aainstitutetext: Department of Mathematics, Uppsala University, Box 480, 751
06 Uppsala, Swedenbbinstitutetext: Institut Mittag-Leffler, Aurav. 17, 182 60
Djursholm, Swedenccinstitutetext: Walter Burke Institute for Theoretical
Physics, California Institute of Technology,
Pasadena, CA 91125, USAddinstitutetext: Faculty of Physics, University of
Warsaw, ul. Pasteura 5, 02-093 Warsaw, Polandeeinstitutetext: Institute for
Theoretical Physics, ETH Zurich, CH - 8093, Zurich, Switzerland

# Knot homologies and generalized quiver partition functions

Tobias Ekholm c,d , Piotr Kucharski e , and Pietro Longhi
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We introduce generalized quiver partition functions of a knot $K$ and
conjecture a relation to generating functions of symmetrically colored HOMFLY-
PT polynomials and corresponding HOMFLY-PT homology Poincaré polynomials. We
interpret quiver nodes as certain basic holomorphic disks in the resolved
conifold, with boundary on the knot conormal $L_{K}$, a positive multiple of a
unique closed geodesic, and with their (infinitesimal) boundary linking
density measured by the adjacency matrix of the generalized quiver. The basic
holomorphic disks that are quiver nodes appear in a certain $U(1)$-symmetric
configuration. We propose an extension of the quiver partition function to
arbitrary, not $U(1)$-symmetric, configurations as a function with values in
chain complexes. The chain complex differential is trivial at the
$U(1)$-symmetric configuration, under deformations the complex changes, but
its homology remains invariant. We also study recursion relations for the
partition functions connected to knot homologies. We show that, after a
suitable change of variables, any (generalized) quiver partition function
satisfies the recursion relation of a single toric brane in $\mathbb{C}^{3}$.

##  1 Introduction

Polynomial knot invariants such as the Jones and HOMFLY-PT polynomials,
originally defined combinatorially, have been interpreted and further
explained from physical and geometric points of view. In physics the
invariants appear through quantum field theory (Chern-Simons theory)
witten1989 , topological strings and M-theory in combination with the conifold
transition Ooguri:1999bv , and in geometry through Gromov-Witten counts of
bare curves ES ; ES2 . Many polynomial knot invariants admit
categorifications, where the polynomial is expressed as the (graded) Euler
characteristic of a chain complex associated to the knot, with homology which
is a knot invariant. The first example is Khovanov’s categorification of the
Jones polynomial Khovanov . Connections between categorified knot invariants
and BPS states in the physical theories underlying the original knot
polynomials have been proposed, see e.g. GSV0412 ; GS1112 .

Following Ekholm:2018eee ; Ekholm:2019lmb , we study connections between knot
invariants and quivers as in Kucharski:2017poe ; Kucharski:2017ogk , where the
motivic generating function of a quiver gives the generating series for
HOMFLY-PT polynomials from the perspective of BPS counts. Here quiver nodes
are basic BPS states, their interactions are governed by the quiver arrows,
and the interacting nodes generate the whole spectrum. Geometrically, basic
BPS states correspond to M2-branes wrapping basic holomorphic disks in the
resolved conifold with boundary on an M5-brane wrapping the knot conormal, and
the quiver adjacency matrix encodes boundary linking data.

The main theme of this paper is a conjectural extension of the correspondence
between knot invariants and quivers to the categorified level, see (64). The
extension involves new types of quiver nodes that geometrically correspond to
certain ‘stretched’ (non-embedded near the boundary) holomorphic disks and a
generalization of the quiver motivic generating function that we conjecture
gives the generating series of the Poincaré polynomial of symmetrically
colored HOMFLY-PT homology, see section 3.5. This conjecture updates
(Ekholm:2018eee, , Conjecture 1.1), see remark 3.9 for details about the
relation and sections 3.4 and 3.5 for motivations. The basic disks that appear
as quiver nodes arise in a (possibly degenerate) $U(1)$-symmetric situation,
in line with earlier proposals originating from string dualities
Aganagic:2011sg .

We then consider deformations that break the $U(1)$ symmetric configuration,
and propose in section 3.6 an extension of the generalized quiver partition
function. We first show that the moduli spaces of stretched basic disks can be
carried along generic 1-parameter families of complex structures that remain
stretched near the Lagrangian, so that these moduli spaces together with
linking information suffices to compute the refined partition function. We
also perform a bifurcation analysis of such deformations. We generalize the
quiver partition function to a function with values in chain complexes, and
show that at any generic parameter in a generic path of deformations there are
differentials on these chain complexes with homology that remains invariant.
Here, for higher symmetric colorings, the chain complexes correspond to
certain subspaces of the homology which nevertheless contain sufficient
information to recover all of the homology, see section 3.6.2 for details.

In the light of the conjectured connection between HOMFLY-PT homology and
generalized quivers, one might expect that structures in one of the theories
are reflected in the other. We first consider structures in HOMFLY-PT homology
that we expect originate from the geometry of the basic disks in the
$U(1)$-symmetric configuration. There are $\mathfrak{sl}_{N}$ differentials
$d_{N}$, which act on the HOMFLY-PT homology with resulting homology being
$\mathfrak{sl}_{N}$ Kohovanov-Rozansky homology. The $\mathfrak{sl}_{1}$
homology is very simple, it has rank one for every knot (in reduced
normalization). We conjecture that this structure is reflected on the level of
generalized quivers and that the quiver of a knot can be obtained from one
‘spectator’ node and a quiver of half the size of the original quiver together
with a ‘universal disk’ that comes from the closed sector and carries the
quantum numbers of the $\mathfrak{sl}_{1}$-differential $d_{1}$. The
application of the multi-cover skein unlinking operation of Ekholm:2019lmb  to
a basic once-around disk and the universal disk gives a pair of generators
connected by $d_{1}$. This pair corresponds to two holomorphic curves which
differ by a unit of winding around the base $\mathbb{C}{\mathbb{P}}^{1}$ of
the resolved conifold, where the heavier disk can be viewed as a bound state
of the lighter and the universal disk. The connection between reduced and
unreduced homologies seems to be reflected in the structure of the generalized
quiver in a similar way.

We also make predictions for the general structure of HOMFLY-PT homology
colored by symmetric representations with $r$ boxes. We view the generators of
colored HOMFLY-PT homology in the $r$-th symmetric representation as
equivariant vortices of vorticity $r$ in the theory $T[L_{K}]$, see section 3.
The connection between quiver nodes and equivariant vortices is somewhat
involved: a basic curve on a knot conormal with boundary that wraps $k$ times
around the circle in the conormal gives $2^{k-1}$ generators in the $k$-th
symmetrically colored homology. At level $r=1$ every generator corresponds to
a node of the generalized quiver. The quiver partition function describes a
tower of contributions from $r=1$ generators to all higher levels with $r>1$,
which is completely determined by homology data of these nodes and their
mutual linking. Knowing the structure of these contributions from level $1$ to
level $2$ allows to separate them from the genuinely new generators that
appear on level $2$. We continue in this way inductively: taking out
contributions from all nodes of levels $<r$ allows us to identify the
genuinely new generators on level $r$. We verify the claim about homology
contributions of level $r$ generators against conjectural expressions for
homologies of knots $9_{42}$ and $10_{132}$ for $r=2$, see section 6.

Further structures appear when we deform away from the $U(1)$-symmetric
configuration. Here the pure level $r$ generators generate a chain complex of
‘Bott-equivariant’ vortices, see (54), which are transformed by chain homotopy
under deformation. The homology of the level $r$ generators is non-vanishing
only for finitely many $r$ and, together with linking data, it recovers the
usual $U(1)$-symmetric HOMFLY-PT homology.

Generating series for knot polynomials satisfy polynomial recursion relations.
Geometrically, such relations originate from counts of punctured holomorphic
curves at infinity with boundary on the knot conormal and asymptotic to Reeb
chords at punctures, see Ekholm:2018iso ; ES2 . From the viewpoint of the
quiver or the basic disks without punctures, the recursion relation is
generated by similar relations localized around individual basic disks. This
leads to an algebraic description of the ensemble of basic disks in terms of
noncommutative variables, see Ekholm:2019lmb . Here we take this further and
show that after a suitable change of noncommutative variables the recursion
relation of _any_ quiver _is_ the recursion relation of the basic toric brane
in $\mathbb{C}^{3}$. We generalize this simple recursion relation to include
also the new basic curves discussed above and look at the implications of such
a relation for both knot polynomials and their categorifications. As the toric
brane has a unique embedded holomorphic disk ending on it, this universal
relation to the toric brane is in line with $U(1)$-symmetric configurations of
knot conormals, where all holomorphic curves have boundaries which are
multiples of a unique simple closed geodesic.

As already mentioned, the form of generalized quiver partition functions that
we introduce for multiply wrapped basic disks is conjectural. We check it
against the few available (also conjectural) results for knot homologies of
$9_{42}$ and $10_{132}$ and it goes without saying that further tests are
important. In particular, calculations of colored knot homologies of knots
with more than eight crossings would provide support or indicate possible
changes to our conjecture. In this context we point out that our conjecture
has both a structural and a technical aspect. The structural aspect says that
the basic generalized holomorphic disks (generalized quiver nodes) and their
boundary linking densities (weighted quiver arrows) contain all information
about symmetrically colored HOMOFLY-PT homology. The technical aspect gives
the ‘change of variables’ for extracting this information. It is thus possible
that the structural part is the correct, even if the technical part needs
modification. As discussed above, there are knots for which standard quiver
partition functions cannot reproduce HOMFLY-PT homologies and, therefore, a
generalization is needed. Our proposal is in a sense a minimal extension,
compatible with available data and with the geometry of holomorphic curves
under SFT-stretching.

##  2 Background

In this section we review earlier results on knot polynomials and knot
homologies and their connections to quivers and (refined) curve counting. This
is the starting point for our study in later sections.

###  2.1 Knot polynomials and homologies

If $K\subset S^{3}$ is a framed knot, then its framed _HOMFLY-PT polynomial_
$P(K;a,q)$ freyd1985 ; PT  is a two-variable polynomial that can be calculated
from a knot diagram (a projection of $K$ with over/under information at
crossings and framing given by the projection direction) via the framed skein
relation, see figure 1.

Figure 1: The framed skein relation

The polynomial is an invariant of framed knots up to framed isotopy. The
standard HOMFLY-PT polynomial of a knot is the framed HOMFLY-PT of that knot
when equipped with the framing for which its self-linking number vanishes. For
$a=q^{2}$, the HOMFLY-PT reduces to the Jones polynomial $J(K;q)$ Jon85 , and
for $a=q^{N}$ to the $\mathfrak{sl}_{N}$ Jones polynomial $J_{N}(K;q)$ RT90 .

More generally, the  _colored_ HOMFLY-PT polynomials $P_{R}(K;a,q)$ are
similar polynomial knot invariants depending also on a representation $R$ of
the Lie algebra ${\mathfrak{u}}(N)$. In this setting, the original HOMFLY-PT
corresponds to the standard representation. Also the colored version admits a
diagrammatic description: it is given by a linear combination of the standard
polynomial of certain satellite links of $K$. From the physical point of view,
the colored HOMFLY-PT polynomial with $a=q^{N}$ is the expectation value of
the knot viewed as a Wilson line in $U(N)$ Chern-Simons gauge theory on
$S^{3}$ witten1989 . In order to simplify the notation, we will write the
HOMFLY-PT polynomial also when we refer to the more general colored version.

HOMFLY-PT polynomials also have an interpretation in terms of open topological
string or holomorphic curve counts. Physically, $P(K;a,q)$ can be interpreted
as the partition function of open topological string of the Lagrangian
conormal of the knot after transition from $T^{\ast}S^{3}$ to the resolved
conifold, where $q=e^{g_{s}}$ and $a$ is the Kähler parameter in the resolved
conifold Ooguri:1999bv . From the mathematical point of view, this results
from invariant counts of open holomorphic curves by the values of their
boundaries in the skein module of the Lagrangian, see ES ; ES2 .

We next consider knot homologies. In Khovanov , Khovanov introduced a knot
invariant which is a far-reaching generalization of the Jones polynomial. To
any knot $K$ he associated a doubly-graded chain complex, the homology of
which is a knot invariant and such that the Jones polynomial arises as its
(graded) Euler characteristic:

$J(K;q)=\sum_{i,j}(-1)^{j}q^{i}\dim\mathcal{H}^{\mathfrak{sl}_{2}}_{i,j}(K)\,.$
(1)

In this sense $\mathcal{H}^{\mathfrak{sl}_{2}}_{i,j}(K)$ is a categorification
of $J(K;q)$. Khovanov-Rozansky homology, which categorifies the
$\mathfrak{sl}_{N}$ polynomial, was defined in KhR1 :

$J_{N}(K;q)=\sum_{i,j}(-1)^{j}q^{i}\dim\mathcal{H}^{\mathfrak{sl}_{N}}_{i,j}(K)\,.$
(2)

In this paper we focus mostly on HOMFLY-PT homology KhR2 , which is a
categorification of the (original) HOMFLY-PT polynomial:

$P(K;a,q)=\sum_{i,j,k}(-1)^{k}a^{i}q^{j}\dim\mathcal{H}_{i,j,k}(K)\,.$ (3)

The corresponing Poincaré polynomial then provides a $t$-refinement of HOMFLY-
PT polynomial called the  _superpolynomial_ DGR0505 :

$P(K;a,q,t)=\sum_{i,j,k}a^{i}q^{j}t^{k}\dim\mathcal{H}_{i,j,k}(K)\,,$ (4)

with $P(K;a,q,-1)=P(K,a,q)$. Categorifications of colored HOMFLY-PT
polynomials were considered in GS1112 , where it was conjectured that there
exists a colored HOMFLY-PT homology, which is invariant under isotopy and such
that

$\begin{split}P_{R}(K;a,q)&=\sum_{i,j,k}(-1)^{k}a^{i}q^{j}\dim\mathcal{H}_{R;i,j,k}(K)\,,\\\
P_{R}(K;a,q,t)&=\sum_{i,j,k}a^{i}q^{j}t^{k}\dim\mathcal{H}_{R;i,j,k}(K)\,.\end{split}$
(5)

There are also colored Khovanov ($\mathcal{H}^{\mathfrak{sl}_{2}}_{R}$) and
Khovanov-Rozansky ($\mathcal{H}^{\mathfrak{sl}_{N}}_{R}$) homologies, which
categorify $J_{R}(K;q)$ and $J_{N,R}(K;q)$ respectively. We point out that
there are many constructions of colored HOMFLY-PT and $\mathfrak{sl}_{N}$
homologies. They do not always give the same results and some work only in
special cases (e.g., for $\Lambda^{r}$ representations), see Cau1611 ; RW1702
; ETW1703 ; QRS1802 ; GW1904 ; OR2010 ; GHM2103  and references therein. Here
we focus on symmetric representations $R=S^{r}$, corresponding to Young
diagrams with a single row of $r$ boxes, and we will write $P_{r}$ and
$\mathcal{H}_{r}$ instead of $P_{S^{r}}$ and $\mathcal{H}_{S^{r}}$ to simplify
the notation. (For example, $\mathcal{H}_{1}$ will denote the original HOMFLY-
PT homology corresponding to the standard representation.)

###  2.2 Observed relations between knots and quivers

In Kucharski:2017ogk ; Kucharski:2017poe , knot polynomials were related with
representations of quivers. A quiver $Q$ is an oriented graph with a finite
number of vertices connected by finitely many signed arrows. We denote the set
of vertices by $Q_{0}$ and the set of arrows by $Q_{1}$. A  _dimension vector_
for $Q$ is a vector in the integral lattice with basis $Q_{0}$,
$\boldsymbol{d}\in{\mathbb{N}}Q_{0}$. We number the vertices of $Q$ by
$1,2,\dots,m=|Q_{0}|$. A quiver representation with dimension vector
$\boldsymbol{d}=(d_{1},\dots,d_{m})$ is the assignment of a vector space of
dimension $d_{i}$ to the node $i\in Q_{0}$ and of a linear map
$\gamma_{ij}\colon{\mathbb{C}}^{d_{i}}\to{\mathbb{C}}^{d_{j}}$ to each arrow
from vertex $i$ to vertex $j$. The  _adjacency matrix_ of $Q$ is the $m\times
m$ integer matrix with entries $C_{ij}$ equal to the algberaic number of
arrows from $i$ to $j$. A quiver is symmetric if its adjacency matrix is.

Quiver representation theory studies moduli spaces of stable quiver
representations (see e.g., kirillov2016quiver  for an introduction to this
subject). While explicit expressions for invariants describing those spaces
are hard to find in general, they are quite well understood in the case of
symmetric quivers Kontsevich:2008fj ; KS1006 ; 2011arXiv1103.2736E . Important
information (such as the intersection homology Betti numbers of the moduli
space of all semi-simple representations of $Q$ of dimension vector
$\boldsymbol{d}$, see MR1411 ; FR1512 ) about the moduli space of
representations of a symmetric quiver with trivial potential is encoded in the
_motivic generating series_ defined as

$P_{Q}(\boldsymbol{x};q)=\sum_{d_{1},\ldots,d_{m}\geq 0}(-q)^{\sum_{1\leq
i,j\leq
m}C_{ij}d_{i}d_{j}}\prod_{i=1}^{m}\frac{x_{i}^{d_{i}}}{(q^{2};q^{2})_{d_{i}}}=\sum_{\boldsymbol{d}}(-q)^{\boldsymbol{d}\cdot
C\cdot\boldsymbol{d}}\frac{\boldsymbol{x}^{\boldsymbol{d}}}{(q^{2};q^{2})_{\boldsymbol{d}}}\,,$
(6)

where the denominator is the so called $q$-Pochhammer symbol:

$(z;q^{2})_{r}=\prod_{s=0}^{r-1}(1-zq^{2s})\,.$ (7)

We will often refer to $P_{Q}(\boldsymbol{x};q)=P_{Q}(x_{1},\ldots,x_{m};q)$
as the quiver partition function. We point out that the quiver representation
theory involves the choice of an element, the potential, in the path algebra
of the quiver and that the trivial potential is the zero element.

