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Topological Strings, D-Model, and Knot Contact Homology

We study the connection between topological strings and contact homology recently proposed in the context of knot invariants. In particular, we establish the proposed relation between the Gromov-Witten disk amplitudes of a Lagrangian associated to a knot and augmentations of its contact homology algebra. This also implies the equality between the Q-deformed A-polynomial and the augmentation polynomial of knot contact homology (in the irreducible case). We also generalize this relation to the case of links and to higher rank representations for knots. The generalization involves a study of the quantum moduli space of special Lagrangian branes with higher Betti numbers probing the Calabi-Yau. This leads to an extension of SYZ, and a new notion of mirror symmetry, involving higher dimensional mirrors. The mirror theory is a topological string, related to D-modules, which we call the"D-model."In the present setting, the mirror manifold is the augmentation variety of the link. Connecting further to contact geometry, we study intersection properties of branches of the augmentation variety guided by the relation to D-modules. This study leads us to propose concrete geometric constructions of Lagrangian fillings for links. We also relate the augmentation variety with the large N limit of the colored HOMFLY, which we conjecture to be related to a Q-deformation of the extension of A-polynomials associated with the link complement.


Introduction
The subject of topological strings has been important in both physics and mathematics. From the physical perspective it has been studied from many different viewpoints and plays a prominent role in the understanding of string dualities. From the mathematical perspective, its underpinnings (Gromov-Witten invariants) are well understood and play a central role in symplectic geometry and related subjects. Moreover, in many cases one can directly compute the amplitudes using either physical methods or mathematical techniques and the two agree.
An interesting application of topological strings involves the study of knot and link invariants. In particular it is known that HOMFLY polynomials for links K can be reformulated in terms of topological strings on T * S 3 where one includes N Lagrangian D-branes wrapping S 3 and some Lagrangian brane L K intersecting S 3 along the link and asymptotic to the conormal of K at infinity. In this setup, it is possible to shift L K off of the 0-section S 3 ⊂ T * S 3 , which suggests that the data of the link is imprinted in how L K intersects the boundary of T * S 3 at infinity. This ideal boundary is the unit cotangent bundle U * S 3 , which is a contact manifold (topologically S 3 ×S 2 ) with contact 1-form given by the restriction of the action form p dq. The intersection of L K with U * S 3 is a Legendrian torus Λ K .
Large N transition relates this theory to a corresponding theory in the geometry where S 3 shrinks to zero size and is blown up with blow-up parameter given by N g s , where g s is the string coupling constant. Under the transition, boundary conditions corresponding to strings ending on branes wrapping S 3 close up and disappear, but the transition does not affect the Legendrian torus Λ K , which is part of the geometry infinitely far away from the tip of the singular cone where S 3 shrinks and gets replaced by S 2 .
Contact homology is a theory that was introduced in a setting correspond-ing exactly to the above mentioned geometry at infinity. More precisely, one studies the geometry of R × Λ K ⊂ R × U * S 3 , which we can naturally view as the far distance geometry of L K , using holomorphic disks. In this particular case of conormal tori Λ K in the unit cotangent bundle of S 3 , the resulting theory is known as knot contact homology. From the viewpoint of string theory, this theory involves a Hilbert space of physical open string states corresponding to Reeb chords (flow segments of the Reeb vector field with endpoints on Λ K ) and a BRST-operator deformed by holomorphic disks in R × U * S 3 with boundary on R × Λ K and with punctures mapping to Reeb chords at infinity. In this language, knot contact homology is the corresponding BRST-cohomology. From a mathematical perspective this structure can be assembled into a differential graded algebra (DGA) which is well studied and in particular can be computed from a braid presentation of a link through a concrete combinatorial description of all relevant moduli spaces of holomorphic disks.
In view of the above discussion it is natural to ask how this DGA is related with the open version of Gromov-Witten theory of L K . In [1] a partial answer for this was proposed: it was conjectured that the augmentation variety associated to a knot, which is a curve that parametrizes one-dimensional representations of the DGA associated to the knot, is the same as the moduli space of a single Lagrangian L K corrected by disk instantons. This moduli space is in turn, according to a generalization of SYZ conjecture, related to the mirror curve, which is the locus of singular fibers in the bundle of conics that constitutes the mirror, and as such determines the mirror. 1 One aim of this paper is to indicate a path to a mathematical proof of this conjecture: in Section 6.3 we show how to get a local parametrization of the augmentation curve in terms of the open Gromov-Witten potential of a Lagrangian filling of the conormal torus of a knot, see Section 6.4 for the definition, and that expression agrees with the local parametrization of the mirror curve derived using physical arguments, see Section 2.4. In particular, for knots with irreducible augmentation variety, using the conormal filling we obtain a proof of the conjecture, see Remark 6.7.
A further aim is to generalize this from one-component knots to manycomponent links, and the generalization turns out to involve interesting new ingredients. We consider new Lagrangian branes that have the same asymp- Our study of the quantum moduli space of all these Lagrangians leads to a new reformulation of mirror symmetry: the moduli space of branes for an n-component link is n-dimensional and the mirror geometry, instead of being a 1-dimensional curve, will be an n-dimensional variety. It turns out that this variety can be viewed as a Lagrangian variety V ⊂ (C * ) 2n in a canonical way and we identify it with the augmentation variety from knot contact homology.
An unusual feature of this structure is that we encounter higher-dimensional geometries as the mirror. In fact, it may appear that one cannot formulate topological string in this context since the critical dimension for that theory is effectively 1 for non-compact Calabi-Yau 3-folds but for n-component links we have effective dimension n > 1. Nevertheless we propose a string theory mirror even in these cases, based on what we call the topological "D-model".
The D-model is the A-model topological string on (C * ) 2n with a Lagrangian D-brane V and a canonical coisotropic brane filling the whole (C * ) 2n . In this context the topological string is exact at the level of the annulus, with the analog of higher genus A-model corrections being captured not by higher genera of strings, but by the choice of a flux turned on along the coisotropic brane. The D-model leads to a natural definition of the theory in terms of D-modules (hence the name), while for n = 1 (in particular for knots), the D-model is already known to be equivalent to the B-model.
The organization of this paper is as follows. In Section 2 we review the relation between topological strings and Chern-Simons theory, large N transition, and knot invariants. Furthermore, we describe, using a generalized SYZ formulation, how any knot gives a mirror geometry. In Section 3 we introduce new Lagrangians associated to knots with the usual asymptotics at infinity but with different interior topology. We then generalize the discussion from knots to n-component links and show how n-dimensional Lagrangian varieties in flat space (C * ) 2n arise in the the study of the moduli space of Lagrangian branes filling the link conormal. In Section 5, we introduce basic elements of knot contact homology; furthermore, we relate augmentation varieties with the (disk instanton corrected) moduli spaces of Lagrangians associated to knots and links, and study intersection properties of branches of the augmentation variety guided by the D-model. In Section 6, we present a mathematical argument relating knot contact homology for links to disk amplitudes in Gromov-Witten theory, and study geometric constructions of Lagrangian fillings for conormals as well as properties and applications of the resulting Lagrangians. In Section 7 we present some examples. In Section 8, we formulate a conjecture about how to quantize higher dimensional augmentation varieties in terms of the D-model, by relating them to D-modules.
Finally, Appendices A and B contain calculations of a more technical nature.

Review
Consider Chern-Simons gauge theory with gauge group G = SU (N ) on a More precisely, we associate a Wilson loop observable in representation R to a knot K by inserting the path ordered exponential where U (K) = P exp i K A is the holonomy of A along K, in the path integral. In fact, to define this we also have to choose a framing of the knot, i.e. a non-vanishing vector field in the normal bundle of the knot K ⊂ M .
The case of many-component links is similar: a link K is a collection In [10], Witten explained how to solve the above theory exactly. Any three dimensional topological theory corresponds to a two dimensional rational conformal field theory (CFT). The Hilbert space of the three dimensional theory and operators acting on it can be constructed from conformal blocks of the CFT and from representations of the corresponding modular group.
In the Chern-Simons case, the relevant conformal field theory is the SU (N ) k Wess-Zumino-Witten (WZW) model, and one finds that knowledge of the S, T and braiding matrices is all that is needed to solve the theory on any 3-manifold.
In this way, invariants of knots and links in the 3-sphere S 3 that arise from Chern-Simons theory can be explicitly computed. In particular, the polynomial knot invariants considered earlier by Jones correspond to the gauge group G = SU (2) and Wilson lines in the fundamental representation.
More generally, for a link K ⊂ S 3 with knot components K = K 1 ∪ · · · ∪ K n , the expectation values From the perspective of Chern-Simons perturbation theory, the numbers F g,h arise by organizing the Feynman graphs in the following way: thicken the graphs into ribbon graphs with gauge index labels on the boundary; the number F g,h then keeps track of the contributions from the graphs that give rise to ribbon graphs corresponding to a genus g Riemann surface with h boundary components. We also point out that the parameter q, in terms of which the Chern-Simons knot invariants become polynomial, is given by q = e gs .

Knots and links can be included in the correspondence between Chern-
Simons and the topological A-model in the following way [13]. To each knot K in S 3 , we associate a Lagrangian L K in X, which we take to be its conormal in T * S 3 consisting of all covectors along the knot that annihilate its tangent vector. In particular, intersecting L K with the zero section, we get the knot itself, For n-component links K = K 1 ∪ · · · ∪ K n in S 3 , we consider the Lagrangian L K which is the union of the conormals of its components We will consider D-branes on L K j and therefore need to include a sector in the theory that corresponds to worldsheets with boundaries both in the branes on L K and in the branes on the zero section S 3 . We write the partition function of the topological string on X with these branes present as Z GW (X, L K 1 , . . . , L Kn ) and note that, in addition to depending on g s and N , it also depends on the moduli of the Lagrangians L K j which in particular keeps track of the class in H 1 (L K j ) represented by the boundaries of the worldsheets.
In the case under consideration, L K j each has the topology of S 1 × R 2 and we get one modulus x j for each Lagrangian L K j . Here, the complex parameter x j can be written as x j = r j + i S 1 A j , where the real part r j can be viewed as coming from the moduli of the deformations of the Lagrangian and the imaginary part is the holonomy of the U (1) gauge field of the brane around the nontrivial cycle S 1 × {0} ⊂ L K j . In later sections, we will study similar Lagrangians for K of first Betti number b 1 = r > 1, in which case the moduli of the Lagrangian is r-dimensional. Giving x j a nonzero value corresponds to lifting the Lagrangian L K j off of the S 3 .
From the Chern-Simons perspective, assuming there is a single D-brane on L K for a knot K, computing the partition function Z GW (X; L K ) corresponds to inserting the operator det(1 − e −x U (K)) −1 .

