Homological algebra of knots and BPS states

# Homological algebra of knots and BPS states

## Abstract.

It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models.

One important application to knot homologies is the existence of “colored differentials” that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable “mirror symmetry” between these triply-graded theories.

## 1. Setting the stage

Quantum knot invariants were introduced in 1980’s [1, 2]: for every representation of a Lie algebra , one can define a polynomial invariant of a knot . Its reduced version is

 (1.1) Pg,R(K)=¯¯¯¯Pg,R(K)¯¯¯¯Pg,R(\raisebox−2.27622pt\includegraphics[width=11.381102pt]unknot),

where denotes the unknot.

A categorification of the polynomial (or its unreduced version ) is a doubly-graded homology theory1 whose graded Euler characteristic is equal to . In other words, if denotes the Poincaré polynomial of , then we have

 Pg,R(K)(q)=Pg,R(K)(q,t=−1).

Unlike , the explicit combinatorial definition of exists for very few choices of and . However, physics insights based on BPS state counting and Landau-Ginzburg theories predict various properties and a very rigid structure of these homology theories.

One of the first results was obtained in [3] for and its fundamental representation . This work builds on a physical realization of knot homologies as spaces of BPS states [4, 5]:

 (1.2) Hknot=HBPS.

Among other things, this relation predicts the existence of a polynomial knot invariant , sometimes called the superpolynomial, such that for all sufficiently large one has

 (1.3)

Moreover, the polynomial has nonnegative coefficients and is equal to the Poincaré polynomial of a triply graded homology theory that categorifies the reduced two-variable HOMFLY polynomial , and similarly for the unreduced invariants. This triply graded theory comes equipped with a collection of differentials , such that the homology of with respect to is isomorphic to .

There are only two triply-graded knot homologies that have been studied in the literature up to now. Besides the above-mentioned HOMFLY homology, the second triply-graded theory, proposed in [6], similarly unifies homological knot invariants for the -dimensional vector representation of and . This triply-graded theory comes with a collection of differentials , such that the homology with respect to for is isomorphic to , while the homology with respect to for even is isomorphic to . Since the graded Euler characteristic of is equal to the (reduced) Kauffman polynomial of , is called the Kauffman homology of a knot .

One way to discover differentials acting on all of these knot homology theories is via studying deformations of the potentials and matrix factorizations in the corresponding Landau-Ginzburg theories (see section 3 for details). In particular, in the case of the Kauffman homology one finds a peculiar deformation that leads to a “universal” differential and its conjugate , such that the homology with respect to these differentials is, in both cases, isomorphic to the triply-graded HOMFLY homology .

A careful reader may notice that most of the existent results reviewed here deal with the fundamental or vector representations of classical Lie algebras (of Cartan type , , , or ). In this paper, we do roughly the opposite: we focus mainly on but vary the representation . In particular, we propose infinitely many triply-graded homology theories associated with arbitrary symmetric () and anti-symmetric () representation of . Moreover, these colored HOMFLY homology theories come equipped with differentials, such that the homology, say, with respect to is isomorphic to , and similarly for .

Remarkably, in addition to the differentials labeled by (for a given ) we also find colored differentials that allow to pass from one triply graded theory to another, thus relating homological knot invariants associated with different representations!

Specifically, for each pair of positive integers with , we find a differential , such that the homology of with respect to is isomorphic to . Similarly, in the case of anti-symmetric representations, we find an infinite sequence of triply-graded knot homologies , one for every positive integer , equipped with colored differentials that allow to pass between two triply-graded theories with different values of .

The colored differentials are a part of a larger algebraic structure that becomes manifest in a physical realization of knot homologies as spaces of BPS states. As it often happens in physics, the same physical system may admit several mathematical descriptions; a prominent example is the relation between Donaldson-Witten and Seiberg-Witten invariants of 4-manifolds that follows from physics of supersymmetric gauge theories in four dimensions [7]. Similarly, the space of BPS states in (1.2) admits several (equivalent) descriptions depending on how one looks at the system of five-branes in eleven-dimensional M-theory [4] relevant to this problem.

Specifically, for knots in a 3-sphere the relevant system is a certain configuration of five-branes in M-theory on , where is a 4-manifold with isometry group and is a non-compact toric Calabi-Yau 3-fold (both of which will be discussed below in more detail). And, if one looks at this M-theory setup from the vantage point of the Calabi-Yau space , one finds a description of BPS states via enumerative geometry of . Furthermore, for simple knots and links that preserve toric symmetry of the Calabi-Yau 3-fold the study of enumerative invariants reduces to a combinatorial problem of counting certain 3d partitions (= fixed points of the 3-torus action [8]), hence, providing a combinatorial formulation of knot homologies in terms of 3d partitions [9, 10].

