_2^{-1},X,g_2,Y,Z_2,g_2bg_3^{-1},X,g_3b^{-1},Y),\]

\[\hat{f}(Z_1,Z_2,X,g_1,g_2,g_3,Y,b)=(X,g_1,Y,g_2,Y,g_3,\Ad_b Y).\]

Thus using the base change in the upper left corner of the diagram we obtain

\[\HC^0(\calD')\star\HC^0(\calD'')=\pi_{13*}\circ\hat{f}_{*}\circ \hat{i}^*_{cnv}(\mathrm{K}_1\boxtimes \mathrm{K}_2\otimes \pi_{\Dr}^*\times\pi_{\Dr}^*\times\pi_B^*(\calD'\boxtimes\calD'')),\]

where \(\mathrm{K}_i\) is the pull-back of the kernel \(\mathrm{K}\) along the projection on the \(i\)-th copy of
\(\scZ^0\) in the product \(\scZ^0\times \scZ^0\times B\).

Next define the maps \(i_C\) and \(j_{cnv}\):

\[i_C(Z,X,g',g'',Y,b,h)=(Z,\Ad_{g'g''}Z,X,g'h,h,(g'')^{-1}hb,Y,b),\]

\[j_{cnv}(Z,X,g',g'',Y,b,h)=(X,Z,g',X,\Ad_{g'g''}Z,g'').\]

Now   because of the explicit formula for the  Koszul matrix factorization \(\mathrm{K}_2\)  and
the construction of the push-forward <cit.> we conclude that

\[\hat{i}^*_{cnv}(\mathrm{K}_1\boxtimes \mathrm{K}_2\otimes \pi_{\Dr}^*\times\pi_{\Dr}^*\times\pi_B^*(\calD'\boxtimes\calD''))=
i_{C*}(\mathrm{K}\otimes j^*_{cnv}(\calD'\boxtimes \calD'')),\]

where \(\mathrm{K}=i_{C}^*(\mathrm{K}_2)\). There is a unique map \(\pi_Y\) that makes our diagram commute. The commutativity of
the diagram implies  the formula

\[
\HC^0(\calD')\star\HC^0(\calD'') =
\tilde{f}_{\Delta*}\circ \pi_{B*}\circ \pi_{Y*}\circ i_{C*}(\mathrm{K}\otimes j^*(\calD'\boxtimes \calD'')).\]

Applying the base change to the lower-left corner in order to obtain another formula

\[
\HC(\calD')\star\HC(\calD'') =
f_{\Delta*}\circ \pi_{B*}(\mathrm{K}\otimes \pi^*_{\Dr}\times 1\circ \pi_{3*}\times \pi_{G*}\circ j_{cnv}^*(\calD'\boxtimes \calD'')),\]

where \(\mathrm{K}\) is \(\mathrm{K}_{\HC}\). To complete proof we observe that

\[\pi_{3*}\times \pi_{G*}\circ j_{cnv}^*(\calD'\boxtimes \calD'')=\pi_{B}^*(\calD'\star\calD'').\]

For any \(\calD\in \MF_{\dr}\) and a pair of Koszul matrix factorizations  \(\calc^+,\calc^-\in \MF\) such that \(\calc^+\star\calc^-\sim 1 \),
there is a homotopy equivalence:

\begin{equation}
\label{eq:dblpr}
\calC^+\star \HC(\calD)\star \calc^-\sim\HC(\calD).
\end{equation}

The space \(\hat{\scX}=\frg\times (G\times \frn)^4\times B\) has natural \(B^4\)-equariant projections
\(\hat{\pi}_{ij}:\hat{\scX}\to \scX \) as well as an embedding \(\tilde{f}_{\Delta}\colon\scZ\times (G\times \frn)^2\times B\)
defined as

\[\tilde{f}_\Delta(Z,g,X,h,Y,g_1,Y_1,g_4,Y_4,b)=(X,g_1,Y_1,gh,Y,hb,\Ad_bY,g_4,Y_4,b).\]

The double product in the statement is equal to \(\hat{\pi}_{14*}(\CE_{\frn^{(2)}\times\frn^{(3)}}(\calC')^{T^{(2)}\times T^{(3)}})\) where \(\calC'\) is the complex on
\(\hat{\scX}\):

\[\calc':=\hat{\pi}_{12}^*(\calc^+)\otimes \hat{\pi}_{34}^*(\calc^-)\otimes \tilde{f}_{\Delta *}\bigl(j^{0*}\circ\pi_{\dr}^*(\calD)\otimes \mathrm{K}_{\CH}\bigr). \]

Since the push-forward \(\tilde{f}_{\Delta*}\) is adjoint to the pull-back \(\tilde{f}_{\Delta}^*\) with respect to the
pairing \(\CE_{\frn^{(2)}\times\frn^{(3)}}(\cdot\otimes\cdot)^{T^{(2)}\times T^{(3)}}\), the of (<ref>) has an expression

\begin{equation}\label{eq:pi14}
\calC^+\star \HC(\calD)\star \calc^- =
\pi_{14*}(\CE_{\frn^{(2)}\times\frn^{(3)}}(\calc'')^{T^{(2)}\times T^{(3)}}),
\end{equation}

where \(\calc''\) is the matrix factorization
on \(\scz\times (G\times \frn)^2\times B\):

\[\calc'':=\pi^*_{1\bullet}(\calc^+)\otimes \pi^*_{\bullet 4}(\calc^-)\otimes j^{0*}\circ\pi^*_{\dr}(\calD)\otimes \mathrm{K}_\CH,\]

and the maps in the last formula are:

\[\pi_{1\bullet}(Z,g,X,h,Y_\bullet,g_1,Y_1,g_4,Y_4,b)=(X,g_1,Y_1,gh,Y_\bullet),\]

\[\pi_{\bullet 4}(Z,g,X,h,Y_\bullet,g_1,Y_1,g_4,Y_4,b)=(X,hb,\Ad_bY_\bullet,g_4,Y_4),\]

\[\pi_{14}(Z,g,X,h,Y_\bullet,g_1,Y_1,g_4,Y_4,b)=(X,g_1,Y_1,g_4,Y_4).\]

These maps satisfy relations: \(\tilde{f}_\Delta\circ \hat{\pi}_{14}=\pi_{14}\), \(\tilde{f}_\Delta\circ\hat{\pi}_{12}=\pi_{1\bullet}\),
\(\tilde{f}_\Delta\circ \hat{\pi}_{34}=\pi_{\bullet 4}\).

Just as in the proof of proposition <ref> we observe that the left \(B\)-action on \(B\) is free, hence the functor
\(\CE_{\frn^{(2)}\times\frn^{(3)}}(\dots)^{T^{(2)}\times T^{(3)}}\) is equivalent to the composition of the restriction to the unity in \(B\) and \(\CE_{\frn}(\dots)^T\) with respect to the diagonal action of \(B\). In other words, (<ref>) is equal to
\( \pi_{14*}(\CE_{\frn}(\calc''|_{b=1})^{T})\), here and everywhere below we use the same notations \(\pi_{ab}\) for the restriction of maps
\(\pi_{ab}\) on the sub locus \(b=1\).