Higher representations and multiple branes
In Section 2.1, we considered knots and links with a single brane on the K (x 1 , . . . , x n ).
has a contribution from worldsheets with at least one boundary component on one of the branes wrapping S 3 that can be computed by inserting in the Chern-Simons path integral, and which describes the effect of integrating out the bifundamental strings between L K and the S 3 , generalizing (2.1) to the case of more than one brane on L K .
Consider instead n distinct Lagrangians that are copies of L K separated from it by moduli corresponding to Re(x i ). More precisely, we take the conormal of the n-component linkK = K ∪ . . . ∪ K n where K j is the knot obtained by shifting K a distance j , where > 0 is very small, along the framing vector field of K used to define the quantum invariants (i.e. the expectation values Tr R U (K) ). Note that the topological string partition function for a single brane on each L K j has exactly the same contribution, as follows e.g. from the simple mathematical fact that which then holds inside the expectation values as well. In fact the system with n branes on L K is physically indistinguishable from the system with single branes on all components of the conormal LK. The expression on the left is more naturally associated to the former, whereas the one on the right is more naturally associated to the latter.
TreatingK as a general link as considered in Section 2.1, disregarding the effects of the branes being very close, we get a corresponding string partition function ΨK(x 1 , . . . , x n ) given by the insertion in the Chern-Simons path integral described above.
However, when computing the string partition for n coincident branes on L K , it is natural to take these effects into account: contributions that are not captured by Chern-Simons theory correspond to worldsheets with no boundary component on S 3 . For parallel branes these come from short strings connecting different branes; such strings only contribute nontrivially to the annulus diagram of the topological string with amplitude k>0 1 n e kx i e −kx j corresponding to strings connecting L K i to L K j , i < j. Exponentiating this contribution, we then find the following relation between the partition function for branes on L K and the partition function of single branes on LK, treated as a general link: Note that Ψ [n] K encodes the HOMFLY of the knot K colored by representations with n rows: using the expansion where the sum ranges over all SU (N ) representations R with at most n rows.

Large N duality
It was conjectured in [14] that SU (N ) Chern-Simons theory on S 3 , or equivalently the topological A-model string on X = T * S 3 with N D-branes on S 3 , has a dual description in terms of the topological A-model on the resolved conifold Y which is the total space of the bundle Note that X and Y both approach the conical symplectic manifold R×U * S 3 , which is topologically R × S 2 × S 3 , at infinity. At the apex of the cone sits an S 3 in X, while in Y , there sits a CP 1 ≈ S 2 . The duality arises from the geometric transition from X to Y that shrinks S 3 and replaces it by CP 1 without altering the geometry at infinity. Furthermore, the N branes on S 3 in X disappear under the transition, but their number is related to the size (symplectic area) t of CP 1 in Y as follows: t = N g s , where g s is the string coupling constant. We write The partition function of the closed topological string on Y counts holomorphic maps into Y . All such maps arise from perturbation of branched covers of the central CP 1 ⊂ Y , and the variable Q keeps track of their degree. In [14], the partition function of the closed topological A-model string on Y was shown to agree with the partition function of SU (N ) Chern-Simons theory on S 3 , and consequently with the partition function of A-model topological string on X in the background of N D-branes on S 3 , provided that the Chern-Simons parameters, k and N , and the string coupling constant g s (for topological strings in both X and Y ) are related as g s = 2πi k+N , and that (2.2) holds. In other words, for parameters related as described: We next describe how to include knots and links in this picture, see [13].
Let K = K 1 ∪ · · · ∪ K n ⊂ S 3 be a link. As described in Section 2.1, adding branes along the Lagrangian conormal L K , we relate the Chern-Simons path integral with insertions corresponding to the link with open topological string on X with boundaries on either the N branes on S 3 or on the branes on L K .
Recall that L K ⊂ X can be pushed off of the zero section S 3 , corresponding to turning on x j = 0 along L K j . We thus assume that L K ⊂ X is disjoint from S 3 ⊂ X and consider the effect of the geometric transition from X to Y . Since the transition affects only a small neighborhood of the tip of the cone, corresponding to small neighborhoods of S 3 ⊂ X and CP 1 ⊂ Y , the Lagrangian L K is canonically pushed through the transition as a Lagrangian in Y . Furthermore, in analogy with the closed string case discussed above, boundary conditions corresponding to worldsheet boundaries on the N branes on S 3 close up and disappear, while worldsheet boundaries on the branes on L K remain unchanged. Thus, the partition function of branes on the components of L K in Y also gives the partition function of branes in X, wrapping the Lagrangian in X corresponding to L K ⊂ Y under transition, and N branes on S 3 ⊂ X, provided (2.2) holds.

SYZ mirror symmetry for knots
Consider the A-model topological string on the resolved conifold Y with a D-brane wrapping the Lagrangian conormal L K ⊂ Y of a knot K ⊂ S 3 , see Section 2.3. There is then a contribution from short strings beginning and ending on L K . Noting that a small neighborhood of L K is symplectomorphic to a neighborhood of the zero section in T * L K and applying the construction of [12], we find that this contribution is given by the partition function of U (1) Chern-Simons theory on L K ≈ S 1 × R 2 . At infinity, L K looks like R × Λ K , with ideal boundary Λ K ≈ T 2 , which means we should study U (1) (or more precisely, GL(1)) Chern-Simons theory on the manifold S 1 ×D 2 with T 2 boundary. Let λ be the longitudinal cycle of the T 2 , which determines the parallel of the knot K that links it trivially, and let µ be the meridional cycle. We denote by x the holonomy along λ, and by p the holonomy along µ: where A is the connection 1-form. In Chern-Simons theory on L K , the holonomies of λ and µ are canonically conjugate: where g s is the string coupling constant. This means in particular that for a D-brane on L K in Y , p and x are not independent; rather, if Ψ K (x) denotes the partition function of L K , we have In particular, we should view Ψ K (x) as a wave function with asymptotics From the perspective of topological strings, in the classical limit g s → 0, maps of the disks dominate the perturbation expansion, with maps of more complicated Riemann surfaces giving contributions of higher order. This means that is the generating function of Gromov-Witten invariants a n,k corresponding to counting holomorphic disks in Y with boundary on L K . Thus, at the level of the disk, We point out that (2.3) is consistent with the fact that, classically in L K , the cycle µ bounds. Here µ bounding in L K means that the holonomy around µ vanishes if we disregard the disk instanton corrections: ∂W K ∂x vanishes up to contributions of instantons. This leads to the interpretation of the moduli space of the brane as a Lagrangian curve V K ⊂ T * T 2 , with coordinates x, p and symplectic form dx ∧ dp, given by the equation for a polynomial A K (e x , e p , Q) (where we view Q as a parameter). In general, for large x, (2.4) may have more than one solution p = p(x) for p in terms of x. This corresponds to the fact that the theory may have more than critical point in this phase; to get the solution corresponding to L K , we need to pick The curve (2.4) sums up disk instanton corrections to the moduli space of a D-brane on L K probing Y . It was argued in [1] that this gives rise to a mirror Calabi-Yau manifold, X K , given by the hypersurface Mirror symmetry exchanges A-branes, i.e. Lagrangian submanifolds, of Y with B-branes, i.e. holomorphic submanifolds, of the mirror X K . Moreover, quantum mirror symmetry implies that for every (special) Lagrangian brane in Y there is a B-brane in X K such that the quantum corrected moduli space of L K in Y is the same as the classical moduli space of the mirror B-brane in X K . Here we see that the quantum dual B-branes are given by the line for (e x , e p ) = (e x 0 , e p 0 ).
In the special case when K is the unknot, we obtain the same mirror of Y as in [15], but for more general knots, we get new mirrors. In fact, for every knot K in S 3 and its associated Lagrangian L K in Y , we get a canonical mirror geometry (2.5), where the mirror of L K is determined by a point on the mirror Riemann surface V K . Of course, we expect each of the mirror Calabi-Yau manifolds (2.5) to contain not just the mirror of the Lagrangian L K corresponding to the knot K, but also, by mirror symmetry, mirrors of all other Lagrangian branes L K for knots K = K. In general, however, these mirrors have more complicated descriptions.

Chern-Simons theory and Lagrangian fillings
Via large N duality, the Chern-Simons path integral on S 3 contains nonperturbative information about the A-model topological string in the resolved conifold Y . Before the transition, in X = T * S 3 , the data that go into specifying the theory are just the knot K, the number of branes on it, and the holonomy at infinity. We will set the number of branes on K to be 1 in this section, and consider the higher rank generalization in the next section.
The data are imprinted at infinity of the ambient Calabi-Yau manifold, and are thus visible both before and after the transition. After the transition, the data at infinity may be compatible with more than one filling in the interior. In Section 2.4, we focused on the filling that gives the Lagrangian conormal L K , but, as we shall see, Chern-Simons theory encodes information about other Lagrangian fillings as well; these correspond to different classical solutions of the topological string.
At infinity, the Calabi-Yau looks like R × U * S 3 ≈ R × S 3 × S 2 and the Lagrangian L K approaches R × Λ K , where Λ K , the conormal of the knot, is naturally identified with the boundary of a tubular neighborhood of K in S 3 and is thus topologically a torus 2 . There are two natural basic 1-cycles in Λ K : the longitude λ, which is a parallel copy of the knot, and the meridian µ, which is the boundary of a fiber disk in the tubular neighborhood.
A vacuum of the topological string with these boundary conditions is determined once we specify the holonomy of the GL(1) connection around a 1-cycle of the torus Λ K at infinity and find a Lagrangian brane that fills in R × Λ K in Y with these moduli. Note that the holonomy of the GL (1) connection encodes both the position of the brane and the U (1)-holonomy on it, see Section 2.4. 3 In the discussion so far we used the filling L K and fixed the holonomy around the longitude cycle x. With a fixed connection at infinity, there may be more than one corresponding filling. Moreover, having found a vacuum with one choice of the filling, there is an SL(2, Z) family of choices of flat connections, related by canonical transformations, that gives rise to the same filling. This corresponds to a choice of the framing of the Lagrangian, which we will discuss in more detail in Section 3.2.
By taking the holonomy x around the longitude to be very large, we always get a choice of filling corresponding to the conormal Lagrangian L K .
The mirror Riemann surface which encodes the geometry of the mirror, also has information about fillings of R × Λ K . As we explained, the Lagrangian L K corresponds to a branch of the Riemann surface where Since p is the holonomy around the meridian of the knot, the fact that it vanishes classically means that the meridian cycle gets filled in in L K , as is indeed the case topologically. The Lagrangian L K has a one-dimensional moduli space, and the branch p ∼ 0 is the branch where x → ∞ and the worldsheet instanton corrections are maximally suppressed. As we vary x, we can probe all of the Riemann surface and go to a region where this is no longer the case.
There can be other ways of filling in. The region of p and x where the mirror geometry is well approximated by should correspond to filling in which the mµ + nλ cycle in Λ K ≈ T 2 bounds.
In all cases, "∼" denotes "up to instanton corrections": if the approximation in For example, filling in the longitude cycle λ gives x ∼ 0.
The Lagrangian M K one obtains in this way is related to the knot complement In the special case when the knot K is fibered, the complement can actually be constructed as a Lagrangian submanifold of Y asymptotic to R × Λ K , see Section 6.5. In the general case we will give a conjectural construction of a Lagrangian filling with the same classical asymptotics, see Section 6.7. We will discuss this with further details in later sections. Here we simply let M K denote a Lagrangian filling of Λ K in Y that has the classical asymptotics x ∼ 0. In this setting, the holonomy p around the meridian cycle is the natural parameter. In fact, the topological string partition function of Y with a single brane on M K turns out to bẽ