On the other hand, if one looks at this M-theory setup from the vantage point of the 4-manifold , one can express the counting of BPS invariants in terms of equivariant instanton counting on . In this approach (see e.g. [11]), the “quantum” -grading and the homological -grading on the space (1.2) originate from the equivariant action of on .

A closely related viewpoint, that will be very useful to us in what follows, is based on the five-brane world-volume theory [12]. Let us briefly review the basic ingredients of this approach that will make the relation to the setup of [4] more apparent. In both cases, knot homology is realized as the space of BPS states and, as we shall see momentarily, the physical realization of the triply-graded knot homology proposed in [4] is essentially the large- dual of the system realizing the doubly-graded knot homology in [12]. This is very typical for systems with gauge symmetry2 which often admit a dual “holographic” description that comprises all in the same package and leads to useful computational techniques [13].

In the case of homological knot invariants, the five-brane configuration described in [12, sec. 6] is the following:

 space-time : R×T∗W×M4 (1.4) N M5-branes : R×W×D M5-brane : R×LK×D

Here, is a 3-manifold and is the “cigar” in the Taub-NUT space . The Lagrangian submanifold is the conormal bundle to the knot ; in particular,

 (1.5) LK∩W=K.

In all our applications, we consider (or, a closely related case of ). Similarly, the setup of [4] can be summarized as

 (1.6) space-time : R×X×M4 M5-brane : R×LK×D

where is the resolved conifold, i.e. the total space of the bundle over . From the way we summarized (1.4) and (1.6), it is clear that they have many identical elements. The only difference is that (1.4) has extra M5-branes supported on , whereas (1.6) has a different space-time (with a 2-cycle in the Calabi-Yau 3-fold ), which is exactly what one expects from a holographic duality or large- transition [14, 15].

Indeed, what is important for the purpose of studying the space of BPS states, , is that both (1.4) and (1.6) preserve the same amount of supersymmetry and have the same symmetries:

• time translations: both systems have a translation symmetry along the time direction (denoted by the factor in (1.4) and (1.6)). Therefore, in both cases, one can ask for a space of BPS-states — on multiple M5-branes in (1.4), and on a single M5-brane in (1.6) — which is precisely what was proposed as a candidate for the knot homology (resp. HOMFLY homology).

• rotation symmetries:

 (1.7) U(1)P×U(1)F

Here, the two factors correspond, respectively, to the rotation symmetry of the tangent and normal bundle of in a 4-manifold . In particular, in both frameworks (1.4) and (1.6), the former is responsible for the -grading of , which corresponds to the conserved angular momentum derived from the rotation symmetry of .

A well-known feature of the large- duality is that the rank of the gauge group turns into a geometric parameter of the dual system (cf. [13] or [14]). In the present case, it is the Kähler modulus of the Calabi-Yau 3-fold :

 (1.8) N∼log(a)=Vol(CP1).

The reason we denote the Kähler parameter by rather than is that with this convention is the standard variable of the HOMFLY polynomial / knot homology.

Another feature familiar to the practitioners of the refined / motivic Donaldson-Thomas theory is that can jump as one varies stability conditions [16, 17, 18, 19, 20, 21]. Thus, in a closely related type IIA superstring compactification on a Calabi-Yau 3-fold , the stability parameters are the Kähler moduli of , and in the present case there is only one Kähler modulus (1.8) given by the volume of the cycle in . Therefore, we conclude that the space (1.2) can jump as one changes the stability parameter .

Luckily, in the case where is the total space of the bundle over relevant to our applications, the wall-crossing behavior of the refined BPS invariants has been studied in the literature [22, 23, 18]. The one-dimensional space of stability conditions is divided into a set of chambers illustrated in Figure 1. In each chamber, is constant and the jumps of closed BPS states occur at the walls characterized by different types of “fragments”:

 (1.9) W1n:D2/D0 fragmentsW−1n:¯¯¯¯¯¯¯D2/D0 fragmentsW0n:D0 fragments

Notice, the set of chambers in this model can be identified with , the set of integer numbers. As we explain in the next section, this is not a coincidence. Namely, as we shall see, every fragment corresponds to a differential acting on the space (1.2), so that in the present example one finds a set of differentials labeled by .