A correspondence between knots and quivers proposed in Kucharski:2017ogk ;
Kucharski:2017poe  associates quivers to knots by equating the motivic
generating series with the generating series of superpolynomials or HOMFLY-PT
generating series in the variable $x$:

$P_{K}(x,a,q,t)=\sum_{r=0}^{\infty}\frac{P_{r}(K;a,q,t)}{(q^{2};q^{2})_{r}}x^{r}\,,\quad
P_{K}(x,a,q)=\sum_{r=0}^{\infty}\frac{P_{r}(K;a,q)}{(q^{2};q^{2})_{r}}x^{r}\,.$
(8)

More precisely, the most basic version of the correspondence states that for
each knot $K$ there exist a symmetric quiver $Q$ and integers
$\\{a_{i},q_{i}\\}_{i\in{Q}_{0}}$, such that

$P_{K}(x,a,q)=P_{Q}(\boldsymbol{x};q)\quad\textrm{for}\quad
x_{i}=xa^{a_{i}}q^{q_{i}-C_{ii}}\,.$ (9)

The refined version conjectures the bijection between $Q_{0}$ and the set of
generators of $\mathcal{H}_{1}$, which fixes $(a_{i},q_{i},C_{ii})$ to be
$(a,q,t)$-degrees of respective generators:

$P_{K}(x,a,q,t)=P_{Q}(\boldsymbol{x};q)\quad\textrm{for}\quad
x_{i}=xa^{a_{i}}q^{q_{i}-C_{ii}}(-t)^{C_{ii}}\,.$ (10)

For $t=-1$, this equation reduces to (9), following the relation between the
superpolynomials and HOMFLY-PT polynomials.

The correspondence between knot invariants and quivers stated above was proved
for all 2-bridge knots in Stosic:2017wno  and for all arborescent knots in
SW2004  but _it does not hold_ for all knots, e.g., the knot $9_{42}$ gives a
counter example, see section 6. We will discuss a modification of the
correspondence below that includes new types of nodes.

###  2.3 Physics and geometry of knots and quivers

####  2.3.1 3d $\mathcal{N}=2$ gauge theories

The physics around the knots-quivers correspondence is a duality between two
3d $\mathcal{N}=2$ theories: one associated to the knot and denoted
$T[L_{K}]$, and the other associated to the quiver and denoted $T[Q]$, see
Ekholm:2018eee .

The theory associated to the knot $K$, which we denote $T[L_{K}]$, arises from
M-theory over the resolved conifold $X$ with a single M5-brane wrapping the
conormal Lagrangian of the knot $L_{K}$ Ooguri:1999bv :

$\begin{split}\text{space-time}:\quad&{\mathbb{R}}^{4}\times S^{1}\times X\\\
&\cup\phantom{\ \times S^{1}\times\ \ }\cup\\\
\text{M5}:\quad&{\mathbb{R}}^{2}\times S^{1}\times L_{K}.\end{split}$ (11)

In general $T[L_{K}]$ does not admit a simple Lagrangian description, but its
vortex partition function is known to count M2-branes wrapping embedded
holomorphic curves ending on $L_{K}$.

In contrast, $T[Q]$ is easier to describe: the gauge group is $U(1)^{\times
Q_{0}}$ and there is one charged chiral for each $U(1)$ factor. Interactions
among the different sectors are mediated by (mixed) Chern-Simons couplings,
encoded by the quiver adjacency matrix. The quiver variables $x_{i}$ encode
exponentiated Fayet-Ilioupoulos couplings. The partition function of vortices
again counts M2-branes wrapping embedded holomorphic disks. But in this case
there are only $|Q_{0}|$ of them that are linked according to the adjacency
matrix. Here, all other holomorphic curves are branched covers of the basic
embedded disks which after perturbation are counted in the $U(1)$-skein of the
Lagrangian projected to homology and linking Ekholm:2019lmb ; ES . The duality
with $T[L_{K}]$ is encoded by the change of variables (9).

An important distinction to keep in mind is that while $T[Q]$ has a simple
Lagrangian description, $T[L_{K}]$ is closer to knot homology, see appendix C.
Indeed, its vortex partition function equals the generating series of colored
HOMFLY-PT polynomials:

$P_{K}(x,a,q)=\sum_{r=0}^{\infty}\frac{P_{r}(K;a,q)}{(q^{2};q^{2})_{r}}x^{r}=\sum_{r\geq
0}Z_{r}^{\text{vortex}}(a,q)\,x^{r}=Z^{\text{vortex}}(x,a,q)\,.$ (12)

The coefficient $Z_{r}^{\text{vortex}}(a,q)$ is a character for the moduli
space ${\mathcal{M}}_{r}$ of $r$ vortices. It takes the form
${Z_{r}^{\text{internal}}(a,q)}/{(q^{2};q^{2})_{r}}$ since schematically
${\mathcal{M}}_{r}\simeq{\mathcal{M}}_{r}^{\text{internal}}\times({\mathbb{C}}^{r}/S_{r})$,
where the second factor parametrizes positions of $r$ vortices in the plane,
and generates the denominator, see (53) below. Therefore,
$Z_{r}^{\text{internal}}(a,q)$ is a polynomial whose coefficients correspond
to net counts (with signs, including cancellations) of cohomology generators
of ${\mathcal{M}}_{r}^{\text{internal}}$. Passing to categorification, this
polynomial corresponds to an index for the Hilbert space arising from
quantization of ${\mathcal{M}}_{r}^{\text{internal}}$, which serves as a model
for $\oplus_{i,j,k}{\mathcal{H}}_{r;i,j,k}(K)$.

####  2.3.2 Basic holomorphic curves

In the previous subsection we saw that $T[L_{K}]$ arises from M-theory as the
effective theory on the surface of the M5-brane, and that its BPS particles
originate from M2-branes ending on the M5. From the symplectic geometric point
of view, BPS states correspond to generalized holomorphic curves with boundary
on the Lagrangian submanifold $L_{K}$.

We recall the definition of generalized holomorphic curves in the resolved
conifold $X$ with boundary on a knot conormal $L_{K}\subset X$ (as defined in
Ekholm:2018iso ; Ekholm:2018eee ) from the skeins on branes approach to open
curve counts in ES . The key observation in ES  is that the count of bare
curves (i.e., curves without constant components) counted by the values of
their boundaries in the skein module remains invariant under deformations. The
count of such curves also requires the choice of a 4-chain $C_{K}$ with
$\partial C_{K}=2L_{K}$. Intersections of the interior of a holomorphic curve
and the 4-chain contribute to the framing variable $a$ in the skein module.
For generalized curves there is a single brane on $L_{K}$, which leads to
$a=q$. Then, the map from the skein module to ‘homology class and linking’ is
well-defined and thus counting curves this way, less refined than the
$U(1)$-skein, also remains invariant. In $L_{K}\simeq
S^{1}\times\mathbb{R}^{2}$ one can define such a map that depends on the
choice of a framing of the torus at infinity. More precisely, one fixes
bounding chains for the holomorphic curve boundaries that agree with multiples
of the longitude at infinity and replace linking with intersections between
curve boundaries and bounding chains. In Ekholm:2018iso  an explicit
construction of such bounding chains and compatible 4-chain $C_{K}$ from a
certain Morse function of $L_{K}$ was described and shown to give invariant
curve counts in 1-parameter families.

Consider now holomorphic disks with boundary in the basic homology class. Such
disks are generically embedded and can never be further decomposed under
deformations. Assuming, in line with Gopakumar:1998ii ; Gopakumar:1998jq ,
that all actual holomorphic curves with boundary on $L_{K}$ lie in
neighborhoods of such holomorphic disks attached to the conormal, it would
then follow that all generalized holomorphic curves are combinations of
branched covers of the basic disks. Using the multiple cover formula the count
of generalized curves then agrees with the quiver partition function with
nodes at the basic disks and with arrows according to linking and additional
contributions to the vertices given by $4$-chain intersections.

From this point of view, the theory $T[Q]$ can be thought of as changing the
perspective, starting from a neighborhood of the Lagrangian in its cotangent
bundle and attaching small neighborhoods of the basic holomorphic disks along
curves near the central $S^{1}$. The resulting neighborhood is then determined
up to symplectomorphism by the framed link of the boundaries of the disks
attached. In this paper we will extend the collection of basic holomorphic
curves to include certain non-standard disks that are not embedded. Such
curves have boundary that wraps the homology generator of the conormal several
times, but are not combinations of curves going once around that generator. As
we will discuss below, the most basic such curves are not embedded disks: an
embedded disk that goes $d$ times around is expressed as $d$ such basic
holomorphic curves. We give a conjectural picture of the contribution from
holomorphic curves in a neighborhood of such basic curves to the partition
function, also on the refined level. From the quiver point of view, these
curves could perhaps be considered as a new type of ‘orbifold’ quiver nodes,
see section 3.1.2 for the underlying geometry.

###  2.4 Omega-background and refined indices

Having reviewed the geometric and physical features of the quiver-like
description of knot invariants in the context of topological strings, we now
turn to the question of refinement. We begin with a review of well-known facts
about refinement in the context of closed topological string theory. We then
give a brief description of the counterpart for the open sector for a knot
conormal $L_{K}$ from the point of view of the theory $T[L_{K}]$. In section
3.3 we discuss the extension of refinement to the open string sector further,
reviewing and clarifying the role of a certain $U(1)$ symmetry and its
interpretation in the context of branes wrapping knot conormal Lagrangians.

####  2.4.1 Closed string sector

One of the motivations leading to refined topological strings was the explicit
evaluation of partition functions of 4d ${\mathcal{N}}=2$ gauge theories via
localization Nekrasov:2002qd ; Nekrasov:2003rj . These gauge theories may be
engineered by type IIA string theory on a Calabi-Yau threefold $X$ times
${\mathbb{R}}^{4}$, denoted $T_{4d}[X]$. This construction may be further
viewed as a circle compactification of M-theory on $X\times
S^{1}_{\beta}\times{\mathbb{R}}^{4}$, which engineers a 5d theory $T_{5d}[X]$.
The eight supercharges of ${\mathcal{N}}=2$ SUSY in 4d transform as spinors
under $SO(4)\simeq SU(2)_{L}\oplus SU(2)_{R}$, and are further charged under a
$U(1)\times SU(2)_{I}$ R-symmetry. This $U(1)$ is often anomalous and we will
ignore it. Supercharges transform under $SU(2)_{L}\times SU(2)_{R}\times
SU(2)_{I}$ as
$(\bar{Q}^{I}_{\dot{\alpha}},{Q_{\alpha}^{I}})={({\bf{2}},{\bf{1}};{\bf{2}})}\oplus{({\bf{1},\bf{2};\bf{2}})}$.

If $S^{1}\times{\mathbb{R}}^{4}$ is replaced by an ${\mathbb{R}}^{4}$-bundle
over $S^{1}$ with holonomy $g\in SO(4)$, in order to preserve some
supersymmetry one can turn on a nontrivial R-symmetry background as follows
witten1988 . Via the identification $SO(4)\simeq SU(2)_{L}\times SU(2)_{R}$,
we may split $g=(g_{L},g_{R})$. One can turn on a holonomy $g_{I}$ of
$SU(2)_{I}$ for the rank-2 R-symmetry bundle when going around $S^{1}$, in
particular one can set $g_{I}=g_{R}$. It is then natural to consider the
diagonal subgroup $SU(2)_{d}\subset SU(2)_{R}\times SU(2)_{I}$. Supercharges
transform under $SU(2)_{L}\times SU(2)_{d}$ as

$({\bf{2},\bf{1};\bf{2}})\to\underbrace{({\bf{2},\bf{2}})}_{\tilde{Q}_{\mu}}\,,\qquad({\bf{1},\bf{2};\bf{2}})\to\underbrace{({\bf{1},\bf{1}})}_{\tilde{Q}}\oplus\underbrace{({\bf{1},\bf{3}})}_{\tilde{Q}_{\mu\nu}}\,.$
(13)

There is a distinguished supercharge $\tilde{Q}$, which is invariant under the
$S^{1}$ holonomy for any choice of $(g_{L},g_{R})$.

Next we restrict to a specific choice of bundle, the so-called Omega-
background, see Nekrasov:2003rj  for an in-depth description. In this case the
holonomies $g_{L},g_{R}$ are restricted to $U(1)_{L}\times U(1)_{R}$
subgroups. Notably, this implies that one only needs a $U(1)_{I}$ subgroup of
$SU(2)_{I}$ to perform a topological twist. This fact will be important later
when introducing branes. After twisting, the holonomy group is then
$U(1)_{L}\times U(1)_{d}$, and acts by rotations
$(g_{L},g_{d})=(e^{\epsilon_{1}+\epsilon_{2}},e^{\epsilon_{1}-\epsilon_{2}})$
of
${\mathbb{R}}^{4}\sim{\mathbb{R}}^{2}_{\epsilon_{1}}\oplus{\mathbb{R}}^{2}_{\epsilon_{2}}$.
Thanks to this restriction there are actually two conserved supercharges in
the Omega-background.

To understand this, let us choose a basis for spinors in which $J^{3}_{L,R}$
are diagonal. Doing the same for $SU(2)_{I}$, the eight supercharges of the 4d
theory have charges

$\underbrace{\left(\pm\frac{1}{2},0;\pm\frac{1}{2}\right)}_{\bar{Q}^{I}_{\dot{\alpha}}}\oplus\underbrace{\left(0,\pm\frac{1}{2};\pm\frac{1}{2}\right)}_{Q_{\alpha}^{I}}\quad\text{under}\quad
U(1)_{L}\times U(1)_{R}\times U(1)_{I}\,,$ (14)

where signs are chosen independently. With the topological twist, one may
preserve the following two linear combinations:

$\tilde{Q}_{\pm}\sim\left(0,\pm\frac{1}{2};\mp\frac{1}{2}\right)\,,$ (15)

where now the  _signs are correlated_. In fact, under $U(1)_{L}\times
U(1)_{d}$ both of these transform as $(0,0)$.111From these, one recovers
$\tilde{Q}\sim\tilde{Q}_{+}-\tilde{Q}_{-}$ being the singlet ${\bf 1\subset
2\otimes 2}$.. With the Omega-background, as opposed to more general $SU(2)$
holonomy, one may in addition preserve
$\tilde{Q}^{\prime}\sim\tilde{Q}_{+}+\tilde{Q}_{-}$. As already hinted by our
parametrization of $(g_{L},g_{d})$ in terms of $\epsilon_{1},\epsilon_{2}$,
$J^{3}_{L,R}$ are diagonal/anti-diagonal combinations of $J_{1,2}$, the latter
being generators of rotations in
${\mathbb{R}}_{\epsilon_{1}}^{2}\oplus{\mathbb{R}}_{\epsilon_{2}}^{2}$. The
surviving supercharges therefore transform as

$\tilde{Q}_{+}:\left(\frac{1}{2},\frac{1}{2};-\frac{1}{2}\right)\,,\qquad\tilde{Q}_{-}:\left(-\frac{1}{2},-\frac{1}{2};\frac{1}{2}\right)\quad\text{under}\quad
U(1)_{1}\times U(1)_{2}\times U(1)_{I}\,.$ (16)

Note that $\tilde{Q}_{-}=\tilde{Q}_{+}^{\dagger}$. Taking into account that
our conventions for R-charges are opposite to those of Aganagic:2011sg , we
can readily identify these supercharges with $Q_{r},\,\bar{Q}_{r}$ there,
respectively.

