Abstract
For each , we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of foams defined by Mackaay, Stošić, and Vaz. We show that through these functors Soergel's category can be obtained from the foams.
1. Introduction
In [1] Soergel categorified the Hecke algebra using bimodules. Just as the Hecke algebra is important for the construction of the HOMFLY-PT link polynomial, so is Soergel’s category for the construction of Khovanov and Rozansky’s HOMFLY-PT link homology [2], as explained by Khovanov in [3]. Elias and Khovanov [4] constructed a diagrammatic version of the Soergel category with generators and relations, which Elias and Krasner [5] used for a diagrammatic construction of Rouquier’s complexes associated to braids.
In [6] Bar-Natan gave a new version of Khovanov’s [7] original link homology, also called the link homology, using 2d-cobordisms modulo certain relations, which we will call foams. Using 2d-cobordisms with a particular sort of singularity modulo certain relations, which we will call foams, Khovanov constructed the link homology [8]. Khovanov and Rozansky [9] then constructed the link homologies, for any , using matrix factorizations. These link homologies are closely related to the HOMFLY-PT link homology by Rasmussen’s spectral sequences [10], with -page isomorphic to the HOMFLY-PT homology and converging to the homology, for any . In [11] Mackaay et al. gave an alternative construction of these link homologies, for , using foams, which are 2d-cobordisms with two types of singularities satisfying relations determined by a formula from quantum field theory, originally obtained by Kapustin and Li [12] and later adapted by Khovanov and Rozansky [13].
Khovanov and Rozansky in [2, 9] and Rasmussen in [10] used matrix factorizations for their constructions. Therefore, the question arises whether their results can be understood in diagrammatic terms and what could be learned from that. In [14] Vaz constructed functors from Elias and Khovanov’s diagrammatic version of Soergel’s category to the categories of and foams. In this paper we construct the analogous functors from the same version of Soergel’s category to the category of foams for . To complete the picture, one would like to construct the analogues of Rasmussen’s spectral sequences in this setting. However for this, one would first have to understand the Hochschild homology of bimodules in diagrammatic terms, which has not been accomplished yet. Hochschild homology plays an integral part of the construction. Nevertheless, there is an interesting result which can already be shown using the functors in this paper. In a certain technical sense, which we will make precise in Proposition 4.2, Soergel’s category can be obtained from the foams, and therefore from the Kapustin-Li formula, using our functors. This result should be compared to Rasmussen’s Theorem in [10].
We thank Catharina Stroppel for pointing out the connection of our work to results in [15]. We quote her directly: In [15] a categorification of "special trivalent" graphs modulo the MOY relations was constructed by exact functors acting between certain blocks of parabolic category . Using Soergel’s functor passing from Lie theory to the combinatorial bimodule category the construction in [15] in fact produces an action of the diagrammatic Soergel category on these various category s.
We have tried to make the paper as self-contained as possible, but the reader should definitely leaf through [4, 5, 11, 14] before reading the rest of this paper.
In Section 2 we recall Elias and Khovanov’s version of Soergel’s category. In Section 3 we review foams, as defined by Mackaay, Stoši , and Vaz. Section 4 contains the new results: the definition of our functors, the proof that they are indeed monoidal, and a statement on faithfulness in Proposition 4.2.
2. Elias and Khovanov’s Version of Soergel’s Category
This section is a reminder of the diagrammatics for Soergel categories introduced by Elias and Khovanov in [4]. Actually we give the version which they explained in [2, Section ] and which can be found in detail in [5].
Fix a positive integer . The category is the category whose objects are finite length sequences of points on the real line, where each point is colored by an integer between and . We read sequences of points from left to right. Two colors and are called adjacent if and distant if . The morphisms of are given by generators modulo relations. A morphism of is a -linear combination of planar diagrams constructed by horizontal and vertical gluings of the following generators (by convention no label means a generic color ) (i) Generators involving only one color are It is useful to define the cap and cup as (ii) Generators involving two colors are –The 4-valent vertex, with distant colors, and the 6-valent vertex, with adjacent colors and ,
read from bottom to top. In this setting a diagram represents a morphism from the bottom boundary to the top. We can add a new colored point to a sequence and this endows with a monoidal structure on objects, which is extended to morphisms in the obvious way. Composition of morphisms consists of stacking one diagram on top of the other.