The differentials are part of the homological algebra of knots / BPS states, depending on whether one prefers to focus on the left or right side of the relation (1.2). For larger representations, in addition to the differentials one finds colored differentials that allow to pass between homology theories associated with different . Even though a combinatorial definition of the majority of such theories, with all the differentials, is still missing, their structure (deduced from physics) is so rigid that enables computation of the homology groups for many knots and passes a large number of consistency checks.

In particular, by computing the triply-graded homologies and for various knots, we find the following surprising symmetry between the two theories:

 (1.10) HΛri,j,∗(K)≅HSri,−j,∗(K).

One of the implication is that and can be combined into a single homology theory!

###### Conjecture 1.1.

For every positive integer , there exists a triply-graded theory together with a collection of differentials , with , such that the homology of with respect to , for , is isomorphic to , while the homology of with respect to , for , is isomorphic (up to a simple regrading) to .

Moreover, it is tempting to speculate that the symmetry (1.10) extends to all representations:

 (1.11) mirror symmetry'' : Hλ(K)≅Hλt(K),

where and are a pair of Young tableaux related by transposition (mirror reflection across the diagonal), e.g.

 λ=\vbox\offinterlineskip\kern 1.0pt\vbox{\hbox{% \vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{% \hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to % 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{% \hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{% \hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0% .4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt heig% ht 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3% pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9% pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3% .9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{% \hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}\vbox{% \hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to% 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt hei% ght 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt% \hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}% \kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}% \hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt% \vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{% \vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{\vbox% {\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox t% o 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt hei% ght 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt% \hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}% \kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}% \hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt% \vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{% \vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{\vbox% {\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox t% o 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{}}\kern 1.0pt\kern 0.4pt⟷λt=\vbox\offinterlineskip\kern 1.0pt\vbox{\hbox{\vbox{\hbox{\vbox to % 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule w% idth 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4% pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt wid% th 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt wid% th 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}% \kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{% \vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}% \vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{% \hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to % 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{% \hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{\vbox{\hbox{\vbox to% 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule w% idth 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4% pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt wid% th 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt wid% th 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}% \kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{% \vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}% }\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{% \vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{% \hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox t% o 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox t% o 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule w% idth 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4% pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt wid% th 4.3pt\vss}}}\kern-0.4pt}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.% 4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9% pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}% \hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0% .4pt}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}% \hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt% \vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{% \vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{\vbox% {\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox t% o 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{}}\kern 1.0pt\kern 0.4pt

The symmetry (1.11) has not been discussed in physical or mathematical literature before.

While we offer its interpretation in section 5.3, we believe the mirror symmetry for colored knot homology (1.11) deserves a more careful study, both in physics as well as in mathematics. In particular, its deeper understanding should lead to the “categorification of level-rank duality” in Chern-Simons theory, which is the origin of the simpler, decategorified version of (1.11):

 (1.12) Pλ(K)(a,q)=Pλt(K)(a,q−1)

for colored HOMFLY polynomials [24, 25, 26, 27], and extends the familiar symmetry of the ordinary HOMFLY polynomial. We plan to pursue the categorification of level-rank duality and to study the new, homological symmetry (1.11) in the future work.

### Organization of the paper

We start by explaining in section 2 that, in general, the space of open BPS states forms a representation of the algebra of closed BPS states. Then, in section 3 we review elements of the connection between string realizations (1.4)–(1.6) of knot homologies and Landau-Ginzburg models that play an important role in mathematical formulations of certain knot homologies based on Lie algebra and its representation . In particular, we illustrate in simple examples how the corresponding potentials can be derived from the physical setup (1.4)–(1.6) and how deformations of these potentials lead to various differentials acting on . This gives another way to look at the algebra acting on (1.2). Based on these predictions, in section 4 we summarize the mathematical structure of the triply-graded homology , together with its computation for small knots. Section 5 lists the analogous properties of the homology associated with anti-symmetric representations, and explains the explicit form of the “mirror symmetry” (1.10) between symmetric and anti-symmetric triply-graded theories. Unreduced triply-graded theory for symmetric and anti-symmetric representations is briefly discussed in section 6. In appendix A we collect the list of our notations, whereas in appendix B we present the computations of the , and Kauffman triply-graded homology for knots and . These particular examples of “thick” knots provide highly non-trivial tests of all the properties of the homologies presented in the paper. Appendix C contains the computation of the and homology of the figure-eight knot . Finally, appendix D collects some notations and calculations relevant to the unreduced colored HOMFLY polynomial of the unknot discussed in section 6.