Since \(\mathrm{K}_{\CH}=[X-\Ad_gX, \Ad_hY-Z]\), we can use row transformations of Koszul matrix factorizations in order to replace \(X\) with \(\Ad_gX\) in \(\pi^*_{1\bullet}(\calc^+)\). Now we combine this computation with the fact that the matrix
factorization  \(\pi^*_{1\bullet}(\calc^+)\) is strongly
\(G\)-equivariant,

to establish the isomorphism

\[
\calc''|_{b=1} \cong
\calc''':=\pi'^{*}_{1\bullet}(\calc^+)\otimes \pi^*_{\bullet 4}(\calc^-)\otimes j^{0*}\circ\pi^*_{\dr}(\calD)\otimes \mathrm{K}'_\CH,\]

where \(\mathrm{K}'_{\CH}=[X-\Ad_gX,*]\) and the map \(\pi'_{1\bullet}\) is

\[\pi'_{1\bullet}(Z,g,X,h,Y_\bullet,g_1,Y_1,g_4,Y_4,b)=(X,g^{-1}g_1,Y_1,h,Y_\bullet).\]

To compute \(*\) in \(\mathrm{K}'_{\CH}\), observe that this matrix factorization has the potential:

\begin{multline*}
\Tr(X(\Ad_{g^{-1}g}Y_1-\Ad_hY_\bullet))+\Tr(X(\Ad_hY_\bullet-\Ad_{g_4}Y_4))+\Tr(X(Z-\Ad_gZ))\\-\Tr(X(\Ad_{g_1}Y_1-\Ad_{g_4}Y_4))=
\Tr((\Ad_gX-X)(\Ad_{g_1}Y_1-Z))
\end{multline*}

Thus we conclude that \(*=\Ad_{g_1}Y_1-Z\) and in \(\calc'''\) only the first two factors depend on the variable \(h,Y_\bullet\) and have
a non-trivial $B$-action. Let \(\pi\) be the projection along
the coordinates \(h,Y_\bullet\). Since \(\pi_{*}(\CE_{\frn}(\pi'^{*}_{1\bullet}(\calc^+)\otimes \pi^*_{\bullet 4}(\calc^-))^T)=\calc^+\star\calc^-\sim
\calc_{\parallel}\)
we have the homotopy between the convolution in (<ref>)

and \(\tilde{\pi}_{14*}(\tilde{\calc})\), where
\(\tilde{\calc}\) is the matrix factorization on the space \(\frg\times G\times \scX\):

\[\tilde{\calc}:=\calc_{\parallel}\otimes  j^{0*}\circ\pi^*_{\dr}(\calD)\otimes \mathrm{K}'_\CH, \]

and \(\tilde{\pi}_{14}\) is determined by \(\pi_{14}=\tilde{\pi}_{14}\circ \pi.\)
Finally,   recall that \(\calC_{\parallel}=j_{\parallel*}(\calO)\), where
\(j_{\parallel}:\frg\times G\times\scX(B) \rightarrow\frg\times G\times\scX\), \(\scX(B)=\frg\times B\times G\times \frn\)
is defined by

\[j_{\parallel}(Z,g,X,b,k,Y)=(Z,g,X,k,Y,kb,\Ad_bY).\]

Hence the convolution in (<ref>)

is equal to \(\pi_{B*}(j_{\parallel*}(j^{0*}\circ\pi^*_{\dr}(\calD)\otimes j_{\parallel}^*(\mathrm{K}'_{\CH})))\), where \(\pi_B\) is the projection along \(B\).
By identifying \(\frg\times G\times \scX(B)\) with \(\scZ^0_{\HC}\times B\) we match \(j_{\parallel}\) with \(\tilde{f}_\Delta\).
Since \(j^*_{\parallel}(\mathrm{K}'_{\CH})=\mathrm{K}_{\CH}\) the statement follows.

In <cit.> we constructed an algebra homomorphism from the affine braid group \(\Phi^{\aff}\colon\Br_{\aff}\to \MF\).
Under this homomorphism the generators of the group become Koszul matrix factorizations, thus the previous proposition

For any \(\beta\in \Br_{\aff}\) and \(\calD\in \MF_{\dr}\) we have:

\[\Phi^{\aff}(\beta)\star \HC(\calD)\sim \HC(\calD)\star \Phi^{\aff}(\beta).\]

The proof of theorem <ref> requires a computation of the co-Chern of the matrix factorization \(\calO\).

The element \(\calO\in \MF_{\Dr}\) is the convolution unit and

\[\HC(\calO)=\calC_{\parallel}.\]

Recall that \(\calC_{\parallel}=j_{\parallel*}(\calO)\), where \(j_{\parallel}\colon \calX(B):=\frg\times B\times G\times\frn\rightarrow\scX\)
is defined by

\[j_{\parallel}(X,b,h,Y)=(X,h,Y,hb,\Ad_bY).\]

There is a unique \(B^2\)-equivariant structure on the space \(\scX(B)\) that makes the map
\(j_\parallel\) equivariant.
On the other hand, \(\calX(B)\) embeds naturally into \(\scZ_{\HC}\times B\):

\begin{equation}
i_{\CH}\colon \scX(B)\to \scZ_{\CH}\times B,\quad   i_{\CH}(X,b,h,Y)=(\Ad_{h}Y, 1,X,h,Y,b).
\end{equation}

Moreover, \(\tilde{f}_\Delta\circ i_{\CH}=
j_\parallel\) and the subvariety \(i_{\CH}(\scX(B))\)
is defined by  equations

\begin{equation}\label{eq:XB}
Z=\Ad_{h}Y,\quad g=1.
\end{equation}

To complete the proof, observe that the  Koszul matrix factorization  \(\pi_{\Dr}^*(\calO)\otimes \mathrm{K}_{\CH}\) is the
Koszul matrix for the ideal with the generators (<ref>). The equations in (<ref>) form a regular sequence, hence
by lemma 3.6 in <cit.> such Koszul matrix factorization is unique  and it coincides with the push-forward
\(i_{\CH *}(\calO)\) by  definition.

§ COHERENT SHEAVES AND DRINFELD CENTER

§.§ Framed and stable matrix factorizations

In this section we introduce the framed and stable enhancements of the matrix factorizations from \(\MF\) and \(\MF_{\Dr}\):

\[\MF^{\rst},\quad\MF^{\rfr},\quad\MF^{\rfs},\quad\MF^{\rst}_{\Dr},\quad\MF^{\rfr}_{\Dr},
\quad\MF^{\rfs}_{\Dr}.\]

The category \(\MF^{\rst}\) appeared in <cit.>, its convolution structure was used to represent the braid group \(\Br_n \) and thus construct the link homology. We recall this construction in the next section.

The categories \(\MF^{\rfr}_{\bullet}\) `lie between' the categories \(\MF^{\rst}_{\bullet}\) and \(\MF^{\rfs}_{\bullet}\).

Both categories \(\MF^{\rst}_{\Dr}\) and \(\MF^{\rfs}_{\Dr}\) are related to the dg category \(\Dper (\Hilb_n(\CC^2))\) by the localization functor defined later:

\[\Loc^{\rst}\colon \Dper(\Hilb_n(\CC^2))\rightarrow \MF^{\rst}_{\Dr},\quad \Loc^{\rfs}\colon \Dper(\Hilb_n(\CC^2))\rightarrow \MF^{\rfs}_{\Dr}.\]

Moreover, we  show that the functor \(\Loc^{\rfs}\) is an isomorphism. Thus we have a well-defined functor

\[\CH^{\rfs}_{\rloc}:=\left(\Loc^{\rfs}\right)^{-1}\circ \CH^{fs}.\]

Define framed versions of the spaces \(\scX\) and \(\scC\)

\[\scX^{\rfr}:=\scX\times V^*\times V\times V,\quad \scC^{\rfr}:=\scC\times V\times V^*,\]

and their potentials

\[W^{\rfr}(X,g_1,Y_1,g_2,Y_2,w,v_1,v_2):=W(X,g_1,Y_1,g_2,Y_2)+\Tr(w(g_1v_1-g_2v_2)),\]

\[W^{\rfr}_{\Dr}(X,g,Z,v,w)=W_{\Dr}(X,g,Z)+\Tr(w(v-gv)).\]

There are natural functors \(i_\bullet^{\rfr}\colon\MF_\bullet\to\MF^{\rfr}_\bullet\)
defined as

\[i_{\bullet}^{\rfr}(\calF):=\pi^*(\calF)\otimes \mathrm{K}^{\rfr}_\bullet,\]

where $\pi$ is a natural projection of a framed spaces into the unframed one, while the Koszul matrix factorizations are

\[\mathrm{K}^{\rfr}:=[w,g_1v_1-g_2v_2],\quad \mathrm{K}_{\Dr}^{\rfr}:=[w,v-gv].\]

The framed categories have a natural convolution algebra structure similar to the unframed case.