Note that $[F,\tilde{Q}_{\pm}]=\pm\tilde{Q}_{\pm}$ with
$F=2(J_{L}^{3}+J_{R}^{3})$ ensures that $\\{\tilde{Q}_{\pm},(-1)^{F}\\}=0$.
Reducing the theory along
${\mathbb{R}}^{2}_{\epsilon_{1}}\oplus{\mathbb{R}}^{2}_{\epsilon_{2}}$ gives a
${\mathcal{N}}=2$ quantum mechanics on $S^{1}$, where
$\\{\tilde{Q}_{+},\tilde{Q}_{-}\\}\sim H$ and $[\tilde{Q}_{\pm},H]=0$. It is
well-known that $\tilde{Q}_{+}$ cohomology only gets contributions from
groundstates with $H=0$, due to cancellations among bosons and fermions for
all excited states. Deformations of the theory may lead some excited states to
become groundstates, or vice versa, however the count of bosonic ($F\in
2{\mathbb{Z}}$) minus fermionic ($F\in 2{\mathbb{Z}}+1$) groundstates remains
invariant. Witten index ${\rm Tr}(-1)^{F}e^{-\beta H}$ coincides with the
invariant difference of dimensions of spaces of groundstates ${\rm
dim}{\mathcal{H}}^{B}_{(0)}-{\rm dim}{\mathcal{H}}^{F}_{(0)}$ Witten:1982im .
By construction $\tilde{Q}_{\pm}$ are invariant under $J^{3}_{L}$, and
$J^{3}_{R}+J^{3}_{I}$, therefore one may introduce additional grading on the
whole Hilbert space (both groundstates and excited states), preserving
cancellations induced by the twisting $(-1)^{F}$ in the trace. This leads to
the following for the 5d theory Nekrasov:2002qd :

$Z(\epsilon_{1},\epsilon_{2})={\rm Tr}_{{\cal
H}[T_{5d}]}(-1)^{F}e^{-(\epsilon_{1}-\epsilon_{2})J^{3}_{L}-(\epsilon_{1}+\epsilon_{2})(J^{3}_{R}+J^{3}_{I})}\,.$
(17)

Although this index will not be an object of primary interest for us, we will
use it as a point of contact to match with conventions of Aganagic:2011sg . If
we turn off R-symmetry and only rotate ${\mathbb{R}}^{4}$, one readily sees
that $J_{1}=J^{3}_{L}+J^{3}_{R}$ and $J_{2}=-J^{3}_{L}+J^{3}_{R}$ are the
generators of rotations of ${\mathbb{R}}^{2}_{\epsilon_{1}}$ and
${\mathbb{R}}^{2}_{\epsilon_{2}}$ respectively. We then identify
$S_{1},S_{2},S_{R},e^{-\epsilon_{1}},e^{-\epsilon_{2}}$ from Aganagic:2011sg
with $J_{1},J_{2},-J^{3}_{I},\mathfrak{q},\mathfrak{t}^{-1}$ in the present
paper. With these identifications, the index takes the form

$Z(\mathfrak{q},\mathfrak{t})={\rm Tr}_{{\cal
H}[T_{5d}]}(-1)^{F}\mathfrak{q}^{S_{1}^{\prime}}\mathfrak{t}^{-S_{2}^{\prime}}\,,$
(18)

where $S_{i}^{\prime}=S_{i}-S_{R}$ for $i=1,2$, as claimed in Aganagic:2011sg
. Adapting to conventions from our earlier work Ekholm:2019lmb ;
Ekholm:2018eee , we will henceforth switch to

$q=\mathfrak{q}^{1/2},\qquad t=-(\mathfrak{t}/\mathfrak{q})^{-1/2}\,.$ (19)

####  2.4.2 Open string sector

Let $L$ be a special Lagrangian submanifold of the resolved conifold
$X=\mathcal{O}(-1)\oplus\mathcal{O}(-1)\to\mathbb{C}{\mathbb{P}}^{1}$. The
low-energy dynamics of an M5-brane on $L\times S^{1}\times{\mathbb{R}}^{2}$ is
described by a 3d ${\mathcal{N}}=2$ theory $T[L]$ on
$S^{1}\times{\mathbb{R}}^{2}$ Dimofte:2011ju ; Terashima:2011qi . From this
viewpoint, $J_{1}$ is a space-time rotation, however $J_{2}$ is now an
R-symmetry. As stressed above, it only makes sense to turn on fugacities
$\mathfrak{q},\mathfrak{t}$ in the index if the corresponding generators
commute with the surviving supercharges. The presence of the M5-brane on
${\mathbb{R}}^{2}$ inside ${\mathbb{R}}^{4}$ breaks Lorentz invariance,
therefore we must re-investigate whether supercharges are preserved or not.

Recall our discussion of the 5d theory in a general background, which led to
the existence of a single preserved supercharge $\tilde{Q}$. Since this
transforms as a scalar in 4d, clearly it is also preserved by the presence of
the M5 that breaks Lorentz invariance. However to define an index and its
deformation by $q,t$, we need two supercharges. Without the brane this was
made possible by the choice of a non-generic background, the Omega-background
which breaks $SO(4)$ to $U(1)_{1}\times U(1)_{2}$. As long as the defect lies
in a plane ${\mathbb{R}}^{2}$, that is either
${\mathbb{R}}^{2}_{\epsilon_{1}}$ or ${\mathbb{R}}^{2}_{\epsilon_{2}}$, it
preserves the group of rotations of the Omega-background. To preserve two
supercharges it was also crucial to perform a topological twist using a
$U(1)_{I}$ subgroup of the $SU(2)_{I}$ R-symmetry. In general there is no
reason to expect that neither $SU(2)_{I}$ nor $U(1)_{I}$ should be preserved
by the presence of M5 on $L\times S^{1}\times{\mathbb{R}}^{2}$. However, one
may hope that there exist specific geometric configurations of $L$ that allow
to preserve at least the $U(1)_{I}$ subgroup required by the topological
twist.

We will assume that $U(1)_{I}\subset SU(2)_{I}$ is preserved by the presence
of M5 on $L$. This is a nontrivial and crucial assumption made in
Aganagic:2011sg , whose geometric significance in our setting will be
clarified in section 3.3.2. With the topological twist, one may preserve the
same two supercharges (16) discussed before introducing the M5. As already
stressed, these coincide precisely with $Q_{r},\,\bar{Q}_{r}$ from
Aganagic:2011sg .

Finally, we come to the object of main interest for us: the partition function
of 3d BPS states and its refinement. These consist of open M2-branes ending on
the M5-brane, which give rise to BPS vortices of the 3d ${\mathcal{N}}=2$
theory $T[L]$. The 3d index counting such BPS states is a Witten index, again
deformed by fugacities coupled to symmetries of the theory that commute with
$\tilde{Q}_{\pm}$ and therefore with $H$:

$Z(\mathfrak{q},\mathfrak{t})={\rm Tr}_{{\cal
H}[T_{3d}]}(-1)^{F}\mathfrak{q}^{S_{1}^{\prime}}\mathfrak{t}^{-S_{2}^{\prime}}a^{\mathcal{Q}}x^{\mathcal{T}}\,.$
(20)

Here ${\mathcal{Q}}$ is the charge of a global $U(1)$ symmetry of $T[L]$
associated to rotations of the base $\mathbb{C}{\mathbb{P}}^{1}$ of $X$
Fuji:2012nx , and ${\mathcal{T}}$ is the charge of the topological $U(1)$
symmetry associated to the gauge $U(1)$ of $T[L]$ Aharony:1997bx .222The fact
that $T[L]$ is a $U(1)$ gauge theory follows from the fact that
$b_{1}(L\approx S^{1}\times{\mathbb{R}}^{2})=1$. The abelian 2-form on M5
reduced along the 1-cycle in $L$ gives rise to an abelian gauge field in
$T[L]$. Contributions to the index by states with $a^{2k}x^{n}$ correspond to
M2-branes wrapping a relative homology cycle labeled by $(k,n)\in H_{2}(X,L)$.
It is natural to switch to a different basis for $\mathfrak{q},\mathfrak{t}$
to distinguish between the spin of a BPS state (along
${\mathbb{R}}^{2}_{\epsilon_{1}}$) and its R-charge (along transverse
directions). Adopting (19) yields

$\begin{split}Z(q,t)&={\rm Tr}_{{\cal
H}[T_{3d}]}(-1)^{2(J_{1}+J_{2}+J_{I})}q^{2(J_{1}-J_{2})}t^{2(J_{2}+J_{I})}\,a^{\mathcal{Q}}x^{\mathcal{T}}\,,\end{split}$
(21)

where we lightened notation $J_{I}^{3}\to J_{I}$. Spin $J_{1}$ along the
direction tangent to M5 is detected only by $q$, not $t$. Conversely, the
R-symmetry charge $J_{I}$ is only detected by $t$, not $q$. When $L$ is a knot
conormal, this 3d index is expected to correspond to the refined HOMFLY-PT
generating series of $K$ (with symmetric colors) Ooguri:1999bv ;
Aganagic:2011sg .

##  3 Generalized quivers and HOMFLY-PT homology

In this section we motivate and state our main conjectures on the relation
between HOMFLY-PT homology and generalized quivers.

######  Remark 3.1.

One of the implicit assumptions in the original formulation of the knots-
quivers correspondence Kucharski:2017ogk ; Kucharski:2017poe  is that each
node of the quiver has an associated change of variables where $x_{i}$ is
directly proportional to $x$. From a geometric perspective, this means that
each one of the basic disks represented by nodes of $Q$ has a boundary that
wraps around the longitude of $L_{K}$ exactly once. It was noticed in
Ekholm:2018eee  that this assumption is not always satisfied (see section 6
for concrete counterexamples), and an extension of the knots quivers
correspondence was formulated (Ekholm:2018eee, , Conjecture 1.1). This
conjecture was erroneously stated on the level of the refined partition
function – the problem is that it does not say what the refined contribution
for higher level nodes would be. In this section we discuss such contributions
and present a new and complete version of the conjecture.

In more physical terms, the basic disks corresponding to the nodes of $Q$ are
mutually linked and interact according to the quiver adjacency matrix.
Together they generate the whole spectrum of BPS states corresponding to
M2-branes wrapping embedded holomorphic curves Ekholm:2018eee ; Ekholm:2019lmb
. The quiver description expresses a BPS state winding $d$ times around as a
bound state of $d$ copies of once-around basic disks, each corresponding to
one of the nodes of $Q$. As mentioned above, for knots that do not admit such
descriptions there are BPS states corresponding to curves whose boundary wraps
$d$ times around the longitude of $L_{K}$, which cannot be realized as bound
states of once-around curves. In these cases the original formulation of the
knots-quivers correspondence fails.

###  3.1 Geometry and combinatorics of multiply-wrapped basic disks

In this section we study the geometry of the basic holomorphic disks that are
the nodes of our generalized quiver. We will first describe their geometry and
then derive their multi-cover formulas. As mentioned above, quiver node curves
appear for a $U(1)$-symmetric configuration, when the knot conormal lies on
top of the unknot conormal. Since the unknot conormal supports no basic higher
genus curves, we expect all quiver nodes to correspond to disks. The simplest
such curves are embedded disks and their multiple-cover formula is well-known.

Consider now an embedded holomorphic disk with connected boundary in class
$x^{\mu}$ for $\mu>1$. We apply so-called Symplectic Field Theory (SFT)
stretching around $L_{K}$, see EGH . This is a degeneration of the complex
structure under which holomorphic curves degenerate into several level
holomorphic buildings, with parts near $L_{K}$ and parts far from $L_{K}$
joined at Reeb orbits in the unit cotangent bundle of $L_{K}$. In the case at
hand, we pick a metric on $L_{K}\approx S^{1}\times\mathbb{R}^{2}$ with only
one simple closed geodesic, and correspondingly only two simple closed Reeb
orbits which are the unit cotangent lifts of this geodesic with its two
orientations. Then, since the unknot conormal has holomorphic curves that go
in only one of the two directions, all nearby curves are multiples of the
basic cylinder stretching from the simple Reeb orbit to the geodesic (with the
positive orientation) in the zero section. In particular, our basic once-
around disk becomes a two level building consisting of an outside sphere with
puncture and an inside disk with puncture. The contribution to the partition
function comes from multiple covers of this curve together with constant
curves attached along it.

Since all curves near the Lagrangian in the stretched limit are multiples of
the basic punctured disk (stretching between the Reeb orbit and the unique
geodesic in $S^{1}\times\mathbb{R}^{2}$), we find that the lower part of the
holomorphic building in the limit is a $\mu$-fold cover of this basic curve.
The outside part of the curve, the upper level, is a once punctured curve
asymptotic to the $\mu$-fold cover of the basic Reeb orbit. The curve before
the limit is assumed not to be a multiple cover. Assuming that it is somewhere
injective also in the limit, this limit curve is generically embedded as well.
The simplest such two-level building has a sphere with one puncture as its
upper level. We define such buildings as our new $\mu$-times around basic
curves, and note that for $\mu=1$ we simply get stretched versions of our
previous basic disks.

####  3.1.1 Generalized curves, the four-chain and deformations of M2-M5
configurations

We next consider how these two-level buildings and their multiple covers glue
and how they contribute to the partition function counting generalized curves.
Consider $d$-fold covers of the levels. The inside piece looks like a disk
with an $U(1)$-action with a fixed point of order $\mu d$ at the Reeb orbit.
The outside piece looks like a $d$-fold cover of a sphere with a single
puncture. In Gromov-Witten counts of connected curves, a $\mu d$-fold cover of
a disk with a single fixed point contributes

$\frac{1}{\mu d}\left(\frac{1}{1-q^{2\mu d}}\right).$ (22)

To see this, we use a deformation argument. Consider first an embedded annulus
stretching between two Lagrangians $L$ and $L^{\prime}$, and with standard
normal bundle. Assume that the boundary of the annulus in $L$ is homologically
essential and lies in homology class $\log x\neq 0\in H_{1}(L)$, and that the
boundary in $L^{\prime}$ is contractible. The $d$-fold cover of $L$ then
contributes

$\frac{1}{d}x^{d}.$

Assume now that the boundary in $L^{\prime}$ shrinks to a point. This leaves a
disk with a $d$-fold branch point. As in the skein count ES , this disk is an
instance of a 1-parameter family of disks that crosses $L^{\prime}$. The
intersection of this disk with the 4-chain of $L^{\prime}$ changes by $2d$ as
they cross ($d$ positive intersections become negative). If the disk
contributes $\xi$, then invariance in the $U(1)$-skein projected to homology
and linking gives the equation

$q^{2d}\xi-\xi=\frac{1}{d}x^{d},$ (23)

and we find

$\xi=\frac{x^{d}}{d(1-q^{2d})},$

as claimed.

Consider now gluing on the outside piece. This outside piece breaks the
symmetry, there are $\mu$ ways of gluing (here we glue first the underlying
curve and then take multiple covers) giving instead the contribution

$\mu\cdot\frac{1}{\mu d}\left(\frac{1}{1-q^{2\mu
d}}\right)=\frac{1}{d}\left(\frac{1}{1-(q^{2\mu})^{d}}\right).$ (24)

Therefore the contribution to the Gromov-Witten partition function from a
single basic two-level configuration with boundary going $\mu$ times around
the generator is

$\exp\left(\sum_{d>0}\frac{1}{d}\frac{(x^{\mu})^{d}}{1-(q^{\mu})^{2d}}\right)=(x^{\mu},q^{2\mu})_{\infty}^{-1}.$
(25)

######  Remark 3.2.

Relation (23) has a suggestive interpretation from a physics viewpoint: the
left-hand side represents contributions of M2-branes wrapping holomorphic
curves belonging to the same family of deformations, and located on either
side of an M5-brane on $L^{\prime}$. On the right hand side, there is a
contribution of an annulus with a boundary on $L^{\prime}$. This is a kind of
skein relation between M2 and M5-branes supported on higher-dimensional
objects of total codimension one in the Calabi-Yau, see figure 2. (The objects
are linking from the viewpoint of the $A$-model on $X$, where M2 is
represented by a holomorphic curve and M5 by a Lagrangian. Lifting to M
theory, the M2 couples to the three-form potential $A_{3}$ sourced by the M5
by $j_{M5}\sim dF_{4}$ where $F_{4}=dA_{3}$ locally. Let $\tilde{F}_{7}=\star
F_{4}$ be the dual field in 11d; this is sourced by the M2 current $j_{M2}$
with a boundary term correction $d\tilde{F}_{7}=j_{M2}+H_{3}\wedge j_{M5}$,
where $H_{3}$ is the self-dual field strength on the M5, which couples to the
M2 boundary Strominger:1995ac .) At the level of (plethystic-)exponentiated
partition functions, this relation is expressed by the well-known property
$(q^{2}x;q^{2})_{\infty}=\frac{1}{1-x}(x;q^{2})_{\infty}$. (This generalizes
to $\mu$-times around curves, see (39)) Tracking 4-chain intersections is key
for establishing this relation via Gromov-Witten counting. Closing the
puncture corresponding to the boundary on $L$, yields a similar relation
between generating functions of M2-branes on spheres and disks
$(q^{2}a^{2};q^{2};q^{2})_{\infty}=(a^{2};q^{2})_{\infty}^{-1}(a^{2};q^{2};q^{2})_{\infty}$,
where $a^{2}$ is the flux of M5-branes on $L$ linked by M2.

Figure 2: Skein-like relation for an M2-brane wrapping an embedded holomorphic
disk: when crossing an M5-brane, the M2 breaks up and leaves behind a new
curve with a new boundary on the M5.

####  3.1.2 Orbifold models of multiply wrapped disks

As discussed above, two-level holomorphic buildings where the inside (the part
near the Lagrangian) is a punctured un-branched cover of a punctured disk and
where the outside (the part far from the Lagrangian) is an embedded punctured
sphere are key objects in our generalized quivers. In order to understand the
behavior of such holomorphic objects, we can view them from inside and move
the embedded sphere part far away. This means shrinking the outside punctured
sphere together with the Reeb orbit where it is attached to a point and that
gives an orbifold disk. In this section we study topological strings on
orbifolds.

We start our discussion from the following identity, relating the partition
function of a $\mu$-times around basic disk to $\mu$ once-around basic disks,
with specific $B$-field fluxes:

$(x^{\mu};q^{2\mu})_{\infty}^{-1}=\prod_{n\geq 0}(1-x^{\mu}q^{2\mu
n})^{-1}=\prod_{k=0}^{\mu-1}\prod_{n\geq
0}(1-\zeta^{k}xq^{2n})^{-1}=\prod_{k=0}^{\mu-1}(\zeta^{k}x;q^{2})_{\infty}^{-1}\,,$
(26)

where $\zeta$ is a primitive $\mu$-th root of unity. We will explain how such
expressions appear from certain ‘fillings’ of cylindrical versions of
punctured orbifold singularities. We start in the simplest non-trivial case,
when $\mu=2$. Here we simply have

$(x^{2};q^{4})_{\infty}=(x;q^{2})_{\infty}(-x;q^{2})_{\infty}\,.$ (27)

This equation appears geometrically from the toric brane in
$\mathcal{O}(0)\oplus\mathcal{O}(-2)\to{\mathbb{C}}{\mathbb{P}}^{1}$ in a
certain limit. Consider the toric diagram in figure 3. The mirror curve is

$F(x,y)=1-y-(1+Q)x+Qx^{2}\,,$ (28)

with a single solution

$y=(1-x)(1-Qx)\,.$ (29)

In the limit $Q\to e^{\pi i}=-1$ (the complexified Kähler modulus is purely
$B$-field) this becomes

$1-y-x^{2}=0\,,$ (30)

which is precisely the classical limit of (103) with $C_{ii}=0$.

Figure 3: Toric diagram of
$\mathcal{O}(0)\oplus\mathcal{O}(-2)\to{\mathbb{C}}{\mathbb{P}}^{1}$.