## 2. Algebra of BPS states and its representations

Differentials in knot homology form a part of a larger algebraic structure that has an elegant interpretation in the geometric / physical framework. Because this algebraic structure has analogs in more general string / M-theory compactifications, in this section we shall consider aspects of such structure for an arbitrary Calabi-Yau 3-fold with extra branes supported on a general Lagrangian submanifold , e.g.

 (2.1) space-time : R×X×M4 M5-brane : R×L×D

For applications to knot homologies, one should take to be the total space of the bundle over and to be the Lagrangian submanifold determined by a knot [15, 28, 29]. Then, (2.1) becomes precisely the setup (1.6), in which homological knot invariants are realized as spaces of refined BPS states, cf. (1.2).

In fact, there are two spaces of BPS states relevant to this particular problem and its variants based on a more general 3-fold . One is the space of refined closed BPS states, denoted as , and the other is called the space of refined open BPS states, . The difference is that, while the latter contains BPS particles in the presence of defects3, the former comprises only those BPS states which are still present in a theory when all defects are removed. In the description [11] via equivariant instanton counting on a 4-manifold , the defect (M5-brane) corresponds to a particular ramification along the divisor , the so-called surface operator.

On the other hand, if one looks at the general setup (2.1) from the vantage point of the Calabi-Yau space , then and can be formulated in terms of enumerative invariants of and that “count”, respectively, closed holomorphic curves embedded in and bordered holomorphic Riemann surfaces with boundary on the Lagrangian submanifold . As a way to remember this, it is convenient to keep in mind that

• depends only on the Calabi-Yau space

• depends on both the Calabi-Yau space and the Lagrangian submanifold

In applications to knots, open (resp. closed) BPS states are represented by open (resp. closed) membranes in the M-theory setup (1.6) or by bound states of D0 and D2 branes in its reduction to type IIA string theory. It is the space of open BPS states that depends on the choice of the knot and, therefore, provides a candidate for homological knot invariant in (1.2).

In general, the space of BPS states is -graded, where is the “charge lattice” and the extra -grading comes from the (half-integer) spin of BPS states, such that . For example, in the case of closed BPS states, the charge lattice is usually just the cohomology lattice of the corresponding Calabi-Yau 3-fold ,

 (2.2) Γ=Heven(X;Z).

In the case of open BPS states, also depends on the choice of the Lagrangian submanifold .

When is the total space of the bundle over and , as in application to knot homologies, the lattice is two-dimensional for both open and closed BPS states. As a result, both and are -graded. In particular, the space of open BPS states is graded by spin and by charge , where the degree is sometimes called the “D2-brane charge” and is the “D0-brane charge.” In relation to knot homologies (1.2), these become the three gradings of the theory categorifying the colored HOMFLY polynomial:

Now, let us discuss the algebraic structure that will help us understand the origin of differentials acting on the triply-graded vector space . The fact that forms an algebra is well appreciated in physics [30] as well as in math literature [31]. Less appreciated, however, is the fact that forms a representation of the algebra :

 (2.4) ∫refined open BPS states:HopenBPS↺refined closed BPS states:HclosedBPS∫

Indeed, two closed BPS states, and , of charge can form a bound state, of charge , as a sort of “extension” of and ,

 (2.5) 0→B2→B12→B1→0,

thereby defining a product on :

 (2.6)
 \raisebox−14.226378pt\includegraphics[width=142.26378pt]closed

Similarly, a bound state of a closed BPS state with an open BPS state is another open BPS state :

 (2.7) (Bclosed1,Bopen2)↦Bopen12
 \raisebox−14.226378pt\includegraphics[width=142.26378pt]open

This defines an action of the algebra of closed BPS states on the space of open BPS states.

The process of formation or fragmentation of a bound state in (2.6) and (2.7) takes place when the binding energy vanishes. Since the energy of a BPS state is given by the absolute value of the central charge4 function this condition can be written as

 (2.8) ∣∣Z(γ1+γ2)∣∣−∣∣Z(γ1)∣∣−∣∣Z(γ2)∣∣=0

for a process that involves either or its inverse . A particular instance of the relation (2.8) is when the central charge of the fragment vanishes:

 (2.9) Z(γfragment)=0

Then, a fragment becomes massless and potentially can bind to any other BPS state of charge . When combined with (2.4), it implies that closed BPS states of zero mass correspond to operators acting on the space of open BPS states . The degree of the operator is determined by the spin and charge of the corresponding BPS state, as in (2.3).