The convolution spaces are

\[\scX_{\rcnv}^{\rfr}:=\scX_{\rcnv}\times V\times V^*\times V^*\times V^*,\quad \scC_{\rcnv}:=\scC_{\rcnv}\times V\times V^*\]

and the extended maps are

\[\pi_{ij}(X,g_1,Y_1,g_2,Y_2,g_3,Y_3,v,w_1,w_2,w_3)=(X,g_i,Y_i,g_j,Y_j,v,w_i,w_j),\]

\[\pi_1(Z,X,g_1,g_2,v,w)=(Z,X,g_1,v,w),\quad \pi_2(Z,X,g_1,g_2,v,w)=(Z,\Ad_{g_1}X,g_1v,w),\]

\[ \pi_3(Z,X,g_1,g_2,v,w)=(Z,X,g_1g_2,v,w).\]

The convolution product is defined by the formulas (<ref>) and (<ref>) and it is easy to check the following:

The functors \(i^{\rfr}_\bullet\)  are homomorphisms of the convolution algebras.

The framed categories are connected by the framed versions of the functors \(\HC\) and \(\CH\). The framed version of the
space \(\scZ_{\CH}\) and maps \(\pi_{\Dr}\), \(f_\Delta\) are defined as

\[\scZ_{\CH}^{\rfr}:=\scZ_{\CH}\times V\times V^*, \quad \pi_{\Dr}\colon\scZ_{\CH}^{\rfr}\rightarrow\scC^{\rfr},\quad f_\Delta\colon \scZ_{\CH}^{\rfr}\rightarrow \scX^{\rfr} \]

\[\pi_{\Dr}(Z,g,X,h,Y,v,w)=(Z,g,X,hv,w),\quad f_\Delta(Z,g,X,h,Y,v,w)=(X,h,Y,gh,Y,v,v,w).\]

We define the framed version of the Chern character functor \(\CH^{\rfr}\) by the formula (<ref>).
The framed version \(\HC^{\rfr}\) is defined by (<ref>), where this time

\[\tilde{f}_\Delta\colon\scZ^{\rfr}_{\CH}\times B\rightarrow \scX^{\rfr}\times B,\quad \tilde{f}_\Delta(Z,g,X,h,Y,v,b)=(X,gh,Y,hb,\Ad_bY,v,v,w,b).\]

The stable versions of the categories are defined on the stable pieces of the corresponding varieties. In particular, we have

\[j_{\rst}\colon\scX^{\rst}\hookrightarrow \scX\times V,\quad \scC^{\rst}\hookrightarrow\scC\times V,
\quad j_{\rfs}\colon \scX^{\rfs}\hookrightarrow \scX^{\rfr},\quad \scC^{\rfs}\hookrightarrow \scC^{\rfr}. \]

Denote by \(v\) the coordinate along the vector space \(V\). The open stability condition for \(\scX^{\rst}\) and \(\scC^{\rst},\scC^{\rfs}\) is defined by the following requirements:

\[\CC\langle X,\Ad_{g_1}^{-1}Y_1\rangle\, v=V,\quad \CC\langle Z,X\rangle\, v=V.\]

The open stability condition for \(\scX^{\rfs}\) is

\[\CC\langle X,\Ad_{g_1}^{-1}Y_1\rangle v_1=V,\]

where \(v_1,v_2\) denote coordinates along the factor \(V^2\) in \(\scX^{\rfr}\).

We also have the following closed

constraint for \(\scC^{\rst}\):
\(gv=v.\)

The stable piece  \(\scX^{\rst}_{\rcnv}\subset\scX_{\rcnv}\times V\) is an open subset

which is an intersection

\[\scX^{\rst}_{\rcnv}:=(\pi_{12}\times 1)^{-1}(\scX^{\rst})\cap (\pi_{23}\times 1)^{-1}(\scX^{\rst}).\]

We define the convolution  product on
\[\MF^{\rst}:=\MF_{G\times B^2}(\scX^{\rst},W)\]
by the same formula as before:

\[\calF\star\calG:=(\pi_{13}\times 1)_{*}(\CE_{\frn^{(2)}}(\pi^*_{12}\times 1(\calF)\otimes \pi^*_{23}\times 1(\calG))^{T^{(2)}}).\]

Similarly, the convolution algebra structure is defined on categories

\[\MF^{\rst}_{\Dr}:=\MF_G(\scC^{\rst},W_{\Dr}),\quad \MF^{\rfs}_{\Dr}:=\MF_G(\scC^{\rfs},W^{\rfr}),\quad \MF^{\rfs}:=\MF_{G\times B^2}(\scX^{\rfs},W^{\rfr})\]

by restricting the corresponding maps to the stable pieces of the convolution spaces.

Moreover, the same argument as in Theorem 1.2.3 of <cit.> implies the following:

The pull-back maps \(j_{\rfs}^*\), \(j_{\rst}^*\circ\pi^*_V\), where \(\pi_V\)  is the projection along \(V\), are convolution algebra

We define the stable version of the space \(\scZ_{\CH}\) as an open locus inside \(\scZ_{\CH}\times V\) defined as an intersection:

\[\scZ_{\CH}^{\rst}:=\pi_{\Dr}^{-1}(\scC^{\rst})\cap f_{\Delta}^{-1}(\scX^{\rst}),\]

where the corresponding maps \(f_{\Delta}\), \(\pi_{\Dr}\) are defined by restricting the framed maps to the locus
where the covectors vanish. Thus we can define the functors \(\CH^{\rst}\) and \(\HC^{\rst}\) by the formulas (<ref>)
and (<ref>).

Similarly we define the framed stable version of the space \(\scZ_{\CH}\) as an intersection:

\[\scZ_{\CH}^{\rfs}:=\pi_{\Dr}^{-1}(\scC^{\rfs})\cap f_\Delta^{-1}(\scX^{\rfs})\]

and we define functors \(\CH^{\rfs}\) and \(\HC^{\rfs}\) by restricting pull-backs and push-forwards in (<ref>) and (<ref>) to the corresponding open sets.

The functors constructed in this subsection are naturally compatible with each other.
Consider two natural inclusions with the same name $i_{\rcv}$:

\begin{gather*}
i_{\rcv}\colon\scX^{\rst}\rightarrow\scX^{\rfs}, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
(X,g_1,Y_1,g_2,Y_2,v)\mapsto(X,g_1,Y_1,g_2,Y_2,g_1v,g_2v,0),\quad (Z,g,X,v)\mapsto(Z,g,X,v,0).
\end{gather*}

They describe the stable varieties as subspaces of the framed stable varieties determined by vanishing of covectors.