Geometric transition between L K and M K
Let us clarify the relation between L K and M K and the nature of the transition between them. We start with N D-branes on the S 3 and a single brane on the conormal bundle L K , intersecting along the knot K The fact that L K and S 3 intersect makes the configuration slightly singular, but one can remedy this by using the fact that L K moves in a one-parameter family, parametrized by x. For any Re(x) = 0, we can then move L K off of S 3 so that they no longer intersect.
However, there is another way to smooth out the singularity by smoothing out the intersection between the S 3 and L K . This requires breaking the gauge symmetry on S 3 by picking one of the N D-branes. Cut out a neighborhood of the knot K from both the D-brane on S 3 and on L K . This gives S 3 − K and L K − K both with a torus boundary. The two manifolds can then be glued together along their boundaries. Topologically, this results in S 3 − K, since gluing in T 2 × R does not change the topology. (This way of obtaining S 3 − K discussed, in a related context, in [16].) As mentioned above, it is not always possible to move the Lagrangian version of S 3 − K off of the zero section, and we have a somewhat more involved construction with similar features to deal with this case. As in the previous section, we will use M K to denote a Lagrangian that classically gives x ∼ 0.
Before the transition, the Lagrangian S 3 −K and L K project with different degrees to S 3 and no transition between them is possible. Also before the transition, N is finite, and hence Q = e N gs = 1 classically. In this limit, the curve factorizes and contains where A K class is the classical A-polynomial of the knot, describing the moduli space of flat connections on the knot complement. The e p − 1 = 0 branch, corresponding to L K , disconnects from the curve describing S 3 − K.
After the transition, the S 3 has shrunk, Q gets an expectation value Q = 1, the curve generally becomes irreducible 0 = A K (e x , e p , Q), (3.5) and the distinction between L K and M K disappears. Since both p ∼ 0 and x ∼ 0 branches lie on the same Riemann surface, M K and L K are smoothly connected once we include disk instanton corrections, with no phase transition between them.
To make this point even clearer, it is useful to consider one very simple example of this when K is the unknot. We will study this both before and after the transition. In the mirror geometry L K is given by choosing a point λ in the curve where λ = e x and µ = e p . When |λ| 1 this gives the Lagrangian mirror of L K for the unknot. Note that this is consistent with µ = 1, i.e., p = 0.
The Lagrangian mirror to M K is given by |µ| 1, which requires λ = 1, i.e., x = 0 as expected. Note that going from L K to M K can be done via a smooth path in the mirror curve, even though in the original geometry they are classically distinct. 4 The transition between L K and M K described above has a simple interpretation in the topological string. At the intersection of the D-branes wrapping L K and S 3 lives a pair of complex scalars A, B transforming in the bifundamental representation (N, −1), (N , 1), corresponding to strings with one boundary on the zero section and the other on L K . As long as L K and S 3 intersect, the bifundamental is massless (the real mass vanishes). In one phase, we move the Lagrangian L K off of the zero section. In this phase, the bifundamental hypermultiplet gets a mass r = Re(x) corresponding to the modulus of moving off. From this perspective, the partition function Ψ K (x) arises as follows. Integrating out the massive bifundamental of mass x generates the determinant [13] is the holonomy along the knot and x is identified with holonomy on L K . In Chern-Simons theory, we integrate over A and thereby compute the expec- which is in turn computed by the colored HOMFLY polynomial as explained earlier. This is in fact the derivation of the partition function given in [13].
The theory has a second phase, where the bifundamental hypermultiplet remains massless and gets an expectation value p. This requires the gauge group on the S 3 to be broken from U (N ) to U (N − 1) × U (1), and then the U (1) factors on the S 3 and on L K to be identified. Giving the expectation value to the bifundamental gives rise to the smoothing of the intersection between L K and S 3 , which we described above. In the next subsection we will show that the partition function in this phase is given by (3.3).

Framing of the Lagrangians
To write down the partition function of the theory, in addition to choosing the vacuum by picking a point on the Riemann surface, we have to choose a framing of the Lagrangian: a flat connection at infinity. Since the holonomies around different cycles of the Λ K ≈ T 2 do not commute, different choices are related by wave-function transforms. In L K , the natural variable is the holonomy along the longitude of the knot x, and the wave function is given by In M K , the natural variable is p, the holonomy around the meridian.
Since x and p are canonically conjugate in the theory on the brane, We can simply deform the contour, at least in perturbation theory, to find In particular, the Gromov-Witten "potentials" W K (x) and U K (p) that we previously associated to L K and M K are simply dual to each other, related by Legendre transformation. In Section 6.9, we will provide a different

Large N duality, mirror symmetry, and HOM-FLY invariant for links
Consider Chern-Simons theory on S 3 with an n-component link K, where K j are the knot components of K. The conormal is the union of the n conormals of the knot components, and L K ∩ S 3 = K. Under large N transition as in Section 2.3, X is replaced by the total space Y of the bundle O(−1) ⊕ O(−1) → CP 1 . As we explained in Section 3.2, it is natural to view the boundary data as the only fixed data, in which case we should consider the geometry of L K far from the apex of the cone. Here the Lagrangian L K approaches R × Λ K , where Λ K is the union of the n conormal tori Λ K j of the components of K that describe the imprint of the corresponding branes at the S 2 × S 3 at infinity of the Calabi-Yau. As in the knot case, there are different ways to fill in Λ K in the interior of Y . As we shall see, the link case is more involved than the case of a single knot: there is a larger number of ways to fill in R × Λ K , connecting different numbers of components of Λ K in the interior.
As in Section 2.4, we are led to consider holonomies of a GL(1) gauge field on the brane around the 1-cycles of Λ K . This gives a phase space of the system, which is the cotangent bundle of a 2n-dimensional torus: The torus is the torus of holonomies of the U (1) gauge field on the brane around the 2n 1-cycles of Λ K . The cotangent direction arises from the moduli of the Lagrangians. Equivalently, the phase space M n arises from holonomies of the GL(1) gauge field on R × Λ K around the cycles of n copies of T 2 that comprise the infinity Λ K . Thus on M n we have (C * ) 2n -coordinates e x i , e p j , i, j = 1, . . . , n associated to the longitudes and the meridians of knots K i , and a symplectic form Note that M n is in fact hyper-Kähler, a fact which we will make use of later in this paper.
Inside M n there is a Lagrangian subvariety V K associated to the link (i.e. V K is half-dimensional and ω| V K = 0). As we shall see, V K is in general reducible: where the subvarieties correspond to different fillings of Λ K labeled by partitions P of {1, . . . , n}. We want to identify the fillings that can be obtained by smoothly varying the holonomies, since these give rise to the same Lagrangian submanifold V K (P ). We will initiate the study of the varieties in this section, using Chern-Simons theory and large N duality. In Sections 5 and 6 we will use another approach based on knot contact homology, where V K will be identified with the augmentation variety of the knot. In Section 4.4, we will study quantum mirror symmetry, where we explain how to quantize this variety.

Conormal bundle L K
In the simplest case, R × Λ K is filled by the conormal bundle of the link This is simply a union of n disconnected Lagrangians L K j , which we move off of S 3 independently and then transition to Y . This was studied previously in [17].
On Y , in the presence of L K , there are no holomorphic disks with boundary on more than one component of L K , as the conormals L K j are disjoint from each other. So the disk amplitude is simply a sum of k contributions, coming from disks with boundaries on one of the L K j : From the perspective of large N dualities, this arises as follows. We start with X = T * S 3 , with the n Lagrangians L K j intersecting the zero section along the link K. Pushing L K j off of S 3 by x j for each j, we give a mass x j to the bifundamental hypermultiplet at the intersection of K j and S 3 .
Integrating these k massive hypermultiplets out, we end up computing the expectation value of  As was shown in [17], the classical limit of this is Ψ K,1 n (x 1 , . . . , x n , g s , Q) = exp 1 g s W K,1 n (x 1 , . . . , x n ) + . . . , where W K,1 n is as in (4.1). In this case, the link indeed gives rise to a n-dimensional variety, but one which is simply a direct product of n 1dimensional curves, We will denote this variety by V K (1 n ), to indicate that it factors as a product of n pieces, each of dimension 1. Equivalently, we can write this as
The quantum corrected moduli space of the brane on M K in Y is an n-dimensional variety V K (n) given by By the above construction, V K (n) can be viewed as a Lagrangian subspace in the 2n-dimensional space of (x i , p i ) ∈ (C) 2n relative to the symplectic form i dx i ∧ dp i . Note that if the link K is a completely split link (i.e., one can find disjoint and V K (1 n ) coincide. This is because the HOMFLY polynomial of the link factor in this case, and consequently the brane partition function, is an exact product of V K i (1).
This suggests that in this case, one can go smoothly from the phase where we have a single Lagrangian M K , to the conormal Lagrangian L K consisting of n disconnected Lagrangians. Since V K (n) would have no new information in this case, we will simply identify it with V K (1 n ) and reserve the notation of V K (n) for a non-split link K of n components: the fact that the link is non-split translates into the fact that V K (n) is not a direct product of some lower-dimensional varieties.
Just as in the case of the knots it is natural to expect V K (n) to correspond to moduli space of SL(2, C) flat connections on S 3 − K. More precisely, we expect this to be a Q-deformed version of it, where SL(2, C) is embedded in the canonical way in SL(N, C). This is natural from the viewpoint of the Higgsing construction discussed before.

General fillings
We will now describe the set of physically distinct ways of filling in the torus Λ K in the interior of Y. As we will explain, the physically distinct fillings are labeled by the ways to partition an n-component link K into sublinks. The different ways to do this are labeled by integer partitions P of {1, . . . , n}, where we view the parts P a of the partition P as giving corresponding sublinks K a of K. An alternative way to label the fillings is by a link K(P ) which is a split union of sublinks K a : (That is, move each of the sublinks away from the others, so that the result is split.) We conjecture that the general distinct fillings are labeled by the primitive partitions P , namely those for which each sublink K a is non-split, or, equivalently, those that have no refinement P such that K(P ) = K(P ).  Given such a primitive partition P and the associated collection of sublinks K(P ), we will denote the corresponding Lagrangian filling by The filling F K(P ) gives rise to an n-dimensional Lagrangian submanifold which is a product of the varieties V Ka associated to the sublinks K a of K(P ).
Each sublink K a is by assumption non-split; suppose that K a consists of n a knot components, where n = a n a . We now explain the meaning of V K (P ).
Consider the disk amplitude obtained by studying the topological A-model string on Y , together with the Lagrangian brane F K(P ) . We define the "D-mirror" variety V K (P ) as We have written U K(P ) in terms of the meridian variables, which is a convenient choice when the meridian cycles get filled in. Since there cannot be any disk instantons ending on disconnected components of a Lagrangian filling, the disk potential is a sum Each disconnected component M Ka of the filling is homologically like the complement of the link K a in the classical limit, and U Ka is the corresponding potential. This together with the fact that the potentials U Ka give rise to varieties V Ka (n a ) gives the result we claimed. This is consistent with what we expect from large N duality: since the link K(P ) is a split link, with K a its non-split sublinks, the corresponding potential should be captured by the classical limit of the HOMFLY polynomial of K(P ): g s log(H p 1 /gs,...,pn a /gs (K(P ))).
It is important to note that the HOMFLY polynomial H K(P ) of the link K(P ) is not the full topological string amplitude for this filling; P and K(P ) only label the filling. The branes in the filling labeled by P do remember the full data of the link K they came from. The partitions P for which we get nontrivial fillings correspond to saddle (critical) points of the HOMFLY polynomial H K of the link, and the full partition function of topological string with the Lagrangian branes on the filling F K(P ) is encoded by the expansion of H K , the HOMFLY polynomial of the original link K, around its saddle point P . We will discuss this in more detail below.
To summarize, for each different splitting of K into non-split sublinks K a , labeled by the primitive partition P , we get a Lagrangian filling F K(P ) of the Legendrian tori Λ K at infinity. This filling is a union of link-complementlike Lagrangians M Ka , one for each sublink of K(P ). The mirror variety corresponding to this filling is V K (P ), which is a product of the corresponding varieties V Ka (n a ).