For example, when is the total space of the bundle over , as in application to knot homologies, we have

 (2.10) exp(Z)=aβqn,

where we used the relation (1.8) between and . Therefore, for special values of and we have the following massless fragments, cf. (1.9):

 (2.11) a=q−N:D2/D0 fragmentsa=qN:¯¯¯¯¯¯¯D2/D0 fragmentsq=1:D0 fragments

Moreover, the D2/D0 fragments obey the Fermi-Dirac statistics (see e.g. [18, 23]) and, therefore, lead to anti-commuting operators (i.e. differentials) on .

To summarize, we conclude that various specializations of the parameters (stability conditions) are accompanied by the action of commuting and anti-commuting operators on . The algebra of these operators is precisely the algebra of closed BPS states . Mathematical candidates for the algebra of closed BPS states include variants of the Hall algebra [32], which by definition encodes the structure of the space of extensions (2.5):

 (2.12) [B1]⋅[B2]=∑B12|0→B2→B12→B1→0|[B12]

In the present case, the relevant algebras include the motivic Hall algebra [17], the cohomological Hall algebra [31], and its various ramifications, e.g. cluster algebras. Therefore, the problem can be approached by studying representations of these algebras, as will be described elsewhere.

## 3. B-model and matrix factorizations

Let us denote by a homology theory of knots and links colored by a representation of the Lie algebra . Many such homology theories can be constructed using categories of matrix factorizations [33, 34, 35, 36, 37, 38, 39, 40]. In this approach, one of the main ingredients is a polynomial function called the potential, associated to every segment of a link (or, more generally, of a tangle) away from crossings. For example, for the fundamental representation of the potential is a function of a single variable,

 (3.1) Wsl(N),\vbox\offinterlineskip\kern 1.0pt\vbox{\hbox{\vbox{% \hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to% 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{}}\kern 1.0pt\kern 0.4pt(x)=xN+1.

In physics, matrix factorizations are known [41, 42, 43, 44, 45, 46] to describe D-branes and topological defects in Landau-Ginzburg models which, in the present context, are realized on the two-dimensional part of the five-brane world-volume in (1.4) or (1.6). More precisely, it was advocated in [6] that reduction of the M-theory configuration (1.4) on one of the directions in and a T-duality along the time direction gives a configuration of intersecting D3-branes in type IIB string theory, such that the effective two-dimensional theory on their common world-volume provides a physical realization of the Landau-Ginzburg model that appears in the mathematical constructions.

In particular, this interpretation was used to deduce potentials associated to many Lie algebras and representations. Indeed, since away from crossings every segment of the knot is supposed to be described by a Landau-Ginzburg theory with potential , we can approximate this local problem by taking and in (1.4). Then, we also have and the reduction (plus T-duality) of (1.4) gives type IIB theory in flat space-time with two sets of D3-branes supported on 4-dimensional hyperplanes in : one set supported on , and another supported on . The space of open strings between these two groups of D3-branes contains information about the potential .

For example, in the case of the fundamental representation of , the first stack consists of D3-branes and the second only contains a single D3-brane. The open strings between these two stacks of D3-branes transform in the bifundamental representation under the gauge symmetry on the D3-branes. The Higgs branch of this two-dimensional theory is the Kähler quotient of the vector space parametrized by the bifundamental chiral multiplets, modulo gauge symmetry of a single D3-brane supported on :

 (3.2) CN//U(1)≅CPN−1.

The chiral ring of this theory on the intersection of D3-branes is precisely the Jacobi ring of the potential (3.1).

Following similar arguments one can find potentials associated to many other Lie algebras and representations [6], such that

 (3.3) Hg,R(\raisebox−2.27622pt\includegraphics[width=11.381102pt]unknot)=J(Wg,R).

For example, the arguments that lead to (3.1) can be easily generalized to , the -th anti-symmetric representation of . The only difference is that, in this case, the corresponding brane systems (1.4) and (1.6) contain coincident M5-branes supported on . Following the same arguments as in the case of the fundamental representation () and zooming in closely on the local geometry of the brane intersection, after all the dualities we end up with a system of intersecting D3-branes in flat ten-dimensional space-time,

 (3.4) N D3-branes : R×W r D3′-branes : R×LK

where, as in the previous discussion, for the purpose of deriving we can approximate and , so that . Now, the open strings between two sets of D3-branes in (3.4) transform in the bifundamental representation under the gauge symmetry on the D3-branes. Here, if we want to “integrate out” open strings ending on the D3-branes, only the second gauge factor should be considered dynamical, while should be treated as a global symmetry of the two-dimensional gauge theory on the brane intersection. In the infrared this theory flows to a sigma-model based on the Grassmannian manifold:

 (3.5) Gr(r,N)=U(N)U(r)×U(N−r).