Switching from the classical to the quantum level, we consider a toric brane
on the horizontal leg. Its partition function can be evaluated by topological
vertex techniques:333Compared to topological vertex conventions, we absorb
inessential powers of $q$ in $x$ and $Q$ for cleaner expressions.

$Z(x)=(x;q^{2})_{\infty}^{-1}(Qx;q^{2})_{\infty}^{-1}\,.$ (31)

This is annihilated by the quantum curve

$\hat{y}=(1-\hat{x})(1-Q\hat{x})\,,$ (32)

whose classical limit recovers (29). Now an orbifold limit $Q\to-1$ turns the
curve into

$\hat{y}=(1-\hat{x})(1+\hat{x})=1-\hat{x}^{2}$ (33)

and the partition function collapses to

$Z(x)\to(x;q^{2})_{\infty}^{-1}(-x;q^{2})_{\infty}^{-1}=(x^{2};q^{4})_{\infty}^{-1}\,.$
(34)

This is precisely the proposal for the unrefined partition function of a
twice-around basic disk. Indeed,

$(x^{2};q^{4})_{\infty}^{-1}=\sum_{k\geq 0}\frac{x^{2k}}{(q^{4};q^{4})_{k}}$
(35)

matches with the form proposed in (64) with $C_{11}=0$, $\mu_{1}=2$ and
$x_{1}=x^{2}$.

In order to see the geometry underlying these calculations, consider the
orbifold $\mathbb{C}\times(\mathbb{C}^{2}/(\mathbb{Z}/2\mathbb{Z}))$ and
remove the codimension four fixed point locus $\mathbb{C}\times[0]$, where
$[0]$ denotes the point in the orbifold quotient which is the image of
$0\in\mathbb{C}^{2}$. We view the resulting symplectic manifold as having a
negative end (the negative half of the symplectization of standard contact
$\mathbb{R}{\mathbb{P}}^{3}$ times $\mathbb{C}$) near the removed locus and
note that there are two fillings: the orbifold itself and
$\mathcal{O}(-2)\oplus\mathcal{O}(0)$. In the latter filling we view the
symplectic area of the sphere as zero since the filling is at negative
infinity. We can now interpret the limit $Q\to-1$ as this negative end
splitting off from $\mathcal{O}(-2)\oplus\mathcal{O}(0)$. When this happens,
punctured versions of the curves remain in the punctured orbifold and these
can be completed by curves in the actual orbifold filling. As the flux through
the sphere at negative infinity is set to $\pi i$, the partition functions
match. Geometrically, this can be interpreted as the fact that upper level
connected curves with odd asymptotics cancel, whereas those with even
asymptotics add. This means that all non-zero curves can be filled with the
orbifold end or the $\pi i$ flux $\mathcal{O}(-2)\oplus\mathcal{O}(0)$ and
that the curve counts agree. Thus, from the viewpoint of curve counts in the
two-level symplectic manifold, the two fillings give the same result.
Moreover, as the filling $\mathcal{O}(-2)\oplus\mathcal{O}(0)$ moves toward
negative infinity, the moduli space of curves in the upper level approaches
the corresponding space for the orbifold. It is in this sense that taking the
limit $Q\to-1$ should be understood.

Figure 4: Toric diagram of the resolution of the orbifold
${\mathbb{C}}\times({\mathbb{C}}^{2}/({\mathbb{Z}}/\mu{\mathbb{Z}}))$.

Higher degree basic disks appear in more complicated orbifold quotients. The
corresponding toric diagram is shown in figure 4 and gives the mirror curve

$y=(1-x)(1-Q_{1}x)\dots(1-Q_{\mu-1}x).$ (36)

In analogy with the case $\mu=2$ above, we take the limit $Q_{k}\to
e^{\frac{2\pi i}{\mu}k}$ and get the mirror curve of the orbifold
$\mathbb{C}\times(\mathbb{C}^{2}/(\mathbb{Z}/\mu\mathbb{Z}))$.

$1-y-x^{\mu}=0.$ (37)

On the quantum level this reproduces the identity (26). The geometric
interpretation is as above: we count two level curves in the two level
symplectic manifold. The upper level of this manifold is the orbifold with the
central $\mathbb{C}\times[0]$ removed, viewed as a symplectic manifold with an
end given by the negative half of the symplectization of the standard contact
lens space $L(\mu,1)$ times $\mathbb{C}$. The filling of the negative end is
either the orbifold itself or the manifold given in figure 4 with zero-area
spheres, since they live at negative infinity, and with $B$-field fluxes
$\frac{2\pi}{\mu}k$ so that curve counts are identical. The convergence of
moduli spaces works in direct analogy with the $\mu=2$ case.

####  3.1.3 Semiclassical consequences of basic disk denominators

It is worth noting that the difference between the standard denominators
$(q^{2};q^{2})_{d_{i}}$ and $(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}$ we propose
in (64) can be seen already at the semiclassical level. Consider a single
basic $\mu$-times around disk with no self-linking. Then the partition
function

$P=\sum_{d\geq
0}\frac{x^{d\mu}}{(q^{2\mu};q^{2\mu})_{d}}=(x^{\mu};q^{2\mu})_{\infty}^{-1}$
(38)

is annihilated by $1-\hat{y}-\hat{x}^{\mu}$, since

$\hat{y}\cdot(x^{\mu};q^{2\mu})_{\infty}^{-1}=(q^{2\mu}x^{\mu};q^{2\mu})_{\infty}^{-1}=(1-x^{\mu})(x^{\mu};q^{2\mu})_{\infty}^{-1}\,.$
(39)

The classical curve is therefore

$1-x^{\mu}-y=0\,.$ (40)

On the other hand, a partition function like

$\tilde{P}=\sum_{d\geq
0}\frac{x^{d\mu}}{(q^{2};q^{2})_{d}}=(x^{\mu};q^{2})_{\infty}^{-1}$ (41)

would behave as follows:

$\hat{y}\cdot(x^{\mu};q^{2})_{\infty}^{-1}=(q^{2\mu}x^{\mu};q^{2})_{\infty}^{-1}=(1-x^{\mu})\dots(1-x^{\mu}q^{2\mu-2})(x^{\mu};q^{2\mu})_{\infty}^{-1}\,.$
(42)

The classical curve in this case is quite different:

$(1-x^{\mu})^{\mu}-y=0\,.$ (43)

Therefore, in order to detect the difference between the two types of
denominator, it is sufficient to compute the unrefined contributions from
genus-zero basic curves to the augmentation polynomial. In section 6 we will
provide concrete examples where the distinguished form of the curve (40)
appears.

###  3.2 Holomorphic curves viewed as BPS generators

The structure of HOMFLY-PT homology seems to indicate that it is more closely
related to the expansion of the refined partition function of the theory
$T[L_{K}]$ in terms of equivariant vortices than to the corresponding
expansion in terms of basic holomorphic curves. This means that in order to
extract homology information from basic holomorphic curves viewed as BPS
generators, we must understand how to expand a configuration of such objects
as a combination of vortices. In this section we discuss a proposal for such
an expansion and its origins. We study the expansions for single generators,
first on the unrefined and then on the refined levels. The discussion here is
in a sense localized near the boundary of the curve and independent of other
charges, which on the unrefined level comes from homology class, 4-chain
intersections, self-linking, and – on the refined level – also on a certain
framing density term that we discuss in section 3.4.

Consider a basic holomorphic curve. As explained in section 3.1, such a curve
consists of a punctured embedded sphere on the outside, asymptotic to a
multiple of the unique Reeb orbit, and an unbranched multiple cover of the
basic cylinder stretching from the simple Reeb orbit to the geodesic in the
zero section on the inside.

In particular, our basic once-around disk becomes a two level building
consisting of an outside sphere with puncture and an inside disk with
puncture. The contribution to the partition function comes from multiple
covers of this curve together with constant curves attached along it. Here we
propose to push the contributions from the constants down to the boundary. We
observe that at the level of generalized curves this can be done in a specific
way: the contributions from constants correspond to very thin holomorphic
annuli with one boundary component linking the basic curve and the other one
not linking it. For a $d$-fold multiple cover of a basic disk, there is one
such annulus on each level of magnification. The first annulus links all
strands, the second all but one, etc., until the last annulus which links only
one strand, see figure 5.

Figure 5: Contributions of pushed down constant annuli to generalized curve
counts. Both boundaries of each annulus follow the longitude, one of the
boundaries also wraps around a small meridian, linking with the boundary of
the $i$-th strand. The overall holonomy is $y_{i}\cdot x\cdot x^{-1}=y_{i}$.

To support this picture, let us explain how it relates to the more familiar
count of multiple covers of a single holomorphic disk with boundary on
$L_{K}$. Recall that $L_{K}$ carries a $U(1)$ local system, with meridian and
longitudinal holonomies at the boundary denoted $x,y$. These holonomies
provide (exponentiated) Darboux coordinates for the moduli space of abelian
flat connections on $T^{2}\simeq\partial L_{K}$. We denote their deformation
quantization by $\hat{x},\hat{y}$, obeying the relation
$\hat{y}\hat{x}=q^{2}\hat{x}\hat{y}$. Any disks ending on $L_{K}$ arise
through the large $N$ geometric transition, descending from a common
holomorphic cylinder stretching between the zero-section in $T^{*}S^{3}$ and
$L_{K}$. As the boundaries of this cylinder are the original knot $K\subset
S^{3}$ and the longitude in $L_{K}$, its skein valued partition function will
be given by (8), where $P_{r}(K;a,q)$ is the HOMFLY-PT polynomial of $K$ in
the $r$-th symmetric representation, and $x^{r}$ is the $U(1)$ HOMFLY-PT skein
element in $L_{K}$ projected to homology and linking. For illustration,
consider the unknot in reduced normalization, for which $P_{r}(0_{1};a,q)=1$.
This gives the following partition function of a holomorphic disk

$\psi(x)=P_{0_{1}}(x,a,q)=\sum_{r\geq
0}\frac{x^{r}}{(q^{2};q^{2})_{r}}=(x;q^{2})_{\infty}^{-1}$ (44)

From the viewpoint of $U(1)$ Chern-Simons on $L_{K}$, boundaries of worldsheet
instantons wrapping the holomorphic disk give rise to infinite series of
Wilson lines Witten:1992fb . A Wilson line wrapping $r\geq 0$ times around
$L_{K}$ with $k$ kinks on it contributes with $q^{2k}x^{r}$. The number of
such Wilson lines is $c_{k,r}$, defined by the expansion

$\psi(x)=\sum_{r\geq 0}\sum_{k\geq 0}c_{k,r}\,q^{2k}{x^{r}}\,.$ (45)

The interpretation of $q^{2k}x^{r}$ as an $r$-times wrapped Wilson line with
$k$ kinks can be seen by noting that kinks are counted by powers of $a$ as in
figure 1, and by noting that in $U(1)$ Chern-Simons $a=q$. Incidentally,
$c_{k,r}$ is also the dimension of the Hilbert space of BPS vortices of
$T[L_{K}]$ with vorticity $r$ and spin $k$ Dimofte:2010tz .

Kinks on the boundary of a Wilson line can be traded with linking with a dual
Wilson line using the $U(1)$ skein algebra of variables $\hat{x},\hat{y}$:

$\frac{x^{r}}{(q^{2};q^{2})_{r}}=\,:\
\underbrace{\frac{1}{1-\hat{y}}\hat{x}\dots\frac{1}{1-\hat{y}}\hat{x}}_{r}\ :$
(46)

where $:\ :$ is the normal ordering operation defined in Ekholm:2019lmb .
Trading kinks with linking loops may be viewed as a ‘half’ of the $U(1)$ skein
relation on $L_{K}$: the $\hat{x}$ Wilson line with a small $\hat{y}$ loop
around it can be viewed as the over-crossing diagram in the first line of
figure 1, while the kink corresponds to $q^{2}$ times the smoothing also on
the first line.

Once again, this specific ensemble of Wilson lines can be given a geometric
interpretation in terms of curve counting. The factor $\frac{1}{1-\hat{y}}$
corresponds to an annulus with a boundary linking the boundary of the
$x$-cylinder in $L_{K}$ once, and with the other boundary linking zero times.
Overall, the partition function resembles an $x$-annulus ending on $L_{K}$,
with a linked $\hat{y}$-annulus on each strand of its multi-cover:

$\psi(x)=\ :\ \sum_{r\geq 0}\left(\frac{1}{1-\hat{y}}\,\hat{x}\right)^{r}\ :\
=\ :\ \frac{1}{1-\frac{1}{1-\hat{y}}\,\hat{x}}\ :$ (47)

The corresponding recursion relation is

$\left(1-\left(\frac{1}{1-\hat{y}}\,\hat{x}\right)\right)\psi=1\,,$ (48)

equivalent, upon left-multiplication by $1-\hat{y}$, to the more familiar

$(1-\hat{y}-\hat{x})\psi=0\,.$ (49)

As remarked above, a single vortex would contribute $q^{2k}x^{r}$
corresponding to a Wilson line. On the geometric side, we repackage Wilson
lines into linked holomorphic annuli. On the gauge theory side of $T[L_{K}]$,
this repackaging corresponds to the definition of equivariant vortices.

Given this, we now propose to think about an equivariant vortex in the theory
$T[L_{K}]$ simply as a configuration of the form above: on level $d$, there
are $d$ parallel copies of the central curve linked in the nested way by basic
annuli, see figure 5.

We next consider the corresponding procedure applied to a $\mu$ times around
generator. We propose that such generators become multiples of $\mu$ with
constants and anti-constants attached on all intermediate covers. The constant
and anti-constant push down to the boundary as almost identical annuli, where
the second annulus has a twist in the trivialization on the boundary
corresponding to $t$ or $t^{-1}$, depending on the underlying half-framing of
the underlying punctured disk. Figure 6 shows the configuration for a twice-
around generator.

Figure 6: A degree 2 generator with constants pushed down.

The configuration for a $\mu$ times around generator is clear and gives the
refined contribution

$:\frac{1}{{1+t^{\pm 1}\hat{y}}}\left(\frac{1+t^{\pm
1}\hat{y}}{1-\hat{y}}\hat{x}\right)^{\mu}:.$ (50)

The recursion relation for the refined partition function of a $\mu$-times
around generator is then

$\left(1-\frac{1}{{1+t^{\pm 1}\hat{y}}}\left(\frac{1+t^{\pm
1}\hat{y}}{1-\hat{y}}\hat{x}\right)^{\mu}\right)\psi=1.$ (51)

To illustrate the procedure, we next consider the second level of a system of
two unlinked basic disks. We must express the configuration of two strands
with an annulus linking each in terms of our standard basis, which leads to
the counting of intersections presented in figure 7.

Figure 7: Contributions on second level from two unlinked basic disks.

We next look at this way of expanding states in terms of vortices and their
Hilbert spaces. Consider an equivariant vortex. Its partition function is
given by

$\psi(x,q)=\sum_{d\geq 0}\frac{x^{d}}{(1-q^{2})\dots(1-q^{2d})}.$ (52)

Expanding the denominators using geometric series, it is convenient to think
of the Hilbert space at level $d$ as follows. Consider the points in the
simplex $x_{1}\geq x_{2}\geq\dots x_{d}\geq 0$ with integer coordinates
$(n_{1},\dots,n_{d})$. The dimension of the Hilbert space of states with
$q$-charge $k$ is then the number of points in the intersection between the
simplex and the plane $\sum_{j=1}^{d}x_{j}=k$.

The Hilbert space of mixed states of two vortices on level two as above looks
like the integral points over the first quadrant and can be expressed as two
copies of the simplex, one shifted by multiplication by $q^{2}$. Similarly,
one generator going $\mu$ times around corresponds to the points along the
diagonal $x_{1}=\dots=x_{\mu}$ and can be expressed as a sum of shifted
simplices. This indicates that refinement applies to the Hilbert spaces
associated to vortices individually.

We give one final perspective on vortices and homology. Viewing a vortex as an
$U(1)$-invariant fixed point in $\mathbb{C}$, we may compute its contribution
to the partition function by Bott localization. Here we change coordinates and
think of the $d$-fold vortex as a point in the configuration space of $d$
points in $\mathbb{C}$ thought of as the space of polynomials using the
relation between roots an coeffcients. The contribution

$\frac{1}{(1-q^{2})\dots(1-q^{2d})}$ (53)

now arises as the equivariant Chern character. For a $\mu$ times around
generator, only the top degree coefficients of the polynomials are fixed,
others are free to vary. This means we have a corresponding
$\mathbb{C}^{\mu-1}$ family of fixed points with similar action on normal
bundles. The contribution from the corresponding Bott manifold is

$\frac{1}{(1-q^{2d})}\cdot[(\mathbb{C}{\mathbb{P}}^{1})^{d-1}],$ (54)

where we have replaced $\mathbb{C}$ with $\mathbb{C}{\mathbb{P}}^{1}$ and a
fixed point also at infinity and where when we pass from Bott to Morse we get
the contribution

$\frac{1+tq^{2}}{1-q^{2}}\dots\frac{1+tq^{2(\mu-1)}}{1-q^{2(\mu-1)}}\frac{1}{1-q^{2d}},$
(55)

corresponding to two fixed points in each $\mathbb{C}{\mathbb{P}}^{1}$-factor.

###  3.3 M2-branes wrapping holomorphic curves

In this section we discuss the physical and geometric interpretations of
refinement, connecting the conjectures of Aganagic:2011sg  to our
interpretation of the knot-quiver correspondence Ekholm:2018eee ;
Ekholm:2019lmb . The 3d ${\mathcal{N}}=2$ index (20), proposed by
Aganagic:2011sg , corresponds to the refined partition function of knot
invariants. However, this index is well-defined if not just one, but _two_
supercharges are preserved in the Omega-background and with an M5-brane
inserted along $L\times S^{1}\times{\mathbb{R}}^{2}$. We will now provide
motivation for the existence of this additional supercharge, based on the
presence of a certain geometric $U(1)$ symmetry, expanding on a suggestion of
Aganagic:2011sg . Eventually, this will lead us to a geometric interpretation
of the R-charges of BPS states in terms of ‘self-linking densities’,
consistent with our previous observations in Ekholm:2018eee . Then, together
with the study in section 3.2, this gives our generalized quiver partition
function.