The functors in the commuting squares:

\[
\begin{tikzcd}
\MF\arrow[r,"i^{\rfr}"]\arrow[d,"j^*_{\st}\circ\pi_V^*"]&\MF^{\rfr}\arrow[d,"j_{\mathrm{\rfs}}^*\circ\pi_V^*"]\\
\MF^{\rst}&\MF^{\rfs}\arrow[l,"i^*_{\rcv}"]
\end{tikzcd},
\quad\quad\quad
\begin{tikzcd}
\MF_{\Dr}\arrow[r,"i^{\rfr}_{\Dr}"]\arrow[d,"j^*_{\st}\circ\pi_V^*"]&\MF^{\rfr}_{\Dr}\arrow[d,"j^*_{\mathrm{\rfs}}\circ\pi_V^*"]\\
\MF^{\rst}_{\Dr}&\MF^{\rfs}_{\Dr}\arrow[l,"i^*_{\rcv}"]
\end{tikzcd}
\]

are monoidal and the functors \(i^*_{\rcv}\) is an isomorphism. Moreover, the functors \(\CH^\bullet\), \(\HC^\bullet\) intertwine the squares and
the functors \(\HC^\bullet\) are monoidal. The functors  \(\CH^\bullet\), \(\HC^\bullet\) are adjoint:

\[\mathcal{E}\mathrm{xt}(\calF,\HC^\bullet(\calD))=\mathcal{E}\mathrm{xt}(\CH^\bullet(\calF),\calD).\]

The commutativity of the squares follows from the direct computation. The pull-back \(i^*_{\rcv}\) is the Knorrer periodicity functor, hence it is an isomorphism. Adjointness of the functors is shown the same way as in <ref>.

§.§ Linear Koszul duality

In this section  we remind the setting of the linear Koszul duality and establish isomorphisms between the
linearized categories \(\MF^\bullet_{\Dr,\rlin}\) and the corresponding derived category \(\Coh^\bullet\).
In the case of \(\bullet=\rfs\) we show that the linearized category is actually equivalent to the
original category, thus we complete the construction of the functor
\(\CH^{\rfs}_{\rloc}\)  introduced at the beginning of the section.

A coordinate substitution $Y = U g^{-1}$
on our main variety \(\scC\) makes the potential tri-linear:

\[\underline{W}_{\Dr}(X,U,g)=\Tr(X[U,g])=W_{\Dr}(X,Ug^{-1},g).\]

The framed potential also linearizes:

\[\underline{W}_{\Dr}^{\rfr}(X,U,g,v,w)=\underline{W}_{\Dr}(X,U,g)+\Tr(w(v-gv)).\]

Thus we introduce linearized categories:

\[\underline{\MF}^\bullet_{\Dr}:=\MF_{G}(\underline{\scC}^{\bullet},\underline{W}^{\bullet}_{\rlin}),\]

where \(\underline{\scC}^\bullet\) is obtained from \(\scC^\bullet\) by taking the closure of \(G\)
inside \(\frg\)
and \(\underline{W}^{\rfs}=\underline{W}^{\rfr}\).

Since \(j_{G}\colon\scC^\bullet\hookrightarrow\underline{\scC}^\bullet\) is an open embedding, the pull-back functor \(j_G^*\)
is a localization functor and we denote

\[\mathrm{loc}^\bullet\colon\MF^\bullet_\Dr\rightarrow\underline{\MF}^\bullet_\Dr\]

for this functor.

The functors \(\mathrm{loc}^{\bullet}\), \(\bullet=\rst,\rfs\) are isomorphisms.

We consider the case of \(\bullet=\rfs\), the proof in the other case is almost identical.
The functor is a localization by the function \(\det=\det(g)\). Then according to the lemma 12.3 from <cit.>,
it is sufficient to show that the zero locus of \(\det\) does not intersect the critical locus of the potential.

The critical locus is given by the system of equations:

\[[X,U]-wv=0,\quad [g,X]=[g,U]=0,\quad vg=v,\quad gv=v.\]

The first equation is the moment map equation from chapter 2 of <cit.> and it is shown there that if
\(\CC\langle X,Y\rangle v=V\) then the moment map equation implies \(w=0\)  and we get a cyclic  triple \(X,Y,v\) that represents a
point in the Hilbert scheme \(\Hilb\). It is also shown in <cit.> that the set of elements \(g\) commuting
with \(X,U\) from a cyclic triple consists of the scales of the identity matrix. Hence the last equation implies that \(g=1\).

§.§ Koszul duality

Recall a general setup of the Koszul duality. We are interested in the version which provides a matrix factorization model for the differential graded category
of the derived complete intersection. The Koszul duality in this context was discussed in <cit.> where the results
of <cit.> and <cit.> are combined. In this note we present slightly more streamlined
construction of the duality functor.

Derived algebraic geometry is explained in many places, here we explain it here in the most elementary setting sufficient for our needs.

Initial data for an affine derived complete  intersection is a collection of elements
\(f_1,\dots,f_m\in\CC[X]\). It determines the differential graded algebra

\[\calR=(\CC[X]\otimes \Lambda^* V,D),\quad D=\sum_{i=1}^mf_i\frac{\partial}{\partial \theta_i},\]

where \(\theta_i\) from a basis of \(U=\CC^m\).

More generally, given a dg algebra \(\calR\) such that \(H^0(\calR)=\calO_Z\) we say that \(\Spec(\calR)\) is a dg
scheme with underlying scheme \(Z\). Respectively, we define dg category of coherent sheaves on
\(\Spec(\calR)\) as

\[\Coh(\Spec(\calR))=\frac{\{\text{bounded complexes of finitely generated  $\calR$  dg modules}\}}{\{\text{quasi-isomorphisms}\}}.\]

Consider a potential on \(X\times U\):

\[W=\sum_{i=1}^m f_i(x)z_i,\]

where \(z_i\) is a basis of \(U^*\) dual to the basis \(\theta_i\). The ring \(\calR\otimes \CC[U]\) is a free finite rank
\(\CC[X\times U]\)-module. Thus \(\calR\)-module with the differential:

\[\mathrm{B}=(\calR\otimes \CC[U], D_B),\quad D_B=\sum_{i=1}^m z_i\theta_i.\]

If \((M,D_M)\) is dg module over \(\calR\), then the tensor product

\[\mathrm{KSZ}_U(M):=M\otimes_{\CC[X]\otimes\Lambda^*(U)} \mathrm{B}\]

is an object of \(\MF(X\times U,W)\) with the differential \(D=D_M\otimes 1+ 1\otimes D_B\). The map \(\mathrm{KSZ}_U\) extends to a functor between triangulated categories:

\[\mathrm{KSZ}_U\colon \Coh(\Spec(\calR))\rightarrow \MF(X\times U,W).\]

The functor in the other direction is based on the dual matrix factorization:

\[\MF(X\times U,-W)\ni B^*=(\calR\otimes \CC[U],D_B^*),\quad D^*_B=-\sum_{i=1}^m z_i\theta_i+f_i\frac{\partial}{\partial \theta_i},\]

\[\mathrm{KSZ}_U^*\colon\MF(X\times U,W)\rightarrow \Coh(\Spec(\calR)),\quad \mathrm{KSZ}^*_U(\calF):=
\Hom_{\calR}(\calF\otimes_{\CC[X\times U]}\mathrm{B}^*,\calR).\]

The functors \(\mathrm{KSZ}_U\) and \(\mathrm{KSZ}_U^*\) are mutually inverse.

To simplify notations we assume that
\(X=\Spec(\CC[x])\) and \(m=1\). First we prove that \(\mathrm{KSZ}^*_U\circ \mathrm{KSZ}_U= 1\).