We consider our diagrams modulo the following relations.
“Isotopy’’ relations are
The relations are presented in terms of diagrams with generic colorings. Because of isotopy invariance, one may draw a diagram with a boundary on the side, and view it as a morphism in by either bending the line up or down. By the same reasoning, a horizontal line corresponds to a sequence of cups and caps.
One color relations are
Relations involving two distant colors are
Relations involving two adjacent colors are
Relations involving three colors are (adjacency is determined by the vertices which appear)
Furthermore, we also have a useful implication of relation (2.12) as follows:
Introduce a -grading on declaring that dots have degree , trivalent vertices have degree , and - and -valent vertices have degree .
Definition 2.1. The category is the category containing all direct sums and grading shifts of objects in and whose morphisms are the grading-preserving morphisms from .
Definition 2.2. The category is the Karoubi envelope of the category .
Elias and Khovanov’s main result in [4] is the following theorem.
Theorem 2.3 (Elias-Khovanov). The category is equivalent to the Soergel category in [1].
From Soergel’s results from [1] we have the following corollary.
Corollary 2.4. The Grothendieck algebra of is isomorphic to the Hecke algebra.
Notice that is an additive category but not abelian and we are using the (additive) split Grothendieck algebra.
In Section 4 we will define a a family of functors from to the category of foams, one for each . These functors are grading preserving, so they obviously extend uniquely to . By the universality of the Karoubi envelope, they also extend uniquely to functors between the respective Karoubi envelopes.
3. Foams
3.1. Prefoams
In this section we recall the basic facts about foams. For the definition of the Kapustin-Li formula, for proofs of the relations between foams, and for other details, see [11, 16]. The foams in this paper are composed of three types of facets: simple, double, and triple facets. The double facets are coloured and the triple facets are marked to show the difference. Intersecting such a foam with a generic plane results in a web, as long as the plane avoids the singularities where six facets meet, such as on the right in Figure 1.
Definition 3.1. Let be a finite oriented closed -valent graph, which may contain disjoint circles and loose endpoints. We assume that all edges of are oriented. A cycle in is defined to be a circle or a closed sequence of edges which form a piecewise linear circle. Let be a compact orientable possibly disconnected surface, whose connected components are simple, double, or triple, denoted by white, coloured, or marked. Each component can have a boundary consisting of several disjoint circles and can have additional decorations which we discuss below. A closed prefoam is the identification space obtained by gluing boundary circles of to cycles in such that every edge and circle in are glued to exactly three boundary circles of and such that for any point , (1) if is an interior point of an edge, then has a neighborhood homeomorphic to the letter Y times an interval with exactly one of the facets being double, and at most one of them being triple; for an example see Figure 1, (2) if is a vertex of , then it has a neighborhood as shown in Figure 1. We call the singular graph, its edges and vertices singular arcs and singular vertices, and the connected components of the facets.
Furthermore the facets can be decorated with dots. A simple facet can only have black dots (), a double facet can also have white dots (), and a triple facet besides black and white dots can have double dots (). Dots can move freely on a facet but are not allowed to cross singular arcs.
Note that the cycles to which the boundaries of the simple and the triple facets are glued are always oriented, whereas the ones to which the boundaries of the double facets are glued are not, as can be seen in Figure 1. Note also that there are two types of singular vertices. Given a singular vertex , there are precisely two singular edges which meet at and bound a triple facet: one oriented toward , denoted as , and one oriented away from , denoted as . If we use the “left-hand rule’’, then the cyclic ordering of the facets incident to and is either or , respectively, or the other way around. We say that is of type I in the first case and of type II in the second case. When we go around a triple facet, we see that there have to be as many singular vertices of type I as there are of type II for the cyclic orderings of the facets to match up. This shows that for a closed prefoam the number of singular vertices of type I is equal to the number of singular vertices of type II.
We can intersect a prefoam generically by a plane in order to get a closed web, as long as the plane avoids the vertices of . The orientation of determines the orientation of the simple edges of the web according to the convention in Figure 2.
Suppose that, for all but a finite number of values , the plane intersects generically. Suppose also that and intersect generically and outside the vertices of . Furthermore, suppose that is a disc in W and its boundary circle such that is a disjoint union of vertical line segments. This means that we are assuming that does not intersect . We call an open prefoam between the open webs and . Interpreted as morphisms, we read open prefoams from bottom to top, and their composition consists of placing one prefoam on top of the other, as long as their boundaries are isotopic and the orientations of the simple edges coincide.