The analysis of preserved supersymmetries that we develop here takes a
different perspective from the one in section 2.4. There we started with
supercharges of the 5d theory engineered by $X$ and examined which ones
survive in presence of the Omega-background, and eventually also in presence
of the M5-brane on $L\times S^{1}\times{\mathbb{R}}^{2}$. Here we shall start
with the worldvolume supersymmetry on the M5 and analyze how it is broken by
introducing the Calabi-Yau background and by placing the brane on $L$. An
interesting novelty of this perspective is that it will clarify a little-
appreciated consequence of the $U(1)$ symmetry postulated in Aganagic:2011sg .
As we will show, the existence of a $U(1)$ symmetry corresponds to a point in
the moduli space of $T[L]$ with enhanced supersymmetry.

####  3.3.1 Conserved supercharges in the generic case

We study M-theory on the resolved conifold
$X=\mathcal{O}(-1)\oplus\mathcal{O}(-1)\to{\mathbb{C}}{\mathbb{P}}^{1}$ times
$S^{1}\times{\mathbb{R}}^{4}$, with an Omega-deformation turned on. This
consists of rotations of the planes ${\mathbb{R}}^{2}\times{\mathbb{R}}^{2}$
by independent phases $\epsilon_{1,2}$ when going around the $S^{1}$
Nekrasov:2002qd ; Nekrasov:2003rj .444Recall that these rotations arise from
the diagonal and anti-diagonal combination of Cartan generators of
$U(1)_{L}\times U(1)_{R}\subset SU(2)_{L}\times SU(2)_{R}\simeq SO(4)$, via
the identification
$(g_{L},g_{R})=(e^{\epsilon_{1}+\epsilon_{2}},e^{\epsilon_{1}-\epsilon_{2}})$.

The Calabi-Yau background preserves eight supercharges. This amount of
supersymmetry is further reduced if we consider an M5-brane wrapped on a
special Lagrangian submanifold $L\subset X$ times
$S^{1}\times{\mathbb{R}}^{2}_{\epsilon_{1}}$. Recall that a special Lagrangian
is a Lagrangian calibrated by the holomorphic top form on $X$, which must have
a constant phase along $L$, the phase determines which supercharges are
preserved and which are broken by the M5-brane. Under a genericity assumption,
such a configuration preserves four supercharges. Let us review how this
works.

In a flat space the theory on an M5-brane would be the abelian 6d (2,0) SCFT,
with 16 supercharges transforming as $({\bf 4,4})$ spinors under
$Spin(1,5)\times Spin(5)$, with the second factor corresponding the R-symmetry
group. With the Calabi-Yau background, the latter is broken to $Spin(3)\times
Spin(2)$, where $Spin(3)$ is a subgroup of the local rotations on $X$ which
leave $L$ fixed, and $Spin(2)\simeq U(1)_{2}$ is the group of rotations of
${\mathbb{R}}^{2}_{\epsilon_{2}}$. The (2,0) 6d supercharges thus decompose as
follows:

$({\bf 4;4})\to({\bf 2,2;2})_{+1/2}\oplus({\bf 2,2;2})_{-1/2}$ (56)

as representations of

$\underbrace{Spin(3)_{TL}\times
Spin(1,2)_{S^{1}\times{\mathbb{R}}^{2}_{\epsilon_{1}}}}_{\subset
Spin(1,5)}\times\underbrace{Spin(3)_{NL}\times
Spin(2)_{{\mathbb{R}}^{2}_{\epsilon_{2}}}}_{{\subset Spin(5)}}\,.$ (57)

The holonomy on $X$ belongs to an $SU(3)\subset SU(4)\simeq Spin(6)$ subgroup.
This induces nontrivial holonomies for both $Spin(3)_{TL}$ and $Spin(3)_{NL}$,
further reducing the number of conserved supercharges. The special Lagrangian
property of $L$ implies that holonomies of the tangent and normal directions
to $L$ are related. This means that to any given loop $\gamma\in\pi_{1}(L)$
one may associate holonomies for both $NL$ and $TL$, denoted respectively
$g_{TL}\in Spin(3)_{TL}$ and $g_{NL}\in Spin(3)_{NL}$. Due to the fact that
$L$ is a special Lagrangian of $X$, these group elements are in fact related
to each other, allowing one to perform a topological twist by considering the
diagonal subgroup $Spin(3)_{dL}\subset Spin(3)_{TL}\times Spin(3)_{NL}$. Then,
spinors in the $({\bf 2,2})$ transform as ${\bf 2\otimes 2=1\oplus 3}$,
implying that one out of four components is invariant under holonomy. This
leaves four conserved supercharges, which transform as

$({\bf 1,2})_{+1/2}\oplus({\bf 1,2})_{-1/2}\quad\text{under}\quad
Spin(3)_{dL}\times
Spin(1,2)_{S^{1}\times{\mathbb{R}}^{2}_{\epsilon_{1}}}\times
Spin(2)_{{\mathbb{R}}^{2}_{\epsilon_{2}}}\,.$ (58)

These are the four supercharges of 3d ${\mathcal{N}}=2$ theory $T[L]$ arising
on the M5-brane worldvolume, along directions
$S^{1}\times{\mathbb{R}}^{2}_{\epsilon_{1}}$. This amount of supersymmetry can
be preserved under generic conditions under our assumptions, namely for an
M5-brane wrapping any special Lagrangian $L$ in any Calabi-Yau threefold $X$.

####  3.3.2 Supersymmetry enhancement from a geometric $U(1)$-symmetry

Following Aganagic:2011sg , the definition of a refined index requires an
additional supercharge in addition to those in (58). In view of this we make
the following assumption.

######  Assumption 3.3.

The Lagrangian $L$ is diffeomorphic to $S^{1}\times{\mathbb{R}}^{2}$ and there
exists a $U(1)$-action on $L$ which rotates the $S^{1}$-factor. The
corresponding vector field $V$ defines a normal plane ${\mathbb{R}}^{2}_{V}$
at each point. Then $\partial{\mathbb{R}}^{2}_{V}$ is the contractible
meridian on the torus $\partial L$.

The embedding $TL\subset TX$ defines a dual vector field $JV$ normal to $L$
($J$ is the complex structure on $X$). We view $JV$ as the direction in which
an M2-brane wrapping a holomorphic curve in $X$ attaches to $L$. (If the M2
boundary lies along a flow line of $V$ in $TL$, and the M2 wraps a holomorphic
curve in $X$, its normal is tangent to a flow line of $JV$ in $J\cdot TL\simeq
NL$, using the local splitting $TX\simeq TL\oplus NL$.) Let
${\mathbb{R}}^{2}_{NL}$ be the family of planes normal to $JV$ within $NL$.
Now we come to the assumption: the existence of a $U(1)$-action implies that
the holonomy $g_{TL}\in Spin(3)_{TL}$ is actually _not generic_ , but belongs
to a subgroup $Spin(2)_{V}$ that rotates the planes ${\mathbb{R}}^{2}_{TL}$
tangent to ${\mathbb{R}}^{2}_{V}$ as one goes around $S^{1}$ corresponding to
the boundary of a holomorphic curve. Since $L$ is special Lagrangian,
$g_{TL}=g_{NL}$ and therefore the holonomy induces a rotation of
${\mathbb{R}}^{2}_{NL}$. The holonomy $g_{V}=g_{TL}=g_{NL}\in
Spin(2)_{dL}\subset Spin(3)_{dL}$, so it rotates both summands in
${\mathbb{R}}^{2}_{TL}\oplus{\mathbb{R}}^{2}_{NL}$ by the same amount, leaving
fixed the ‘origin’ $S^{1}\times{\mathbb{R}}\subset TL\times NL$, corresponding
to the tangent and normal directions of a holomorphic curve wrapped by M2
attaching to $L$.

Our proposal is to identify $U(1)_{I}\equiv Spin(2)_{NL}$ as the R-symmetry of
the 5d theory $T_{5d}[X]$ that is necessary to perform a topological twist to
preserve additional supercharges leading to a refined index, recall the
discussion from section 2.4. We will elaborate on the details shortly.555This
is a subgroup $U(1)_{I}\subset SU(2)_{I}$ of the R-symmetry group of the 4d
${\mathcal{N}}=2$ theory on ${\mathbb{R}}^{4}$ arising from reduction on the
M-theory circle. It corresponds to the R-symmetry employed by Nekrasov:2002qd
; Nekrasov:2003rj  in the topological twist to define refined partition
functions of 4d ${\mathcal{N}}=2$ gauge theories.

First, we wish to stress that $U(1)_{I}$ is an artifact of the string theory
setup, and not a property of a garden-variety 3d ${\mathcal{N}}=2$ theory. On
the one hand, while surviving 3d ${\mathcal{N}}=2$ spinors are invariant under
$Spin(2)_{dL}$, they transform nontrivially under $U(1)_{I}$. On the other
hand, $U(1)_{I}$ is _not_ an automorphism of the 3d ${\mathcal{N}}=2$ super-
Poincaré algebra, i.e., it is not an R-symmetry of the 3d theory. To establish
these claims, recall that surviving spinors arise from the branching rule
${\bf 2\otimes 2=1\oplus 3}$ for $Spin(3)_{dL}\subset Spin(3)_{TL}\times
Spin(3)_{NL}$. Also recall that, following the above assumption, we identified
a distinguished $Spin(2)_{TL}\times Spin(2)_{NL}\subset Spin(3)_{TL}\times
Spin(3)_{NL}$, whose generators will be denoted $J^{3}_{TL},J^{3}_{NL}$.
Working in a basis where $J^{3}_{TL},J^{3}_{NL}$ are diagonal, we may
reclassify the spinors in (56) as follows:

$({\bf 4;4})\to(\pm 1/2,{\bf 2};\pm 1/2)_{+1/2}\oplus(\pm 1/2,{\bf 2};\pm
1/2)_{-1/2}$ (59)

as representations of of the following subgroup of (57)

$Spin(2)_{TL}\times
Spin(1,2)_{S^{1}\times{\mathbb{R}}^{2}_{\epsilon_{1}}}\times
Spin(2)_{NL}\times Spin(2)_{{\mathbb{R}}^{2}_{\epsilon_{2}}}\,.$ (60)

Topological twisting with reduced holonomy $Spin(2)_{TL}\times Spin(2)_{NL}$
implies retaining those supercharges that are invariant under the diagonal
subgroup. In this case, these are the ones with opposite charges
$j_{V}=-j_{JV}$ under $Spin(2)_{TL}\times Spin(2)_{NL}$:

$(\pm 1/2,{\bf 2};\mp 1/2)_{+1/2}\oplus(\pm 1/2,{\bf 2};\mp 1/2)_{-1/2}\,.$
(61)

This leaves a total of four plus four conserved supercharges, twice the
generic amount in (58). To lighten notation, we will sometimes omit the
representation labels under
$Spin(1,2)_{S^{1}\times{\mathbb{R}}^{2}_{\epsilon_{1}}}\times
Spin(2)_{{\mathbb{R}}^{2}_{\epsilon_{2}}}$, and simply use $(j_{V},j_{JV})$.

Among these, we can recognize the singlet ${\bf 1}$ that survived in the
general case (58). This must be a linear combination proportional to
$(1/2,-1/2)-(-1/2,1/2)$, since it arose as the second anti-symmetric power of
${\bf 2}$ for the diagonal subgroup $Spin(3)_{dL}$. Overall, there are four
supercharges in the singlet: a ${\bf 2}$ of $Spin(1,2)$ with $j_{2}=+1/2$ and
another with $j_{2}=-1/2$. There is also a second combination of supercharges
that is conserved, namely $(1/2,-1/2)+(-1/2,1/2)$. While this is part of the
triplet ${\bf 3}$ and therefore not invariant under the generic
$Spin(3)_{dL}$, it is nonetheless invariant under the reduced $Spin(2)_{dL}$.
Again, this corresponds to four supercharges. Therefore, our assumptions on
the geometry of $L$ and the existence of a $U(1)$-action imply doubling the
supersymmetry from 3d ${\mathcal{N}}=2$ in the generic setting to 3d
${\mathcal{N}}=4$ in presence of $U(1)$.666This fact was observed also in
Aganagic:2011sg  albeit somewhat implicitly. From a physics viewpoint, this
enhancement is also plausible from the viewpoint of the theory $T[L]$ of the
unknot conormal. This is $N_{f}=1$ SQED, which up to a (neutral) adjoint
chiral can indeed be viewed as a 3d ${\mathcal{N}}=4$ theory in disguise. It
is unclear if this enhancement is visible in the theory $T[L]$ for other knot
conormals.

For Lagrangians $L=S^{1}\times\mathbb{R}^{2}$ as above this has the following
consequences. From the point of view of the theory $T[L]$ in the generic case,
without the additional $U(1)$-symmetry, the Hilbert space of states is
generated by vortices and contains states of fixed vorticity $x^{r}$ and fixed
spin in the $\mathbb{R}^{2}_{\epsilon_{1}}$-plane, $q^{2s}$. The partition
function is a (super)trace which for fixed vorticity $x^{r}$ gives a Laurent
series in $q^{2}$. As the state space is generated by vortices, the
coefficients stabilize as the power of $q^{2}$ goes to infinity, and the
coefficient of $x^{r}$ can be expressed as $P_{r}(q^{\pm
2})(q^{2};q^{2})_{r}^{-1}$, where $P_{r}$ is a polynomial with integer
coefficients. With additional $U(1)$-symmetry, there is an additional
$t$-charge that refines this theory and its states. The Hilbert space is now a
sum of finite dimensional vector spaces of fixed charges $x^{r}q^{2s}t^{l}$.
As before, after fixing the vorticity $x^{r}$ the Poincaré polynomial of the
vector spaces stabilizes and we can write it as $\mathcal{P}_{r}(q^{\pm
2},t^{\pm 1})(q^{2};q^{2})_{r}^{-1}$, where $\mathcal{P}_{r}$ is a polynomial
with positive integer coefficients and such that $\mathcal{P}_{r}(q^{\pm
2},-1)=P_{r}(q^{\pm 2})$.

####  3.3.3 Omega-background and 3d index

Now we come back to the definition a refined 3d index (20). First we note
that, while surviving supercharges are not neutral under $Spin(2)_{NL}$,
neither do they transform in a representation of $Spin(2)_{NL}$. In this sense
$U(1)_{I}$ is not a symmetry of the 3d ${\mathcal{N}}=2$ superalgebra. For
this reason we consider linear combinations of the two conserved supercharges.
In particular, one may view $(1/2,-1/2)$ and $(-1/2,1/2)$ as being separately
conserved. With this change of basis, supercharges transform as
representations of $U(1)_{I}$: the first batch with charge $-1/2$, the second
batch with charge $+1/2$. Overall, the surviving supercharges are $({\bf
2})_{j_{V},j_{2}}$, where $j_{V}=-j_{JV}$ and $j_{V},j_{2}\in\\{-1/2,+1/2\\}$
are chosen independently. Note that $j_{I}\equiv j_{JV}=-j_{V}$ is the charge
under $U(1)_{I}$.

The point of having supercharges with a well-defined charge under $U(1)_{I}$
is that one can turn on the Omega-background and use the $U(1)_{I}$ remnant of
the 5d R-symmetry to perform a topological twist. The Omega-background means
that as we go around the M-theory circle, we perform a rotation of
$U(1)_{1}\times U(1)_{2}\times U(1)_{I}$. In particular, this breaks
$Spin(1,2)_{S^{1}\times{\mathbb{R}}^{2}}$ to $U(1)_{1}$ and spinors $({\bf
2})_{j_{V},j_{2}}$ get labeled by their $U(1)$ charges $(j_{1},j_{2},j_{I})$.
In this notation, the surviving supercharges are those with
$j_{1}=j_{2}=-j_{I}$, as derived in (16). This can be shown to match precisely
with the supercharges considered in Aganagic:2011sg .

###  3.4 R-charge, homological degree, and self-linking

We summarize the geometry near the Lagrangian considered above. The Lagrangian
itself is $S^{1}\times\mathbb{R}^{2}$ and a neighborhood of it in the Calabi-
Yau looks like its cotangent bundle $T^{\ast}(S^{1}\times{\mathbb{R}}^{2})$.
We furthermore have a $U(1)$-action that rotates the $S^{1}$-direction, and
which splits the normal bundle of the Lagrangian as
$NL=\mathbb{R}\times{\mathbb{R}}^{2}_{NL}$. We can therefore write a
neighborhood of the locus of the M5-brane as

$(S^{1}\times{\mathbb{R}})\times({\mathbb{R}}^{2}_{TL}\times\mathbb{R}^{2}_{NL})\times{\mathbb{R}}^{2}_{q}\times{\mathbb{R}}^{2}_{q^{-1}t}\times
S^{1},$ (62)

where normal directions lie in
${\mathbb{R}}\times{\mathbb{R}}^{2}_{NL}\times{\mathbb{R}}^{2}_{q^{-1}t}$.
Here ${\mathbb{R}}^{2}_{q}$ and ${\mathbb{R}}^{2}_{q^{-1}t}$ come from (21):
they are the planes that are rotated by $J_{1}$ and $J_{2}$, respectively. The
topological twist reduces the structure group of this bundle to a diagonal
$U(1)$ rotating both factors
${\mathbb{R}}^{2}_{NL}\times{\mathbb{R}}^{2}_{q^{-1}t}$ in the same way.