The potential of the corresponding Landau-Ginzburg model [47] is a homogeneous polynomial of degree ,

 (3.6) Wsl(N),Λr(z1,…,zr)=xN+11+…+xN+1r,

where the right-hand side should be viewed as a function of the variables of degree , , which are the elementary symmetric polynomials in the ,

 zi=∑j1

We shall return to the discussion of the potential later in this section. In the case of more general representations, one needs to consider various sectors of the gauge theory on labeled by non-trivial flat connections (Wilson lines) around the codimension-2 locus where D3-branes meet D3-branes, cf. [15, 48].

In this paper we are mostly interested in knots colored by symmetric and anti-symmetric representations of , even though much of the present discussion can be easily generalized to other Lie algebras and representations. Thus, for a symmetric representation of one finds that the corresponding potential is a homogeneous polynomial of degree in variables of degree , much like (3.6). Moreover, the explicit form of such potentials can be conveniently expressed through a generating function [6]:

 (3.7) ∑N(−1)NtN+rWsl(N),Sr(z1,…,zr)=(1+r∑i=1tizi)log(1+r∑i=1tizi),

which in the basic case gives

 (3.8) Wsl(2),\vbox\offinterlineskip\kern 1.0pt\vbox{\hbox{\vbox{% \hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to% 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt hei% ght 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt% \hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}% \kern-0.4pt}}}\hbox{}}\kern 1.0pt\kern 0.4pt=z41−6z21z2+6z22.

Instead of going through the derivation of this formula we can use a simple trick based on the well known isomorphism under which a vector representation of is identified with the adjoint representation of . Indeed, it implies that (3.8) should be identical to the well known potential

 (3.9) Wso(3),V=x2+xy2

in the homology theory, cf. [6, 37]. It is easy to verify that the potentials (3.8) and (3.9) are indeed related by a simple change of variables.5

Moreover, the fact that the adjoint representation of is identical to the vector representation of implies that

 (3.10) ∫Hsl(2),\vbox\offinterlineskip\kern 1.0pt\vbox{\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt wi% dth 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}% }\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{% \vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}% \vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{% \hbox to 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to % 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{% \hrule height 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{}}\kern 1.0pt\kern 0% .4pt(K)≅Hso(3),V(K)∫

should hold for every knot . In particular, it should hold for the unknot. And, since is 3-dimensional, it follows

 (3.11)

This is indeed what one finds in physical realizations of knot homologies reviewed in section 1. In the framework of [4] the colored homology of the unknot was computed in [9] using localization with respect to the toric symmetry of the Calabi-Yau space . Similarly, in the gauge theory framework [12] the moduli space of solutions on with a single defect operator in the adjoint representation of the gauge group is the weighted projective space (= the space of Hecke modifications [49], see also [50]). In this approach, the colored homology is given by the cohomology of the moduli space which is 3-dimensional, in agreement with (3.11).

### 3.1. Colored differentials

One of the reasons why we carefully reviewed the properties of the potentials is that they hold a key to understanding the colored differentials. Namely, in doubly-graded knot homologies constructed from matrix factorizations differentials that relate different theories are in one-to-one correspondence with deformations of the potentials [51, 52, 53, 54, 55, 6]:

 (3.12) ∫differentials on Hg,R⟺deformations of Wg,R∫

For example, deformations of the potential (3.1) of the form with , correspond to differentials that relate and knot homologies (with ).

More generally, one can consider deformations of the potential such that and

 (3.13) Wg,R+ΔW≃Wg′,R′

for some Lie algebra and its representation . Here, the symbol “” means that the critical point(s) of the deformed potential is locally described by the new potential . A deformation of this form leads to a spectral sequence that relates knot homologies and . With the additional assumption that the spectral sequence converges after the first page one arrives at (3.12). Moreover, the difference

 (3.14) degWg,R−degΔW

gives the -grading of the corresponding differential. Notice, the condition implies that this -grading is positive.

For example, among deformations of the degree-4 potential (3.8) one finds , which leads to a differential of -degree 1 that relates and . This deformation has an obvious analog for higher rank -colored homology; it deforms the homogeneous polynomial of degree in such a way that the deformed potential has a critical point described by the potential of degree . Therefore, it leads to a colored differential of -degree 1, such that

 (3.15)

In section 4 we present further evidence for the existence of a differential with such properties not only in the doubly-graded theory but also in the triply-graded knot homology that categorifies the colored HOMFLY polynomial.

