# Matrix factorizations and $\mathfrak{gl}(m|k)$-quantum invariants

A. Oblomkov  A. Oblomkov
Department of Mathematics and Statistics
University of Massachusetts at Amherst
Lederle Graduate Research Tower
710 N. Pleasant Street
Amherst, MA 01003 USA<EMAIL_ADDRESS>and  L. Rozansky  L. Rozansky
Department of Mathematics
University of North Carolina at Chapel Hill
CB # 3250, Phillips Hall
Chapel Hill, NC 27599 USA<EMAIL_ADDRESS>
###### Abstract.

In our previous papers we used the Hilbert scheme of points on
$\mathbb{C}^{2}$ in order to construct a triply graded link homology and its
$\mathfrak{gl}(m)$ version. Here we extend the $\mathfrak{gl}(m)$ construction
to super-algebras $\mathfrak{gl}(m|k)$.

##  1\. Introduction

In our previous papers [OR18c] we developed a matrix factorization approach to
the categorification of the Ocneanu-Jones trace. The construction relies on
the homomorphism

$\Phi_{n}:\mathfrak{Br}_{n}\to\mathrm{MF}_{n}^{\textup{st}}=\mathrm{MF}_{\textup{GL}_{n}}\bigl{(}(\mathfrak{gl}_{n}\times\mathrm{T}^{*}\mathrm{Fl}\times\mathrm{T}^{*}\mathrm{Fl}\times\mathbb{C}^{n})^{\textup{st}},W\bigr{)},$
$W(X,z_{1},z_{2},v)=\textup{Tr}\bigl{(}X(\mu(z_{1})-\mu(z_{2}))\bigr{)},$

where $\mathrm{MF}_{\textup{GL}_{n}}(\ldots,W)$ is the category of
$\textup{GL}_{n}$-equivariant matrix factorizations of $W$,
$\mu:\mathrm{T}^{*}\mathrm{Fl}\to\mathfrak{gl}_{n}$ is the moment map, and the
stable locus $(\ldots)^{\textup{st}}$ consists of the points
$(X,z_{1},z_{2},v)$ such that $\mathbb{C}\langle
X,\mu(z_{i})\rangle\,v=\mathbb{C}^{n}$. The scaling of $\mathfrak{gl}_{n}$ and
of the cotangent fibers of $\mathrm{T}^{*}\mathrm{Fl}$ factors yields an
additional $T_{qt}=\mathbb{C}^{*}\times\mathbb{C}^{*}$-equivariant structure
on the matrix factorizations.

In the subsequent paper [OR18b] we defined the Chern character $\mathsf{CH}$
and its right adjoint $\mathrm{HC}$:

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where $\mathscr{D}_{T_{qt}}^{\textup{per}}(\mathrm{Hilb}_{n})$ is a derived
category of two-periodic $T_{qt}$-equivariant complexes on the Hilbert scheme
of $n$ points on $\mathbb{C}^{2}$.

In the last diagram $\mathsf{CH}$ is a trace functor and $\mathrm{HC}$ is
monoidal. In particular, in [OR18b] we define a categorification of the
Ocneanu-Jones trace by

(1.1)
$\mathcal{T}r(\beta)=\mathsf{CH}(\Phi(\beta))\otimes\Lambda^{*}\mathcal{B},$

where $\mathcal{B}$ is the tautological vector bundle on the Hilbert scheme,
the fiber of the dual vector bundle being defined by
$\mathcal{B}^{\vee}|_{I}=\mathbb{C}[x,y]/I$. We show in [OR20] that vector
space of the derived global sections $\mathbb{H}(\mathcal{T}r(\beta))$ is the
triply-graded homology [KR08a],[KR08b], [Kho07] $\mathrm{HHH}(\beta)$ of the
closure $L(\beta)$.

A polynomial $f\in\mathbb{C}[x,y]$ determines a Koszul differential

$D_{f}:\mathcal{S}\otimes\Lambda^{\bullet}\mathcal{B}\to\mathcal{S}\otimes\Lambda^{\bullet-1}\mathcal{B}$

for any $\mathcal{S}\in\mathrm{D}^{\textup{per}}_{T_{qt}}(\mathrm{Hilb}_{n})$.
Here we show that the choice $f=x^{m}y^{k}$ and an insertion of the
corresponding differential into the categorified Ocneanu-Jones trace (1.1)
leads to a categorification of the quantum $\mathfrak{gl}(m|k)$ invariants of
links.

######  Theorem 1.0.1.

For any $\beta\in\mathfrak{Br}_{r}$ the total homology of the complex:

$\mathrm{HH}_{m|k}(\beta):=\mathbb{H}(\mathcal{T}r(\beta),D_{x^{m}y^{k}})$

is a doubly graded vector space that categorifies $\mathfrak{gl}(m|k)$-quantum
invariant of $L(\beta)$.

The (triple) $(q,t,a)$-grading structure of the homology $\mathrm{HHH}(\beta)$
is defined by assigning degrees to the generators $x$ and $y$ of the Hilbert
scheme related ring $\mathbb{C}[x,y]$ and to the exterior powers of the
tautological bundle $\mathcal{B}$:

$\deg x=(1,0,0),\qquad\deg
y=(-1,2,0),\qquad\deg\Lambda^{i}\mathcal{B}=i(0,1,1i).$

The differential $D_{x^{m}y^{k}}$ respects only the double $(Q,T)$ grading,
which is defined by similar relations:

$\deg x=(1,0),\qquad\deg
y=(-1,2),\qquad\deg\Lambda^{i}\mathcal{B}=i(m-k,1,1).$

Theorem 1.0.1 was already anticipated by many authors. In particular, in
[GORS14] conjectural relation between the rational DAHAs and triply-graded
homology of torus knot $T_{n,l}$ was proposed. The Gordon-Stafford theory
[GS06] translates the conjecture to the relation between the sheaves on
$\mathrm{Hilb}_{n}(\mathbb{C}^{2})$ and triply-graded homology of the torus
knots. In this context the differential $D_{x^{m}}$ appears naturally and a
precise conjecture for isomorphism between $\mathrm{HH}_{m}(T_{n,l})$ and
$\mathfrak{gl}(n)$ homology [KR08a] is given in [GORS14].

In the case of $m=2,k=0$ the action of the differential $d_{2|0}$ on
$\lim_{l\to\infty}\mathrm{HHH}(T_{n,l})$ is explored in [GOR13] and [BGK19].

The exploration of the differential similar to $D_{x^{m}y^{k}}$ for general
$m,k$ was initiated in [GGS18]. Also in [QRS18] a proposal for
$\mathfrak{gl}(-m)$ homology is presented. Finally, it is generally
anticipated that $\mathfrak{gl}(1|1)$-homology are related to the Hegaard-
Floer homology [Ras15].

Acknowledgments 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.

##  2\. Matrix factorization technology

For a proof of the main theorem we need to use some subtle details of the
categorification of the braid group from our previous work [OR18c]. We relied
on theory of matrix factorizations and the current section provides a short
introduction to the matrix factorization technique of the paper.

###  2.1. Braid group functor

In our earlier papers [OR18c], [OR19], [OR20] we defined several monoidal
functors from the braid group $\mathfrak{Br}_{n}$ to categories of matrix
factorizations (see the next subsection for more details on matrix
factorizations). In this note we concentrate on the functor studied in
[OR18c]. We outline the construction of the braid group construction in this
subsection and provide more technical details in the later subsections. The
key player in the construction is the space with the potential:

$\mathcal{X}_{n}=\mathfrak{g}_{n}\times G_{n}\times\mathfrak{n}_{n}\times
G_{n}\times\mathfrak{n}_{n},\quad
W(X,g_{1},Y_{1},g_{2},Y_{2})=\textup{Tr}(X(\textup{Ad}_{g_{1}}Y_{1}-\textup{Ad}_{g_{2}}Y_{2})).$

Here and everywhere in the paper we use short-hand notations for the groups
and Lie algebras:

$G_{n}=\textup{GL}_{n},\quad\mathfrak{g}_{n}=\mathfrak{gl}_{n},\quad\mathfrak{h}_{n}\subset\mathfrak{b}_{n}\subset\mathfrak{g}_{n},\quad\mathfrak{b}_{n}=\textup{Lie}\
(B_{n}),\quad\mathfrak{h}_{n}=\textup{Lie}\
(T_{n})\quad\mathfrak{n}_{n}=[\mathfrak{b}_{n},\mathfrak{b}_{n}].$

When the rank of the group is clear from the context the subindex is omitted.

The category of $G_{n}\times B_{n}\times B_{n}$-equivariant matrix
factorizations has a natural monoidal structure:

(2.1) $\mathrm{MF}_{n}=\mathrm{MF}_{G_{n}\times
B^{2}}(\mathcal{X}_{n},W)\ni\mathcal{F}_{1},\mathcal{F}_{2},\quad\mathcal{F}_{1}\star\mathcal{F}_{2}=\pi_{13*}(\mathrm{CE}_{\mathfrak{n}_{n}}(\pi_{12}^{*}(\mathcal{F}_{1})\otimes\pi_{23}^{*}(\mathcal{F}_{2})))^{T_{n}},$

where $\mathrm{CE}_{\mathfrak{n}}(\cdot)$ is a functor of derived
$\mathfrak{n}$-invariants (see section 2.3 and discussion in the papers
[OR18c],[OR19],[OR20]) and the maps in the definition of the convolution
$\star$ are the projections
$\pi_{ij}:\mathcal{X}^{3}_{n}=\mathfrak{g}_{n}\times(G_{n}\times\mathfrak{b}_{n})^{3}\to\mathcal{X}_{n}$.

######  Proposition 2.1.1.

[OR18c],[OR18a] For any $n$ there is a monoidal functor from the affine braid
group $\mathfrak{Br}_{n}^{\textup{aff}}$ to the monoidal category of matrix
factorizations:

$\Phi_{n}^{\textup{aff}}:\mathfrak{Br}_{n}^{\textup{aff}}\to(\mathrm{MF}_{n},\star).$

The finite braid group $\mathfrak{Br}_{n}$ is naturally a subgroup of
$\mathfrak{Br}_{n}^{\textup{aff}}$. In particular, we can restrict the
monoidal functor $\Phi_{n}^{\textup{aff}}$ to the finite braid group to obtain
monoidal functor $\Phi_{n}:\mathfrak{Br}_{n}\to\mathrm{MF}_{n}$ The inclusion
of the braid groups is a section of the projection homomorphism
$\mathrm{fgt}:\mathfrak{Br}_{n}^{\textup{aff}}\to\mathfrak{Br}_{n}$. By
introducing stability constraints for matrix factorizations [OR18a] we
categorify the projection homomorphism.