Definition 3.2.
Let be the category whose objects are webs and whose morphisms are -linear combinations of isotopy classes of prefoams with the obvious identity prefoams and composition rule.
We now define the -degree of a prefoam. Let be a prefoam,
,
, and the disjoint union of its simple, double, and marked facets, respectively, and its singular graph. Furthermore, let
,
, and be the number of simple, double, and marked vertical boundary edges of
, respectively. Define the partial -gradings of as
where is the Euler characteristic and denotes the boundary.
Definition 3.3. Let be a prefoam with dots of type , dots of type , and dots of type . The -grading of is given by
The following result is a direct consequence of the definitions.
Lemma 3.4. is additive under the gluing of prefoams.
We denote a simple facet with dots by Recall that the two-variable Schur polynomial can be expressed in terms of the elementary symmetric polynomials and . By convention, the latter correspond to and on a double facet, respectively, so that is defined to be the linear combination of dotted double facets corresponding to the expression of in terms of and . Analogously we can express the three-variable Schur polynomial in terms of the elementary symmetric polynomials , , and . By convention, the latter correspond to , , and on a triple facet, respectively, so we can make sense of
3.2. Foams
In [11, 16] we gave a precise definition of the Kapustin-Li formula, following Khovanov and Rozansky’s work [13]. We will not repeat that definition here, since it is complicated and unnecessary for our purposes in this paper. The only thing one needs to remember is that the Kapustin-Li formula associates a number to any closed prefoam and that those numbers have very special properties, some of which we will recall below. By , we denote the Kapustin-Li evaluation of a closed prefoam .
Definition 3.5. The category Foam N is the quotient of the category Pfoam by the kernel of , that is, by the following identifications: for any webs , and finite sets and we impose the relations for any fixed way of closing the , denoted by . By “fixed’’ we mean that all the are closed in the same way. The morphisms of are called foams.
In the next proposition we recall those relations in that we need in the sequel. For their proofs and other relations we refer to [11].
Proposition 3.6.The following identities hold in
The dot conversion relations are
The dot migration relations are
The neck cutting relations are (these were called cutting neck relations in [11, 16])
The sphere relations are
The -foam relations are Inverting the orientation of the singular circle of inverts the sign of the corresponding foam. A theta-foam with dots on the double facet can be transformed into a theta-foam with dots only on the other two facets, using the dot migration relations.
The Matveev-Piergalini relation is
4. The Functors
Let and be arbitrary but fixed. In this section we define a monoidal functor between the categories and .
On Objects. sends the empty sequence to and the one-term sequence to : with given by the vertical composite .
On Morphisms. (i) The empty diagram is sent to parallel vertical sheets: (ii) The vertical line coloured is sent to the identity cobordism of : The remaining vertical parallel sheets on the r.h.s. are not shown for simplicity, a convention that we will follow from now on. (iii) The StartDot and EndDot morphisms are sent to the zip and the unzip, respectively: (iv) Merge and Split are sent to cup and cap cobordisms: (v) The 4-valent vertex with distant colors. For we have The case is given by reflexion around a horizontal plane. (vi) For the 6-valent vertices we have The case with the colors switched is given by reflection in a vertical plane. Notice that respects the gradings of the morphisms.
Proposition 4.1. is a monoidal functor.
Proof.
The assignment given by clearly respects the monoidal structures of and
. So we only need to show that is a functor, that is, it respects the relations (2.5) to (2.22) of Section 2.
“Isotopy Relations’’. Relations (2.5) to (2.9) are straightforward to check and correspond to isotopies of their images under
. For the sake of completeness we show the first equality in (2.5). We have
One Color Relations. For relation (2.10) we have
where the first equivalence follows from relations (2.5) and (2.7) and the second from isotopy of the foams involved.
For relation (2.11) we have
Relation (2.12) requires some more work. We have
where the second equality follows from the
relation. We also have
Using (3.12), we obtain
and, therefore, we have that
Two Distant Colors. Relations (2.13) to (2.15) correspond to isotopies of the foams involved and are straightforward to check.