A dg \(\calR\)-module $M$
has a free resolution \(M^\bullet\) since
\(X\) is an affine space. Thus the composition \(\mathrm{KSZ}^*_U\circ \mathrm{KSZ}_U(M)\) is
equal to

\begin{equation}
\label{eq:BB1}
\Hom_\calR(B^*\otimes_{\CC[x,z]}B,\calR)\otimes_{\calR}M^\bullet=
(\Hom_{\calR}(\CC[x,z,\theta_1,\theta_2],\calR),D^\vee_{BB})\otimes_{\calR} M^\bullet,\end{equation}

where the differential \(D^\vee_{BB}\) is dual to the differential

\[D_{BB}=z\theta_-+f(x)\frac{\partial}{\partial\theta_+},\quad \theta_\pm=\theta_1\pm\theta_2 \]

acting on \(B\otimes_{\CC[x,z]}B=\CC[x,z,\theta_1,\theta_2]\). There is a homotopy equivalence of the
complexes of \(\CC[\theta_-]\)-modules

\[(\CC[z,\theta_-],z\theta_-)\sim [
\CC[\theta_-]\xrightarrow{\theta_-}\CC[\theta_-]\xrightarrow{\theta_-}\CC[\theta]\xrightarrow{\theta_-}\dots].\]

The duality functor \(\Hom_{\CC[\theta_-]}(\bullet,\CC[\theta_-])\) inverts the arrow, and we obtain a resolution of
the simple \(\CC[\theta_-]\)-module:

\[[\CC[\theta_-]\xleftarrow{\theta_-}\CC[\theta_-]\xleftarrow{\theta_-}\CC[\theta_-]\xleftarrow{\theta_-}\dots]\sim \CC.\]

Hence the complex (<ref>) is quasi-isomorphic to \(M^\bullet\) and the statement follows.

The analysis of the composition in other direction \(\mathrm{KSZ}_U\circ\mathrm{KSZ}^*_U\) is even easier, because it is reduced to the

study of the dg module

\[(B\otimes_{\CC[x,\theta]}B^*,D'_{BB}),\quad D'_{BB}=\theta(z_1+z_2), \]

which is quasi-isomorphic to the dg module \(\CC[x,z_1,z_2]/(z_1+z_2)\) with the trivial differential. Hence the statement follows.

If we have reductive group \(G\) acting on \(X\) and acting linearly on \(U\) and on \(W\in \CC[X\times U]^G\) then the above functors extend to the isomorphism between the equivariant
version of the categories:

\[\Coh_{G}(\Spec(\calR))\simeq \MF_{G}(X\times U,W).\]

§.§ Localization functors

In our setting \(U=\frg\), while \(X=\mathscr{CV}^\bullet\) is a version of the commuting variety:

\begin{equation}\label{eq:CV}
\mathscr{CV}\subset \frg^2,\quad \mathscr{CV}^{\rst}\subset \frg^2\times V,
\quad \mathscr{CV}^{\rfr}, \mathscr{CV}^{\rfs}\subset \frg^2\times V\times V^*.\end{equation}

The closed conditions describing the spaces \(\mathscr{CV},\mathscr{CV}^{\rst}\) are

\[[X,Y]=0, \quad X,Y\in \frg,\]

for the spaces \(\mathscr{CV}^{\rfr},\mathscr{CV}^{\rfs}\) are

\[ [X,Y]-vw=0, \quad X,Y\in \frg,\quad v\in V, w\in V^*,\]

the open condition for the stable spaces is:
\(\CC[X,Y]v=V.\)
Thus we define the dg algebra \(\calR^\bullet\) as the algebra \(\CC[\mathscr{CV}^\bullet]\otimes\Lambda^* U\) with the differntial

\begin{equation}\label{eq:DG}
D^\bullet=\sum_{ij}f_{ij}^\bullet\frac{\partial}{\partial \theta_{ij}},
\end{equation}

where \(\theta_{ij}\) is a basis of matrix units in \(\frg\) and \(f^\bullet_{ij}\) is the \(ij\)-entry of the corresponding
version of the defining equation (<ref>). To simplify our notations we denote

\[\Coh^\bullet:=\Coh_{G}(\Spec{\calR^\bullet}).\]

Since the potential \(\underline{W}^\bullet\) is linear as a function of \(g\in\frg\) and the scaling torus \(T_{sc}\) does not act on
\(g\), we obtain a pair of mutually inverse functors:

\[
\begin{tikzcd}
\underline{\MF}^\bullet_{\Dr}\arrow[rr,bend left,"\mathrm{KSZ}_{\frg}^*"]&&\Coh^\bullet\arrow[ll,bend left,"\mathrm{KSZ}_\frg"]
\end{tikzcd}
\]

The tensor product of dg modules gives categories \(\Coh^\bullet\) monoidal structure, and the functor \(\mathrm{KSZ}_{\frg}\) is
compatible with it.

The functor \(\mathrm{KSZ}_\frg\) is monoidal.

Since \(\mathrm{loc}^{\bullet}\), \(\bullet=\rfs,\rst\) is invertible, we can
use the Koszul duality functor in order to construct the functor

\[\CH^{\bullet}_{\rloc}:=\CH^{\bullet}\circ (\mathrm{loc}^{\bullet})^{-1}\circ \mathrm{KSZ}^*_\frg: \MF^{\bullet}\rightarrow \mathrm{D}^{\rper}(\Hilb_n(\CC^2)).\]

The localization functor does not seem to be invertible in  cases of  \(\bullet=\emptyset,\rfr\), however a construction of the
functor in the opposite direction does not require invertibility of the localization:

\[\HC^\bullet_{\rloc}:=\HC^\bullet\circ \mathrm{loc}^\bullet\circ \mathrm{KSZ}_\frg\colon\Coh^\bullet\rightarrow\MF^\bullet.\]

As shown in <cit.>, if \(\bullet=\emptyset,\rfr\), then there is a homomorphism
\(\Phi^{\raff}\colon\Br^{\raff}\to \MF^\bullet\) and similarly if \(\bullet=\rst,\rfs\) then there is a homomorphism
\(\Phi\colon \Br\to \MF^{\bullet}\). In all cases the generators of the braid group are mapped to
Koszul matrix factorizations, hence the previous argument implies the following:
The functor \(\HC^\bullet_{\rloc}\) is monoidal. Moreover,
its image is central:

for any \(\calD\in \Coh^\bullet\)  and \(\beta\in \Br^\bullet\)

\[\HC^\bullet_{\rloc}(\calD)\star \Phi^\bullet(\beta)\sim \Phi^\bullet(\beta)\star \HC^{\bullet}_{\rloc}(\calD).\]

§ KNOT INVARIANTS

In this section we discuss the relation between the Chern character and the functor of the braid closure that was used in our previous papers <cit.>. There we constructed a link invariant \(\HHHb(\beta)\) of the closure of a braid
\(\beta\in \Br_n\). Our construction of the invariant is based on the homomorphism:

\[\Br_n\to \MF^{\bullet},\quad \beta\mapsto \calC^{\bullet}_\beta,\]

\(\bullet=\rst,\rfs\),
and the braid closure procedure

\[\mathbb{L}\colon \MF^{\rst}\to \MFB,\quad \calC\mapsto j^*_{e,G}(\calC)^G,\]

where \(\MFB\) is the category of coherent sheaves on the quotient \(\widetilde{\FHilb}^{\rfree}/B\), where

\[\widetilde{\FHilb}^{\rfree}=\{(X,Y,v)\in \frb\times\frn\times V\;|\;\CC\langle X,Y\rangle\, v=V\}.\]

The space \(\widetilde{\FHilb}^{\rfree}\times G\) embeds inside \(\scX\) by a \(G\times B^2\)-equivariant map \(j_{e,G}\) used in the
construction of \(\mathbb{L}\):

\[j_{e,G}\colon\widetilde{\FHilb}^{\rfree}\times G\rightarrow \scX,\quad (X,Y,g)\mapsto (X,g,Y,g,Y).\]

It is shown in <cit.> that the triply-graded space

\[\HHH^\bullet(\beta):=\CE_{\frn}(\mathbb{L}(\calC^{\rst}_\beta)\otimes \Lambda^\bullet V)^{T},\]

is an isotopy invariant of the closure \(L(\beta)\) of the braid \(\beta\) (after an explicit
grading shift that is expressed in terms of wreath of the braid).