The stable sub-variety of $\mathcal{X}_{n}\times\mathbb{C}^{n}$ is defined by
the open condition:

$\left(\mathcal{X}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}}=\\{(X,g_{1},Y_{1},g_{2},Y_{2},v)|\mathbb{C}\langle
X,\textup{Ad}_{g_{i}}\rangle v=\mathbb{C}^{n},i=1,2\\}.$

The natural projection
$\pi_{v}:\left(\mathcal{X}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}}\to\mathcal{X}_{n}$
allows us to pull back the potential $W$ to the stable space
$\left(\mathcal{X}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}}$ and use $W$
for the pull-back $\pi_{v}^{*}(W)$. Thus we define the stable category [OR18c]
as:

$\mathrm{MF}_{n}^{\textup{st}}=\mathrm{MF}_{G_{n}\times
B^{2}_{n}}(\left(\mathcal{X}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}},W).$

The monoidal structure on the category $\mathrm{MF}_{n}^{\textup{st}}$ and
functor $\Phi_{n}$ are defined similarly to the case of $\mathrm{MF}_{n}$.
Moreover, the monoidal structure on $\mathrm{MF}_{n}^{\textup{st}}$ is
compatible with the monoidal structure of $\mathrm{MF}_{n}$ in the following
sense. An open inclusion

$j_{\textup{st}}:\left(\mathcal{X}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}}\to\mathcal{X}_{n}\times\mathbb{C}^{n}$

induces functor the pull-back functor $j_{\textup{st}}^{*}$. In [OR18a] we
show that diagram of morphisms:

${\mathfrak{Br}_{n}^{\textup{aff}}}$${\mathrm{MF}_{n}}$${\mathfrak{Br}_{n}}$${\mathrm{MF}_{n}^{\textup{st}}}$$\scriptstyle{\Phi_{n}^{\textup{aff}}}$$\scriptstyle{\mathrm{fgt}}$$\scriptstyle{j_{\textup{st}}^{*}\circ\pi_{v}^{*}}$$\scriptstyle{\Phi_{n}}$

is a commuting diagram of monoidal functors.

###  2.2. Matrix Factorizations

Matrix factorizations were introduced by Eisenbud [Eis80] and later the
subject was further developed by Orlov [Orl04]. Below we present only the
basic definitions and do not present any proofs.

Let us remind that for an affine variety $\mathcal{Z}$ and a function
$F\in\mathbb{C}[\mathcal{Z}]$ there exists a triangulated category
$\mathrm{MF}(\mathcal{Z},F)$. The objects of the category are pairs

$\mathcal{F}=(M_{0}\oplus M_{1},D),\quad D:M_{i}\rightarrow M_{i+1},\quad
D^{2}=F,$

where $M_{i}$ are free $\mathbb{C}[\mathcal{Z}]$-modules of finite rank and
$D$ is a homomorphism of $\mathbb{C}[\mathcal{Z}]$-modules.

Given $\mathcal{F}=(M,D)$ and $\mathcal{G}=(N,D^{\prime})$ the linear space of
morphisms $\textup{Hom}(\mathcal{F},\mathcal{G})$ consists of homomorphisms of
$\mathbb{C}[\mathcal{Z}]$-modules $\phi=\phi_{0}\oplus\phi_{1}$,
$\phi_{i}\in\textup{Hom}(M_{i},N_{i})$ such that $\phi\circ
D=D^{\prime}\circ\phi$. Two morphisms
$\phi,\rho\in\textup{Hom}(\mathcal{F},\mathcal{G})$ are homotopic if there is
homomorphism of $\mathbb{C}[\mathcal{Z}]$-modules $h=h_{0}\oplus h_{1}$,
$h_{i}\in\textup{Hom}(M_{i},N_{i+1})$ such that $\phi-\rho=D^{\prime}\circ
h-h\circ D$.

In the paper [OR18c] we introduced a notion of equivariant matrix
factorizations which we explain below. First let us remind the construction of
the Chevalley-Eilenberg complex.

###  2.3. Chevalley-Eilenberg complex

Suppose that $\mathfrak{h}$ is a Lie algebra. Chevalley-Eilenberg complex
$\mathrm{CE}_{\mathfrak{h}}$ is the complex $(V_{\bullet}(\mathfrak{h}),d)$
with
$V_{p}(\mathfrak{h})=U(\mathfrak{h})\otimes_{\mathbb{C}}\Lambda^{p}\mathfrak{h}$
and differential $d_{ce}=d_{1}+d_{2}$ where:

$d_{1}(u\otimes x_{1}\wedge\dots\wedge
x_{p})=\sum_{i=1}^{p}(-1)^{i+1}ux_{i}\otimes
x_{1}\wedge\dots\wedge\hat{x}_{i}\wedge\dots\wedge x_{p},$ $d_{2}(u\otimes
x_{1}\wedge\dots\wedge x_{p})=\sum_{i<j}(-1)^{i+j}u\otimes[x_{i},x_{j}]\wedge
x_{1}\wedge\dots\wedge\hat{x}_{i}\wedge\dots\wedge\hat{x}_{j}\wedge\dots\wedge
x_{p},$

Let us denote by $\Delta$ the standard map
$\mathfrak{h}\to\mathfrak{h}\otimes\mathfrak{h}$ defined by $x\mapsto x\otimes
1+1\otimes x$. Suppose $V$ and $W$ are modules over the Lie algebra
$\mathfrak{h}$ then we use notation $V\stackbin[]{\Delta}{\otimes}W$ for the
$\mathfrak{h}$-module which is isomorphic to $V\otimes W$ as a vector space,
the $\mathfrak{h}$-module structure being defined by $\Delta$. Respectively,
for a given $\mathfrak{h}$-equivariant matrix factorization
$\mathcal{F}=(M,D)$ we denote by
$\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}\mathcal{F}$ the
$\mathfrak{h}$-equivariant matrix factorization
$(CE_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}\mathcal{F},D+d_{ce})$. The
$\mathfrak{h}$-equivariant structure on
$\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}\mathcal{F}$ originates
from the left action of $U(\mathfrak{h})$ that commutes with right action on
$U(\mathfrak{h})$ used in the construction of $\mathrm{CE}_{\mathfrak{h}}$.

A slight modification of the standard fact that $\mathrm{CE}_{\mathfrak{h}}$
is the resolution of the trivial module implies that
$\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}M$ is a free resolution
of the $\mathfrak{h}$-module $M$.

###  2.4. Equivariant matrix factorizations

Let us assume that there is an action of the Lie algebra $\mathfrak{h}$ on
$\mathcal{Z}$ and $F$ is a $\mathfrak{h}$-invariant function. Then we can
construct the following triangulated category
$\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},W)$.

The objects of the category are triples:

$\mathcal{F}=(M,D,\partial),\quad(M,D)\in\mathrm{MF}(\mathcal{Z},W)$

where $M=M^{0}\oplus M^{1}$ and $M^{i}=\mathbb{C}[\mathcal{Z}]\otimes V^{i}$,
$V^{i}\in\textup{Mod}_{\mathfrak{h}}$,
$\partial\in\oplus_{i>j}\textup{Hom}_{\mathbb{C}[\mathcal{Z}]}(\Lambda^{i}\mathfrak{h}\otimes
M,\Lambda^{j}\mathfrak{h}\otimes M)$ and $D$ is an odd endomorphism
$D\in\textup{Hom}_{\mathbb{C}[\mathcal{Z}]}(M,M)$ such that

$D^{2}=F,\quad D_{tot}^{2}=F,\quad D_{tot}=D+d_{ce}+\partial,$

where the total differential $D_{tot}$ is an endomorphism of
$\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}M$, that commutes with
the $U(\mathfrak{h})$-action.

Note that we do not impose the equivariance condition on the differential $D$
in our definition of matrix factorizations. On the other hand, if
$\mathcal{F}=(M,D)\in\mathrm{MF}(\mathcal{Z},F)$ is a matrix factorization
with $D$ that commutes with $\mathfrak{h}$-action on $M$ then
$(M,D,0)\in\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},F)$.

There is a natural forgetful functor
$\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},F)\to\mathrm{MF}(\mathcal{Z},F)$ that
forgets about the correction differentials:

$\mathcal{F}=(M,D,\partial)\mapsto\mathcal{F}^{\sharp}:=(M,D).$

Given two $\mathfrak{h}$-equivariant matrix factorizations
$\mathcal{F}=(M,D,\partial)$ and
$\tilde{\mathcal{F}}=(\tilde{M},\tilde{D},\tilde{\partial})$ the space of
morphisms $\textup{Hom}(\mathcal{F},\tilde{\mathcal{F}})$ consists of homotopy
equivalence classes of elements
$\Psi\in\textup{Hom}_{\mathbb{C}[\mathcal{Z}]}(\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}M,\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}\tilde{M})$
such that $\Psi\circ D_{tot}=\tilde{D}_{tot}\circ\Psi$ and $\Psi$ commutes
with $U(\mathfrak{h})$-action on
$\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}M$. Two maps
$\Psi,\Psi^{\prime}\in\textup{Hom}(\mathcal{F},\tilde{\mathcal{F}})$ are
homotopy equivalent if there is

$h\in\textup{Hom}_{\mathbb{C}[\mathcal{Z}]}(\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}M,\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}\tilde{M})$

such that $\Psi-\Psi^{\prime}=\tilde{D}_{tot}\circ h-h\circ D_{tot}$ and $h$
commutes with $U(h)$-action on
$\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}M$.

Given two $\mathfrak{h}$-equivariant matrix factorizations
$\mathcal{F}=(M,D,\partial)\in\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},F)$ and
$\tilde{\mathcal{F}}=(\tilde{M},\tilde{D},\tilde{\partial})\in\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},\tilde{F})$
we define
$\mathcal{F}\otimes\tilde{\mathcal{F}}\in\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},F+\tilde{F})$
as the equivariant matrix factorization
$(M\otimes\tilde{M},D+\tilde{D},\partial+\tilde{\partial})$.

###  2.5. Push forwards, quotient by the group action

The technical part of [OR18c] is the construction of push-forwards of
equivariant matrix factorizations. Here we state the main results, the details
may be found in section 3 of [OR18c]. We need push forwards along projections
and embeddings. We also use the functor of taking quotient by group action for
our definition of the convolution algebra.

The projection case is more elementary. Suppose
$\mathcal{Z}=\mathcal{X}\times\mathcal{Y}$, both $\mathcal{Z}$ and
$\mathcal{X}$ have $\mathfrak{h}$-action and the projection
$\pi:\mathcal{Z}\rightarrow\mathcal{X}$ is $\mathfrak{h}$-equivariant. Then
for any $\mathfrak{h}$ invariant element
$w\in\mathbb{C}[\mathcal{X}]^{\mathfrak{h}}$ there is a functor
$\pi_{*}\colon\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},\pi^{*}(w))\rightarrow\mathrm{MF}_{\mathfrak{h}}(\mathcal{X},w)$
which simply forgets the action of $\mathbb{C}[\mathcal{Y}]$.

We define an embedding-related push-forward in the case when the subvariety
$\mathcal{Z}_{0}\xhookrightarrow{j}\mathcal{Z}$ is the common zero of an ideal
$I=(f_{1},\dots,f_{n})$ such that the functions
$f_{i}\in\mathbb{C}[\mathcal{Z}]$ form a regular sequence. We assume that the
Lie algebra $\mathfrak{h}$ acts on $\mathcal{Z}$ and $I$ is
$\mathfrak{h}$-invariant. Then there exists an $\mathfrak{h}$-equivariant
Koszul complex
$K(I)=(\Lambda^{\bullet}\mathbb{C}^{n}\otimes\mathbb{C}[\mathcal{Z}],d_{K})$
over $\mathbb{C}[\mathcal{Z}]$ which has non-trivial homology only in degree
zero. Then in section 3 of [OR18c] we define the push-forward functor

$j_{*}\colon\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z}_{0},W|_{\mathcal{Z}_{0}})\longrightarrow\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},W),$

for any $\mathfrak{h}$-invariant element
$W\in\mathbb{C}[\mathcal{Z}]^{\mathfrak{h}}$.