####  3.4.1 Once-wrapped disks

We next read off the $t$-charge of an M2-brane wrapped on
$(S^{1}\times{\mathbb{R}})\times S^{1}$. In the Calabi-Yau space this is a
holomorphic curve along $S^{1}\times{\mathbb{R}}$. By holomorphicity, its
normal variation along the boundary is given by a vector field
$(v,Jv)\in{\mathbb{R}}^{2}_{TL}\times{\mathbb{R}}^{2}_{NL}$. This implies
that, going around a loop in $S^{1}$, the vector field $v$ rotates in
${\mathbb{R}}^{2}_{TL}$ by an amount that equals that of $Jv$ in
${\mathbb{R}}^{2}_{NL}$. The former furthermore equals the self-linking
$C_{ii}$ in $L$. By the topological twist the rotation of $Jv$ contributes to
the $t$-degree and it follows that the corresponding $t$-charge of a $d$-fold
cover is $q^{-C_{ii}d}(-t)^{C_{ii}d}$, in line with the conjectural
identification of $C_{ii}$ and the homological $t$-degree Kucharski:2017poe ;
Kucharski:2017ogk .

We point out that the origin of the contribution $q^{C_{ii}d^{2}}$ to the
partition function is different: it comes from counting $q$-power in the
$U(1)$-skein and then passing to homology and (self-)linking. In other words,
the quadratic growth of powers of $q$ corresponds to the contribution to the
Gromov-Witten invariant from counting generalized holomorphic curves. In
physics this corresponds to the open topological string partition function
with an $A$-brane on $L$, where $q$ arises through the M-theory interpretation
of the string coupling constant Ooguri:1999bv . There are $C_{ii}$ crossings
between $d$-fold covers of an underlying once-around curve which gives
$q^{C_{ii}d^{2}}$. Overall we have contribution
$q^{C_{ii}d^{2}-C_{ii}d}(-t)^{C_{ii}d}$ for the $d$-fold cover of a once-
around disk. We stress that this reasoning is based on a picture of M2-branes
wrapping $d$ times around, viewed as tightly packed copies of an underlying
once-around disk.

Our prescription should recover the contribution to the unrefined partition
function in the limit $t=-1$. On this level we should perturb multi-covers of
the underlying curve and then count generalized curves. The geometry behind
the count is the following. A contribution to $C_{ii}$ can be thought of as an
infinitesimal kink. After perturbation we find an actual kink with $d^{2}$
crossings; in order for this to be $d$ times a curve in framing $C_{ii}$, we
deform to make all innermost kinks very small and then pull them out, making
them infinitesimal again and trading them for framing. This leaves $d(d-1)$
crossings, and summing contributions from all $C_{ii}$ infinitesimal twists we
eventually get $q^{C_{ii}d(d-1)}$. This agrees with the specialization to
$t=-1$.

######  Remark 3.4.

To round off our discussion on the geometric interpretation of the homological
degree of BPS states, we offer yet another reason why such geometric
information (the ‘wiggling’ of an M2-brane boundary) should play a role in the
definition of the BPS partition function. M2-branes wrapping holomorphic
curves produce Wilson lines for the $U(1)$ Chern-Simons theory on $L$. From
the viewpoint of the full 6d ${\mathcal{N}}=$(2,0) theory describing the
worldvolume dynamics of a single M5-brane, such Wilson lines correspond to
reduction of Wilson surfaces. By supersymmetry, these involve integration
along the boundary of the M2 of the abelian 2-form connection, as well as the
five scalars. In particular, the latter transform as a vector under the
$Spin$(5) R-symmetry of the (2,0) theory. Recalling that R-symmetry
corresponds to rotations transverse to the brane, the winding number of these
scalars around the brane must correspond to their charge under R-symmetry. At
the same time, the scalars parametrize embeddings of M5 in the transverse
directions, and in particular they parametrize the direction along which the
M2 ends on M5. Yet again, this suggests that the wiggling of the M2 boundary
is captured by the R-charge of a BPS state, which in the setup involving
$L\subset X$ is eventually identified with the homological degree.

####  3.4.2 Multiply-wrapped disks

Our suggestion for refinement for multiply wrapped disks have two sources. The
first arises when we express the contribution of the curve in terms of
standard vortices, or geometrically in terms of a specific link as described
in section 3.2. Here, internal $t$-charge of the M2-brane is transferred to
thin annuli and appears as a difference in framing along their two boundary
components. The second comes from an external framing analogous to the framing
of the standard basic disks discussed above.

To describe this external $t$-degree, we refer to section 3.1.2 where multiply
wrapped disks are related to disks in orbifolds and consider an M2-brane
wrapped on a disk in a degree $\mu$ orbifold corresponding to (37). The
$t$-charge arises from the $U(1)$-action on $L$, and the M2-brane is invariant
under this action (rotation along $S^{1}$) only if its boundary remains a
tightly wrapped $\mu$-fold cover of $S^{1}\times\text{point}$. This means that
its boundary gives a single Wilson line of charge $\mu$ rather than a
$\mu$-times around Wilson line of charge $1$. As in section 3.4.1, by
holomorphicity and the topological twist that identifies rotations in
${\mathbb{R}}_{NL}$ and ${\mathbb{R}}^{2}_{q^{-1}t}$, the $t$-degree is the
framing of this charge-$\mu$ Wilson line, which in turn is determined by its
self-linking. We define $C_{ii}$ to be the framing density along the
$\mu_{i}$-fold boundary of the M2-brane, thus
$C_{ii}\in\frac{1}{\mu_{i}}\mathbb{Z}$. This means that a single twist of the
underlying simple curve corresponds to $C_{ii}=\frac{1}{\mu_{i}}$. The total
twist in the normal bundle is then $C_{ii}\mu_{i}\in{\mathbb{Z}}$, which leads
to the charge $x_{i}^{d}\sim q^{-C_{ii}\mu_{i}d}(-t)^{C_{ii}\mu_{i}d}$ for a
$d$-fold cover of the $\mu$-wrapped M2-brane. Next we consider the self-
linking contribution to $q$-degree for generalized curves invariant under the
covering transformation. We think of the framing $C_{ii}\mu_{i}$ of the
underlying curve as a collection of infinitesimal kinks. These lift to
$C_{ii}\mu_{i}^{2}$ kinks, giving the total contribution to the $q$-degree
$q^{C_{ii}\mu_{i}^{2}d^{2}}$ at the $d$-fold level of the brane. In summary we
then have contributions proportional to
$q^{C_{ii}\mu_{i}^{2}d^{2}-C_{ii}\mu_{i}d}(-t)^{C_{ii}\mu_{i}d}$ at the
$d$-fold covering level.

Similarly to the once-around case, we consider the $t=-1$ specialization of
this formula and the contribution to generalized curves after perturbation. As
above we consider the lift to the $d$-fold cover. A $\mu_{i}$-wrapped curve
with $C_{ii}\mu_{i}^{2}$ infinitesimal kinks gives $C_{ii}\mu_{i}^{2}d^{2}$
crossings at the $d$-fold covering level, after perturbation. In order to lift
a curve $d$ times around a curve with framing $C_{ii}\mu_{i}$, we trade the
innermost kinks at $C_{ii}\mu_{i}$ of the crossings for framing, leaving a net
$C_{ii}\mu_{i}^{2}d^{2}-C_{ii}\mu_{i}d$ crossings and hence an overall
contribution of $q^{C_{ii}\mu_{i}d(\mu_{i}d-1)}$.

We next consider a degree $\mu_{i}$ basic disk on a more general Lagrangian of
topology $S^{1}\times\mathbb{R}^{2}$. Here, as before, we get a
$U(1)$-symmetric boundary of the disk by SFT-stretching. In the limit where
the embedded disk is split off, the lower piece close to the Lagrangian can be
identified with the complement of the orbifold point in an orbifold disk. The
addition of the orbifold point corresponds to adding the embedded disk ‘at
infinity’. Again, the existence of a $U(1)$-action on $L$ implies that all
boundaries of holomorphic curves should sit very close to each other, within
an infinitesimal neighborhood of the orbits $S^{1}\times\text{point}$. The
$t$-degree is then obtained from the framing of the once-around charge-$\mu$
boundary in the plane field normal to the orbits, and the contribution to the
refined partition function at level $d$ is
$(-t)^{C_{ii}\mu_{i}d}q^{C_{ii}\mu_{i}^{2}d^{2}-C_{ii}\mu_{i}d}$, exactly as
above.

###  3.5 Multiply-wrapped basic disks and generalized quiver partition
functions

In this section we collect the results of the arguments in section 3.4 and
define a partition function for generalized quivers.

A _generalized quiver_ $Q$ is an oriented graph with a finite number of
vertices connected by finitely many signed arrows (signed oriented edges). We
denote the set of vertices by $Q_{0}$ and the set of arrows by $Q_{1}$. A
_dimension vector_ for $Q$ is a vector in the integral lattice with basis
$Q_{0}$, $\boldsymbol{d}\in{\mathbb{Z}}Q_{0}$. We number the vertices of $Q$
by $1,2,\dots,m=|Q_{0}|$. Each vertex $i$ has a positive integral
_multiplicity_ $\mu_{i}$. The  _adjacency matrix_ of $Q$ is the $m\times m$
rational matrix with entries $C_{ij}$ defined as follows. Let
$N_{ij}\in\mathbb{Z}$ denote the number of signed arrows from $i$ to $j$. Then

$C_{ij}=\begin{cases}N_{ij}&\text{ if }\mu_{i}\neq\mu_{j},\\\
\frac{1}{\mu}\cdot N_{ij}&\text{ if }\mu_{i}=\mu_{j}=\mu.\end{cases}$ (63)

A generalized quiver is symmetric if its adjacency matrix is.

Let $Q$ be a generalized quiver with nodes $1,2,\dots,m$ of multiplicities
$\mu_{1},\dots,\mu_{m}$ and adjacency matrix $C_{ij}$. We associate variables
$x_{i}$ to the generalized quiver nodes $i$ and define two partition functions
as follows.

######  Definition 3.5.

The partition function of $Q$ is

${P_{Q}(\boldsymbol{x};q)=\sum_{\boldsymbol{d}}(-1)^{\sum_{i}C_{ii}\mu_{i}d_{i}}q^{\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\prod_{i=1}^{|Q_{0}|}\frac{x_{i}^{d_{i}}}{(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}}}\,,$
(64)

and the refined partition function of $Q$ is

${P_{Q,t}(\boldsymbol{x};q)=\sum_{\boldsymbol{d}}(-1)^{\sum_{i}C_{ii}\mu_{i}d_{i}}q^{\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\prod_{i=1}^{|Q_{0}|}\frac{x_{i}^{d_{i}}}{(q^{2};q^{2})_{\mu_{i}d_{i}}}\frac{(-q^{2}t^{\sigma_{i}};q^{2})_{d_{i}\mu_{i}}}{(-q^{2\mu_{i}}t^{\sigma_{i}};q^{2\mu_{i}})_{d_{i}}}}\,,$
(65)

where $\boldsymbol{d}$ ranges over all dimension vectors and where
$\sigma_{i}=\pm 1$.

Note that $P_{Q,t=-1}(\boldsymbol{x};q)=P_{Q}(\boldsymbol{x};q)$ and all
exponents of $q$ and $t$ appearing in the formulas are integers.

Definition 3.5 is the starting point for our main conjecture on HOMFLY-PT
homology. Consider a knot $K$ and its conormal $L_{K}$ in the resolved
conifold. We conjecture that it (after degeneration onto the unknot conormal)
admits a $U(1)$-symmetry and that there is a finite number of basic disks that
we label $1,\dots,m$. We view these disks as generalized quiver nodes, where
the multiplicity of a node is the multiplicity of the boundary of the
corresponding basic disk. Attaching data for the normal bundles of the basic
disks give a quiver adjacency matrix $C_{ij}$. Here, for $i\neq j$, $C_{ij}$
is the infinitesimal mutual linking (a relative framing). In particular
$C_{ij}$ is integral if $\mu_{i}\neq\mu_{j}$. For $\mu_{i}=\mu_{j}=\mu$,
$C_{ij}$ is $\frac{1}{\mu}\cdot N_{ij}$, where $N_{ij}$ is the self- or
mutual-linking, depending on whether $i=j$ or $i\neq j$. This defines a
symmetric generalized quiver $Q$ associated to $K$. Let $\log x$ be the
positive generator of $H_{1}(L_{K})$ and let $\log a^{2}$ denote homology
class of $\mathbb{CP}^{1}$ in the resolved conifold. Let the homology class of
the generalized disk $i$ be $\mu_{i}\log x+a_{i}\log a$ and its invariant
self-linking (or 4-chain intersectiond far from the boundary) be $q_{i}$.

######  Conjecture 3.6.

If we substitute

$x_{i}=x^{\mu_{i}}q^{q_{i}-C_{ii}\mu_{i}}a^{a_{i}}(-t)^{C_{ii}\mu_{i}},$ (66)

for $i=1,2,\dots,m$, then the following hold.

  * •

The generalized quiver partition function
$P_{Q}(\boldsymbol{x};q)=P_{Q,t=-1}(\boldsymbol{x};q)$ equals the generating
function of the Gromov-Witten invariants counting generalized holomorphic
curves, or equivalently the generating function of the symmetrically colored
HOMFLY-PT polynomials of $K$.

  * •

The refined generalized quiver partition function $P_{Q,t}(\boldsymbol{x};q)$
equals the generating function of Poincaré polynomials of HOMFLY-PT homology
in the symmetric represtentions.

######  Remark 3.7.

The powers $\sigma_{i}=\pm 1$ in the refined partition function are
interpreted as follows: view the $\mu_{i}$ of the boundary of the generator
$x_{i}$ as strands of a braid over the underlying once-around curve; we obtain
a connected $\mu_{i}$-times around loop by gluing the endpoint of the last
strand to the beginning of the first one, and this can be done passing above
$\sigma_{i}=+1$ or below $\sigma_{i}=-1$ all other intermediate strands,
according to the half framing of the attaching data of the neighborhood of the
basic disk.

######  Remark 3.8.

The change of framing operation, counterpart to changing variables $x\to
x(-y)^{f}$ with $f\in{\mathbb{Z}}$ in the augmentation curve, is given by the
usual shift of the linking matrix $C_{ij}\to C_{ij}+f$, both for diagonal and
off-diagonal elements, in the partition function.

######  Remark 3.9.

Equation (65) completes and corrects (Ekholm:2018eee, , Conjecture 1.1) as
follows. There, a $d$ times around generator in framing zero was assumed to be
an embedded disk that contributes $(x^{d};q^{2})_{\infty}$ to the partition
function. Its contribution to the refined partition function was not specified
as can be seen already on level $d$. To do so, one would have to explain how
to express the current denominator $(q^{2};q^{2})_{1}$ in terms of
$(q^{2};q^{2})_{d}$. Here the basic object is different: it is an embedded
disk outside completed by a multiply covered disk inside. It contributes
$(x^{d};q^{2d})_{\infty}$ to the partition function and on the refined level
its denominators are corrected to $(q^{2};q^{2})_{kd}$ in (64).

Equations (64) and (65) differ from the proposed knots-quivers correspondence
of Kucharski:2017ogk ; Kucharski:2017poe  in several ways. They are not quiver
partition functions, nor are they characters of a cohomological Hall algebra
Kontsevich:2008fj ; 2011arXiv1103.2736E . Nevertheless, both expressions
reflect the idea that the whole spectrum of BPS states is generated by a
finite set of basic objects, interacting in a certain way. The basic objects
may be viewed as nodes of a ‘generalized quiver’, whose adjacency matrix
encodes their interactions. The main difference with the knots-quivers
correspondence is that we allow for nodes with $\mu>1$ and with new
denominators.

We list some arguments that show that some properties of our proposed
partition function are necessary:

  1. 0.

In any quiver-like description, nodes with $\mu>1$ are necessary, see
Ekholm:2018eee  and the examples in section 6.

  2. 1.

A basic disk wrapping $\mu$-times around $L_{K}$ contributes to the multi-
cover formula with $q^{2}$ replaced by $q^{2\mu}$. See section 3.1 for
motivation and section 6 for examples. This leads to replacing denominators
$(q^{2};q^{2})_{d_{i}}$ in (6) by $(q^{2\mu_{i}},q^{2\mu_{i}})_{d_{i}}$ in
(64).

  3. 2.

Poincaré polynomials of knot homologies seem to agree with the refinement of
vortex Hilbert spaces, as discussed in section 3.2 where denominators of
$x^{\mu d}$ correspond to contributions from standard vortices,
$(q^{2},q^{2})_{\mu d}$. The change of variables then requires multiplication
and division by the finite polynomial: $(q^{2},q^{2})_{\mu
d}/(q^{2\mu_{i}},q^{2\mu_{i}})_{d_{i}}$.

  4. 3.

In the next step we consider refinement. The correction introduced in the
numerator by $(q^{2},q^{2})_{\mu d}/(q^{2\mu_{i}},q^{2\mu_{i}})_{d_{i}}$
should arise from a similar correction by polynomials in $(q,t)$ with positive
coefficients, where negative signs in the original expression come from (odd)
powers of $t$. Algebraically, there are several possibilities. Our proposal is
based on the passage from Bott to Morse localization in (53)–(55). We check it
in concrete calculations of colored superpolynomials for knots $9_{42}$ and
$10_{132}$ in section 6.

###  3.6 HOMFLY-PT homology and geometric deformations

Sections 3.3.2 and 3.4 explain how holomorphic disks in a $U(1)$-symmetric
setting, via branched covers with constant ghost bubbles attached, generate
symmetrically colored HOMFLY-PT homology. In this section we isolate the
ingredients in this description and show how they give rise to a deformation
invariant version: holomorphic curves for generic almost complex structures
(constraint near the Lagrangian only) give a chain complex with a
differential, the homology of which remains invariant and equals certain
filtered quotients of HOMFLY-PT homology that recover all of the original
HOMFLY-PT homology. Naturally, as a chain complex is not determined by its
homology, the chain complexes we propose ar not uniquely determined. The
guiding principle for our definition is that it should reflect the underlying
geometry and that it should model the initial $U(1)$ invariant situation as
closely as possible.