Similar colored differentials exist in other knot homologies associated with more general Lie algebras and representations. Basically, a knot homology associated to a representation of the Lie algebra comes equipped with a set of colored differentials that, when acting on , lead to homological invariants associated with smaller representations (and, possibly, Lie algebras),

 (3.16) dimR′

While it would be interesting to perform a systematic classification of such colored differentials using the general principle (3.12), in this paper we limit ourselves only to symmetric and anti-symmetric representations of .

As we already discussed earlier, when is the -th anti-symmetric representation of the corresponding Landau-Ginzburg potential (3.6) is a homogeneous polynomial of degree . Equivalently, the potentials with a fixed value of can be organized into a generating function, analogous to (3.7):

 (3.17) ∑N(−1)NtN+1Wsl(N),Λr(z1,…,zr)=log(1+r∑i=1tizi).

For example, in the first non-trivial case of there are only two variables, and . For one finds a “trivial” potential of degree 3, which corresponds to the fact that the anti-symmetric representation (also denoted ) is trivial in . For , the existence of the anti-symmetric tensor identifies the second anti-symmetric representation with the fundamental representation of . The next case in this sequence, , is the first example where the second anti-symmetric representation is not related to any other representation of . According to (3.6) and (3.17), the corresponding potential is a homogeneous polynomial of degree 5,

 (3.18) Wsl(4),\vbox\offinterlineskip\kern 1.0pt\vbox{\hbox{\vbox{% \hbox{\vbox to 0.0pt{\hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to% 0.0pt{\vrule width 0.4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{% \vrule width 0.4pt height 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule h% eight 0.4pt width 4.3pt\vss}}}\kern-0.4pt}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{% \hrule height 0.4pt width 3.9pt\vss}}\hbox{\vbox{\hbox to 0.0pt{\vrule width 0% .4pt height 3.9pt\hss}}\kern 3.9pt\vbox{\hbox to 0.0pt{\vrule width 0.4pt heig% ht 3.9pt\hss}}}\hbox{\vbox{\hbox{\vbox to 0.0pt{\hrule height 0.4pt width 4.3% pt\vss}}}\kern-0.4pt}}}\hbox{}}\kern 1.0pt\kern 0.4pt=z515−z31z2+z1z22.

Before studying deformations of this potential, we note that by a simple change of variables it is related to the potential

 (3.19) Wso(6),V=x5+xy2

associated to a vector representation of . This is a manifestation of the well known isomorphism under which the six-dimensional anti-symmetric representation of is identified with the vector representation of . This isomorphism can help us understand deformations of the potential . Indeed, the deformations of were already studied in [6]; they include several deformations which lead to canceling differentials and a deformation by that leads to a universal differential .

In view of the relation , these deformations (and the corresponding differentials) should be present in the theory as well. In particular, there are deformations of that lead to canceling differentials and, more importantly, there is a deformation by that leads to the universal differential which relates and . Note, from the viewpoint of the knot homology, this is exactly the colored differential that does not change the rank of the Lie algebra, but changes the representation. Making use of (3.17) it is easy to verify that, for all values of , the potential admits a deformation by terms of degree that leads to and, therefore, to the analog of (3.15):

 (3.20)

Much like in the case of the symmetric representations, this colored differential as well as canceling differentials come from the triply-graded theory that categorifies the                 -colored HOMFLY polynomial (see section 5 for details).

## 4. Colored HOMFLY homology

In this section we propose structural properties of the triply-graded theory categorifying the colored version of the reduced HOMFLY polynomial. The central role in this intricate network of structural properties belongs to the colored differentials, whose existence we already motivated in the previous sections.

### 4.1. Structural properties

Let and be positive integers, and let denote a reduced doubly-graded homology theory categorifying , the polynomial invariant of a knot labeled with the -th symmetric representation of . denotes the Poincaré polynomial of . Motivated by physics, we expect that such theories with a given value of have a lot in common.

###### Conjecture 4.1.

For a knot and a positive integer , there exists a finite polynomial such that

 (4.1) PSrN(K)(q,t)=PSr(K)(a=qN,q,t),

for all sufficiently large .

Since the left-hand side of (4.1) is a Poincaré polynomial of a homology theory, all coefficients of must be nonnegative. This suggests that there exists a triply-graded homology theory whose Poincaré polynomial is equal to , and whose Euler characteristic is equal to the normalized -colored two-variable HOMFLY polynomial.