D-mirror variety
Given the link K, it is natural to define the mirror variety so that it depends on the data at infinity of the Calabi-Yau alone, and not the specific filling.
With this in mind, we propose to identify as the mirror variety to the link K. This is a natural proposal as it contains information both about knots that comprise the link, and the way they are linked together. Moreover, this is exactly the same data that goes into defining the dual Chern-Simons partition function on S 3 with the link K. In In general, different saddle points contribute to Ψ K : where Ψ P K is a wave function canonically associated to the corresponding saddle point P , and the coefficients c P are integers. Which saddle point dominates the Chern-Simons path integral depends on the values of the parameters x i , g s , and N , and the classical action at the saddle point . , x n ) + . . . , 5 Non-perturbative aspects of topological strings have been studied in [18,19,20].
where W K (P ) is the potential defining the variety V K (P ) via In the regime where one of the saddle points dominates the others, are exponentially suppressed.
The above way of writing Ψ K is a little bit crude because it neglects the fact that in general, a single P may give rise to more than one wave function, as V K (P ) may give rise to several vacua. We will disregard this fact for two reasons: notational simplicity, and the fact that it is V K (P ) that plays the crucial role in quantizing the theory. (While the subtlety affects the possible The existence of a single function Ψ K (x 1 , . . . , x n ) that has all these different classical limits imposes constraints on the structure of V K , viewed as a reducible variety. As we will discuss in Section 8, quantization of the variety V K gives rise to a D-module on M n . The wave function Ψ K (x 1 , . . . , x n ) is a section of this D-module. The fact that the D-module that arises in this way is irreducible (it corresponds to a single irreducible quantum system, rather than being a direct sum of super-selection sectors) implies that different components V K (P ) of V K must fit together in a specific way, encoded by a graph that we now describe.
To every link K we can associate the graph Γ K as follows. The vertices of the graph correspond to the primitive partitions P of {1, . . . , n}, corresponding to the ways of partitioning K into non-split sublinks. The set of all such partitions is a subset of partitions of the set {1, . . . , n}. Two vertices P and P of the graph are connected by an edge if P is a refinement of P and there is no other graph vertex P between the two (where P is a refinement of P and P is a refinement of P ).
We claim that the graph Γ K obtained in this way captures the geometry of V K as follows: for any pair of vertices of the graph that are connected by an edge, we conjecture that where codimension is counted inside either of the n-dimensional varieties V K (P ), V K (P ). Note that, based on counting dimensions, a generic intersection would be over points (codimension n). The presence of an edge means that the intersection of V K (P ) and V K (P ) is as non-generic as possible without them coinciding. In a generic situation thus the graph would consist of vertices alone. The existence of a single function Ψ K implies that this is never the case for multi-component links. Furthermore, it is natural to generalize our conjecture as follows: for any two primitive partitions P and P , the codimension of the intersection of V K (P ) and V K (P ) is at most the distance between the partitions, defined as the minimum number of edges in Γ K needed to pass from P to P : If all partitions are primitive and thus give vertices of the graph, then we can connect the coarsest partition 6 P c = (12 · · · n) with the finest partition P f = (1) · · · (n) by a sequence of partitions P c = P 1 ,P 2 , . . . , P n = P f , where P i = (1) · · · (i − 1)(i · · · n). Passing successively from P 1 to P n involves removing the knot components of K one by one, starting with the first component, until we have removed all of them. We expect that for each i, V K (P i ) and V K (P i+1 ) intersect in codimension 1, and this allows us to move between components of V K in such a way that each move is between components whose intersection has codimension 1. See Section 5.4 for further discussion for general links and a reformulation of this discussion in terms of the augmentation variety.
Let us now explain why the irreducibility of the D-module implies the codimension condition. A way to characterize the fact that Ψ K corresponds to an irreducible D-module is that, first of all, it satisfies a set of differential (or, more precisely, difference) equations generated by a finite set of operators where each of A α is of the form where a k 1 ,...,kn are polynomials in e x i . The fact that the equation (4.7) is linear implies that not only Ψ K but also each of Ψ P K satisfy the equation. It was proven in [21] that the colored HOMFLY polynomial of any link satisfies such a finite set of difference equations, and that the set is holonomic (or more precisely, q-holonomic). What this means is that, in the classical limit , when x i , p j = g s ∂ x j become numbers, the set of equations (4.7) parametrize a Lagrangian submanifold of M n . This Lagrangian submanifold is just what The fact that irreducible components of V K = P V K (P ) intersect over the varieties of codimension 1 is a consequence of a general theory of systems of differential equations of this type [22,23,24]. 7 . We will discuss this point further in Section 8, where we discuss quantization of the system from the physical perspective.

An example: Hopf link
The simplest nontrivial 2-component link is the Hopf link K = Hopf = In this case, we expect two distinct fillings, corresponding to partitions P = (12) (which we will abbreviate as 2) and P = (1)(2) (which we will abbreviate as 1 2 ). In the first case P = (12), Lagrangian filling of Λ Hopf is the complement of the entire Hopf link in S 3 , and this gives rise to V Hopf (2). The filling is easy to see from the toric diagram, before and after the transition. The Hopf link is realized by taking two Lagrangian branes on opposite toric legs before the transition. They can each glue to one copy of the brane on S 3 to move off the S 3 . The topology of the resulting Lagrangian is T 2 × R. This has b 1 = 2, so the moduli space is 2-dimensional. Moreover, there are no disk instanton corrections to it 8 , so the potential is determined classically. It is easy to see that since the critical points associated to this simply state that the meridian of one knot in the Hopf link gets identified with the longitude of the other, once we glue M K from pieces, as we described. Namely, in terms of The other filling corresponding to P = (1)(2) has two branes L K i , each simply a conormal to the unknot, probing the conifold geometry after the transition. At the level of the disk potential, they cannot talk to each other, leading to a direct product of two copies of Riemann surfaces mirror to the unknot: The variety (2) coincides with the augmentation variety of the Hopf link, as we will see in the next section.
The same result can be obtained in a number of different ways, two of which we mention here. First, the result can be obtained by calculating the HOMFLY polynomials of the Hopf link, colored by totally symmetric representations, and taking the classical limit. We will show this in Appendix A in detail. Second, as we will brush on in Section 8, one can obtain the same result by considering a pair of branes on the Riemann surface mirror to the unknot.
These two approaches allow one not only to recover the classical variety V Hopf , but also give a prediction for its quantization. Namely, Here S m 1 m 2 is the matrix element corresponding to the totally symmetric representations with m 1 , m 2 boxes.
Using the well-known explicit expressions (see Appendix A), one can show that Ψ Hopf satisfies a set of difference equations , In the classical limit, these reduce to A α = 0, where The Lagrangian solutions to this are precisely the variety V Hopf = V Hopf (1 2 ) ∪ V Hopf (2). (There are other solutions that do not lead to Lagrangians. These are not of interest, as they do not describe saddle points.) Finally, note that given by where µ 1 , λ 1 lie on the unknot curve The equations are simple enough that we can solve them exactly. Two linearly independent solutions, with different asymptotics, can be obtained by considering two different contours of integration C (2) , where Ψ (x) is the partition function of the unknot. 9 The wave function corresponding to V Hopf (2) is obtained by taking a contour C (2) which is a small circle p 2 = x 1 ; this gives The wave function corresponding to V Hopf (1 2 ) can be obtained by taking the contour C (1 2 ) to run along the real axis. Using the fact that Fourier transform

Knot parallels and higher rank representations
Consider a linkK obtained by taking n parallels of a single knot K. This is closely related to studying n branes on a single Lagrangian associated to the knot K. As explained in Section 2.2, the partition functions are related by arises from integrating out short strings, with boundary on L K alone and with no boundaries on the S 3 . This is a sum over annuli and hence has no This relation between ZK and Z  K (x 1 , . . . , x n ), where by assumption these strings are massive and we have integrated them out to 9 We have simplified things slightly by shifts of variables. For details see Appendix A.
get ∆(x). We will give a rigorous mathematical proof of this statement in Section 5, from the perspective of knot contact homology. It is satisfying that the same picture emerges from Chern-Simons theory and large N duality.

Knot contact homology
In this section we discuss the connection between knot contact homology and topological strings in the context of knot and link invariants. We first give a brief discussion of this connection in the next subsection, before giving a more detailed discussion in the following subsections and in Section 6.

Brief summary of relation between topological strings and knot contact homology
Knot contact homology uses the Lagrangian brane L K in X = T * S 3 , but with the zero section S 3 ⊂ T * S 3 deleted. In other words, one considers the geometry far from the apex of the cone where the geometry is R × U * S 3 ≈ R × S 3 × S 2 ; this shares the basic feature of the geometry of Y after the transition, where S 2 does not bound.
For simplicity of notation, we temporarily assume that K is a singlecomponent knot. Then L K is R × Λ K ≈ R × T 2 and it is natural to view the R-factor as the "time" direction. Here the torus Λ K can be viewed as the geometry of the Lagrangian brane at infinity. The knot leaves its imprint on how the Legendrian submanifold Λ K is embedded in the contact manifold Consider the physical open string states ending on Λ K . Among these curves with endpoints on Λ K , the paths that are stationary for the action  Using the large N duality, this should be the same as the Q-deformed Apolynomial we defined earlier, which is the quantum corrected moduli space of the single brane L K . The fact that we find agreement between these two setups can be interpreted as further evidence for large N duality.
The same setup applies to the case of links. The main novelty there is that we have n Legendrian tori and thus we have We now turn our attention to a more mathematical description of knot contact homology. We begin with a brief general introduction to contact homology, an object associated to contact manifolds and to their Legendrian submanifolds. Next we describe knot contact homology, which is the contact homology associated to the Legendrian torus Λ K ⊂ U * R 3 (which can be viewed as a patch of U * S 3 = S 2 × S 3 ), first geometrically and then algebraically. We then present augmentations from an algebraic perspective and use these to define a polynomial knot invariant, the augmentation polynomial, or a variety in the case of a multi-component link. The augmentation variety is conjectured to agree with the mirror-symmetry variety V K from previous sections. Besides agreeing with V K in a number of examples, we will see that the augmentation variety shares many properties of V K . This will be discussed further in Section 6.