Finally, let us discuss the quotient map. The complex
$\mathrm{CE}_{\mathfrak{h}}$ is a resolution of the trivial
$\mathfrak{h}$-module by free modules. Thus the correct derived version of
taking $\mathfrak{h}$-invariant part of the matrix factorization
$\mathcal{F}=(M,D,\partial)\in\mathrm{MF}_{\mathfrak{h}}(\mathcal{Z},W)$,
$W\in\mathbb{C}[\mathcal{Z}]^{\mathfrak{h}}$ is

$\mathrm{CE}_{\mathfrak{h}}(\mathcal{F}):=(\mathrm{CE}_{\mathfrak{h}}(M),D+d_{ce}+\partial)\in\mathrm{MF}(\mathcal{Z}/H,W),$

where $\mathcal{Z}/H:=\mathrm{Spec}(\mathbb{C}[\mathcal{Z}]^{\mathfrak{h}})$
and use the general definition of $\mathfrak{h}$-module $V$:

$\mathrm{CE}_{\mathfrak{h}}(V):=\textup{Hom}_{\mathfrak{h}}(\mathrm{CE}_{\mathfrak{h}},\mathrm{CE}_{\mathfrak{h}}\stackbin[]{\Delta}{\otimes}V).$

##  3\. Markov moves

To prove our main result we need to show that the doubly-graded vector space
is invariant with respect to the Markov moves and that compute the character
of doubly-graded vector space for the unknot. In this section we work compute
deal with the Markov moves. Other words the main result of this section is a
pair of propositions.

######  Proposition 3.0.1.

For any $\alpha,\beta\in\mathfrak{Br}_{n}$ and $m,k$ we have

$\mathrm{HH}_{m|k}(\alpha\beta)=\mathrm{HH}_{m|k}(\beta\alpha).$

The braid group $\mathfrak{Br}_{n}$ has standard generators $\sigma_{i}$,
$i=1,\dots,n-1$ and
$\mathfrak{Br}_{n+1}=\langle\mathfrak{Br}_{n},\sigma_{n}\rangle$:

######  Proposition 3.0.2.

For any $\beta\in\mathfrak{Br}_{n}$ and any $m,k$ we have

$\mathrm{HH}_{m|k}(\beta\sigma_{n}^{\epsilon})=\mathrm{HH}_{m|k}(\beta),\quad\epsilon=\pm
1.$

It is convenient to introduce a super-group specialization functor to encode
the properties of the differential $d_{m|k}$.

###  3.1. Super-group specialization

Let us introduce two functors on category $\mathrm{MF}_{n}^{\textup{st}}$ and
$\mathscr{D}_{T_{qt}}(\mathrm{Hilb}_{n})$:

$\mathcal{R}_{m|k}:\quad\mathrm{MF}_{n}^{\textup{st}}\to\mathrm{MF}_{n}^{\textup{st}},\quad\quad\mathscr{R}_{m|k}:\quad\mathscr{D}_{T_{qt}}(\mathrm{Hilb}_{n})\to\mathscr{D}_{T_{qt}}(\mathrm{Hilb}_{n}).$

The functor is defined is almost identically for both categories. Indeed, let
us fix coordinates on space $\mathcal{X}_{n}\times\mathbb{C}^{n}$ as
$(X,g_{1},Y_{1},g_{2},Y_{2},v)$. We describe $\mathrm{Hilb}_{n}$ as
$\textup{GL}_{n}$ quotient of the stable locus of
$T^{*}\mathfrak{gl}_{n}\times\mathbb{C}^{n}$ and the coordinates on the space
$T^{*}\mathfrak{gl}_{n}\times\mathbb{C}^{n}$ are $(X,Y,v)$. The functors then
defined as

$\mathcal{R}_{m|k}(\mathcal{F})=\mathcal{F}\otimes(\Lambda^{\star}\mathbb{C}^{n},d_{m|k}),\quad
d_{m|k}=\sum_{i=1}(X^{m}\textup{Ad}_{g_{2}}(Y_{2})^{k}v)_{i}\frac{\partial}{\partial\theta_{i}},$
$\mathscr{R}_{m|k}(\mathcal{C})=\mathcal{C}\otimes(\Lambda^{\star}\mathbb{C}^{n},d_{m|k}),\quad
d_{m|k}=\sum_{i=1}(X^{m}Y^{k}v)_{i}\frac{\partial}{\partial\theta_{i}},$

where $\theta_{i}$, $i=1,\dots,n$ are coordinates along
$\Lambda^{\star}\mathbb{C}^{n}$.

The similarity of the functors translates into the intertwining property:

######  Proposition 3.1.1.

For any $m,k,n$ we have

$\mathsf{CH}^{\textup{st}}_{\textup{loc}}\circ\mathcal{R}_{m|k}=\mathscr{R}_{m|k}\circ\mathsf{CH}^{\textup{st}}_{\textup{loc}}.$

###### Proof.

Let us recall the construction of the functor
$\mathsf{CH}^{\textup{st}}_{\textup{loc}}$ from [OR18b]. First we recall the
non-localized functor $\mathsf{CH}^{\mathrm{fr}}$ for framed categories:

$\mathsf{CH}^{\mathrm{fr}}:\mathrm{MF}^{\mathrm{fr}}\to\mathrm{MF}^{\mathrm{fr}}_{\mathrm{Dr}},$

where the framed categories are defined as follows.

Define framed versions of the spaces $\mathscr{X}$ and $\mathscr{C}$

$\mathscr{X}^{\mathrm{fr}}:=\mathscr{X}\times V^{*}_{G}\times
V_{B^{(1)}}\times
V_{B^{(2)}},\quad\mathscr{C}^{\mathrm{fr}}:=\mathscr{C}\times V_{G}^{*}\times
V_{G},\quad\mathscr{C}=\mathfrak{g}\times G\times\mathfrak{g}$

where the subindexes indicate the equivariant structure of the vector space
$V=\mathbb{C}^{n}$ and potentials are

$W^{\mathrm{fr}}(X,g_{1},Y_{1},g_{2},Y_{2},w,v_{1},v_{2}):=W(X,g_{1},Y_{1},g_{2},Y_{2})+\textup{Tr}(w(g_{1}v_{1}-g_{2}v_{2})),$
$W^{\mathrm{fr}}_{\mathrm{Dr}}(X,g,Z,w,v)=W_{\mathrm{Dr}}(X,g,Z)+\textup{Tr}(w(v-gv)),\quad
W_{\mathrm{Dr}}(X,g,Z)=\textup{Tr}(Z(X-\textup{Ad}_{g}X)).$

Respectively, we define the framed categories as

$\mathrm{MF}^{\mathrm{fr}}:=\mathrm{MF}_{G\times
B^{2}}^{\mathbb{T}_{q,t}}(\mathscr{X}^{\mathrm{fr}},W^{\mathrm{fr}}),\quad\mathrm{MF}^{\mathrm{fr}}_{\mathrm{Dr}}:=\mathrm{MF}_{G}^{\mathbb{T}_{q,t}}(\mathcal{C}^{\mathrm{fr}}).$

There two auxiliary spaces that are needed for the construction of
$\mathsf{CH}^{\mathrm{fr}}$:

$\mathscr{Z}_{\mathsf{CH}}^{0,\mathrm{fr}}=\mathfrak{g}\times
G\times\mathfrak{g}\times G\times\mathfrak{n}\times V^{*}_{G}\times
V_{G},\quad\mathscr{Z}_{\mathsf{CH}}=\mathfrak{g}\times
G\times\mathfrak{g}\times G\times\mathfrak{b}\times V^{*}_{G}\times V_{B}$

The action of $G\times B$ on these spaces is

$(k,b)\cdot(Z,g,X,h,Y,u,v)=(\textup{Ad}_{k}(Z),\textup{Ad}_{k}(g),\textup{Ad}_{k}(X),khb,\textup{Ad}_{b^{-1}}(Y),ku,kv).$

The spaces $\mathscr{C}$ and $\mathscr{X}$ are endowed with the standard
$G\times B^{2}$-equivariant structure, the action of $B^{2}$ on $\mathscr{C}$
is trivial. The following maps

$\pi_{\mathrm{Dr}}(Z,g,X,h,Y,w,v)=(Z,g,X,w,v),$
$f_{\Delta}(Z,g,X,h,Y,w,v)=(X,gh,Y,h,Y,w,h^{-1}g^{-1}(v),h^{-1}(v)).$

are fully equivariant if we restrict the $B^{2}$-equivariant structure on
$\mathscr{X}$ to the $B$-equivariant structure via the diagonal embedding
$\Delta:B\rightarrow B^{2}$. Let us denote by $j^{0}$ is the inclusion map of
$\mathscr{Z}_{\mathsf{CH}}^{0,\mathrm{fr}}$ to
$\mathscr{Z}_{\mathsf{CH}}^{\mathrm{fr}}$.

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

$\mathrm{K}_{\mathsf{CH}}:=[X-\textup{Ad}_{g^{-1}}X,\textup{Ad}_{h}Y-Z]\in\mathrm{MF}(\mathscr{Z}_{\mathsf{CH}},\pi_{\mathrm{Dr}}^{*}(W_{\mathrm{Dr}})-f^{*}_{\Delta}(W)).$

and we define the Chern functor:

(3.1)
$\mathsf{CH}^{\mathrm{fr}}(\mathcal{C}):=\pi_{\mathrm{Dr}*}(\mathrm{CE}_{\mathfrak{n}}(\mathrm{K}_{\mathsf{CH}}\otimes(j^{0}_{*}\circ
f^{*}_{\Delta}(\mathcal{C})))^{T}).$

Here and everywhere below we use notation $(-)^{T}$ for $T$-invariants. From
the formulas we see that
$j^{0*}\circ\pi^{*}_{\mathrm{Dr}}(d_{m|k})=f^{*}_{\Delta}(d_{m|k})$. Thus we
have intertwining relation:

$\mathsf{CH}^{\mathrm{fr}}\circ\mathcal{R}_{m|k}=\mathscr{R}_{m|k}\circ\mathsf{CH}^{\mathrm{fr}}.$

The categories the open subspaces
$\mathscr{X}^{\mathrm{fs}}\subset\mathscr{X}^{\mathrm{fr}}$ and
$\mathscr{X}^{\mathrm{fs}}_{\mathrm{Dr}}\subset\mathscr{X}^{\mathrm{fr}}_{\mathrm{Dr}}$
are defined by the natural stability conditions. The respective categories of
matrix factorizations are $\mathrm{MF}^{\mathrm{fs}}$ and
$\mathrm{MF}_{\mathrm{Dr}}^{\mathrm{fs}}$ and the Chern functor
$\mathsf{CH}^{\mathrm{fs}}$ between the categories is defined by the same
formula as before. Thus the intertwining relation holds:

$\mathsf{CH}^{\mathrm{fs}}\circ\mathcal{R}_{m|k}=\mathscr{R}_{m|k}\circ\mathsf{CH}^{\mathrm{fs}}.$

The Knörrer periodicity allows us to eliminate the quadratic term
$W^{\mathrm{fr}}-W$ and the corresponding vector space factors
$V_{G},V_{B^{(1)}}$ in the space $\mathscr{X}^{\mathrm{fr}}$. Thus we have an
equivalence between $\mathrm{MF}^{\mathrm{fs}}$ and
$\mathrm{MF}^{\textup{st}}$ and the functor
$\mathsf{CH}^{\textup{st}}:\mathrm{MF}^{\textup{st}}\to\mathrm{MF}_{\mathrm{Dr}}^{\mathrm{fs}}$.

The Knörrer periodicity does not interacts with the factor $V_{B^{(2)}}$ in
the space $\mathscr{X}^{\mathrm{fr}}$ hence the corresponding equivalence
respects the differential $d_{m|k}$. Thus the intertwining relation for the
functor $\mathsf{CH}^{\textup{st}}$ holds.