As in the case of ordinary HOMFLY homology [3] (that, in fact, corresponds to ) and in the case of Kauffman homology [6], this triply-graded theory comes with the additional structure of differentials, that will imply Conjecture 4.1. In particular, for each positive integer we have a triply-graded homology theory of a knot . Moreover, these theories come with additional structure of differentials that, as in (3.16), allow us to pass from the homology theory with to theories with and .

Thus, we arrive to our main conjecture that describes the structure of the triply-graded homology categorifying the -colored HOMFLY polynomials:

###### Conjecture 4.2.

For every positive integer there exists a triply-graded homology theory that categorifies the reduced two-variable -colored HOMFLY polynomial of . It comes with a family of differentials , with , and also with an additional collection of universal colored differentials , for every , satisfying the following properties:

Categorification: categorifies

 χ(HSr∗(K))=PSr(K).

Anticommutativity: The differentials anticommute6:

 dSrNdSrM=−dSrMdSrN.

Finite support:

 dim(HSr∗)<+∞.

Specializations: For , the homology of with respect to is isomorphic to :

 (HSr∗(K),dSrN)≅Hsl(N),Sr(K).

Canceling differentials: The differentials and are canceling: the homology of with respect to the differentials and is one-dimensional, with the gradings of the remaining generators being simple invariants of the knot .

Vertical Colored differentials: The differentials , for , have -degree , and the homology of with respect to the differential is isomorphic, after simple regrading that preserves - and -gradings, to the -colored homology .

Universal Colored differentials: For any positive integer , with , the differentials have -degree zero, and the homology of with respect to the colored differential is isomorphic (after regrading) to the -colored homology :

 (HSr∗(K),dr→m)≅HSm∗(K).

A combinatorial definition of a triply-graded theory with the structure given in Conjecture 4.2, as well as of the homologies for and , still does not exist in the literature.

Even though there is no such combinatorial definition, one can use any combination of the above axioms as a definition, and the remaining properties as consistency checks. In particular, one can obtain various consequence of the Conjecture 4.2 and properties of the triply-graded homology , along with the predictions for the triply-graded homology of simple knots.

In the rest of this section we give a summary of these properties, including some non-trivial checks.

### 4.2. A word on grading conventions

So far we summarized the general structural properties of the colored knot homology. Now we are about to make it concrete and derive explicit predictions for colored homology groups of simple knots. This requires committing to specific grading conventions, as well as other choices that may affect the form of the answer. It is important to realize, however, that none of these affect the very existence of the structural properties, which are present with any choices and merely may look different. While some of these choices will be discussed in section 6.2, here we focus on

• choices that associate various formulae to a Young tableaux versus its transpose ;

• choices of grading, e.g. grading conventions used in this paper (that we sometimes refer to as “old”) and grading conventions used in most of the existent literature [3, 58, 59] (that we sometimes call “new” in view of the forthcoming work [60] based on this choice).

The first choice here breaks the symmetry (“mirror symmetry”) between representations and . Indeed, since in view of the Conjecture 1.1 the triply-graded homologies associated with these representations are essentially identical and can be packaged in a single theory , one has a choice whether homologies arise for or .

The second choice listed here starts with different grading assignments, but turns out to be exactly the same as the first choice. In other words, the “old” gradings and “new” gradings are related by “mirror symmetry.” Another way to describe this is to note, that in grading conventions of this paper the -colored superpolynomials are related (by a simple change of variables) to the -colored invariants that one would find by following the same steps in grading conventions of e.g. [3, 58, 59]:

 (4.2) PSrhere=PΛr% elsewhere

Note, that the -colored invariant is related to the -colored invariant, and vice versa. The explicit change of variables in this transformation is sensitive to even more elementary redefinitions, such as and which is ubiquitous in knot theory literature. For example, with one of the most popular choices of - and -grading, the transformation of variables / gradings looks like:

 A ↦ at3, (4.3) q ↦ 1qt2, t ↦ 1q.

The moral of the story is that, besides the grading conventions used in the earlier literature, the present paper offers yet another choice of grading conventions consistent with all the structural properties. And the relation between the two grading conventions can be viewed as a manifestation of mirror symmetry (1.10). Keeping these words of caution in mind, now let us take a closer look at the structure of the colored knot homology.

### 4.3. Consequences of Conjecture 4.2

First of all, our main Conjecture 4.2 implies the Conjecture 4.1. Indeed, in order to be consistent with the specialization from (4.1), the -degree of the differential must be proportional to . Since has finite support, this leads to the Conjecture 4.1, with being the Poincaré polynomial of .

More precisely, the differentials