Mathematical overview of knot contact homology
Here we provide a summary of the mathematics behind knot contact homology and augmentations. This goes into more detail than the previous subsection and also places the construction in the context of contact and symplectic geometry. The full story is rather long, and we will omit many technical details in the interest of readability. As a result, some of the discussion below is rather imprecise, but we will provide references to more detailed treatments in the literature for the interested reader. A slightly less brief overview can be found in [26].
Knot contact homology is a special case of Legendrian contact homology, which is itself a small part of the more elaborate Symplectic Field Theory package in symplectic topology introduced in [27]. The general setup begins with a contact manifold, which for our purposes is a (2m − 1)-dimensional Legendrian contact homology is associated to the contact manifold V along with a Legendrian submanifold Λ ⊂ V , which is a submanifold along which α is identically 0, of maximal dimension; the contact condition on α forces this maximal dimension to be m−1. In this case, R×Λ is a Lagrangian Define a Reeb chord of Λ to be a flowline of R α that begins and ends on Λ. We define the algebra A = A(Λ) to be the free (tensor) algebra over the We can construct a differential map ∂ : A → A by counting certain Jholomorphic curves in R × V with boundary on R × Λ. More precisely, for where the second sum is over all rigid (i.e., 0-dimensional moduli space) Jholomorphic maps ∆ from a disk with k + 1 punctures along its boundary to R × V , such that the boundary of the disk is mapped to R × Λ and the punctures are mapped in order to a neighborhood of a i near t = ∞, and neighborhoods of a j 1 , . . . , a j k near t = −∞; (sgn ∆) is a sign associated to ∆; and [∆] is the homology class of ∆ in H 2 (V, Λ). See Figure 1.
In appropriate circumstances, ∂ 2 = 0, ∂ lowers degree by 1, and the homology of the differential graded algebra (A, ∂) is an invariant of the Legendrian submanifold Λ, called the Legendrian contact homology of Λ. (See e.g. [28,29]. From a physical perspective, the main necessary condition is We now apply this general setup to one particular case of interest. Let V be the bundle U * S 3 consisting of unit-length cotangent vectors in S 3 ; this is a five-dimensional contact manifold with α = p dq. If K ⊂ S 3 is a link, then define the unit conormal bundle Λ K ⊂ U * S 3 to be the subset consisting of all unit covectors in T * S 3 lying over K and annihilating T K. It is an exercise to check that Λ K is Legendrian in U * S 3 . We can then define the knot contact homology of K to be the homology of the differential graded algebra (A, ∂) defined above with Λ = Λ K .
If K is an n-component link K 1 ∪ · · · ∪ K n , then Λ K = Λ K 1 ∪ · · · ∪ Λ Kn is a disjoint union of n two-dimensional tori, and we can write Here t is the generator of H 2 (S 2 ) and to the standard Seifert framing (so that x i and K i have linking number 0).
Correspondingly, we can write the coefficient ring for A as Given a braid with m strands such that K is the closure of the braid (i.e., the result of gluing together corresponding ends of the braid in S 3 ), there is a combinatorial formula for the differential graded algebra (A, ∂) associated to K. Outside of computations, we will not need the precise formula, and we omit it here; please see the Appendix of [26] for the full definition of the version of the invariant that we use in this paper, noting that our Q is denoted by U in that paper. (The algebra given in [26] is in turn based on work that originally appeared in the series of papers [30,31,32,33].) For our purposes, it suffices to note that A is finitely generated as an algebra over , and the generators of A are all of degree ≥ 0. These generators include m(m − 1) generators of degree 0 that we write as a ij for 1 ≤ i, j ≤ m and i = j.
As an example, consider the Hopf link, which is represented by the 2strand braid σ 2 1 . For this link, A is generated by the following generators: a 12 , a 21 of degree 0; b 12 , b 21 , c 11 , c 12 , c 21 , c 22 of degree 1; and a number of other generators of degree 1 and 2 that are irrelevant for our purposes. From the formula from [26], the differential ∂ is given by: We will use this computation in the next subsection.

Augmentations, the augmentation variety, and the A-polynomial
Having defined the differential graded algebra for knot contact homology, we now turn to augmentations. In general, an augmentation of a differential graded algebra (A, ∂) is an algebra map (preserving multiplication) from A to a ring S, such that (1) = 1, • ∂ = 0, and vanishes on elements of nonzero degree. Here we will take our augmentations to be graded, where we think of the ring S as a DGA with trivial differential concentrated in degree zero. Thus, our augmentations are nonzero only on generators of degree zero.
Augmentations arise naturally in symplectic topology in the context of symplectic fillings. Suppose that V is a contact manifold and W is a symplectic filling of V ; this is a symplectic manifold whose boundary is V , satisfying a certain compatibility condition (V should be a convex end of W ). We further assume that W is an exact symplectic filling if the contact 1-form α on V extends to a 1-form α on W such that ω = dα, where ω is the symplectic form on W . Next let Λ ⊂ V be Legendrian, and suppose that L ⊂ W is a Lagrangian filling of Λ; this is an embedded Lagrangian submanifold whose boundary is Λ (and which looks like R × Λ near V ).
A special case of this construction is when L is an exact Lagrangian filling of Λ, i.e., when α| L has a primitive on L. In this case, we can associate to L a canonical augmentation of the DGA for Λ, obtained by counting holomorphic disks in W with boundary on L and asymptotic to a Reeb chord for Λ. More generally, if L is non-exact, we can use L to deduce certain structures on the space of abstract augmentations of Λ. This perspective relating augmentations to Lagrangian fillings is central to the mathematical side of this paper, and we return to it in detail in Section 6.
For now, we examine the space of augmentations in our particular case.
When K is an n-component link and (A, ∂) is its differential graded algebra , an augmentation to C is a (graded) algebra map : A → C such that (∂(a)) = 0 for all Reeb chords a. Note that an augmentation restricts to a ring homomorphism from the coefficient ring of A to C.
With this in mind, define the augmentation variety of the n-component link K to be the subset of (C * ) 2n+1 = (C − {0}) 2n+1 given by (More precisely, the augmentation variety is the closure of the maximaldimensional piece of this set.) This variety is the set of ways to assign nonzero complex numbers to µ 1 , . . . , µ n , λ 1 , . . . , λ n , Q such that a collection of polynomials in m(m − 1) variables a ij (with coefficients involving µ 1 , . . . , µ n , λ 1 , . . . , λ n , Q), namely the images under ∂ of the generators of degree one, has a common root. Note that the augmentation variety is algebraic (it is cut out in (C * ) 2n+1 by the zero locus of some collection of polynomials), and that it is a link invariant since the DGA for knot contact homology is invariant up to homotopy. Also, in line with previous sections, one can view the augmentation variety as a one-parameter family of varieties in (C * ) 2n with parameter given by Q.
As an example, for the Hopf link, the expression for the differential from Section 5.2 implies that any augmentation must satisfy either (a 12 ) = (a 21 ) = 0, in which case or (a 12 ), (a 21 ) = 0, in which case (λ 2 ) = (µ 1 ) and (λ 1 ) = (µ 2 ). It follows that Further examples of augmentation varieties are given in Section 7.
As observed in [30], there is a close connection between the augmentation variety and the A-polynomial, which we now discuss. First assume that K is a single-component knot, and let m, l ∈ π 1 (S 3 − K) denote the meridian and longitude of K. For a representation ρ : simultaneously diagonalize ρ(m) and ρ(l), and let (µ, λ) and (µ −1 , λ −1 ) be the corresponding eigenvalues. Then the closure of the (highest-dimensional When K is a knot, the augmentation variety V K ⊂ (C * ) 3 is also the vanishing locus of a polynomial, the augmentation polynomial Aug K (λ, µ, Q).
The relation between the augmentation variety and the A-polynomial is then as follows [30,Prop. 5.9]: Here "|" denotes "divides". In particular, (µ − 1) and (λ − 1) are both factors of Aug K (λ, µ) for any knot K, where the latter comes from reducible SL(2, C) representations. These two factors have a nice interpretation in terms of Lagrangian fillings: (µ − 1) comes from the filling of Λ K by the conormal Lagrangian L K , while (λ − 1) comes from the filling by the knot We now generalize to the case where K is an n-component link. There is a well-known generalization of the A-polynomial to links, as follows. As before, let ρ : π 1 (S 3 − K) → SL(2, C) be a representation, and let m i , l i denote the meridian and longitude for component i. For any single i = 1, . . . , n, ρ(m i ) and ρ(l i ) can be simultaneously diagonalized, with eigenvalues (µ i , λ i ) and Then (the closure of the highest-dimensional part of) is a variety in (C * ) 2n , which we denote by V A K and call the A-polynomial variety of K. 10 Note that for any fixed i = 1, . . . , n, one can abelianize π 1 (S 3 − K) and set all meridians but m i to 0, to obtain Z. It follows that for any µ i ∈ C * , the curve is contained in V K . This is the analogue of the fact that (λ − 1) divides the A-polynomial for knots.
Arguing along the lines of Proposition 5.9 from [30] then yields the fol- Here we defineλ i = λ i µ k i1 1 · · · µ k in n , where the k ij are integers determined by . Thus the A-polynomial variety of K can be viewed as a subset at Q = 1 of the augmentation variety, for links as well as for knots.
We conclude this section with a remark about the nature of the augmentation variety. For all examples where the variety has been computed, the following holds: for fixed Q, In x, p coordinates, this says that the augmentation variety is Lagrangian with respect to the symplectic form n i=1 dx i ∧ dp i , in accordance with the notion that the variety is locally given by a potential. (In particular, V K is n-dimensional.) The underlying geometric reason for this observation is related to Lagrangian fillings and will be discussed further in Section 6 from the viewpoint of knot contact homology; it agrees well with corresponding predictions from large N duality and the relation with Chern-Simons theory.
See Section 4. In the rest of the subsection we will provide evidence for the conjecture.

Components of the augmentation variety
First, we will prove that every partition P of {1, . . . , n} leads to a (possibly empty) component of the augmentation variety of K, which we label V K (P ) in accordance with Section 4.4. Next, we will provide evidence that only the primitive partitions P lead to nonempty components of augmentation variety, and that moreover these are all components of V K .
If we view a link K as a union of two sublinks K 1 ∪ K 2 , then the fiber Here µ 1 = (µ 1 , . . . , µ n 1 ), µ 2 = (µ n 1 +1 , . . . , µ n ), and similarly for λ 1 and λ 2 , where the first n 1 components of K belong to K 1 and the remainder to K 2 . This follows from considering augmentations of the DGA for K that send all "mixed Reeb chords" a ij , i.e., Reeb chords with one endpoint in Λ K 1 and the other in Λ K 2 , to 0. The idea of separating the DGA for a Legendrian link into the DGAs of sublinks by sending mixed chords to 0 is well-established in contact geometry; see e.g. [35].
By iterating this process, one sees that if P is a partition of {1, . . . , n} consisting of subsets, then augmentations of (the DGAs for) the corresponding sublinks of K produce an augmentation of K, where mixed Reeb chords connecting different sublinks are sent to 0. Say that an augmentation of K obtained in this way is associated to the partition P . Note that if P is a refinement of P , then any augmentation associated to P is also associated to P , and in particular that all augmentations are associated to the partition (12 · · · n).
Say that an augmentation associated to a partition P is irreducible if it is not associated to any refinement of P . Then we can define V K (P ) to be the subset of the augmentation variety V K corresponding to irreducible augmentations associated to P . If P consists of subsets, then V K (P ) is a fiber product of the augmentation varieties associated to the relevant sublinks. As in Section 4.4, we sometimes abbreviate For example, for the Hopf link, from the computation in Section 5.3, we Under certain circumstances, V K (P ) may be forced to be empty: Then any augmentation of the DGA of K splits into augmentations of the DGAs of K 1 and K 2 , and Proof. If K is split, then it can be given as the closure of a braid B that is also split. In this case, the matrices Φ L , Φ R from the definition of the DGA for knot contact homology are block-diagonal, and the result follows from the formula for the differential.
A more geometric proof is as follows: the split link is a connected sum (S 3 , K 1 )#(S 3 , K 2 ) and the unit disk cotangent bundle is then obtained by joining the two unit disk cotangent bundles with the corresponding Weinstein 1-handle. This shows that there exists a contact form on U * S 3 for which there are no Reeb chords connecting Λ K 1 to Λ K 2 , and the result follows immediately.
By Proposition 5.2, if K = K 1 ∪ K 2 is split and P is the corresponding two-set partition of {1, . . . , n}, then V K (P ) = ∅. More generally, if K is a general link and P is an -set partition for which one of the sublinks is split, then V K (P ) = ∅. We are not currently aware of any other circumstance in which V K (P ) is empty; see also the discussion in Section 4.4.