The functor $\mathsf{CH}^{\textup{st}}_{\textup{loc}}$ is obtained from the
functor $\mathsf{CH}^{\textup{st}}$ by post composing with the pull-back along
the open inclusion map $\textup{loc}:\mathfrak{g}^{3}\times V_{G}\times
V_{G}^{*}\to\mathscr{C}^{\mathrm{fr}}$ and the Koszul duality along the vector
space $\mathfrak{g}$.

The composition of the Koszul duality functor with the pull-back
$\textup{loc}^{*}$ is equivalent to the pull-back along the inclusion
$j_{e}:\mathfrak{g}^{2}\times V_{G}\times
V^{*}_{G}\to\mathscr{C}^{\mathrm{fr}}$ which is induced by the inclusion of
the identity to the group $G$. Since $j^{*}_{e}(d_{m|k})=d_{m|k}$ the
statement of the proposition follows. ∎

The functor $\mathcal{R}_{m|k}$ has monoidal properties. More precisely, the
functor is bi-modular:

######  Proposition 3.1.2.

For any $m,k,n$ and any $\mathcal{F}_{i}\in\mathrm{MF}_{n}^{\textup{st}}$ we
have

$\mathcal{R}_{m|k}(\mathcal{F}_{1}\star\mathcal{F}_{2})=\mathcal{F}_{1}\star\mathcal{R}_{m|k}(\mathcal{F}_{2})=\mathcal{R}_{m|k}(\mathcal{F}_{1})\star\mathcal{F}_{2}.$

###### Proof.

The convolution in $\mathrm{MF}^{\textup{st}}$ is defined by the formula

$\mathcal{F}_{1}\star\mathcal{F}_{2}=\pi_{13*}(\mathrm{CE}_{\mathfrak{n}_{n}}(\pi_{12}^{*}(\mathcal{F}_{1})\otimes\pi_{23}^{*}(\mathcal{F}_{2})))^{T_{n}},$

where $\pi_{ij}:\mathfrak{g}\times(G\times\mathfrak{n})^{3}\times
V\to\mathfrak{g}\times(G\times\mathfrak{n})^{2}\times V$ are natural
projections.

Since $\pi_{13}^{*}(d_{m|k})=\pi_{23}^{*}(d_{m|k})$, the first equality is
immediate. To show the second equality we observe that
$\pi_{13}^{*}(d^{\prime}_{m|k})=\pi_{12}^{*}(d^{\prime}_{m|k})$ where

$d^{\prime}_{m|k}=\sum_{i=1}(X^{m}\textup{Ad}_{g_{1}}(Y_{1})^{k}v)_{i}\frac{\partial}{\partial\theta_{i}}.$

Thus to complete the proof we need to show that for any
$\mathcal{F}\in\mathrm{MF}^{\textup{st}}$ matrix factorizations
$\mathcal{F}\otimes\mathcal{K}_{m|k}$ and
$\mathcal{F}\otimes\mathcal{K}^{\prime}_{m|k}$ are homotopic here
$\mathcal{K}_{m|k}=(\Lambda^{*}\mathbb{C}^{n},d_{m|k})$ and
$\mathcal{K}_{m|k}=(\Lambda^{*}\mathbb{C}^{n},d^{\prime}_{m|k})$.

Let $D$ be the differential of the matrix factorization then the partial
derivative $\frac{\partial D}{\partial X}$ is homotopy between
$\textup{Ad}_{g_{1}}Y_{1}$ and $\textup{Ad}_{g_{2}}Y_{2}$. Since $D$ commutes
with $d_{m|k}$ and the matrix factorization with differentials
$D+d^{\prime}_{m|k}$ and $D+d_{m|k}$ are homotopic. ∎

######  Corollary 3.1.3.

For any $\mathcal{F}_{1},\mathcal{F}_{2}\in\mathrm{MF}^{\textup{st}}_{n}$ we
have

$\mathscr{R}_{m|k}(\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{F}_{1}\star\mathcal{F}_{2}))=\mathscr{R}_{m|k}(\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{F}_{2}\star\mathcal{F}_{1}))$

###### Proof.

$\mathscr{R}_{m|k}(\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{F}_{1}\star\mathcal{F}_{2}))=\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{R}_{m|k}(\mathcal{F}_{1}\star\mathcal{F}_{2}))=\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{R}_{m|k}(\mathcal{F}_{1})\star\mathcal{F}_{2})=\\\
\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{F}_{2}\star\mathcal{R}_{m|k}(\mathcal{F}_{1}))=\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{R}_{m|k}(\mathcal{F}_{2}\star\mathcal{F}_{1}))=\mathscr{R}_{m|k}(\mathsf{CH}_{\textup{loc}}^{\textup{st}}(\mathcal{F}_{2}\star\mathcal{F}_{1})).$

∎

The functor $\mathscr{R}_{m|k}$ provides another presentation for the doubly
graded space from the introduction

$\mathrm{HH}_{m|k}(\beta)=\mathbb{H}(\mathscr{R}_{m|k}(\mathsf{CH}_{\textup{loc}}^{\textup{st}}\circ\Phi_{n}(\beta))).$

Thus proposition 3.0.1 follows immediately from the corollary 3.1.3. The rest
of the section is dedicated to the proof of proposition 3.0.2.

We need to recall the construct of the homology from our previous work. In
particular, we need to recall the construction for the closure, induction and
inclusion of braids.

###  3.2. Reduced convolution

To explain induction and inclusion functors we need to introduce a smaller
‘reduced’ space $\overline{\mathcal{X}}^{\ell}_{n}:=\mathfrak{b}_{n}\times
G^{\ell-1}_{n}\times\mathfrak{n}_{n}$ with the $B^{\ell}_{n}$-action:

$(b_{1},\dots,b_{\ell})\cdot(X,g_{1},\dots,g_{\ell-1},Y)=(\textup{Ad}_{b_{1}}(X),b_{1}g_{1}b_{2}^{-1},b_{2}g_{2}b_{3}^{-1},\dots,\textup{Ad}_{b_{\ell}}(Y)).$

In particular the space
$\overline{\mathcal{X}}_{n}=\overline{\mathcal{X}}_{n}^{2}$ has the following
$B^{2}$-invariant potential:

$\overline{W}(X,g,Y)=\textup{Tr}(X\textup{Ad}_{g}(Y)).$

The proposition 5.1 from [OR18c] provides a functor:

$\Phi:\mathrm{MF}_{B^{2}_{n}}(\overline{\mathcal{X}}_{n},\overline{W})\rightarrow\mathrm{MF}_{B^{2}_{n}}(\mathcal{X}_{n},W)$

which is an embedding of the categories. Without the $B^{2}$-equivariant
structure the functor is an ordinary Knörrer functor [Kno87], the equivariant
version of the Knörrer functor is defined as composition of the equivariant
pull-back and push-forward (see section 5 of [OR18c]):

$\Phi_{\mathrm{kn}}:=j^{x}_{*}\circ\pi^{*}_{y},$

where $\pi_{y}:\widetilde{\mathcal{X}}\rightarrow\overline{\mathcal{X}}_{n}$,
$\widetilde{\mathcal{X}}:=\mathfrak{b}_{n}\times
G_{n}\times\mathfrak{n}_{n}\times G_{n}\times\mathfrak{n}_{n}$ is the
projection $\pi_{y}(X,g_{1},Y_{1},g_{2},Y_{2})=(X,g_{1}^{-1}g_{2},Y_{2})$ and
$j^{x}$ is the natural embedding of $\widetilde{\mathcal{X}}$ into
$\mathcal{X}_{n}$.

Let us also introduce a convolution algebra structure on the category of
matrix factorizations
$\mathrm{MF}_{B^{2}_{n}}(\overline{\mathcal{X}}_{n},\overline{W})$. There are
the following maps
$\bar{\pi}_{ij}:\overline{\mathcal{X}}^{3}_{n}\to\overline{\mathcal{X}}_{n}$:

$\bar{\pi}_{12}(X,g_{12},g_{13},Y)=(X,g_{12},\textup{Ad}_{g_{23}}(Y)_{++}),\quad\bar{\pi}_{13}(X,g_{12},g_{13},Y)=(X,g_{12}g_{23},Y),$
$\bar{\pi}_{23}(X,g_{12},g_{13},Y)=(\textup{Ad}_{g_{12}}^{-1}(X)_{+},g_{23},Y).$

Here and everywhere below $X_{+}$ and $X_{++}$ stand for the upper and
strictly-upper triangular parts of $X$. The map
$\bar{\pi}_{12}\times\bar{\pi}_{23}$ is $B^{2}$-equivariant but not
$B^{3}$-equivariant. However in section 5.4 of [OR18c] we show that for any
$\mathcal{F},\mathcal{G}\in\mathrm{MF}_{B^{2}_{n}}(\overline{\mathcal{X}}_{n},\overline{W})$
there is a natural element

(3.2)
$(\bar{\pi}_{12}\otimes_{B}\bar{\pi}_{23})^{*}(\mathcal{F}\boxtimes\mathcal{G})\in\mathrm{MF}_{B^{3}}(\overline{\mathcal{X}}_{3},\bar{\pi}_{13}^{*}(W)),$

such that we can define the binary operation on
$\mathrm{MF}_{B^{2}}(\overline{\mathcal{X}},\overline{W})$:

$\mathcal{F}\bar{\star}\mathcal{G}:=\bar{\pi}_{13*}(\mathrm{CE}_{\mathfrak{n}^{(2)}}((\bar{\pi}_{12}\otimes_{B}\bar{\pi}_{23})^{*}(\mathcal{F}\boxtimes\mathcal{G}))^{T^{(2)}})$

and $\Phi$ intertwines the convolution structures:

$\Phi_{\mathrm{kn}}(\mathcal{F})\star\Phi_{\mathrm{kn}}(\mathcal{G})=\Phi_{\mathrm{kn}}(\mathcal{F}\bar{\star}\mathcal{G}).$

As we mentioned in the introduction, it is natural to consider the framed
version of our basic spaces. The framed version of the non-reduced space is an
open subset
$\mathcal{X}_{\textup{st}}^{\ell}\subset\mathcal{X}^{\ell}_{n}\times V$,
$V=\mathbb{C}^{n}$ defined by the stability condition:

$\mathbb{C}\langle\textup{Ad}_{g_{i}}^{-1}(X),Y_{i}\rangle
g^{-1}_{i}(u)=V,\quad i=1,\dots,\ell-1.$

Similarly, we define the framed reduced space
$\left(\overline{\mathcal{X}}_{n}\times
V\right)^{\textup{st}}=\overline{\mathcal{X}}_{\textup{st}}^{2}\subset\overline{\mathcal{X}}^{2}_{n}\times
V$ with the stability condition

(3.3) $\mathbb{C}\langle X,\textup{Ad}_{g}(Y)\rangle u=V.$

Let us also define $\overline{\mathcal{X}}_{\textup{st}}^{3}$ to be the
intersection
$\bar{\pi}_{12}^{-1}(\overline{\mathcal{X}}_{\textup{st}}^{2})\cap\bar{\pi}_{23}^{-1}(\overline{\mathcal{X}}_{\textup{st}}^{2})$
where $\bar{\pi}_{ij}$ are the maps $\overline{\mathcal{X}}^{3}_{n}\times
V\rightarrow\overline{\mathcal{X}}_{n}\times V$ which are just extensions of
the previously discussed maps by the identity map on $V$. Similarly we have
the natural maps
$\pi_{ij}:\mathcal{X}_{\textup{st}}^{3}\rightarrow\mathcal{X}_{\textup{st}}^{2}$
and both reduced and non-reduced spaces have natural convolution algebra
structure defined by the formulas (2.1) and (3.2)

We denote by $j_{\textup{st}}$ the maps
$\mathcal{X}_{\textup{st}}^{2}\rightarrow\mathcal{X}_{n}$,
$\overline{\mathcal{X}}_{\textup{st}}^{2}\rightarrow\overline{\mathcal{X}}_{n}$
that forget the framing. Lemma 12.3 of [OR18c] says that the corresponding
pull-back morphism is an homomorphism of the convolution algebras:

$j_{\textup{st}}^{*}(\mathcal{F}\star\mathcal{G})=j_{\textup{st}}^{*}(\mathcal{F})\star
j_{\textup{st}}^{*}(\mathcal{G}).$

Finally, let us mention that we can restrict the Knörrer functor
$\Phi_{\mathrm{kn}}$ on the open set
$\overline{\mathcal{X}}_{\textup{st}}^{2}$ to obtain the functor

$\Phi_{\mathrm{kn}}:\mathrm{MF}_{B^{2}_{n}}(\overline{\mathcal{X}}_{\textup{st}}^{2},\overline{W})\rightarrow\mathrm{MF}_{B^{2}_{n}}(\mathcal{X}_{\textup{st}}^{2},W).$

This functor intertwines the convolution algebra structures on the reduced and
non-reduced framed spaces.