####  3.6.1 Tracing the refined partition function under deformation

Recall the data that gave the generalized quiver partition function from a
$U(1)$-symmetric version of the knot conormal: a collection of numbered basic
disks $i\in\\{1,\dots,n\\}$ together with linking densities $C_{ij}$ measuring
linking and self linking between their boundaries. Here the holomorphic curves
are in fact two-level holomorphic buildings in an SFT-stretched complex
structure: the outside parts are embedded once punctured spheres, while the
inside parts are unbranched covers of the trivial cylinder over the unique
geodesic in $S^{1}\times\mathbb{R}^{2}$. In this limit all curve boundaries
are multiples of the unique geodesic, and the linking densities $C_{ij}$
appear as the (normalized, relative) framing of these boundary components. We
point out that the only curves near the Lagrangian that are invariant under
the $U(1)$-action are these unbranched covers of the punctured disk.

We next consider a class of deformations for which the picture above persists.
To this end we express the data we have as follows. A collection of
$\mathcal{M}_{i}$, moduli spaces of holomorphic punctured spheres asymptotic
to Reeb orbits, and trivial cylinders near $L_{K}$ with linking densities
$C_{ij}$, where $C_{ij}$ are integers if $\mu_{i}\neq\mu_{j}$ and
$C_{i,j}\in\frac{1}{\mu}\mathbb{Z}$ if $\mu_{i}=\mu_{j}=\mu$. We will now keep
the trivial cylinders and linking densities $C_{ij}$, but allow for
deformations of the outside curves. More precisely, we consider the moduli
spaces $\mathcal{M}_{i}$ for varying almost complex structure $J_{s}$ on the
outside part, while we keep $J$ fixed and equal to the stretched structure
near the negative end of the symplectic manifold (i.e., near the Lagrangian).

To organize this, we note that the homology data of each punctured sphere in
$\mathcal{M}_{i}$ is characterized by its Reeb multiplicity $\mu_{i}$, its
homology class $a_{i}$, and its $4$-chain intersection $q_{i}$. We write
$\mathcal{M}(\mu,\alpha,\nu;\chi=1)$ for the moduli space of all embedded
punctured spheres $i$ with $\mu_{i}=\mu$, $a_{i}=\alpha$, and $q_{i}=\nu$.
Then

$\mathcal{M}(\mu,\alpha,\nu;\chi=1)\qquad=\quad\bigcup_{\\{i\colon\mu_{i}=\mu,a_{i}=\alpha,q_{i}=\nu\\}}\mathcal{M}_{i}$

is a space with a finite number of points, one for each disk $i$ that has
these quantum numbers. Furthermore, if their orientations are twisted by their
orientation data at the boundary, all the points come with a positive sign.

We next study the moduli spaces $\mathcal{M}_{s}(\alpha,\nu,\mu;\chi=1)$
corresponding to a 1-parameter family of almost complex structures $J_{s}$. We
argue that a generic $1$-parameter family gives a cobordism of moduli spaces
with the following properties. At generic times $s$ the actual spheres in the
moduli space admit standard neighborhoods such that any other holomorphic
curve inside such a neighborhood must be a branched cover of the basic sphere.
This is analogous to what happens in the neighborhood of the central sphere in
the resolved conifold: here any other holomorphic curve must lie completely
over the central sphere. There are finitely many instances where there are
Morse modifications in our 1-dimensional cobordism. Such instances correspond
to birth/death of curves. Also for this phenomenon, there is a standard
neighborhood of the sphere at the critical moment, that contains only the
central curve and the two redundant curves for nearby times, while all other
curves are multiple covers of these. We mention that also moduli spaces
$\mathcal{M}(\alpha,\mu,\nu;\chi=\chi_{0})$ of punctured embedded curves of
Euler characteristic $\chi_{0}<1$ can be handled in the the same way and that
our conjecture says that these moduli spaces are empty in the $U(1)$-symmetric
situation.

We next discuss how one might establish such a structural result, and which
objects the corresponding moduli spaces actually contain. One starts with the
simplest punctured spheres with once-around boundary. Here the result on
moduli spaces is straightforward: by minimality there can be no splitting and
the moduli space gives a cobordism. We next consider embedded punctured
spheres with boundaries that go twice around. The entire moduli space for
these must also include certain two-level curves: two once-around spheres on
the outside, with a branched covered trivial cylinder in the symplectization
level. We do not perturb the cylinder, but equip it with both an obstruction
bundle and with gluing data to the spheres above it, which then allows us to
count. Once we check that these configurations contain all possible limits, we
would get the required cobordism. The only configuration not accounted for in
the SFT-compactification is then the one given by a branch-covered cylinder
glued to two copies of the same once-around sphere. However, since the once-
around sphere has a standard neighborhood, the whole curve would have to agree
with this configuration at earlier times and then already from the first step,
which contradicts our sphere starting out as embedded. After establishing the
existence of neighborhoods for twice around disks, the argument proceeds to
the next Reeb orbit covering level. It is clear how to continue: include all
several level curves with branched covered cylinders at the negative end, the
existence of the standard neighborhoods at earlier steps shows that the moduli
space gives a 1-dimensional cobordism. Thus if we start from a finite number
of moduli spaces, the cobordism class never changes as we deform $J_{s}$. The
same argument works to show that the algebraic count of embedded higher genus
curves equals zero since there are no such curves at the starting
$U(1)$-symmetric configuration. We point out that an important property of the
knot conormal used in this argument is that it can be placed so that all
outside curves go positively around the basic geodesic. This is arranged by
placing the conormal close to the toric Lagrangian on an external leg of the
conifold toric diagram.

####  3.6.2 Partition functions with values in chain complexes

As explained in section 3.6.1, we can find the symmetrically colored HOMFLY-PT
homology from the moduli spaces of embedded curves
$\mathcal{M}(\alpha,\nu,\mu;\chi=1)$ for any generic almost complex structure
with standard negative end near the Lagrangian, provided we know the linking
information. For an actual deformation this is easy to trace, but it may also
be useful to have a bifurcation analysis version of this. The behavior of the
full chain complex is somewhat complicated, see section 3.7, but certain
filtered quotients, from which the full homology can be recovered, are simpler
to trace. In this section we introduce these quotients and study them under
homotopy.

We start by considering the filtration we will use for the $U(1)$-symmetric
configuration. Let $K$ be a knot and $Q$ its associated generalized quiver
with set of nodes $Q_{0}$. We subdivide the nodes according to their
multiplicity and write $Q_{0}=\bigcup_{\mu=1}^{M}Q_{0}(\mu)$, where
$Q_{0}(\mu)$ is the set of nodes corresponding to disks of multiplicity $\mu$.
To each node $j$ we associate a quiver variable $x_{j}$ and basic charges

$x_{j}=x^{\mu_{j}}q^{q_{j}}a^{a_{j}}q^{C_{jj}\mu_{j}}(-t)^{C_{jj}\mu_{j}}\,.$
(67)

The refined partition then has the form

$P_{Q}(x_{1},\dots,x_{m},q,t)=\sum_{r}h_{r}(a,q,t)\frac{x^{r}}{(q^{2};q^{2})_{r}},$
(68)

where $h_{r}$ is a Laurent polynomial with positive integer coefficients that
– we conjecture – equals the Poincaré polynomial of HOMFLY-PT homology in the
$r^{\rm th}$ symmetric coloring. Here we will categorify this in a trivial
way. We view the monomials in $h_{r}(a,q,t)$ together with the denominators as
generators of a vector space of equivariant vortices of corresponding charges.
Explicitly, a to summand $n\frac{a^{k}q^{s}t^{v}}{(q^{2};q^{2})_{r}}$, $n>0$,
we associate a vector space $\mathbb{C}^{n}$ spanned by $n$ equivariant
vortices of charges $a^{k}q^{s}t^{v}x^{r}$. We further think of these vector
spaces as a chain complex with the trivial differential. We write the chain
complex $\mathcal{C}_{0}=\bigoplus_{r}\mathcal{C}_{0,r}$ and its homology
$\mathcal{H}=\bigoplus_{r}\mathcal{H}_{r}$,where the extra subscript $0$
refers to the starting point of the deformation we will consider. As the
differential $\delta_{0}$ on $\mathcal{C}_{0}$ is trivial, we have
$\mathcal{C}_{0}=\mathcal{H}$.

We next consider a filtration on $\mathcal{C}_{0}$. The vector space
$\mathcal{C}_{0,r}$ is spanned by equivariant vortices. Each such vortex is a
bound state of one of more basic disks $x_{j}$: it arises in the refined
partition function as a coefficient of a unique monomial of the form
$x_{j_{1}}^{k_{1}}\dots x_{j_{l}}^{k_{l}}$ and we define its filtration degree
as the maximal multiplicity $\mu_{j_{i}}$ of the quiver variables $x_{j_{i}}$
in this monomial. We then define
$\mathcal{C}_{0,r}(k)\subset\mathcal{C}_{0,r}$ as the subspace spanned by
vortices of filtration degree $\leq k$, and
$\mathcal{C}_{0}(k)=\bigoplus_{r}\mathcal{C}_{0,r}(k)$. Then

$\mathcal{C}_{0}(1)\subseteq\mathcal{C}_{0}(2)\subseteq\dots\subseteq\mathcal{C}_{0}(M)=\mathcal{C}_{0}.$
(69)

Consider now the quotient $\mathcal{C}_{0}(r)/\mathcal{C}_{0}(r-1)$. It is
rather complicated, spanned by all bound states of nodes of multiplicity $\leq
r$ that contain at least one node of multiplicity $r$. However, if we restrict
to the degree $r$ part of the complex
$\mathcal{C}_{0,r}(r)/\mathcal{C}_{0,r}(r-1)$, we find something simpler; here
any monomial is a single multiplicity $r$ generator and we can write the
contribution as a sum of collections of vortices of the following form:

$(1+t^{\pm 1}q^{2})(1+t^{\pm 1}q^{4})\dots(1+t^{\pm
1}q^{2r-2})\frac{x_{j}}{(q^{2};q^{2})_{r}}.$ (70)

We define a new chain complex of equivariant vortices generated only by the
$r$ times around nodes. We take $\mathcal{D}_{0}(r)$ to be the vector space of
equivariant vortices corresponding to $r$ times around disks. Recall that such
vortices have partition functions

$\frac{x^{rd}}{(q^{2r};q^{2r})_{d}}.$ (71)

We define $\mathcal{D}_{0}(r)$ to be the chain complex with trivial
differential associated to the unrefined partition function, except for
substituting the variables $x_{i}$ by the above powers of $x,a,q,t$:

$\check{P}_{Q(r)}(x,q,t)=P_{Q(r)}(\boldsymbol{x};q^{\mu})\,,$ (72)

where $q^{\mu}$ denotes the substitution of denominators $(q^{2};q^{2})_{d}$
by $(q^{2\mu};q^{2\mu})_{d}$ in the quiver partition function (6) and where
$Q_{K}(r)$ denotes the multiplicity $r$ part of the quiver $Q_{K}$, i.e., all
the degree $r$ nodes and all weighted arrows connecting them. We let
$\mathcal{G}(r)$ denote the homology of $\mathcal{D}_{0}(r)$ with the trivial
differential. Geometrically, this is the vector space generated by the Bott
equivariant vorticity $r$ vortices as in (54).

We will show below that for a generic path of complex structures $J_{s}$ we
can define a family of complexes $\mathcal{D}_{s}$ with differentials
$\delta_{s}$ such that the homology of $\delta_{s}$ is isomorphic to
$\mathcal{G}$ for each generic $s$.

######  Remark 3.10.

We have $\mathcal{D}_{0}(1)=\mathcal{C}_{0}(1)$. Furthermore, since the quiver
is finite, there is $R>0$ such that $\mathcal{G}_{s}(r)=0$ for $r>R$ and if we
know $\mathcal{G}_{s,r}(r)$ for all $r\leq R$, then together with the framing
density matrix $C_{ij}$ and half-framing information for each node of
multiplicty $\geq 1$ we can recover $\mathcal{H}$.

We next consider redundant nodes. Consider a quiver node $i$ of multiplicity
$\mu_{i}=\mu$ with associated quiver variable $x_{i}$ and linking density
$C_{ii}=0\in\frac{1}{\mu}\mathbb{Z}$. Its $\mu$-refined partition function is
given by

$\check{P}_{(i)}(x,q,t)=\sum_{d_{i}}\frac{x_{i}^{d_{i}}}{(q^{2\mu};q^{2\mu})_{d_{i}}}.$
(73)

We define the  _redundant node_ of $i$ as the quiver node $i^{*}$ of
multipicity $\mu_{i^{*}}=\mu$ and with linking density
$C_{i^{*}i^{*}}=\frac{1}{\mu}$. Then the $\mu$-refined partition function of
$i^{*}$ in the quiver variable $x_{i^{*}}$ reads

$\check{P}_{(i^{*})}(x,q,t)=\sum_{d_{i^{*}}}(-1)^{d_{i^{*}}}q^{\mu^{2}d_{i^{*}}^{2}}\frac{x_{i^{*}}^{d_{i^{*}}\mu}}{(q^{2\mu};q^{2\mu})_{d_{i^{*}}}},$
(74)

and setting $x_{i^{*}}=q^{-\mu}(-t)x_{i}$ we find that $P_{i,t=-1}(x_{1})\cdot
P_{i^{*},t=-1}(x_{i^{*}})=1$. We call the pair $(i,i^{*})$ the degree $\mu$
redundant pair of linking density zero. We think of the redundant pair as a
two node generalized quiver with nodes $i$ and $i^{*}$ with linking density
matrix

$\left(\begin{matrix}0&0\\\ 0&\tfrac{1}{\mu}\end{matrix}\right).$ (75)

We define the degree $\mu$ redundant pair of linking density
$f\in\frac{1}{\mu}{{\mathbb{Z}}}$ by a fractional framing change on the
linking density zero pair. Thus its linking density matrix is instead

$\left(\begin{matrix}f&f\\\ f&f+\tfrac{1}{\mu}\end{matrix}\right).$ (76)

Since framing change alters the partition function of a quiver according to
the substitution $x_{i}\mapsto q^{\mu_{i}^{2}f}(-t)^{f\mu_{i}}x_{i}$, the
unrefined partition function of a redundant pair is still $1$.

The degree $1$ redundant pair is a ‘standard’ redundant pair noticed first in
Kucharski:2017ogk  and explained in terms of multi-cover skein relations in
Ekholm:2019lmb .

######  Remark 3.11.

We comment on the geometry of the redundant nodes. Think of the node $1_{0}$
as an orbifold disk. A redundant disk should have underlying framing different
by one twist. We can imagine undoing the twist and picking up $\mu_{1}$
4-chain intersection, the shift in quiver variable $q^{-\mu_{1}}$ can be
thought of as 4-chain intersections canceling these and returning the original
disk.

The fractional framing change of a redundant pair corresponds to framing in
the $\mu_{1}$-fold cover where the lift lives and indicates that a framed
redundant pair cannot be equivariantly separated.

We now return to the definition of our chain complex at generic complex
structures. For each node $i$ of the generalized quiver corresponding to an
embedded disk $i$ pick a redundant node $i^{\ast}$ with
$C_{i^{\ast}i^{\ast}}=C_{ii}+\frac{1}{\mu_{i}}$, $C_{ii^{\ast}}=C_{ii}$, and
$C_{ij}=C_{i^{\ast}j}$. We will define chain complexes along a path $J_{s}$ of
complex structures using a level by level construction.

Consider the chain complexes $\mathcal{D}_{0}(r)$ for integers $r>0$. This
complex is generated by the punctured spheres in the moduli spaces
$\mathcal{M}(\mu,\alpha,\nu)$ where $\mu=r$. Let the linking matrix be
$C_{ij}$. Consider the $1$-parameter family $J_{s}$ of almost complex
structure and the corresponding moduli spaces
$\mathcal{M}_{s}(\mu,\alpha,\nu)$. When $J_{s}$ moves away from the
$U(1)$-symmetric configuration, the moduli spaces undergo Morse modifications
at birth/death instances and is otherwise unchanged. We will describe
corresponding changes in the chain complex.

At a birth moment $s_{0}$, the chain complex at times slightly before $s_{0}$,
$\mathcal{D}_{s_{0}-\epsilon}(r)$, changes by addition of of a redundant pair
to a new chain complex $\mathcal{D}_{s_{0}+\epsilon}(r)$ with differential
$\delta_{s_{0}+\epsilon}$ such that the homology of $\delta_{s_{0}+\epsilon}$
is equal to that of $\delta_{s_{0}-\epsilon}$ (and hence still equal to
$\mathcal{G}(r)$). Below we show that such a differential exists. At death
moments the chain complex changes by removal of a redundant pair, and
similarly the homology is unchanged.

####  3.6.3 The differential on $\mathcal{D}_{s}$

We show that the required differential exists. Let $Q$ be a generalized quiver
with $m$ nodes of vorticity $\mu$. The birth of a redundant pair of nodes
involves modifying $Q$ by adding a pair of nodes which link with all previous
$m$ nodes in the same way, resulting in a new quiver $Q^{\prime}$. The signs
of the two spheres in the redundant pairs are opposite. This means that their
framings differ by one unit when the orientation data is pushed to the
boundary, and the disks must form a redundant pair as the partition function
stays unchanged at the unrefined level.