Adjacent Colors. We prove the case where “blue’’ corresponds to and “red’’ corresponds to
. The relations with colors reversed are proved the same way. To prove relation (2.16) we first notice that using the
move we get
Apply
to the simple-double square tube perpendicular to the triple facet to obtain two terms. The first term contains a double-triple digon tube which is the left-hand side of the
relation rotated by around a vertical axis. Next apply the
relation and use
to remove the four singular vertices, which results in simple-triple bubbles (with dots) in the double facets. Using (3.19) to remove these bubbles gives
which is
. The second term contains
behind a simple facet with dots (notice that all dots are on simple facets). Using the
relation to get a simple-triple bubble in the double facet, followed by
and we obtain
which equals
.
We now prove relation (2.17). We have an isotopy equivalence
Notice that
is the l.h.s. of the relation. The first term on the r.h.s. of is isotopic to
. For the second term on the r.h.s. of we notice that
contains
Applying
followed by
to remove the singular vertices creating simple-simple bubbles on the two double facets and using (3.20) to remove these bubbles, we conclude that
is the second term on the r.h.s. of
.
We now prove relation (2.18) in the form
The image of the l.h.s. also contains a bit like the one in (4.21). Simplifying it like we did in the proof of (2.17), we obtain that reduces to
For the r.h.s. we have
Using
on the vertical digon, followed by
and the Bubble relation (3.20), we obtain (4.23).
Relation (2.19) follows from straightforward computation and is left to the reader.
Relations Involving Three Colors. Relations (2.20) and (2.21) follow from isotopies of the foams involved. To show that respects relation (2.22), we use a different type of argument. First of all, we note that the images under of both sides of relation (2.22) are multiples of each other, because the graded vector space of morphisms in between the bottom and top webs has dimension one in degree zero. Verifying this only requires computing the coefficient of (this includes the necessary shift!) in the MOY polynomial associated to the web
which is a standard calculation left to the reader. To see that the multiplicity coefficient is equal to one, we close both sides of relation (2.22) simply by putting a dot on each open end. Using relations (2.14) and (2.16) to reduce these closed diagrams, we see that both sides give the same nonzero sum of disjoint unions of coloured StartDot-EndDot diagrams. Note that we have already proved that respects relations (2.14) and (2.16). By applying foam relation (4.12) to the images of all nonzero terms in the sum, we obtain a nonzero sum of dotted sheets. This implies that both sides of (2.22) have the same image under
.
We have now proved that is a monoidal functor for all . Our main result about the whole family of these functors, that is, for all together, is the proposition below. It implies that all the defining relations in Soergel’s category can be obtained from the corresponding relations between foams, when all are considered, and that there are no other independent relations in Soergel’s category corresponding to relations between foams.
Proposition 4.2. Let , be two arbitrary objects in and let be arbitrary. If for all , then .
Proof.
Let us first suppose that
. Suppose also that has degree and that
. Recall that, as shown in [3, Corollary ], we know that is the free commutative polynomial ring generated by the StartDot-EndDots of all possible colors. So is a polynomial in StartDot-EndDots, and therefore a sum of monomials. Let be one of these monomials, no matter which one, and let denote the power of the StartDot-EndDot with color in
. Close by gluing disjoint discs to the boundaries of all open simple facets (i.e., the vertical ones with corners in the pictures). For each color
, put dots on the left simple open facet corresponding to and also put dots on the rightmost simple open facet. Note that, after applying
, we get a linear combination of dotted simple spheres. Only one term survives and is equal to
, because only in that term each sphere has exactly dots. This shows that
, because it admits a nonzero closure.
Now let us suppose that and is arbitrary. By [2, Corollaries and ], we know that is the free -module of rank one generated by the disjoint union of StartDots coloured by
. Closing off the StartDots by putting dots on all open ends gives an element of
, whose image under is nonzero for big enough by the above. This shows that the generator of has nonzero image under for big enough, because is a functor.
Finally, the general case, for and arbitrary, can be reduced to the previous case by [2, Corollary ].
Acknowledgments
The authors were supported by the Fundação para a Ciência e a Tecnologia (ISR/IST plurianual funding) through the programme “Programa Operacional Ciência, Tecnologia, Inovação” (POCTI) and the POS Conhecimento programme, cofinanced by the European Community fund FEDER. Pedro Vaz was also financially supported by the Fundação para a Ciência e a Tecnologia through the post-doctoral fellowship SFRH/BPD/46299/2008.