###  3.3. Induction functors

The standard parabolic subgroup $P_{k}$ has Lie algebra generated by
$\mathfrak{b}$ and $E_{i+1,i}$, $i\neq k$. Let us define space
$\overline{\mathcal{X}}(P_{k}):=\mathfrak{b}\times P_{k}\times\mathfrak{n}$
and let us also use notation $\overline{\mathcal{X}}(G_{n})$ for
$\overline{\mathcal{X}}_{n}$. There is a natural embedding
$\bar{i}_{k}:\overline{\mathcal{X}}(P_{k})\rightarrow\overline{\mathcal{X}}_{n}$
and a natural projection
$\bar{p}_{k}:\overline{\mathcal{X}}(P_{k})\rightarrow\overline{\mathcal{X}}(G_{k})\times\overline{\mathcal{X}}(G_{n-k})$.
The embedding $\bar{i}_{k}$ satisfies the conditions for existence of the
push-forward and we can define the induction functor:

$\overline{\textup{ind}}_{k}:=\bar{i}_{k*}\circ\bar{p}_{k}^{*}:\mathrm{MF}_{B_{k}^{2}}(\overline{\mathcal{X}}_{k},\overline{W})\times\mathrm{MF}_{B_{n-k}^{2}}(\overline{\mathcal{X}}_{n-k},\overline{W})\rightarrow\mathrm{MF}_{B_{n}^{2}}(\overline{\mathcal{X}}_{n},\overline{W})$

Similarly we define the space
$\overline{\mathcal{X}}_{\textup{st}}(P_{k})\subset\mathfrak{b}\times
P_{k}\times\mathfrak{n}\times V$ as an open subset defined by the stability
condition (3.3). The last space has a natural projection map
$\bar{p}_{k}:\overline{\mathcal{X}}_{\textup{st}}(P_{k})\rightarrow\overline{\mathcal{X}}(G_{k})\times\overline{\mathcal{X}}_{\textup{st}}(G_{n-k})$
and the embedding
$\bar{i}_{k}:\overline{\mathcal{X}}_{\textup{st}}(P_{k})\rightarrow\overline{\mathcal{X}}_{\textup{st}}(G_{n})$
and we can define the induction functor:

$\overline{\textup{ind}}_{k}:=\bar{i}_{k*}\circ\bar{p}_{k}^{*}:\mathrm{MF}_{B_{k}^{2}}(\overline{\mathcal{X}}(G_{k}),\overline{W})\times\mathrm{MF}_{B_{n-k}^{2}}(\overline{\mathcal{X}}_{\textup{st}}(G_{n-k}),\overline{W})\rightarrow\mathrm{MF}_{B_{n}^{2}}(\overline{\mathcal{X}}_{\textup{st}}(G_{n}),\overline{W})$

It is shown in section 6 (proposition 6.2) of [OR18c] that the functor
$\overline{\textup{ind}_{k}}$ is the homomorphism of the convolution algebras:

$\overline{\textup{ind}}_{k}(\mathcal{F}_{1}\boxtimes\mathcal{F}_{2})\bar{\star}\overline{\textup{ind}}_{k}(\mathcal{G}_{1}\boxtimes\mathcal{G}_{2})=\overline{\textup{ind}}_{k}(\mathcal{F}_{1}\bar{\star}\mathcal{G}_{2}\boxtimes\mathcal{F}_{2}\bar{\star}\mathcal{G}_{2}).$

To define the non-reduced version of the induction functors one needs to
introduce the space $\mathcal{X}^{\circ}(G_{n})=\mathfrak{g}\times
G_{n}\times\mathfrak{n}\times\mathfrak{n}$ which is a slice to the
$G_{n}$-action on the space $\mathcal{X}(G_{n})$. In particular, the potential
$W$ on this slice becomes:

$W(X,g,Y_{1},Y_{2})=\textup{Tr}(X(Y_{1}-\textup{Ad}_{g}(Y_{2}))).$

Similarly to the case of the reduced space, one can define the space
$\mathcal{X}^{\circ}(P_{k}):=\mathfrak{g}\times
P_{k}\times\mathfrak{n}\times\mathfrak{n}$ and the corresponding maps
$i_{k}:\mathcal{X}^{\circ}(P_{k})\rightarrow\mathcal{X}^{\circ}(G_{n})$,
$p_{k}:\mathcal{X}^{\circ}(P_{k})\rightarrow\mathcal{X}^{\circ}(G_{k})\times\mathcal{X}^{\circ}(G_{n-k})$.
Thus we get a version of the induction functor for non-reduced spaces:

${\textup{ind}}_{k}:=i_{k*}\circ
p_{k}^{*}:\mathrm{MF}_{B_{k}^{2}}(\mathcal{X}(G_{k}),W)\times\mathrm{MF}_{B_{n-k}^{2}}(\mathcal{X}(G_{n-k}),W)\rightarrow\mathrm{MF}_{B_{n}^{2}}(\mathcal{X}(G_{n}),W)$

It is shown in proposition 6.1 of [OR18c] that the Knörrer functor is
compatible with the induction functor:

$\textup{ind}_{k}\circ(\Phi_{\mathrm{kn}}\times\Phi_{\mathrm{kn}})=\Phi_{\mathrm{kn}}\circ\textup{ind}_{k}.$

###  3.4. Generators of the finite braid group action

Let us define $B^{2}_{n}$-equivariant embedding
$i:\overline{\mathcal{X}}(B_{n})\rightarrow\overline{\mathcal{X}}_{n}$,
$\overline{\mathcal{X}}(B_{n}):=\mathfrak{b}_{n}\times
B_{n}\times\mathfrak{n}_{n}$. The pull-back of $\overline{W}$ along the map
$i$ vanishes and the embedding $i$ satisfies the conditions for existence of
the push-forward
$i_{*}:\mathrm{MF}_{B^{2}_{n}}(\overline{\mathcal{X}}(B_{n}),0)\rightarrow\mathrm{MF}_{B^{2}}(\overline{\mathcal{X}}(G_{n}),\overline{W})$.
We denote by
$\underline{\mathbb{C}[\overline{\mathcal{X}}(B_{n})]}\in\mathrm{MF}_{B^{2}}(\overline{\mathcal{X}}(B_{n}),0)$
the matrix factorization with zero differential that is homologically non-
trivial only in even homological degree. As it is shown in proposition 7.1 of
[OR18c] the push-forward

$\bar{\mathds{1}}_{n}:=i_{*}(\underline{\mathbb{C}[\overline{\mathcal{X}}(B_{n})]})$

is the unit in the convolution algebra. Similarly,
$\mathds{1}_{n}:=\Phi_{\mathrm{kn}}(\bar{\mathds{1}}_{n})$ is also a unit in
non-reduced case.

Let us first discuss the case of the braids on two strands. The key to
construction of the braid group action in [OR18c] is the following
factorization in the case $n=2$:

$\overline{W}(X,g,Y)=y_{12}(2g_{11}x_{11}+g_{21}x_{12})g_{21}/\det,$

where $\det=\det(g)$ and

$g=\begin{bmatrix}g_{11}&g_{12}\\\ g_{21}&g_{22}\end{bmatrix},\quad
X=\begin{bmatrix}x_{11}&x_{12}\\\ 0&x_{22}\end{bmatrix},\quad
Y=\begin{bmatrix}0&y_{12}\\\ 0&0\end{bmatrix}$

Thus we can define the following strongly equivariant Koszul matrix
factorization:

$\bar{\mathcal{C}}_{+}:=(\mathbb{C}[\overline{\mathcal{X}}(G_{2})]\otimes\Lambda\langle\theta\rangle,D,0,0)\in\mathrm{MF}_{B^{2}_{2}}(\overline{\mathcal{X}}(G_{2}),\overline{W}),$
$\quad
D=\frac{g_{12}y_{12}}{\det}\theta+\left[g_{11}(x_{11}-x_{22})+g_{21}x_{12}\right]\frac{\partial}{\partial\theta},$

where $\Lambda\langle\theta\rangle$ is the exterior algebra with one
generator.

This matrix factorization corresponds to the positive elementary braid on two
strands. The negative elementary braid is defined with twisting of the
$B^{2}_{2}$-action. We define:

$\bar{\mathcal{C}}_{-}:=\mathcal{C}_{+}\langle-\chi_{1},\chi_{2}\rangle.$

Using the induction functor we can extend the previous definition on the case
of the arbitrary number of strands. For that we introduce an insertion
functor:

$\overline{\textup{Ind}}_{k,k+1}:\mathrm{MF}_{B_{2}^{2}}(\overline{\mathcal{X}}(G_{2}),\overline{W})\rightarrow\mathrm{MF}_{B_{n}^{2}}(\overline{\mathcal{X}}(G_{n}),\overline{W})$
$\overline{\textup{Ind}}_{k,k+1}(\mathcal{F}):=\overline{\textup{ind}}_{k+1}(\overline{\textup{ind}}_{k-1}(\bar{\mathds{1}}_{k-1}\times\mathcal{F})\times\bar{\mathds{1}}_{n-k-1}),$

and similarly we define non-reduced insertion functor

$\textup{Ind}_{k,k+1}:\mathrm{MF}_{B_{2}^{2}}(\mathcal{X}(G_{2}),W)\rightarrow\mathrm{MF}_{B_{n}^{2}}(\mathcal{X}(G_{n}),W).$

Thus we define the generators of the braid group as follows:

$\bar{\mathcal{C}}_{\epsilon}^{(k)}:=\overline{\textup{Ind}}_{k,k+1}(\bar{\mathcal{C}}_{\epsilon}),\quad\mathcal{C}_{\epsilon}^{(k)}:=\textup{Ind}_{k,k+1}(\mathcal{C}_{\epsilon}).$

For an element $\beta\in\mathfrak{Br}_{n}$, if we can choose a presentation
$\beta=\sigma_{i_{1}}^{\epsilon_{1}}\dots\sigma_{i_{l}}^{\epsilon_{l}}$ then
the element
$\bar{\mathcal{C}}_{\beta}:=\bar{\mathcal{C}}_{\epsilon_{1}}^{(i_{1})}\bar{\star}\dots\bar{\star}\bar{\mathcal{C}}_{\epsilon_{l}}^{(i_{l})}$
does not depends on the presentation of the braid. Respectively, we use
notation $\mathcal{C}_{\beta}=\Phi_{\mathrm{kn}}(\bar{\mathcal{C}}_{\beta})$.
Respectively, the finite part $\Phi_{n}$ of the homomorphism
$\Phi_{n}^{\textup{aff}}$ from proposition 2.1.1 is defined by
$\Phi_{n}(\beta)=\mathcal{C}_{\beta}$.