Therefore $P_{Q^{\prime}}=P_{Q}\cdot 1$ where the identity is represented as
follows

$\begin{split}1&=\sum_{r\geq 0}\sum_{d=0}^{r}(-1)^{(\rho+1)r+d}q^{\mu\rho
r^{2}+\mu(r-d)^{2}+2\mu
r(\alpha_{1}d_{1}+\dots+\alpha_{m}d_{m})}\frac{x_{m+1}^{d}x_{m+2}^{r-d}}{(q^{2\mu};q^{2\mu})_{d}(q^{2\mu};q^{2\mu})_{r-d}}\end{split}.$
(77)

Here $\mu$ is the multiplicity of the redundant pair of disks $x_{m+1}\sim
x_{m+2}\sim x^{\mu}$, while $\rho=\mu C_{m+1,m+1}=\mu C_{m+2,m+2}-1=\mu
C_{m+1,m+2}$ is the integer self and mutual linking of the pair of new nodes,
and $\alpha_{i}=C_{i,m+1}\mu_{i}=C_{i,m+2}\mu_{i}$ is the linking between the
pair and the $i^{\text{th}}$ pre-existing node. The identity is true provided
that $x_{m+2}=q^{-\mu}x_{m+1}$, and reduces to (Ekholm:2019lmb, , eq. (4.19))
when $\mu=1$.

Recall that we allow $C_{m+1,m+2}$ to be valued in $\frac{1}{\mu}{\mathbb{Z}}$
since $\mu_{m+1}=\mu_{m+2}=\mu$. In connection with this, note that at a
redundant pair of disks the $U(1)$-symmetry has necessarily been broken, disk
boundaries are no longer tightly packed, and single strands of a $\mu$-times
around disk can link with strands of other disks individually. For
perturbations not too far from the the $U(1)$-symmetric setting, we need to
relax integrality of mutual linking $C_{ij}$ only for pairs of disks with the
same homology data (the same $\mu$\- and $a$-degree). More general
perturbations may however require further relaxing integrality of the linking
matrix. Understanding this would require a further study of the geometry of
perturbations, which we leave to future work.

At the refined level (77) ceases to hold. By our general rules on the
assignment of homological degree, we refine the partition function by setting
$x_{m+2}=-tq^{-\mu}x_{m+1}$, which reflects the fact that $\mu C_{m+2,m+2}=\mu
C_{m+1,m+1}+1$. This deforms (77) as follows:

$\begin{split}&\sum_{r\geq
0}x_{m+1}^{r}\sum_{d=0}^{r}(-1)^{(\rho+1)r+d}q^{\mu\rho
r^{2}+\mu(r-d)^{2}+2\mu
r(\alpha_{1}d_{1}+\dots+\alpha_{m}d_{m})}\frac{(-tq^{-\mu})^{r-d}}{(q^{2\mu};q^{2\mu})_{d}(q^{2\mu};q^{2\mu})_{r-d}}\\\
=&\sum_{r\geq 0}x_{m+1}^{r}(-1)^{\rho r}q^{\mu(\rho+1)r^{2}-\mu r+2\mu
r(\alpha_{1}d_{1}+\dots+\alpha_{m}d_{m})}\frac{(1+tq^{2\mu(r-1)})\dots(1+q^{2\mu}t)(1+t)}{(q^{2\mu};q^{2\mu})_{r}}\,.\end{split}$
(78)

While this is not equal to $1$ anymore, it is an expression of the form

$1+(1+t)\sum_{r\geq 1}\frac{x^{\mu r}}{(q^{2\mu};q^{2\mu})_{r}}\cdot
f_{r}(a,q,t)\,,$ (79)

where the Laurent polynomial $f_{r}(a,q,t)$ is a sum of positive integer
multiples of monomials. We define the differential to map a vortex
corresponding to a monomial $q^{s}a^{r}t^{k}$ in $1\cdot f_{r}(a,q,t)$ to
$q^{s}a^{r}t^{k+1}$ in $t\cdot f_{r}(a,q,t)$ and to be $0$ on all other
vortices. This gives a differential of degree $(a,q,t)=(0,0,1)$ with desired
properties.

######  Remark 3.12.

It is crucial to keep the denominators $(q^{2\mu};q^{2\mu})_{r}$ of Bott-
equivariant vortices here. Switching to the usual denominator
$(q^{2};q^{2})_{\mu r}$ of standard equivariant vortices spoils the positivity
of coefficient in the numerator. The positivity of the coefficients is what
allows us to define the required differential, which then allows us to
interpreted the partition function as the Poincaré polynomial of the complex
$\mathcal{D}$ of Bott-vortices defined above. In order to carry out the same
program for a chain complex generated by standard vortices, one would have to
introduce other types of generators, see section 3.7 for an analogous
discussion.

######  Remark 3.13.

The differential $\delta_{s}$ defined in this way depends on the path of
almost complex structures $J_{s}$ as well as choices of near which basic disks
the redundant pair should be located. It would be interesting to consider the
construction of chain homotopies of different differentials on
$\mathcal{D}_{s}$ that came from different paths. (Note that any two such
paths are homotopic since the space of almost complex structures is
contractible.)

###  3.7 Unlinking and multiple disks

When defining the differentials $\delta_{s}$ in the complexes
$\mathcal{D}_{s}$ in section 3.6.2, we worked with a stretched almost complex
structure that allowed us to keep holomorphic disk boundaries fixed. This was
essential when demonstrating the existence of chain complexes still generated
by equivariant vortices and with invariant homology, along paths of almost
complex structures. Similar constructions not keeping all symmetry require
additional generators of new types. To illustrate this, we consider the basic
unlinking operation in the presence of nodes with multiplicity $>1$.

Recall that in the case of embedded disks there are, besides the birth/death
of redundant pairs of nodes, also other operations on quivers that do not
affect the partition function Ekholm:2019lmb . The most basic such move is
unlinking, where two linked nodes are replaced by two nodes with mutual
linking decreased by one unit, together with a new node obtained by gluing the
other two. We discuss the counterpart of this for multiply wrapped disks, and
show that such an identity involving multiply wrapped disks cannot exist,
unless we introduce certain additional holomorphic objects. This then
indicates that $U(1)$-invariant configurations are essential for defining
HOMFLY-PT homology, and that a much more involved description is required if
one allows to break that symmetry near the boundary. To see how unlinking
works for multiply covered disks, we consider crossing a simple disk in class
$x_{1}$ with a $\mu$-times around disk in class $x_{2}$. Recall that we can
write the contribution to the partition function by a $\mu$-times around disk
as a product

$\Psi_{q^{\mu}}(x_{2})=\Psi_{q}(\zeta)\Psi_{q}(e^{\frac{2\pi
i}{\mu}}\zeta)\dots\Psi_{q}(e^{\frac{2\pi i(\mu-1)}{\mu}}\zeta),$ (80)

where here $\zeta$ is a root of $\zeta^{\mu}=x_{2}$, and $\Psi_{q}$ is defined
in (84). We next assume that there is a disk $x_{1}$ that links once
equivariantly with $x_{2}$. To see the partition function after unlinking, we
average over all possible unlinkings, and get

$\Psi_{q}(x_{1})\Psi_{q}(x_{2})\cdot\left(\frac{1}{\mu}\sum_{k=0}^{\mu-1}\Psi_{q}^{-1}(qe^{\frac{2\pi
ik}{\mu}}\zeta
x_{1})\right)=\Psi_{q}(x_{1})\Psi_{q}(x_{2})\phi(x_{2}x_{1}^{\mu}),$ (81)

where

$\phi(x_{2}x_{1}^{\mu})=\sum_{d=0}^{\infty}(-q)^{\mu^{2}d^{2}}\frac{(qx_{2}x_{1}^{\mu})^{d}}{(q^{2};q^{2})_{\mu
d}}\,.$ (82)

This latter function is a new contribution from a $\mu$-times around disk with
a simple disk attached. For unlinking of more complicated disks we would
similarly get new objects, and more of them when we unlink further. To
categorify these changes, we would need to find contributions on the refined
level as well as necessary differentials.

##  4 Recursion relations for generalized quiver partition functions

A fundamental property of partition functions of knot invariants is the fact
that they obey certain types of recursion relations. In Ekholm:2018eee ;
Ekholm:2019lmb  we showed that quiver partition functions share this property,
in fact leading to a decomposition of (quantum) $A$-polynomials (and
generalizations thereof) into universal building blocks. Here we review these
properties, and extend them to include generalized partition functions with
multiply wrapped disk nodes.

###  4.1 Noncommutative variables

In Ekholm:2019lmb  it was shown that the partition function of a quiver $Q$
can be conveniently expressed by introducing noncommutative variables

${\mathbb{P}}_{Q}=\Psi_{q}(X_{m})\cdot\Psi_{q}(X_{m-1})\cdots\Psi_{q}(X_{1})\,,$
(83)

where

$\Psi_{q}(\xi)=\sum_{k\geq
0}\frac{q^{k}}{(q^{2};q^{2})_{k}}\xi^{k},\qquad\quad
X_{i}=(-1)^{C_{ii}}q^{C_{ii}-1}\hat{x}_{i}\hat{y}_{i}^{C_{ii}}\prod_{j<i}\hat{y}_{j}^{C_{ij}}\,.$
(84)

The noncommuting variables obey quantum torus relations
$X_{i}X_{j}=q^{2A_{ij}}X_{j}X_{i}$, where $A$ is an antisymmetrization of the
quiver matrix determined by a choice of ordering of nodes. This is based on
the convention that
$\hat{y}_{i}\hat{x}_{j}=q^{2\delta_{ij}}\hat{x}_{j}\hat{y}_{i}$. The quiver
partition function is obtained from ${\mathbb{P}}_{Q}$ by normal ordering,
defined as taking all the $\hat{y}_{i}$ variables to the right:

$P_{Q}=\ :\,{\mathbb{P}}_{Q}\,:$ (85)

####  4.1.1 Generalization to quivers with multiply wrapped basic disks

We can apply this idea to generalized partition function and write (64) in
operator form, as a generalization of (83). We single out a node, whose
variable we denote $x_{0}$, and rewrite the partition function as follows:

$\begin{split}P_{Q}&=\sum_{d_{0},\boldsymbol{d}}\Bigg{[}(-1)^{C_{00}\mu_{0}d_{0}+\sum_{i}C_{ii}\mu_{i}d_{i}}q^{C_{00}\mu_{0}^{2}d_{0}^{2}+2\sum_{i}C_{0i}\mu_{0}\mu_{i}d_{0}d_{i}+\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\\\
&\qquad\quad\times\left(\frac{x_{0}^{d_{0}}}{(q^{2\mu_{0}};q^{2\mu_{0}})_{d_{0}}}\right)\left(\prod_{i>0}\frac{x_{i}^{d_{i}}}{(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}}\right)\Bigg{]}\\\
&=\left[\sum_{d_{0}}(-1)^{C_{00}\mu_{0}d_{0}}q^{C_{00}\mu_{0}^{2}d_{0}^{2}}\left(\frac{x_{0}^{d_{0}}}{(q^{2\mu_{0}};q^{2\mu_{0}})_{d_{0}}}\right)\left(\prod_{i>0}\hat{y}_{i}^{\mu_{i}C_{0i}}\right)^{\mu_{0}d_{0}}\right]\\\
&\qquad\quad\times\left[\sum_{\boldsymbol{d}}(-1)^{\sum_{i}C_{ii}\mu_{i}d_{i}}q^{\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\left(\prod_{i>0}\frac{x_{i}^{d_{i}}}{(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}}\right)\right]\,,\end{split}$
(86)

where we employed quantum torus variables with the usual $q$-commutator
$\hat{y}_{i}\hat{x}_{j}=q^{2\delta_{ij}}\hat{x}_{j}\hat{y}_{i}$. Next, we
rewrite the first sum as an operator using
$(x_{0}\hat{y}_{0}^{k})^{n}=x_{0}^{n}\hat{y}_{0}^{kn}q^{(n^{2}-n)k}$, with
$n=d_{0}$ and $k=C_{00}\mu_{0}^{2}$. This gives

$\begin{split}P_{Q}&=\left[\sum_{d_{0}}(-1)^{C_{00}\mu_{0}d_{0}}q^{C_{00}\mu_{0}^{2}d_{0}}\left(\frac{(x_{0}\hat{y}_{0}^{C_{00}\mu_{0}^{2}})^{d_{0}}}{(q^{2\mu_{0}};q^{2\mu_{0}})_{d_{0}}}\right)\left(\prod_{i>0}\hat{y}_{i}^{\mu_{i}C_{0i}}\right)^{\mu_{0}d_{0}}\right]\\\
&\qquad\qquad\times\left[\sum_{\boldsymbol{d}}(-1)^{\sum_{i}C_{ii}\mu_{i}d_{i}}q^{\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\left(\prod_{i>0}\frac{x_{i}^{d_{i}}}{(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}}\right)\right]\\\
&=\left[\sum_{d_{0}}\frac{((-1)^{C_{00}\mu_{0}}q^{C_{00}\mu_{0}^{2}}\,x_{0}\hat{y}_{0}^{C_{00}\mu_{0}^{2}})^{d_{0}}}{(q^{2\mu_{0}};q^{2\mu_{0}})_{d_{0}}}\left(\prod_{i>0}\hat{y}_{i}^{\mu_{i}C_{0i}}\right)^{\mu_{0}d_{0}}\right]\\\
&\qquad\qquad\times\left[\sum_{\boldsymbol{d}}(-1)^{\sum_{i}C_{ii}\mu_{i}d_{i}}q^{\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\left(\prod_{i>0}\frac{x_{i}^{d_{i}}}{(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}}\right)\right]\\\
&=\Psi_{q^{\mu_{0}}}\left((-1)^{C_{00}\mu_{0}}q^{C_{00}\mu_{0}^{2}-\mu_{0}}\,x_{0}\hat{y}_{0}^{C_{00}\mu_{0}^{2}}\prod_{i>0}\hat{y}_{i}^{\mu_{0}\mu_{i}C_{0i}}\right)\\\
&\qquad\qquad\times\left[\sum_{\boldsymbol{d}}(-1)^{\sum_{i}C_{ii}\mu_{i}d_{i}}q^{\sum_{i,j}C_{ij}(\mu_{i}d_{i})(\mu_{j}d_{j})}\left(\prod_{i>0}\frac{x_{i}^{d_{i}}}{(q^{2\mu_{i}};q^{2\mu_{i}})_{d_{i}}}\right)\right]\,.\end{split}$
(87)

Iterating on all remaining nodes, we eventually rewrite the generalized
partition function as $P_{Q}=:{\mathbb{P}}_{Q}:$, the normal ordering of a
suitable operator

${{\mathbb{P}}_{Q}=\Psi_{q^{\mu_{m}}}(X_{m})\cdot\Psi_{q^{\mu_{m-1}}}(X_{m-1})\cdots\Psi_{q^{\mu_{1}}}(X_{1})\,,}$
(88)

with $\Psi$ defined in (84), and with variables defined as follows:

$X_{i}=(-1)^{C_{ii}\mu_{i}}q^{C_{ii}\mu_{i}^{2}-\mu_{i}}x_{i}\hat{y}_{i}^{C_{ii}\mu_{i}^{2}}\prod_{j<i}\hat{y}_{j}^{\mu_{i}\mu_{j}C_{ij}}\,.$
(89)

####  4.1.2 Including refinement

Let us now turn to the operator form of the refined generalized partition
function (65). For simplicity, we shall illustrate this for the case of a
single $d$-times around basic disk with framing zero (i.e. trivial adjacency
matrix), whose partition function is

$P_{Q}=\sum_{n}\frac{(a^{\alpha}q^{\beta}t^{\delta}x^{d})^{n}}{(q^{2};q^{2})_{dn}}\frac{(-q^{2}t^{\pm
1};q^{2})_{dn}}{(-q^{2d}t^{\pm 1};q^{2d})_{n}}\,.$ (90)

The main task will be to rewrite this as a product of $q$-Pochhammers with
suitable variables and steps. To this end, note that we may rewrite
$x^{dn}\,(-q^{2}t^{\pm 1};q^{2})_{dn}/(-q^{2d}t^{\pm 1};q^{2d})_{n}$ as

$\displaystyle x^{dn}\,$ $\displaystyle(-q^{2}t^{\pm
1};q^{2d})_{n}(-q^{4}t^{\pm 1};q^{2d})_{n}\dots(-q^{2d-2}t^{\pm
1};q^{2d})_{n}$ $\displaystyle=x^{dn}\,\frac{(-q^{2}t^{\pm
1};q^{2d})_{\infty}}{(-q^{2+2dn}t^{\pm 1};q^{2d})_{\infty}}\frac{(-q^{4}t^{\pm
1};q^{2d})_{\infty}}{(-q^{2+2dn}t^{\pm
1};q^{2d})_{\infty}}\dots\frac{(-q^{2d-2}t^{\pm
1};q^{2d})_{\infty}}{(-q^{2+2dn}t^{\pm 1};q^{2d})_{\infty}}$ (91)
$\displaystyle=\frac{1}{(-q^{2}t^{\pm
1}\hat{y};q^{2d})_{\infty}\dots(-q^{2}t^{\pm
1}\hat{y};q^{2d})_{\infty}}x^{dn}\,(-q^{2}t^{\pm
1}\hat{y};q^{2d})_{\infty}\dots(-q^{2d-2}t^{\pm 1}\hat{y};q^{2d})_{\infty}\,,$

where we made use of $\hat{y}\,x^{dn}=q^{2dn}\,x^{dn}\,\hat{y}$ (notice that
in this case $\hat{y}=\hat{y}_{i}^{d}$, so this is more generally an instance
of
$\hat{y}_{i}^{\mu_{i}}\,x_{i}^{n}=q^{2\mu_{i}n}\,x_{i}^{n}\,\hat{y}_{i}^{\mu_{i}}$).
Similarly, we may also rewrite the denominator as follows:

$\displaystyle\frac{x^{dn}}{(q^{2};q^{2})_{dn}}$
$\displaystyle=\frac{x^{dn}}{(q^{2d};q^{2d})_{n}}\,\frac{1}{(q^{2};q^{2d})_{n}\dots(q^{2d-2};q^{2d})_{n}}$