The link graph Γ K and the augmentation variety
As in Section 4.4, to an n-component link K we can associate a graph Γ K whose vertices are the primitive partitions of the link K, and where P and P are connected if one is a refinement of the other, and there is no intermediate refinement P between the two such that P is also primitive. Based on computations of the augmentation variety for a number of links, as well as the D-module argument from Section 4.4, we make the following conjecture.
Conjecture 5.3. The graph associated to a link is connected, and if P and P are connected by an edge, then More generally, if P 1 , . . . , P r are vertices such that P i+1 is a refinement of P i and P i , P i+1 are joined by an edge for all There is one important special case where we can recast this conjecture in terms of "Lagrangian reduction". (In Section 6.7 we discuss Conjecture 5.3 and Lagrangian reduction from a geometric perspective that physically corresponds to adding an anti-brane to a filling brane.) Suppose that P = (1 · · · n) and P = (1 · · · (n − 1))(n). (1) are nonempty. According to the remark at the end of Section 5.3, for fixed Q, V K (n) is a Lagrangian submanifold of (C * ) 2(n−1) with the appropriate symplectic structure, as is Suppose more generally that we have a product symplectic manifold W = W 1 × W 2 , where W 1 and W 2 are symplectic, and Lagrangian submanifolds is also a Lagrangian submanifold that we call the reduction of L along L 2 .
In our case (with , we have the following conjecture. (1). Concretely, we have: (1) and

Conjecture 5.4 has been verified in all computed examples. Note that it
is not necessarily the case that V K−Kn (n − 1) is precisely equal to, rather than just a subset of, the reduction of V K (n) along V Kn (1): see Section 7.1.
For our particular choice of P = (1 · · · n) and P = (1 · · · (n − 1))(n), it would follow immediately from Conjecture 5.4 that V K (P ) and V K (P ) have is a key special case of Conjecture 5.3.

Higher rank representations of knot contact homology
Here we trace out the mathematical story that parallels the discussion of multiple branes in Section 2.2. Augmentations of a differential graded algebra (A, ∂), such as the DGA for knot contact homology, are maps : A → C, and can thus be viewed as one-dimensional representations of the DGA. One can also consider a generalization to arbitrary dimension: define a rank n representation of a DGA (A, ∂) to be an algebra map ρ : A → End C n such that ρ • ∂ = 0, ρ sends any element of nonzero degree to 0, and ρ(Q) is a scalar multiple of the identity. (Here End C n is the space of endomorphisms of C n , or n × n matrices.) As a technical note, it is important for rank n > 1 representations that we replace the DGA (A, ∂) described in previous subsections, in which λ, µ commute with all Reeb chords, with a slight variant, the fully noncommutative DGA, in which λ, µ commute with each other (and with Q) but not with Reeb chords. This allows ρ(λ) and ρ(µ) to be something besides scalar multiples of the identity, even in an irreducible representation. See [26] for a description of the fully noncommutative DGA associated to a knot.
Since λ and µ commute with each other, the matrices ρ(λ) and ρ(µ) can be simultaneously diagonalized. Say that a rank n representation of (A, ∂) is diagonal if ρ(λ) and ρ(µ) are both diagonal matrices.
Our goal here is to relate diagonal higher rank representations of the knot contact homology of a single-component knot K to augmentations of a link consisting of multiple parallel copies of K. This follows from a more general result about higher rank representations of Legendrian submanifolds, which we now describe.
Let Λ be a Legendrian submanifold in a contact manifold V . The Legendrian neighborhood theorem states that a tubular neighborhood of Λ is contactomorphic to (i.e., diffeomorphic to with the same contact structure as) the 1-jet space J 1 (Λ) = T * Λ × R. Here J 1 (Λ) is equipped with the contact 1-form dz − p dq, where z is the R coordinate and p dq is the Liouville 1-form on T * Λ. Define a parallel to Λ to be a Legendrian Λ ⊂ V such that Λ lies in a small neighborhood of Λ and Λ corresponds to the graph of a Finally, let n parallel copies of Λ denote an n-component Legendrian submanifold of V given by the union of Λ and n − 1 parallels of Λ, where these n components are pairwise disjoint. (This is a generalization of the notion of an N -copy from [35]. Note that our notion of parallel copies is not unique.) of Λ that begins on Λ j 1 , ends on Λ j 2 , and limits to a i as the neighborhood shrinks to zero.
In the case where K ⊂ S 3 is a knot and Λ K is its conormal bundle, one can consider the conormal bundle Λ K to n parallel copies K of K. (These parallel copies of K can be given by pushoffs of K with respect to any framing, or more generally by any collection of n disjoint closed curves in a tubular neighborhood of K that are each the graph of a section of the normal bundle to K, viewed as the tubular neighborhood.) It is an exercise in local coordinates (see [33]) to show that Λ K comprises n parallel copies of Λ K : write Then the knot contact homology of the link K is the Legendrian contact homology of Λ K . Our main result in this subsection is now as follows. and suppose that in the DGA for Λ K , ∂(a i ) = γ 0 a j 1 γ 1 · · · a jn γ n , where γ 0 , . . . , γ n are words in Q, λ, µ. Then by the above argument, the differential in the DGA for Λ K , omitting short chords, is given by where ∂(A i ) is the matrix (∂(a j 1 ,j 2 i )), multiplication on the right is matrix multiplication, and Γ j is the result of replacing λ, µ in γ j by diagonal matrices diag(λ 1 , . . . , λ n ), diag(µ 1 , . . . , µ n ). The result follows.
We conclude with two examples. First consider the case where K is the unknot. In this case, a diagonal rank n representation of the knot contact homology of K is a map ρ with values in n × n matrices such that ρ(λ) = diag(λ 1 , . . . , λ n ), ρ(µ) = diag(µ 1 , . . . , µ n ), and Any such representation is reducible and a product of n 1-dimensional representations. The resulting variety in (C * ) 2n+1 is the same as the augmentation variety V K (1 n ) for any link K consisting of n parallel copies of the unknot.
Note that K could well have augmentations besides the ones in but these extra augmentations do not send short Reeb chords to 0.
Next, suppose that K is the right-handed trefoil. In this case, by a computation of knot contact homology (omitted here), a rank n representation of the knot contact homology of K is a way to assign n × n matrices to λ, µ, a such that the following relations hold: A diagonal rank n representation further sends λ, µ to diagonal matrices as above.
When n = 2, diagonal rank 2 representations for the trefoil fall into two classes. The reducible representations satisfy This corresponds to at least a portion of VK (2), namely the subset corresponding to augmentations that send all short Reeb chords to 0. We do not currently know if this must give all of VK(2) for anyK, due to the difficulty of computing the augmentation variety for parallel copies of the trefoil.
When n = 3, diagonal rank 3 representations for the trefoil are either reducible, in which case they come from a direct sum of a rank 1 representation (i.e., an augmentation of K) and a rank 2 representation, or irreducible.
The latter can be shown to satisfy and correspond to a portion of VK(3) whereK consists of three parallel copies of K.

Augmentations and Lagrangian fillings
In this section we will consider augmentations of knot contact homology, in- Connecting to the physical perspective, this relation between knot contact homology and the Gromov-Witten potential shows that near augmentations that are geometrically induced by a Lagrangian filling, the augmentation variety of a link as defined through knot contact homology agrees with the physically defined mirror variety V K ; in particular, in the case of knots, the Q-deformed A-polynomial and the augmentation polynomial have zero sets that agree locally. As above, "locally" is in the algebro-geometric sense, so the subsets where the two agree are rather large. In fact, for many knots the augmentation variety is known to be irreducible; in these cases, since there is always the conormal filling, we know that the two polynomials must agree.
Here is an outline of the remainder of this section. In Section 6.1, we discuss the most well-studied case of augmentations of contact homology DGAs that are induced by exact Lagrangian fillings. This gives a geometric explanation of the trivial branches of the augmentation variety for any link.
We then generalize this to certain non-exact Lagrangian fillings, which are the central objects for a geometric understanding the augmentation variety: in Section 6.2 we describe the class of non-exact filling that we use; in Sections 6.3 and 6.4, following [36], we introduce obstruction chains that allow us to generalize the augmentation maps in the exact case to more involved chain maps induced by our class of non-exact Lagrangian fillings.
We then turn to geometric constructions of Lagrangian fillings. In particular, we aim at constructing connected Lagrangian fillings of the conormal of a n-component link K, that leads augmentations in the component V K (n) corresponding to the trivial partition, and that would as for knots above explain why the physical and the mathematical varieties agree. In Sections 6.5 and 6.6, we discuss these geometric constructions of Lagrangian fillings in connection with the conjecture on codimension-1 intersection between augmentation varieties of non-split links from Section 4.4. In Section 6.7 we discuss a gluing operation for Lagrangian fillings that from the physical perspective corresponds to adding an anti-brane along a Lagrangian and that conjecturally leads to unobstructed connected Lagrangian fillings of the conormal of any link. Such Lagrangians would, as mentioned above, serve as actual geometric sources of the branch V K (n) of the augmentation variety corresponding to the trivial partition of a n-component link K. In Section 6.8 we discuss geometric constructions and gluing explicitly in a simple example and in Section 6.9 we look at Legendre duality of potentials from a geometric viewpoint. Finally, in Sections 6.10 and 6.11 we study other ways of obtaining augmentations from Lagrangian fillings by coupling to flat connections on the fillings in the exact and the non-exact case, respectively.
We would like to point out that many of the arguments given here are at best outlines of possible proofs, and indeed all of them need more details. When we state a result as a theorem we believe that it can be proved with existing technology. Rather than providing proofs, the main purpose of this section is to give some idea of the geometry behind the behavior of augmentation varieties.