###  3.5. Braid closure

The braid closure functor $\mathbb{L}$ is very natural in context of the
reduced spaces:

$\mathbb{L}:\overline{\mathrm{MF}_{n}^{\textup{st}}}\to\mathscr{D}_{B}(\left(\mathfrak{b}\times\mathfrak{n}\times\mathbb{C}^{n}\right)^{\textup{st}}),\quad\mathbb{L}(\bar{\mathcal{F}})=j_{e}^{*}(\bar{\mathcal{F}}).$

where $j_{e}:\left(\mathfrak{b}_{n}\times
1\times\mathfrak{n}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}}\to\left(\overline{\mathcal{X}}_{n}\times\mathbb{C}^{n}\right)^{\textup{st}}$
is the inclusion map.

As it is shown in [OR18c] the $B$ acts freely on
$(\mathfrak{b}\times\mathfrak{n}\times\mathbb{C}^{n})^{\textup{st}}$ and the
corresponding quotient is a smooth manifold that we call free flag Hilbert
scheme:

$\mathrm{FHilb}^{\mathrm{free}}_{n}=(\mathfrak{b}\times\mathfrak{n}\times\mathbb{C}^{n})^{\textup{st}}/B.$

Let $(X,Y,v)$ be coordinates on the product
$\mathfrak{b}\times\mathfrak{n}\times\mathbb{C}^{n}$ then similarly, to the
previous case we define super-group restriction functor:

$\mathscr{R}^{\prime}_{m|k}(\mathcal{C})=\mathcal{C}\otimes(\Lambda^{\star}\mathbb{C}^{n},d_{m|k}),\quad
d_{m|k}=\sum_{i=1}(X^{m}Y^{k}v)_{i}\frac{\partial}{\partial\theta_{i}}.$

The closure functor intertwines the functors $\mathscr{R}^{\prime}_{m|k}$ and
$\mathscr{R}_{m|k}$ because of base change relation:

$\mathscr{R}^{\prime}_{m|k}\circ\mathbb{L}\circ\Phi=\mathbb{L}\circ\Phi\circ\mathscr{R}_{m|k}.$

It is shown in [OR18b] that for any
$\mathcal{F}\in\mathrm{MF}^{\textup{st}}_{n}$ we have:

$\textup{Hom}(\mathcal{F},\mathcal{C}_{1})=\mathbb{H}(\mathsf{CH}^{\textup{st}}_{\textup{loc}}(\mathcal{F})).$

On the other hand in [OR18c] it was shown that

$\textup{Hom}(\Phi(\bar{\mathcal{F}}),\mathcal{C}_{1})=\mathrm{CE}_{\mathfrak{n}}(\mathbb{L}(\bar{\mathcal{F}}))^{T}.$

Combining all these observations we obtain another construction for the
homology from the introduction

$\mathrm{HH}_{m|k}(\beta)=\mathrm{CE}_{\mathfrak{n}}(\mathscr{R}^{\prime}_{m|k}\circ\mathbb{L}(\bar{\mathcal{C}}_{\beta}))^{T}.$

To complete our proof of the second Markov move we need to combine the above
description of the homology together with a technical lemma from [OR18c] (also
see [OR20] for further discussion of this lemma). To state the lemma we need
to set up some notations.

The $B^{2}_{n}$-equivariant projection

$\pi:\mathfrak{n}_{n}\times\mathfrak{b}_{n}\to\mathfrak{b}_{n-1}\times\mathfrak{b}_{n-1},$

that projects matrices to the subspace spanned by the matrix units $E_{ij}$,
$i,j>1$ descends to the projection $\pi$ between the free flag Hilbert scheme:
$\mathrm{FHilb}_{n}^{\mathrm{free}}\to\mathrm{FHilb}_{n-1}^{\mathrm{free}}$.
On the other hand the projection

$\pi_{\geq
k}:\mathfrak{n}_{n}\times\mathfrak{b}_{n}\to\mathfrak{n}_{k}\times\mathfrak{b}_{k}.$

that projects matrices to the subspace spanned by the matrix units $E_{ij}$,
$i,j\leq k$ does not descend to to the map between the free flag Hilbert
schemes. The failure is due to the fact that the stability condition is
incompatible with the projection. However, we can use this projection to
understand the homology of the graphs that appear in the the Markov move
theorem:

######  Lemma 3.5.1.

Let $\overline{\mathcal{F}}\in\overline{\mathrm{MF}}_{n-1}$,
$\overline{\mathcal{G}}\in\overline{\mathrm{MF}}_{2}$ and
$V=\mathbb{C}^{n-2}\langle\chi_{1}\rangle$ is a vector space with the
$B_{n}$-action given by the character $\chi_{1}=\exp(\epsilon_{1})$. Suppose
we have

$\mathbb{L}(\overline{\mathcal{F}})=(M,d)\in\mathrm{D}^{\textup{per}}_{B^{2}}(\mathfrak{n}_{n-1}\times\mathfrak{b}_{n-1})$

then there is a deformed $\mathbb{Z}_{2}$-graded complex

$\overline{\mathcal{F}}^{\prime}=(\pi^{*}M\otimes\Lambda^{even}V\oplus\pi^{*}M\otimes\Lambda^{odd}V,D_{V}+\pi^{*}d)\in\mathrm{D}^{\textup{per}}_{B}(\mathfrak{n}_{n}\times\mathfrak{b}_{n})$
$D_{V}:\leavevmode\hbox to325.97pt{\vbox
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such that

$\mathbb{L}\bigg{(}\overline{\textup{ind}}_{1}(\overline{\mathcal{F}})\star\overline{\mathrm{Ind}}_{1,2}(\overline{\mathcal{G}})\bigg{)}=\overline{\mathcal{F}}^{\prime}\otimes\pi^{*}_{\geq
2}(\mathbb{L}(\overline{\mathcal{G}})).$

###  3.6. Proof of the second Markov move

To show proposition 3.0.2 we apply the previous lemma for
$\bar{\mathcal{F}}=\bar{\mathcal{C}}_{\beta}$ and
$\bar{\mathcal{G}}=\bar{\mathcal{C}}_{\epsilon}$. First we observe that
$\mathbb{L}(\bar{\mathcal{C}}_{\epsilon})$ is a Koszul complex

$K(x_{11}-x_{22})\langle\frac{\epsilon-1}{2}\chi_{1}\rangle,\quad
K(x_{11}-x_{22})=[R\xrightarrow{x_{11}-x_{22}}R]$

here and everywhere below
$R=\mathbb{C}[(\mathfrak{b}\times\mathfrak{n}\times\mathbb{C}^{n})^{\textup{st}})]$.

Next, observe for any
$\mathcal{C}\in\mathscr{D}_{B}^{\textup{per}}((\mathfrak{b}_{n-1}\times\mathfrak{n}_{n-1}\times\mathbb{C}^{n})^{\textup{st}})$
we have

$\mathscr{R}^{\prime}_{m|k}(\pi^{*}(\mathcal{C}))=\pi^{*}(\mathscr{R}^{\prime}_{m|k}(\mathcal{C}))\otimes[R\to
R\langle\chi_{1}\rangle]$

The morphism $\pi$ is $B_{n}$-equivariant and we denote by $\hat{\pi}$ the
induced morphism on the quotients. The $B_{n}$-action on
$(\mathfrak{b}\times\mathfrak{n}\times\mathbb{C}^{n})^{\textup{st}}$ is free
and it shown in [OR18c] that the fibers of the projection

$\hat{\pi}:(\mathfrak{b}_{n}\times\mathfrak{n}_{n}\times\mathbb{C}^{n})^{\textup{st}}/B_{n}\to(\mathfrak{b}_{n-1}\times\mathfrak{n}_{n-1}\times\mathbb{C}^{n-1})^{\textup{st}}/B_{n-1}$

are products $\mathbb{C}\times\mathbb{P}^{n-1}$ where the coordinate along the
first component is $x_{11}-x_{22}$.

The functor $\mathrm{CE}_{\mathfrak{n}_{n}}(\cdot)^{T_{n}}$ is a composition
of two functors. The first functor is the push-forward along $\pi_{\geq
1}\times\hat{\pi}$ and the second functor is
$\mathrm{CE}_{\mathfrak{n}_{n-1}}(\cdot)^{T_{n-1}}$. Thus to complete our
proof we need to combine the projection formula for $(\pi_{\geq
1}\times\hat{\pi})_{*}$ with the computation of the push-forward $(\pi_{\geq
1}\times\pi)_{*}$ applied to

$(\Lambda^{*}V,D_{V})\otimes[R\to
R\langle\chi_{1}\rangle]\langle\frac{\epsilon-1}{2}\chi_{1}\rangle$

If $\epsilon=+$ the last push-forward is $\mathcal{O}$ since the only term of
the last tensor product that contribute non-trivially to the push-forward is
$\Lambda^{0}V\otimes R$. If $\epsilon=-$ the last push-forward is
$\mathcal{O}[n-1]$ since the only term of the last tensor product that
contribute non-trivially to the push-forward is $\Lambda^{n-2}V[n-2]\otimes
R[1]\langle\chi_{1}\rangle$.

##  4\. Evaluation of unknot and final remarks

###  4.1. Unknot

To complete proof of the main theorem we need to evaluate the homology of the
unknot. For that we observe that

$(\overline{\mathcal{X}}_{1}\times\mathbb{C})^{\textup{st}}=\mathbb{C}_{x}\times\mathbb{C}^{*}\times\\{0\\}\times\mathbb{C}^{*}_{v}.$

Since for $n=1$ we have $(\mathfrak{b}\times\mathfrak{n}\times
V)^{\textup{st}}=\mathbb{C}_{x}\times\mathbb{C}^{*}_{v}$,
$(\mathfrak{b}\times\mathfrak{n}\times
V)^{\textup{st}}/\mathbb{C}^{*}=\mathbb{C}_{x}$ the closure functor results in

$\mathbb{L}(\Phi_{1}(1))=\mathbb{C}[\mathbb{C}_{x}\times\mathbb{C}^{*}_{v}],\quad\mathcal{T}r(1)=\mathbb{C}[\mathbb{C}_{x}]=\mathbb{C}[x].$

The study the differential $D_{x^{m}y^{k}}$ we split on two cases. The first
case $k=0$ then homology $\textup{H}_{m}(1)$ is computed by the complex

$\mathbf{Q}^{m}\mathbf{T}\cdot\mathbb{C}[x]\xrightarrow{x^{m}}\mathbb{C}[x].$

In the second case $k\geq 0$ and the differential vanishes for $n=1$, that is
$\textup{H}_{m|k}(1)$ is homology of

$\mathbf{Q}^{m-k}\mathbf{T}\cdot\mathbb{C}[x]\xrightarrow{0}\mathbb{C}[x].$

To sum it up we have

$\dim_{\mathbf{Q},\mathbf{T}}\mathrm{HH}_{m}(1)=(\mathbf{Q}^{m}-1)/(\mathbf{Q}-1),\quad\dim_{\mathbf{Q},\mathbf{T}}\mathrm{HH}_{m|k}(1)=(\mathbf{T}\mathbf{Q}^{m-k}+1)/(1-\mathbf{Q}).$

Thus indeed, $\mathrm{HH}_{m|k}$ categorifies $\mathfrak{gl}(m|k)$ invariant
since

$\dim_{\mathbf{Q},\mathbf{T}=-1}\mathrm{HH}_{m|k}(1)=(1-\mathbf{Q}^{m-k})/(1-\mathbf{Q}).$

Let us also remark that in the case $m=k$ the last expression vanishes and
some renormalization is needed to obtain a homology theory that specializes to
the Alexander polynomial. We will that discussion for the future work. We also
expected the technology of dualizable homology from [OR19] provide a path to
proving a $qt$-symmetry conjecture.