The quotient of the space \(\widetilde{\FHilb}^{\rfree}\) by \(B\) is the free flag Hilbert space \(\FHilb^{\rfree}\).
This space is smooth, an explicit smooth atlas is presented in <cit.>. The condition
of commutativity $[X,Y]=0$

defines an embedding of the flag Hilbert scheme
\(\FHilb(\CC)\) inside \(\FHilb^{\rfree}\) (here we follow conventions of <cit.>).

The vector space \(V\)
descends to the vector bundle \(\calB^\vee\) and the dual vector space descends to the dual vector bundle
\(\calB\). We use the same notations for the corresponding vector bundles over the spaces \(\FHilb(\CC)\) and \(\Hilb\).
The following is a simple consequence of the definition of the Chern functor:

For any \(m\) the functor \(\CH^{\rfs}_{\rloc}\) intertwines the tensor powers of \(V^*\) and \(\calB\):

\[\CH^{\rfs}_{\rloc}(\calC\otimes (V^*)^{\otimes m})=\CH^{\rfs}_{\rloc}(\calC)\otimes \calB^{\otimes m},\]

and this functor intertwines the action of the symmetric group \(S_m\). The analogous statement holds for the
dual bundles: \(V\) and \(\calB^\vee\).

Thus we can interpret the homology \(\HHH(\beta)\) geometrically in terms of sheaves on the flag Hilbert schemes, see <cit.> for a detailed discussion.
One unpleasant aspect of the flag Hilbert scheme is that the flag Hilbert scheme \(\FHilb\) is very singular and it is
hard to use the standard tools like localization for computations on it. Below we show how one can fix this issue
with the Chern functor.

For any \(\beta\in \Br_n\) there is an isomorphism

\[\HHH^\bullet(\beta)\cong\mathcal{E}\mathrm{xt}(\calO, \CH^{\rfs}_{\rloc}(\calC_\beta)\otimes \Lambda^\bullet \calB).\]

First, observe an isomorphism of complexes of
\(\CC[\frg]^G\)-modules:

\[\mathcal{E}\mathrm{xt}(\calC^{\rst}_{\parallel},\calC^{\rst}_\beta\otimes \Lambda^\bullet V^*)\simeq \HHH^\bullet(\beta).\]

Indeed, the matrix factorization \(\calC^{\rst}_{\parallel}\) is
the Koszul matrix factorization for the defining ideal of the \(B\)-orbit of the image \(j_{e,G}(\widetilde{\Hilb}^{\rfree})\), and
since the action of \(B\) on the orbit is free, the statement follows (a more detailed the argument can be found in the section 13 of  <cit.>).

Next observe that the units in \(\MF^{\rst}\) and \(\MF^{\rfs}\) are related by the push-forward: \(\calC_{\parallel}^{\rfs}=i_{\rcv,*}(\calC_{\parallel}^{\rst}).\) Hence we can use adjoint property of the push-forward to show that:

\[\mathcal{E}\mathrm{xt}(\calC^{\rfs}_\parallel,\calC^{\rfs}_\beta\otimes \Lambda^\bullet\calB )=
\mathcal{E}\mathrm{xt}(i_{\rcv,*}(\calC^{\rfr}_\parallel),\calC^{\rfs}_\beta\otimes \Lambda^\bullet \calB)=
\mathcal{E}\mathrm{xt}(\calC^{\rfr}_\parallel,\calC^{\rfr}_\beta\otimes \Lambda^\bullet \calB).
\]

Finally, we use the adjointness of the functors \(\CH\) and \(\HC\):

\[\mathcal{E}\mathrm{xt}(\calC^{\rfs}_\parallel,\calC^{\rfs}_\beta\otimes \Lambda^\bullet\calB)=\mathcal{E}\mathrm{xt}(\HC^{\rfs}_{\rloc}(\calO),\calC^{\rfs}_\beta\otimes\Lambda^\bullet V^*)=
\mathcal{E}\mathrm{xt}(\calO,\CH_{\rloc}^{\rfs}(\calC^{\rfs}_\beta)\otimes \Lambda^\bullet \calB).\]

§ COMPUTATIONS

In this section we apply our methods to the homology of the closure of a sufficiently positive JM braid.

§.§ Localization formula for partial twists

In this subsection we derive explicit formulas for the closure of the JM braids mentioned in Theorem <ref>.

The  main tool in our computation is the localization formula, which computes the equivariant Euler characteristic of the complex of coherent
equivariant sheaves. Of course, generally, the Euler characteristic does not determine the homology, however

On the other hand, our link homology is the total sum of all degree, not an alternating sum of the characters of the
homology groups. Thus if we  want to use the localization formula we need to show that the complex under our considerations has non-trivial
homology of fixed parity. Such parity statement could be secured by considering sufficiently positive powers of the JM elements.

Recall that the variety \(\scX^{\rfs}\) has a collection of the line bundles
\(\calL_i\), \(i=1,\dots,n-1\) which are the structure sheaves of \(\scX^{\rfs}\) with twisted \(B^2\)-equivariant
structure. More precisely, if \(\chi_i\) is the character of \(B\) that evaluates the \(i\)-th diagonal element, then
\(\calL_i=\calO_{\scX^{\rfs}}\langle\chi_i,0\rangle\) where the characters in the brackets are the twisting characters of
\(B^2\). Denote

\[
\cLva

We showed in <cit.> that
$\calC^{\rfr}_{\delta^{\vec{a}}} =  \calC_{\parallel}^{\rfr} \otimes \calL^{\vec{a}}$, hence

our main theorem implies that the homology of the closure of \(\delta^{\vec{a}}\)

is computed
in terms of sections of \(\CH^{\rfs}_{\rloc}(\calC_\parallel\otimes \calL^{\vec{a}})\). From this place
till the end of the section we assume that the parity conjecture <ref> from the introduction is true.

Now return to our old method for computing knot homology and recall that the result of the geometric closure operation
is the two periodic complex

$$\calS_{L^{\vec{a}}}:=j_{e,G}^*(\calc_{\parallel}^{\rfr}\otimes \cLva)^G=\mathrm{K}([X,Y]_{--}\otimes \cLva)\in \Dper(\FHilb^{\rfree}),$$

so the link homology is the hypercohomology
$\HHH^i(L^{\vec{a}})=\HHH(\mathrm{K}([X,Y]_{--}\otimes\cLva\otimes\Lambda^i\calB))$. We would like to point out a subtle point
of the knot homology construction from <cit.>: the differential in the complex
\(\mathrm{K}([X,Y]_{--})\) has degree $\mathbf{t}$ with respect to the torus \(T_{sc}\). But we just proved that the hyper-homology
of the \( \mathrm{K}([X,Y]_{--})\otimes \cLva\) vanishes in the odd degrees, hence we can adjust
the $T_{sc}$-weights of the modules in
the complex to obtain $T_{sc}$-equivariant complex \(\mathrm{K}([X,Y]_{--})^{\reven}\) without changing of
the total homology space. Finally, since the push-forward of $\calO(-1):=\prod\calL_i$
is the ample line bundle $\det(\calB)$ on $\Hilb$, we conclude that for sufficently large $m$

\[\HHH^i(
\Lva
)=\chi_{q,t}(\mathrm{K}([X,Y]_{--})^{\reven}\otimes \cLva\otimes \calO(m)),\]

and the last Euler characterics could be computed with the localization technique.

§.§ Combinatorics of the torus fixed points

In this section we discuss the localization on the free flag Hilbert scheme \(\FHilb^{\rfree}\). As was established in <cit.>
this Hilbert scheme is smooth and  has a natural atlas of affine charts.