Exact Lagrangian fillings and chain maps
As always, let K be a knot in S 3 and let Λ K denote its conormal torus.
An exact Lagrangian filling of Λ K is an exact Lagrangian submanifold F K ⊂ T * S 3 that agrees with the Lagrangian conormal L K of K outside some finiteradius disk subbundle of T * S 3 . For grading reasons we also assume that the Maslov class of F K vanishes. If a is a Reeb chord of Λ K , let M τ (a) denote the moduli space of holomorphic disks u : D → T * S 3 with boundary on F K and with one positive puncture where the disk is asymptotic to the Reeb chord a, and which (after capping) represents the homology class τ ∈ H 2 (T * S 3 , F K ). Then the dimension of the moduli space is given by the grading of a, dim(M τ (a)) = |a|. Define the map F K : by counting holomorphic disks with one positive puncture: where |a| = 0 and |M τ (a)| denotes the algebraic number of disks in M τ (a).
Theorem 6.1. The map F K is an augmentation of A, which we call "the augmentation induced by F K ".
Proof. We need to check the chain map condition F K • ∂ = 0. By SFT compactness [37], the two level broken curves that contributes to F K • ∂ are in one to one correspondence with the boundary of the compact oriented see Figure 3. the graph Γ dg of the differential of g. Here we recover the fiber coordinates y j uniquely by where (x 1 , . . . , x m ) are coordinates on U . The front has certain restricted singularities along lower dimensional strata that allows us to patch the solutions for y j corresponding to different sheets of the front. See Figure 4 for an example. We will sometimes make use of a specific non-generic front that we call the Lagrangian cone, see Figure 5.
Using fronts we can represent L K and M K rather concretely. A front of the conormal L K is shown in Figure 6.     complement filling M K takes p to p and x to 0 (or in multiplicative notation e x → e x and e p → 1 respectively e x → 1 and e p → e p ). Both augmentations take Q to 1.
We note that this explains the geometric origin of the trivial branches e x = 1 and e p = 1 of the augmentation variety of K at Q = 1.
The construction of the Lagrangian complement M K discussed above generalizes immediately to the case when K is a several-component link In this case the map on homology is a bit more involved and we get the following result.
Theorem 6.3. If K = K 1 ∪ · · · ∪ K n is an n-component link, then the augmentation induced by the exact knot complement filling M K takes the meridian class p j of the j th component to p j and the corresponding longitude class x j to k =j lk kj p k , where lk kj is the linking number of K k and K j , and as above takes Q to 1.

Non-exact Lagrangian fillings
Our next goal is to generalize the discussion of Section 6.1 to a certain class of non-exact Lagrangian fillings. We discuss concrete constructions of nonexact Lagrangian fillings of this type in Section 6.5. Here we concentrate on basic properties that are crucial for the augmentations to be constructed in Section 6.3.
We will consider Lagrangian submanifolds in T * S 3 or R × U * S 3 , or in the resolved conifold Y . For the general discussion all these ambient spaces can be treated on equal footing and we will denote the ambient symplectic manifold E and have either one of these in mind. In particular, outside a compact set, E agrees with the complement of a finite-radius disk bundle of T * S 3 , which we can think of as the symplectization of U * S 3 , and we write E R ⊂ E for the complement of a disk bundle of radius R.
Consider the conormal Lagrangian L K of a knot K, represented as a front, see Figure 6. Then L K ∩ E R projects to a punctured tubular neighborhood U of K in S 3 . If η is a closed 1-form in U then the shift map is a symplectomorphism.
We will consider non-exact Lagrangian submanifolds F K and adjusted almost complex structures J on E that satisfy the following three conditions: • The Maslov class of F K vanishes.
• For any A > 0 there is a compact subset C A ⊂ E such that any Jholomorphic disk with boundary on F K of area at most A lies in C A .
• There exists R 0 such that F K ∩ E R 0 is the image of a part of L K under a shift map S η for some uniformly bounded closed 1-form η on U .
We call such Lagrangians augmentation admissible fillings of Λ K .
Lemma 6.4. If F K is an augmentation admissible filling of Λ K , then the following version of SFT compactness holds. Any holomorphic disk of energy ≤ A < ∞ either is contained in the compact subset C A or has at least one positive puncture where it is asymptotic to a Reeb chord. Furthermore, any sequence of holomorphic disks of energy ≤ A has a subsequence that converges to a several-level holomorphic building with a first level in E and higher levels consisting of holomorphic disks in the symplectization of U * S 3 .
Proof. This is a small extension of [37]. Consider the case when a non-trivial part of the disk goes to infinity. In order to recover the limit, we use the invariance of the almost complex structure under rescaling. Take t to t − T for some large T > 0, so that level T maps to level 0 in the symplectization.
Under this rescaling the image of S η maps to the image of S T = S e −T η .
For such small shift we find that the adjusted almost complex structure J is at distance of the order of magnitude e −T from an almost complex structure J =S T • J •S −1 T , whereS T is a diffeomorphism that agrees with S T and is suitably cut off in T * U , and the holomorphic disks for the latter structure agree with the ordinary holomorphic disks with boundary on L K .
Standard bootstrap arguments show that J-and J -holomorphic disks lie within distance of order of magnitude e −T .
The condition on vanishing Maslov class of F K and the fact that E is Calabi-Yau shows that the formal dimension of any closed holomorphic disk with boundary on F K equals zero. Assume that a perturbation scheme for rigid holomorphic disks with boundary F K has been fixed. The moduli space M(F K ) of rigid disks in E with boundary on F K then constitutes a weighted branched 0-manifold, see [38,36].

Quantum corrected chain maps
We consider first our main geometric example: let K be a knot and take F K to be the conormal filling L K ≈ S 1 × R 2 shifted off of the zero section, L K ⊂ E. In this case H 2 (E, L K ) is generated by the class of t in H 2 (E), represented by the fiber 2-sphere, and the longitude class x ∈ H 1 (Λ K ). (In case E = T * S 3 , the fiber class is trivial and Q = 1 below.) In order to motivate the use of obstruction chains below let us try to carry over the construction of augmentations from the case of an exact filling to the current more general case in the most naive way. Define : A → C[H 2 (E, L K )] as in (6.1). As there, we try to prove the chain map equation Here, however, not all boundary components contribute to • ∂ because of boundary bubbling with nodal disks with two components, one in M(b) and the other in M(L K ), see Figure 8. Consequently as defined here is not a chain map. In order to remedy this we will introduce a change of variables on the coefficients A, guided by obstruction chains much like in [36]. Consider a holomorphic disk u ∈ M(L K ). Then u : ∂D → L K is a cycle that represents the homology class kx ∈ H 1 (L K ), where k is some integer and x is the longitude class in Λ K . Fix a circle ξ in Λ K representing the class x. For each u as above fix a smooth obstruction chain σ u that connects the cycle u(∂D) to the fixed circle in Λ K traversed k times. More precisely, recall that outside a compact subset of E, L K is the image of R × Λ K under a shift map S η ; we take the chain σ u to agree with R × ξ k , where ξ k denotes the curve ξ traversed k times, with the appropriate weight, reflecting the weight of the curve u, in this region, see Figure 9. and with additional boundary marked points ζ 1 , . . . , ζ s ∈ ∂D such that u(ζ j ) ∈ σ u j for some u j ∈ M(L K ). Furthermore, any disk u j attached in this first level also has additional boundary marked points which map to σ u j for disks u j distinct from the ones already used, which are the disks attached at the second level. If the r th level disks has been defined the (r + 1) th level is obtained by attaching obstruction chains at marked points in the r th level.
A quantum disk is such an object with finitely many levels, see Figure 10.
Schematically, we can view a quantum corrected disk as a rooted tree, where the root is the holomorphic disk u and the other vertices are the disks u j . If we orient this tree so that all edges point away from the root, then an edge from u j to u j corresponds to a way to map a boundary marked point on u j to a point in the cycle σ u j .
We define the homology class represented by the quantum corrected disk Proof. This is a standard application of obstruction chains: Figure 11 indicates why boundary bubbling no longer is to be considered as a boundary of the moduli space, while in Figure 12 a typical boundary configuration is depicted. Figure 11: Obstruction chains turn boundary bubbles to interior points in the moduli space.

The GW-potential and chain maps
We define the GW-potential as the generating function of the holomorphic disk configurations that are attached to a quantum corrected disk. More precisely, we count trees whose vertices are rigid holomorphic disks u i with boundary on L K . To each such tree, we associate a weight as follows. First, each vertex u i in a tree has a weight given by the weight of u i viewed as a point in the moduli space of holomorphic disks. Next, let u i 1 and u i 2 be disks connected by an edge, and let σ u i 1 and σ u i 2 be the corresponding obstruction Write M(L K , σ) for the moduli space of (unrooted) trees, and define the homology class of a tree in H 2 (E, L K ) to be the sum of the homology classes of the disks in the tree. We now define the GW-potential as follows: where C k,r is the sum of the weights of the tree configurations in M(L K ; σ) that represent the homology class kx + rt.
We claim that this GW-potential parametrizes a part of the augmentation Proof. This follows from the fact that contributions to L K •∂(b) correspond to two-level disks in the boundary of M(b; σ). To see this, consider a holomorphic disk u in the symplectization R × U * S 3 with positive puncture at b and negative punctures at a 1 , . . . , a m which (after capping) represents the homology class kx + mp + rt in H 1 (Λ K ). To obtain a two-level structure as in Figure 12, we attach a quantum corrected disk in E with positive puncture at a i (i.e., a point in scM (a i )) to each negative puncture a 1 , . . . , a m , and a tree of holomorphic disks in E at marked points on the boundary of u (note that such a tree can be rooted and oriented by choosing the root to be the disk whose obstruction cycle contains the marked point). Now in the boundary of the one-dimensional moduli space M(b; σ), this precisely gives a contribution of e kx e m ∂W ∂x Q r : first note that each rooted tree of disks in E is counted with multiplicity given by the weight of the tree, and second recall that at infinity the obstruction cycles are products of the form R × ξ l that intersect any curve in the class kx + mp, m · l times. (In the formula m is explicitly visible and l comes from the differentiation.) By definition of the specialized differential, this contribution (summed over k and r) is L K (∂(b)), which must then be equal to 0. In general it is difficult to find augmentation admissible Lagrangian fillings with the topology of a link complement that do not intersect the zero section, but e.g. for fibered links it is possible; see Section 6.7, where we will discuss alternative constructions. Here we will discuss such Lagrangian fillings in order to see the main properties of the more general fillings in the simplest possible context. Thus, let K = K 1 ∪ · · · ∪ K n be an n-component link. Let M K ⊂ E be an augmentation admissible Lagrangian filling with the topology of the link complement M K ≈ S 3 − K. Then H 2 (E, M K ) is generated by the meridian classes p j ∈ H 1 (Λ K j ) (j = 1, . . . , n) of the components and the fiber class t. In analogy with the case of the conormal discussed above, we fix circles η j in Λ K j that represent the class p j , j = 1, . . . , n, and for each u ∈ M(M K ) we fix an obstruction chain σ u connecting u(∂D) to a linear combination of multiples of the fixed curves η j in Λ K j , where as above we take the obstruction chains to be (weighted) cylinders on the fixed curves outside a compact subset. Let again σ denote the collection of all the fixed obstruction chains σ u , u ∈ M(M K ).
Let p = (p 1 , . . . , p n ) and letŨ (p, Q) denote the GW-potential of M K , defined as a sum over tree configurations exactly as above: where B k,r is the sum of the weights of the configurations in M k·p+rt (M K ; σ). Remark 6.9. It follows from Theorem 6.8 that the equations give a parametrization of an n-dimensional branch of the augmentation variety of the link K, in agreement with (4.5). In general this branch involves mixed augmentations (where Reeb chords between different components are augmented) and hence appears to correlate with V K (n). Remark 6.10. As an example of the above, consider the Hopf link K 1 ∪ K 2 .
It follows from (4.8) that the part of the augmentation variety corresponding to mixed augmentations is given by the equations In terms of potentials this corresponds to U (p 1 , p 2 ) = p 1 p 2 , which means that U (p 1 , p 2 ) = 0. This is related to the fact that the Hopf link admits a exact Lagrangian filling M Hopf in T * S 3 − S 3 , on which no nonconstant holomorphic disks can form. To see this, note that there exists a Bott function on S 3 with maxima along K 1 and minima along K 2 and no other critical points. Such a Bott function can be obtained by pulling back a Morse function on S 2 with exactly two critical points by the Hopf map S 3 → S 2 . Then adding cones along these Bott manifolds, we get a front that gives an exact Lagrangian M Hopf that does not intersect the 0-section.