###  4.2. Conjectures

We have shown that that the homology in the note decategorify to the
$\mathfrak{gl}(n|k)$ link invariant. In the case $k=0$ a categorification of
the knot invariant was constructed by Khovanov [Kho00] and Khovanov-Rozansky
homology $\mathrm{H}_{\mathfrak{gl}(n)}^{\mathrm{KhR}}$, [KR08a].

It is also shown in [OR20] that $\mathcal{T}r$ provides a categorification of
Jones-Ocneanu trace that matches the categorification by Khovanov [Kho07].
Thus we expect that a combination of the results from [OR20] and known results
on the relations between the Soergel category and the categories related to
Khovanov-Rozansky $\mathfrak{gl}(k)$-invariants would yield a proof of the
following

######  Conjecture 4.2.1.

For $k\geq 2$ we have for any $\beta\in\mathfrak{Br}_{n}$

$\mathrm{H}_{0|k}(\beta)=\mathrm{H}_{\mathfrak{gl}(k)}^{\mathrm{KhR}}(L(\beta)).$

In more details, the category related to the Khovanov-Rozansky
$\mathfrak{gl}(k)$-invariants is a quotient of the category of singular
Soergel bimodules $\mathrm{SBim}_{n}^{\mathrm{s}}$ by a subcategory generated
by objects

$B^{(r_{1},r_{2})}=R_{n}\otimes_{R_{m}^{S([r_{1},r_{2}])}}R_{n},\quad
r_{1}-r_{2}\geq k,$

where $R_{n}=\mathbb{C}[x_{1},\dots,x_{n}]$, $S([r_{1},r_{2}])\subset S_{n}$
is a the Young subgroup that fixes $1,\dots,r_{1}-1$ and $r_{2}+1,\dots,n$. On
the other hand in [OR20] we construct fully-faithful monoidal functor from
some natural subcategory
$\mathrm{MF}^{\flat}_{n}\subset\mathrm{MF}_{n}^{\textup{st}}$ category of
matrix factorizations to Soergel bimodules

$\mathbb{B}:\mathrm{MF}_{n}^{\flat}\to\mathrm{SBim}_{n}.$

This later category can be enlarged to category $\mathrm{MF}_{n}^{\flat}$ to
slightly larger category
$\mathrm{MF}_{n}^{\flat,\mathrm{s}}\subset\mathrm{MF}_{n}^{\textup{st}}$ so
that we could extend the functor $\mathbb{B}$ to the functor from
$\mathrm{MF}_{n}^{\flat,\mathrm{s}}$ to $\mathrm{SBim}_{n}^{\mathrm{s}}$. Let
us describe the objects that corresponding to the bimodules
$B^{(k_{1},k_{2})}$.

First, let us recall that used a category that is equivalent to the category
$\mathrm{MF}_{n}^{\textup{st}}$ in [OR20]:

$\mathrm{MF}_{n}^{\textup{st}}=\mathrm{MF}_{G\times B^{2}}((\mathfrak{g}\times
G\times\mathfrak{b}\times G\times\mathfrak{b}\times V_{B})^{\textup{st}},W),$

where the potential $W$ and the stability condition is the same as before, see
section 2.1.

The matrix factorizations
$\mathcal{C}_{\bullet}^{(1,r)}\in\mathrm{MF}_{r}^{\textup{st}}$ are defined as
Koszul matrix factorizations:

$(R\otimes\Lambda^{*}(\theta_{ij}),D,0,0),\quad
R=\mathbb{C}[(\mathfrak{g}\times G\times\mathfrak{b}\times
G\times\mathfrak{b}\times V_{B})^{\textup{st}}],$
$D=\sum_{ij}X_{ij}\frac{\partial}{\partial X_{ij}}+Z_{ij}\theta,\quad
W=\sum_{ij}X_{ij}Z_{ij},$

where we used coordinates $(X,g_{1},Y_{1},g_{2},Y_{2},v)$ on
$(\mathfrak{g}\times G\times\mathfrak{b}\times G\times\mathfrak{b}\times
V_{B})^{\textup{st}}$.

The objects $\mathcal{C}_{\bullet}^{(r_{1},r_{2})}$, $r_{2}-r_{1}=r-1$ are
obtained from $\mathcal{C}_{\bullet}^{(1,r)}$ by means of the induction
functors discussed in the section 3.3.

The key observation toward a proof of the conjecture is the following
statement for
$\mathcal{C}_{\bullet}^{(r_{1},r_{2})}\in\mathrm{MF}^{\flat,\mathrm{s}}_{n}$:

$\mathbb{B}\circ\mathcal{R}_{0|k}(\mathcal{C}_{\bullet}^{(r_{1},r_{2})})=0,\quad
r_{2}-r_{1}\geq k.$

It is a straight forward to check that the last equality is compatible with
the induction functors. Thus it is enough to check the statement for
$r_{1}=1,r_{2}=n$. So let us recall a construction of the functor $\mathbb{B}$
from [OR20]:

$\mathbb{B}(\mathcal{C})=\pi_{*}(j_{x=0}^{*}(\mathcal{C})^{G\times B^{2}}),$

where $j_{x=0}$ is the embedding of $((G\times\mathfrak{b})^{2}\times
V_{B})^{\textup{st}}$ inside
$(\mathfrak{g}\times(G\times\mathfrak{b})^{2}\times V_{B})^{\textup{st}}$ and
$\pi$ is the projection from $((G\times\mathfrak{b})^{2}\times
V_{B})^{\textup{st}}$ to $\mathfrak{h}^{2}$.

Thus we the pull back along $j_{x=0}$ results into the Koszul complex:

$j^{*}_{x=0}(\mathcal{C}_{\bullet}^{(1,n)})=K[Z_{ij}].$

Since $Z_{ij}=(\textup{Ad}_{g_{1}}Y_{1}-\textup{Ad}_{g_{2}}Y_{2})$, to
complete our computations of $\mathbb{B}(\mathcal{C}_{\bullet}^{(1,n)})$ we
need to $G\times B^{2}$ invariant part of

$i_{z=0}^{*}((\Lambda^{*}V_{B},d_{0|k})),\quad
i_{z=0}:\\{Z_{ij}=0\\}\to((G\times\mathfrak{b})^{2}\times
V_{B})^{\textup{st}}.$

The $G\times B^{2}$-action on the subspace defined by the equations $Z_{ij}=0$
is free and a slice to the action has the following parametrization:

$g_{1}=g_{2}=1,\quad Y_{1}=Y_{2}=\mathrm{diag}(\vec{y})+J,\quad v=e_{n},$

where $(e_{n})_{i}=\delta_{n-i}$ and $J_{ij}=\delta_{j-i-1}$.

On the slice the differential $d_{0|k}$ defines the complete intersection
variety defined by the equation $Y_{1}^{k}v=0$. Finally, we can observe that
$(\mathrm{diag}(\vec{y})+J)^{k}_{n-k,n}=1$ hence the equation $Y_{1}^{k}v=0$
cuts out empty set on the slice.

Let us also remark that in [OR19] we prove $q\to-q/t$ symmetry of HOMFLYPT
homology. We expect that the method of the mentioned paper yields a proof of
the following

######  Conjecture 4.2.2.

Let $\pi_{0}(L(\beta))=1$ then

$\mathrm{HH}_{m|m}(\beta)=\mathrm{HH}_{m|m}(\beta)|_{\mathbf{Q}\to\mathbf{T}/\mathbf{Q}}.$

## References

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$\mathsf{vars}({q})$ and maps additionally $\mu^{+}(s_{k+1})=c$ and
$\mu^{+}(s_{k+2})=d$. We thus have $\mu^{+}(p)\subseteq\mathbf{r}$. It is
correct to conclude that $\mathbf{db}\cup\\{N(\underline{c},d)\\}$ is a
“yes”-instance for $\mathsf{CERTAINTY}({p})$.

$\impliedby$ Assume that $\mathbf{db}\cup\\{N(\underline{c},d)\\}$ is a
“yes”-instance for the problem $\mathsf{CERTAINTY}({p})$. Let $\mathbf{r}$ be
any repair of $\mathbf{db}$. Then $\mathbf{r}\cup\\{N(\underline{c},d)\\}$ is
a repair of $\mathbf{db}\cup\\{N(\underline{c},d)\\}$, and thus there exists
some valuation $\theta$ with
$\theta(p)\subseteq\mathbf{r}\cup\\{N(\underline{c},d)\\}$. Since
$\mathbf{db}$ contains only one $N$-fact, we have $\theta(s_{k+1})=c$. It
follows that $\theta(q)\subseteq\mathbf{r}$, as desired. ∎

######  Lemma 27.

Let $q$ be a generalized path query with

$q=\\{R_{1}(\underline{s_{1}},s_{2}),R_{2}(\underline{s_{2}},s_{3}),\dots,R_{k}(\underline{s_{k}},s_{k+1})\\}$

where $s_{1}$ is a constant, and each $s_{i}$ is either a constant or a
variable for all $i\in\\{2,\dots,k+1\\}$. Then the problem
$\mathsf{CERTAINTY}({q})$ is in $\mathbf{FO}$.

###### Proof.

Let the $1=j_{1}<j_{2}<\dots<j_{\ell}\leq k+1$ be all the indexes $j$ such
that $s_{j}$ is a constant for some $\ell\geq 1$. Let $j_{\ell+1}=k+1$. Then
for each $i\in\\{1,2,\dots,\ell\\}$, the query

$q_{i}=\bigcup_{j_{i}\leq j<j_{i+1}}\\{R_{j}(\underline{s_{j}},s_{j+1})\\}$

is a generalized path query where each $s_{j_{i}}$ is a constant.

We claim that $\mathsf{CERTAINTY}({q_{i}})$ is in $\mathbf{FO}$ for each
$1\leq i\leq\ell$. Indeed, if $s_{j_{i+1}}$ is a variable, then the claim
follows by Lemma 12; if $s_{j_{i+1}}$ is a constant, then the claim follows by
Lemma 26 and Lemma 12.

Since by construction, $q=q_{1}\cup q_{2}\cup\dots\cup q_{\ell}$, we conclude
that $\mathsf{CERTAINTY}({q})$ is in $\mathbf{FO}$ by Lemma 25. ∎

The proof of Lemma 21 is now simple.

###### Proof of Lemma 21.

If $q$ contains no constants, the lemma holds trivially. Otherwise,
$\mathsf{CERTAINTY}({p})$ is in $\mathbf{FO}$ by Lemma 27. ∎

###  B.2 Elimination of Constants

In this section, we show how constants can be eliminated from generalized path
queries. The _extended query_ of a generalized path query is defined next.