First, we describe the combinatorial data used for labeling  the charts.
Denote by \(\NSn\) the set of the nested pairs of sets with the following properties. An element \(S\in \NSn\) is a pair of nested sets:

\[S_x^1\supset S_x^2\supset\dots\supset S_x^{n-1}\supset S_x^n=\emptyset,\]

\[S_y^1\supset S_y^2\supset\dots\supset S_y^{n-1}\supset S_y^n=\emptyset,\]

such that

\[ S_x^k,S_y^k\subset \{ k+1,\dots,n\},\quad |S_x^i|+|S_y^i|=n-i.\]

Define the sets  of pivots of \(S\) as sets \(P_x(S),P_y(S)\) consisting of the pairs

\[ P_x(S)=\{(ij)|j\in S_x^i\setminus S_x^{i+1}\},\quad P_y(S)=\{(ij)|j\in S_y^i\setminus S_y^{i+1}\}.\]

To an element \(S\in \NSn\) we assign the following affine space \(\mathbb{A}_S\subset \frn\times \frn\):

\[(X,Y)\in\mathbb{A}_S,\mbox{ if } x_{ij}=1, ij\in P_x(S)\quad y_{ij}=1, ij\in P_y(S) \mbox{ and }\]

\[x_{i-1,j}=0,\mbox{ if } j\in S_x^i,\quad y_{i-1,j}=0,\mbox{ if } j\in S_y^i.\]

For a given \(S\) denote by \(N_x(S)\) and \(N_y(S)\) the indices \((ij)\) such that  such that the corresponding entries \(x_{ij}\) (respectively \(y_{ij}\)) are not constant on
\(\mathbb{A}_S\). From the construction we see that \(|N(S)|=n(n-1)/2.\)

Denote by \(\frh\) the subspace of the diagonal matrices inside \(\frb\). The sum \(\frh+\mathbb{A}_S\) is
an affine subspace inside \(\frb\times\frn\) and we show in <cit.>:
The space \(\widetilde{\Hilb}_{1,n}^{\rfree}\subset \frb\times\frn\) is covered by the \(B\)-orbits of the affine spaces  \(B(\frh+\mathbb{A}_S)\), \(S\in NS_n\).
Morevorer, the points in \(\frh+\mathbb{A}_S\) have trivial stabilizers.

§.§ Integration over \(\Hilb^{\rfree}\)

In this section we discuss a formula for the push-forward functor for the projection map from
\(\FHilb^{\rfree}_n\) to \(\FHilb^{\rfree}_{n-1}\). We use the methods similar to those of <cit.>.

The key geometric observation of <cit.> is that the flag Hilbert scheme is an iterated tower of projective spaces appearing as a projectivization of some explicit two step complexes of vector bundles \([V\xrightarrow{\phi} W]\).
In the setting of <cit.> the rank of the map \(\phi\) varies and the author is forced to work with
a DG scheme structure. Our setting is more elementary since \(\FHilb_n^{\rfree}\) is a smooth \(\mathbb{P}^{n-1}\) fibration over
\(\FHilb_n^{\rfree}\). We present this fibration as a projectivization of the a two step complex as in <cit.> in order to use the technology of this paper.

Denote by \(\pi\) the projection from \(\widetilde{\FHilb}^{\rfree}_n\) to \(\widetilde{\FHilb}^{\rfree}_{n-1}\times \cc\) and the induced
maps on the \(B\)-quotients. To state our push-forward formula we also need a slight modification of the function \(\zeta\) from the introduction: define
\[\tzeta(x)=(1-x)/((1-Qx)(1-Tx)\]

For any rational function \(r(\calL_n)\) with coefficients being rational functions of \(\calL_i\), \(i<n\),
the K-theory push-forward is given by

\[\pi_*(r(\calL_n))=\int \frac{r(z)}{(1-z^{-1})}\prod_{i=1}\tzeta(\calL_i/z)\frac{dz}{z},\]

where the contour of integration separates the set \(\mathrm{Poles}(r(z))\cup \{0,\infty\}\) from the poles of the rest of the integrant.

There are three \(B_n\)-equivariant vector bundles \(\und{\frn}_n\), \(\und{\frb}_n\) and
\(\und{V}_n\) over the space \(\widetilde{\FHilb}^{\rfree}_n\).

The  derivative of the \(U_n\)-action can be encoded by the two-step complex

\[\calw_n=[\und{\frn}_n\xrightarrow{a} \mathbf{Q}\und{\frb}_n\oplus \mathbf{T}\und{\frn}\oplus \und{V}_n], \quad a(\eta)_{(X,Y,v)}=([a,X],[a,Y],a\cdot v)_{(X,Y,v)}.\]

where \(\mathbf{Q}\) and \(\mathbf{T}\) denote the shift of the weight of \(\mathbb{T}_{q,t}\)-action.

The  complexes \(\calw_n\) are filtered \(\calw_1\subset \calw_2\subset\dots\subset \calw_n\), and the filtration is respected
by the filtration of the groups \(U_1\subset U_2\subset \dots\subset U_n\). Thus the quotient \(\calw_n/\calw_{n-1}\) is a two-step
\(B_{n-1}\)-equivariant complex over the \(B_{n-1}\)-quotient \(\FHilb_{n-1}^{\rfree}\times \cc\). Moreover, from our explicit description of
the charts on \(\FHilb_n^{\rfree}\) we see that it is fibered over \(\FHilb_{n-1}^{\rfree}\times \cc\) with the fibers
\(\mathrm{Proj}(\calw_n/\calw_{n-1})\).

The line bundle \(\calL_n\) is the tautological line bundle over the last projective fibration, hence the push-forward of
\(c_1(\calL_n)^m\) along \(\pi\) is the Serge class of \(\calw_n/\calw_n\). This observation was incapsulated into
<cit.>, the main formula of the last lemma is exactly our integral formula from the statement.

§.§ Proof of theorem <ref>

In this section we abbreviate the even Koszul complex for the relations
\([X,Y]_{--}\) by \(\mathrm{K}^{\reven}\). The complex \(\mathrm{K}^{\reven}\) is  \(\Tqt\)-equivariant
and  the \((q,t)\)-character of \(\mathrm{K}^{\reven}\otimes \Lambda^\bullet\calB\) is equal to the iterated \(K\)-theoretic
push-forward of the complex along the map that projects \(\FHilb_n^{\rfree}\) to \(\cc^n\). This projection is
the iterated application of the projection \(\pi\) from the previous section.

On the other hand, the \(K\)-theoretic image of the complex \(\mathrm{K}^{\reven}\) is the product
\(\prod_{1\le i<j\le n}(1-QT\calL_i/\calL_j)\). Thus the result of the previous section implies that
\(\chi_{aqt}(\HHH(\mJMa
))\) is equal to the iterated residue integral:

\[\int\dots\int  \prod_i\frac{z_i^{b_i}(1+az_i^{-1})}{1-z^{-1}}\prod_{1\le i<j\le n}\zeta(\frac{z_i}{z_j})\frac{d z_1}{z_1}\dots\frac{d z_n}{z_n}.\]

This integral was studied in the work of Negut. In particular, the delicate analysis of the residues of
the integral in <cit.> implies that the non-trivial resudues are naturally labeled by SYT
and the sum of the sum of the residues is exactly the formula in the statement of the theorem.

§ DISCUSSION AND FURTHER DIRECTIONS

In this section we state various conjectures concerning Drinfeld centers of the categories \(\MF^\bullet\) and
we also discuss a version of the Chern character that might be helpful in clarifying connections with
the work of Bezrukavnikov.

§.§ Weak flag Hilbert scheme

The key observation of this section is that we can introduce an intermediate space between the flag Hilbert scheme
and the usual Hilbert scheme and work with the coherent sheaves on this space instead of the usual Hilbert scheme.
We call the space weak flag Hilbert scheme \(\whilb_n\). Let us first define it as a scheme and later we define it
as dg-scheme, since the derived version of the space is the most relevant for us.

Recall that \(\Hilb_n(\CC^2)\) carries the tautological bundle \(\calB\) with the fiber \(\CC[x,y]/I\) over the
ideal \(I\). The total space of the vector bundle \(\xTot(\calB)\) is locally a direct product \(\CC^n\times U\)
where \(U\subset\Hilb_n(\CC^2)\)  is an affine set inside \(\Hilb_n(\CC^2)\). Thus we
define \(\Fl(\calB)\) as the space of full flags  in the vector bundle \(\calB\), this  space locally is a direct product
\(U\times \Fl(\CC^n)\). The coordinate ring \(\CC[x,y]\) acts on \(\calB\) and we  define

\[\whilb:=\{(F_\bullet,I)|xF_\bullet\subset F_\bullet\}.
\]

This space is highly singular, it has many irreducible components and we actually work with its derived version.
The space \(\whilb_n\) is the \(GL_n\)-quotient of the space:

\[\whilbt:=\{(F_\bullet,X,Y,v)\in \tilde{\frg}\times\frg\times V\;|\; [X,Y]=0,\CC[X,Y]\,v=V\}.\]

The total space \(\tilde{\frg}\times \frg\) is smooth and the singularities of \(\whilbt\) are caused by the commutativity relation, so we define a dg scheme \(\whilbt_n\) to be a dg subscheme with the structure sheaf
\((\calO_{\tilde{\frg}\times\frg}\otimes\Lambda(\frg),D)\) with the differential

\[D=\sum_{ij}[X,Y]_{ij}\frac{\partial}{\partial \theta_{ij}},
\]

where \(\theta_{ij}\) are the coordinates on  \(\frg\).

The dg scheme \(\whilb\) is defined as a \(G\)-quotient of $\whilbt$. We denote by $\Coh(\whilb_n)$ the dg category of coherent
sheaves over the dg scheme $\whilb_n$. The space $\whilb$ is very singular but probably more approachable  than famous
flag Hilbert scheme (see for example <cit.> for a discussion of singularities of the flag
Hilbert scheme): the fibers of the projection $\pi_{\Hilb}$ are Springer fibers, on the other hand, the fibers of
the analogous projection of the flag Hilbert scheme are intersection of Springer fibers.

The advantage of the $\whilb$ over the usual Hilbert scheme is that the later has a large Picard group.
We denote the line bundle over the flag variety $\Fl$ with the fiber $V_{i+1}/V_i$ by $\calL_i$ and we use the same notation
for the corresponding line bundle over $\whilb$. These line bundles have topological counter-parts in elementary JM braids \(\delta_k\)

The category $\Coh(\whilb)$ has a matrix factorization incarnation because of the linear Koszul duality. Indeed, the space
$\tilde{\frg}\times\frg$ maps to the  space $\frg\times\frg$ via the moment map $\mu$ and it is a resolution of singularities.
Thus we define the potential on the space $\tilde{\scC}^{\rfr}=\tilde{\frg}\times\frg\times  G\times V\times V$:

where $\mu:\tilde{\scC}^{\rfr}\to\scC^{\rfr}$ is a natural extension of the previously discussed   map $\mu$. Next we define the
stable locus
$\tilde{\scC}^{\rfr}:=\mu^{-1}(\scC^{\rfs})$. Then the argument similar to  the propposition <ref> implies:

The linear Koszul duality functor provides an equivalence:

$$\mathrm{KSZ}_{\frg}\circ\mathrm{loc}\colon \mf_{G}(\tilde{\scC}^{\rfc},\tilde{W}^{\rfr})\to \Coh(\whilb).  $$

As in the previous case, we introduce the localized version of our main space $\underline{\tilde{\scc}}^{\rfc}:=\mu^{-1}(\underline{\scc}^{\rfc}) $
and the corresponding potential $\underline{\tilde{W}}^{\rfr}$  that is multi-linear.
To complete our proof we need to show   that the localization morphism $\mathrm{loc}$
from $\mf_G(\tilde{\scc}^{\rfc},\tilde{W}^{\rfr})$ to \(\mf_G(\underline{\tilde{\scc}^{\rfc}},\underline{\tilde{W}}^{\rfr})\). The argument here is analogous to the argument of
<ref>, since the  critical locus inside $\tilde{\scc}^{\rfc}$ is the presage of the critical locus inside
$\scc^{\rfc}$ under the map $\mu$. Thus the group element coordinate is equal to $1$,  hence the restriction to an open neighborhood  of
the unit is an isomorphism of the corresponding categories of matrix factorizations.

The category of matrix factorizations from the previous proposition is  related to the $B$-equivariant category $\mf_{G\times B }(\tilde{\scz},\tilde{W}_{\dr}^{\rfc})$, where
$$ \tilde{\scz}=\frg\times G\times\frb\times G\times V\times V^*,\quad \tilde{W}_{\dr}^{\rfc}(X,g,Y,h,v,u)=

by the averaging functor $\av$. Geometrically the averaging corresponds to the
quotient map $\av\colon\scz\to \scz/B=\tilde{\scc}$. We describe this functor in a local charts $U\subset \tilde{\scc} $ and
Given a matrix factorization $\calf\in\mf_{B}(\tilde{\scz},\tilde{W}_{\dr}^{\rfc})$, denote by $\calf_{\tilde{U}}$ its restriction
to the open set. Similarly define

as the corresponding matrix factorization in the local chart $U$.

Combining the previous proposition with  a simple modification of the previous argument in the construction of the functor $\CH$ we
obtain the following proposition:

There is a functor

$$\ch\colon \MF^{\rfs}_{}\to \Coh(\whilb_n) $$
such that

$$\CH=\pi_*\circ \ch.$$

Moreover, for a braid $\beta\in \Br_n$ and for any $\vec{a}\in \ZZ^{n-1}$ we have:

$$\ch(\calC_{\beta\cdot \mJMa})=\ch(\calC_\beta)\otimes\cLva
% \calL^{\otimes k}

where $\mJMa=\delta_1^{a_1}\cdots \delta_{n-1}^{a_{n-1}}$ and
%\calL^{\otimes k}

The embedding of $\tilde{\scz}$ inside the space $\scz$ defined by:

$$\tilde{i}(X,g,Y,h,v,u)=(\Ad_h Y, g,X,h,Y,v,u)$$

and there is a homotopy equivalence of matrix factorizations on $\tilde{\scz}$:

$$\mathrm{K}_{\CH}\otimes f^*_\Delta(\calc)\sim \tilde{i}^*\circ f^*_\Delta(\calc).$$

Thus we define the functor in the statement as:

$$\ch(\calc):=\mathrm{KSZ}_{\frg}\circ\mathrm{loc}\circ\av\circ \tilde{i}^*\circ f_{\Delta}^*(\calc).$$

Then the first statement follows from our construction of the averaging functor $\av$ and the last statement is immediate
from  the main result of <cit.>.

§.§ Drinfeld centers

The propositions <ref> motivate  us to propose the following conjecture:

The functor \(\HC^\bullet\) could be enhanced to the functor to the Drinfeld center
of the corresponding category:

\[\MF^\bullet_{\dr}\to \mathfrak{Z}(\MF^\bullet)\]

We do not expect the enhanced functor to be an isomorphism, but we expect that the Chern character is
injective when restricted to central elements of \(\MF^\bullet\). The sheaves \(\CH^{\rfs}_{\rloc}(\calD)\) are supported
on the Lagrangian subvariety inside \(\Hilb\). Thus we propose

There is an isomorphism between the categories
\(\MF^0_{\dr}\) and \(\mathfrak{Z}(\MF).\)