######  Definition 22 (Extended query).

Let $q$ be a generalized path query. The _extended query_ of $q$, denoted by
${\mathsf{ext}}({q})$, is defined as follows:

  * •

if $q$ does not contain any constant, then ${\mathsf{ext}}({q})\vcentcolon=q$;

  * •

otherwise, ${\mathsf{char}}({q})=\\{R_{1}(\underline{x_{1}},x_{2})$,
$R_{2}(\underline{x_{2}},x_{3})$, …, $R_{\ell}(\underline{x_{\ell}},c)\\}$ for
some constant $c$. In this case, we define

${\mathsf{ext}}({q})\vcentcolon=\\{R_{1}(\underline{x_{1}},x_{2}),\dots,R_{\ell}(\underline{x_{\ell}},x_{\ell+1}),N(\underline{x_{\ell+1}},x_{\ell+2})\\},$

where $x_{\ell+1}$ and $x_{\ell+2}$ are fresh variables and $N$ is a fresh
relation name not occurring in $q$. ∎

By definition, ${\mathsf{ext}}({q})$ does not contain any constant.

######  Example 10.

Let
$q=R(\underline{x},y),S(\underline{y},0),T(\underline{0},1),R(\underline{1},w)$
where $0$ and $1$ are constants. We have
${\mathsf{ext}}({q})=R(\underline{x},y),S(\underline{y},z),N(\underline{z},u)$.
∎

We show two lemmas which, taken together, show that the problem
$\mathsf{CERTAINTY}({q})$ is first-order reducible to
$\mathsf{CERTAINTY}({{\mathsf{ext}}({q})})$, for every generalized path query
$q$.

######  Lemma 28.

For every generalized path query $q$, there is a first-order reduction from
$\mathsf{CERTAINTY}({q})$ to $\mathsf{CERTAINTY}({{\mathsf{char}}({q})})$.

###### Proof.

Let $p\vcentcolon=q\setminus{\mathsf{char}}({q})$. Since
$\mathsf{vars}({{\mathsf{char}}({q})})\cap\mathsf{vars}({p})=\emptyset$,
Lemmas 25 and 27 imply that the following are equivalent for every database
instance $\mathbf{db}$:

  1. 1.

$\mathbf{db}$ is a “yes”-instance for $\mathsf{CERTAINTY}({q})$; and

  2. 2.

$\mathbf{db}$ is a “yes”-instance for
$\mathsf{CERTAINTY}({{\mathsf{char}}({q})})$ and a “yes”-instance for
$\mathsf{CERTAINTY}({p})$.

To conclude the proof, it suffices to observe that $\mathsf{CERTAINTY}({p})$
is in $\mathbf{FO}$ by Lemma 27. ∎

######  Lemma 29.

For every generalized path query $q$, there is a first-order reduction from
$\mathsf{CERTAINTY}({{\mathsf{char}}({q})})$ to
$\mathsf{CERTAINTY}({{\mathsf{ext}}({q})})$.

###### Proof.

Let $q$ be a generalized path query. If $q$ contains no constants, the lemma
trivially obtains because ${\mathsf{char}}({q})={\mathsf{ext}}({q})=q$. If $q$
contains at least one constant, then there exists a first-order reduction from
$\mathsf{CERTAINTY}({{\mathsf{char}}({q})})$ to
$\mathsf{CERTAINTY}({{\mathsf{ext}}({q})})$ by Lemma 26. ∎

###  B.3 Complexity Upper Bounds in Theorem 4

######  Lemma 30.

Let $q$ be a generalized path query that contains at least one constant. If
$q$ satisfies $\mathcal{D}_{3}$, then $q$ satisfies $\mathcal{D}_{2}$ and
${\mathsf{ext}}({q})$ satisfies $\mathcal{C}_{2}$.

###### Proof.

Assume that $q$ satisfies $\mathcal{D}_{3}$. Let
${\mathsf{char}}({q})=[\\![{p},{c}]\\!]$ for some constant $c$. We have
${\mathsf{ext}}({q})=p\cdot N$ where $N$ is a fresh relation name not
occurring in $p$.

We first argue that ${\mathsf{ext}}({q})$ is a factor of every word to which
${\mathsf{ext}}({q})$ rewinds. To this end, let ${\mathsf{ext}}({q})=uRvRwN$
where $p=uRvRw$. Since $q$ satisfies $\mathcal{D}_{3}$, there exists a
homomorphism from ${\mathsf{char}}({q})=[\\![{uRvRw},{c}]\\!]$ to
$[\\![{uRvRvRw},{c}]\\!]$, implying that $uRvRw$ is a suffix of $uRvRvRw$. It
follows that $uRvRwN$ is a suffix of $uRvRvRwN$. Hence ${\mathsf{ext}}({q})$
satisfies $\mathcal{C}_{3}$.

The remaining test for $\mathcal{C}_{2}$ is where
${\mathsf{ext}}({q})=uRv_{1}Rv_{2}RwN$ for consecutive occurrences of $R$. We
need to show that either $v_{1}=v_{2}$ or $RwN$ is a prefix of $Rv_{1}$ (or
both). We have $p=uRv_{1}Rv_{2}Rw$. Since $q$ satisfies $\mathcal{D}_{3}$,
there exists a homomorphism from
${\mathsf{char}}({q})=[\\![{uRv_{1}Rv_{2}Rw},{c}]\\!]$ to
$[\\![{uRv_{1}Rv_{2}Rv_{2}Rw},{c}]\\!]$. Since $c$ is a constant, the
homomorphism must map $Rv_{1}$ to $Rv_{2}$, implying that $v_{1}=v_{2}$. It is
correct to conclude that $q$ satisfies $\mathcal{D}_{2}$ and
${\mathsf{ext}}({q})$ satisfies $\mathcal{C}_{2}$. ∎

######  Lemma 31.

For every generalized path query $q$,

  * •

if $q$ satisfies $\mathcal{D}_{1}$, then ${\mathsf{ext}}({q})$ satisfies
$\mathcal{C}_{1}$;

  * •

if $q$ satisfies $\mathcal{D}_{2}$, then ${\mathsf{ext}}({q})$ satisfies
$\mathcal{C}_{2}$; and

  * •

if $q$ satisfies $\mathcal{D}_{3}$, then ${\mathsf{ext}}({q})$ satisfies
$\mathcal{C}_{3}$.

###### Proof.

The lemma holds trivially if $q$ contains no constant. Assume from here on
that $q$ contains at least one constant.

Assume that $q$ satisfies $\mathcal{D}_{1}$. Then ${\mathsf{char}}({q})$ must
be self-join-free. In this case, ${\mathsf{ext}}({q})$ is self-join-free, and
thus ${\mathsf{ext}}({q})$ satisfies $\mathcal{C}_{1}$.

For the two remaining items, assume that $q$ satisfies $\mathcal{D}_{2}$ or
$\mathcal{D}_{3}$. Since $\mathcal{D}_{2}$ logically implies
$\mathcal{D}_{3}$, $q$ satisfies $\mathcal{D}_{3}$. By Lemma 30,
${\mathsf{ext}}({q})$ satisfies $\mathcal{C}_{2}$. Since $\mathcal{C}_{2}$
logically implies $\mathcal{C}_{3}$, $q$ satisfies $\mathcal{C}_{3}$. ∎

We can now prove the upper bounds in Theorem 4.

###### Proof of upper bounds in Theorem 4.

Since first-order reductions compose, by Lemmas 28 and 29, there is a first-
order reduction from the problem $\mathsf{CERTAINTY}({q})$ to
$\mathsf{CERTAINTY}({{\mathsf{ext}}({q})})$. The upper bound results then
follow by Lemma 31. ∎

###  B.4 Complexity Lower Bounds in Theorem 4

The complexity lower bounds in Theorem 4 can be proved by slight modifications
of the proofs in Sections 7.1 and 7.2. We explain these modifications below
for a generalized path query $q$ containing at least one constant. Note
incidentally that the proof in Section 7.3 needs no revisiting, because, by
Lemma 30, a violation of $\mathcal{D}_{2}$ implies a violation of
$\mathcal{D}_{3}$.

In the proof of Lemma 18, let ${\mathsf{char}}({q})=[\\![{uRvRw},{c}]\\!]$
where $c$ is a constant and there is no prefix homomorphism from
${\mathsf{char}}({q})$ to $[\\![{uRvRvRw},{c}]\\!]$. Let
$p=q\setminus{\mathsf{char}}({q})$. Note that the path query $uRv$ does not
contain any constant. We revise the reduction description in Lemma 18 to be

  * •

for each vertex $x\in V\cup\\{s^{\prime}\\}$, we add $\phi_{\bot}^{x}[u]$;

  * •

for each edge $(x,y)\in E\cup\\{(s^{\prime},s),(t,t^{\prime})\\}$, we add
$\phi_{x}^{y}[Rv]$;

  * •

for each vertex $x\in V$, we add $\phi^{c}_{x}[Rw]$; and

  * •

add a canonical copy of $p$ (which starts in the constant $c$).

An example is shown in Figure 11. Since the constant $c$ occurs at most twice
in $q$ by Definition 16, the query $q$ can only be satisfied by a repair
including each of $\phi_{\bot}^{x}[u]$, $\phi_{x}^{y}[Rv]$,
$\phi^{c}_{y}[Rw]$, and the canonical copy of $p$. $\mathbf{NL}$-hardness can
now be proved as in the proof of Lemma 18.

$s^{\prime}$$s$$a$$t$$t^{\prime}$$c$$u$$u$$u$$u$$Rw$$Rw$$Rw$$p$$Rv$$Rv$$Rv$$Rv$
Figure 11: Database instance for the revised $\mathbf{NL}$-hardness reduction
from the graph $G$ with $V=\\{s,a,t\\}$ and $E=\\{(s,a),(a,t)\\}$.
$(x_{2}\vee\overline{x_{3}})$$(x_{1}\vee\overline{x_{2}})$$x_{3}$$x_{2}$$x_{1}$$c$$u$$uRv$$u$$uRv$$p$$+$$Rw$$-$$RvRw$$+$$Rw$$-$$RvRw$$+$$Rw$$-$$RvRw$
Figure 12: Database instance for the revised $\mathbf{coNP}$-hardness
reduction from the formula
$\psi=(x_{1}\vee\overline{x_{2}})\wedge(x_{2}\vee\overline{x_{3}})$.

In the proof of Lemma 19, let ${\mathsf{char}}({q})=[\\![{uRvRw},{c}]\\!]$
where $c$ is a constant and there is no homomorphism from
${\mathsf{char}}({q})$ to $[\\![{uRvRvRw},{c}]\\!]$. Let
$p=q\setminus{\mathsf{char}}({q})$. Note that both path queries $uRv$ and $u$
do not contain any constant. We revise the reduction description in Lemma 19
to be

  * •

for each variable $z$, we add $\phi_{z}^{c}[Rw]$ and $\phi_{z}^{c}[RvRw]$;

  * •

for each clause $C$ and positive literal $z$ of $C$, we add $\phi_{C}^{z}[u]$;

  * •

for each clause $C$ and variable $z$ that occurs in a negative literal of $C$,
we add $\phi_{C}^{z}[uRv]$; and

  * •

add a canonical copy of $p$ (which starts in the constant $c$).

An example is shown in Figure 12. Since the constant $c$ occurs at most twice
in $q$, the query $q$ can only be satisfied by a repair $\mathbf{r}$ such that
either

  * •

$\mathbf{r}$ contains $\phi_{C}^{z}[uRv]$, $\phi_{z}^{c}[Rw]$, and the
canonical copy of $p$; or

  * •

$\mathbf{r}$ contains $\phi_{C}^{z}[u]$, $\phi_{z}^{c}[RvRw]$, and the
canonical copy of $p$.

$\mathbf{coNP}$-hardness can now be proved as in the proof of Lemma 19.

###  B.5 Proof of Theorem 5

###### Proof of Theorem 5.

Immediate consequence of Theorem 4 and Lemma 30. ∎