Embedded augmentation admissible Lagrangians
In this section we consider basic constructions of embedded Lagrangian fill- Proof. We must check the second property for augmentation admissibility.
Let R > 0 be large and write E R for the complement in T * S 3 of a radius R disk bundle. We argue using monotonicity: if a disk does not lie entirely outside E R 2 but still leaves E R then its area is bounded below by CR 2 for some constant C > 0, which is a contradiction for R sufficiently large. Thus the entire disk must lie outside E R 2 . However, in this region the symplectic form is exact and the area is given by some constant times the number of times its boundary goes around S 1 . Assuming that the link does not lie in a plane, the length of such a curve is bounded below by δR for some δ > 0, and monotonicity for holomorphic disks again gives a lower area bound of the form Cδ 2 R 2 , which gives a contradiction.
We next consider moving link complements off of the 0-section.
Lemma 6.12. If a link K is fibered then Λ K admits an augmentation ad- Proof. Since K is fibered, there exists a non-vanishing 1-form on S 3 − K that can be taken to agree with the standard circular form dθ of the meridian circles in a small neighborhood of a knot. Consider a small Bott function with maximum along the link, and add in the conormal along the Bott maximum as in Figure 7. Then the exact Lagrangian is parametrized as follows over the fiber disks: x(r, ξ) = rξ, y(r, ξ) = α(r)ξ, where (r, ξ) ∈ (−δ, δ) × S 1 and α(r) > 0. Consequently, the Lagrangian shifted by c · dθ is given by where R π 2 denotes rotation by π 2 . Thus adding the conormal along the Bott maximum does not introduce any intersections with the 0-section. Finally, the argument confining closed disks is a repetition of the proof of Lemma 6.11.
If a link K is not fibered then we can move off the complement M K of a link K = K ∪ K 0 where K 0 is a braid axis for K. More precisely, fix a braid presentation of K and let K 0 be a braid axis. Let K be a non-split link with n components K = K 1 ∪ · · · ∪ K n . As in Lemma 6.13, let K = K ∪ K 0 , where K 0 is a braid axis for K. Then also K is non-split and for an appropriate choice of orientation of K 0 , lk 0j = j > 0 for all j > 0. Write x = (x 1 , . . . , x n ), p = (p 1 , . . . , p n ) and = ( 1 , . . . , n ).
Let U K (p 0 , p) denote the superpotential determined by M K , As M K is connected, this gives rise to the following local parametrization of V K (n + 1): Consider now the intersection V (n + 1) ∩ (V (1) × V (n)). Assuming Conjecture 5.3, this intersection has dimension n. On the other hand K 0 is an unknot, we know the first factor V (1) explicitly, and equating the x 0coordinates of a point in the intersection gives Solving for p 0 we get solutions p 0 = g(p), say. Letting U K (p) be a potential for V K (n), we find, by equating the x-coordinates of a point in the intersection, Now, our assumption on the dimension of the intersection implies that this equation must be trivially satisfied for all p since otherwise the intersection would have dimension < n. This thus gives the following formula for the potential of K corresponding to V K (n): which is an example of "Lagrangian reduction" (see Section 5.4), rephrased in the language of potentials.

Immersed Lagrangian fillings
In this section we first discuss immersed Lagrangians, and then indicate a geometric counterpart of intersecting augmentation varieties that physically corresponds to adding an anti-brane along a Lagrangian. This conjecturally yields a connected geometric Lagrangian which is a source for the augmentations in the branch of the augmentation variety corresponding to the trivial partition of any link, and indicates a geometric origin for Conjecture 5.3.
We use the notion E for the ambient symplectic manifold as in Section 6.2.
Consider a generically immersed Lagrangian submanifold F K which satisfies the Maslov class condition and the condition at infinity of an augmentation admissible Lagrangian. Then F K has a finite number of transverse double points in a compact part of E. Let q be a double point of E; then q is the transverse intersection of two local branches of F K . We write q ± for q considered as an ordered intersection of the two branches. One can associate a Maslov type index |q ± | to q ± , see [28], such that the dimension of the moduli space M(q ± ) of holomorphic disks with boundary on F K and one puncture where the disk is asymptotic to q ± , with the boundary orientation inducing the order of the sheets corresponding to the decoration, equals dim(M(q ± )) = |q ± | − 1, Figure 13. We call an immersed Lagrangian F K as above, which can be equipped with an obstruction chain σ so that it has this property with respect to rigid once-punctured disks, unobstructed. In these terms the above arguments can be summarized in the following way.

Geometric Lagrangian fillings and anti-branes
We next construct a connected Lagrangian filling of any link (or more precisely, of the split union of the link with an unknot). Let K be a link and consider the auxiliary link K = K 0 ∪ K, where K 0 is a braid axis for K as in Lemma 6.13. We use notation as there. Let M K ⊂ E denote the Lagrangian complement of K shifted off the 0-section. Recall that the braid axis is an unknot which links all components of K positively. Consider a Hopf link The next step of our construction is to glue M K to M K along Λ K 0 .
From a physical perspective this corresponds to adding an anti-brane along M K . In Figure 15, the gluing operation is described in terms of fronts. The result of this gluing is an immersed Lagrangian filling N K∪K −1 . Since the complement of the unknot is diffeomorphic to S 1 × R 2 , we find that N K∪K −1 has the topology of the complement of a link K ∪ K −1 in S 1 × S 2 . Here S 2 is the union of two hemispheres  or M K . However, M K is the complement of a Hopf link which supports algebraically zero disks, see Remark 6.10. Noting that x −1 , p 1 , . . . , p n generate H 1 (N K∪K −1 ) and that p 0 is homologous to x −1 , we then find that Furthermore, since p −1 is homologous to · p in N K∪K −1 , we get the corresponding superpotential We consider finding conditions that guarantee that the immersed Lagrangian filling N K∪K −1 of Λ K∪K −1 is unobstructed. Assuming that is the case, we find that On the other hand, since K ∪ K −1 is split, we know that V K∪K −1 = V K × V K −1 , and since K −1 is an unknot, we conclude that Solving this equation, we get x −1 = g(p), and in complete agreement with (6.7), we find that is a potential for V K (n).
This leads us to conjecture that the immersed Lagrangian filling N K∪K −1 is unobstructed provided that ∂W ∂x in the case that K is an non-split link. This conjecture gives a connected unobstructed Lagrangian filling N K∪ of K ∪ for any link K, where is an unknot split from K. The corresponding potential U K for K is then obtained by subtracting the unknot potential from the geometrically defined potential U K∪ and gives a geometric source for augmentations in V K (n). We next consider the classical limit of U K . In this limit, ∂U K ∂p 0 = · p = 0 and (6.8) becomes where (x −1 , p −1 ) is a point on the augmentation variety of the unknot. But in the classical limit the augmentation variety of the unknot is {x −1 = 0} ∪ {p −1 = 0}, so x −1 = 0 and the potential U K limits to i<j lk ij p i p j , which from the viewpoint of augmentations agrees with the Lagrangian link complement in T * S 3 .
We will often denote N K∪K −1 equipped with the potential U K (p) = U K∪K −1 (g(p), p) − U (g(p)) by M K , see Section 4.2.
We next consider a possible geometric source of Conjecture 5.3 in a similar spirit. Consider a link K = K 1 ∪ K 2 where K 1 is a knot and K 2 a link such that both K and K 2 are non-split. Let M K and M K 1 be Lagrangian fillings of K ∪ K −1 and K 1 ∪ K −1 , respectively. Here we use the same braid axis K 0 for K and K 1 and for suitably values of p 1 , the shifts of Λ K 1 in M K and M K 1 agree. We next glue M K 1 to M K along Λ K (add an anti-brane along M K 1 ).
Denote the resulting Lagrangian filling M K/K 1 . Then M K/K 1 is a filling of Assuming that M K/K 1 is unobstructed, we consider conditions under which it would define an augmentation. As above, the holomorphic disks with boundary on M K/K 1 are exactly the holomorphic disks in M K and those in M K 1 (the latter counted with opposite signs), and we have obstruction chains from the two pieces. In M K/K 1 , however, the obstruction chains going to p 1 no longer go off to infinity, but end at a finite distance. In order to have the chain map property (see the proof of Theorem 6.6), we must deal with the discontinuity that may appear when a disk crosses the reference cycle p 1 .
This is however straightforward: by equating the one-point functions of the potential of the pieces, Note that in (6.10) we can replace U K∪K −1 with U K since and similarly for U K 1 . We thus find that the augmentation variety V K 2 contains a branch given by the potential where p 1 (p 2 ) is a solution of the equation This is exactly "Lagrangian reduction" rephrased in terms of potentials, see

Geometric constructions in a simple example
In this section, we study a basic example that illustrates Conjecture 5.3 as well as the geometric constructions in Section 6.7.
Consider two knots K 1 and K 2 and their connected sum K 1 #K 2 . Assume that we have Lagrangian fillings M 1 = M K 1 and M 2 = M K 2 with GW-potentials U (p 1 , Q) and U 2 (p 2 , Q) respectively. Then we construct a Lagrangian filling M 12 = M K 1 #K 2 with potential U 12 (p, Q). Studying the augmentations of Λ K 1 #K 2 presented with a short Reeb chord near the connecting point, one can show that the GW-potentials are related by where U (p, Q) is the potential of the Lagrangian complement of the unknot (which is isomorphic to the conormal). To see this, recall from Section 5 how the augmentation polynomial behaves under connected sum, and note that is a restatement of that in terms of GW-potentials.
We now consider the example K of an (n+1)-component link consisting of an unlink K on n components with a braid axis K 0 , see Figure 16. Consider we can compute the superpotential U K for K as Adding an anti-brane along M K 0 introduces the homology relation p 1 + · · · + p n = 0 and gives the superpotential U K/K 0 (x 1 , p 2 , . . . , p n ) = x 1 (p 1 + · · · + p n ) − nU (x 1 ), Figure 17: Left, the configuration of K 0 and K −1 ; middle, the front of N K ∪K −1 over a fiber; right, the front of N K ∪K −1 seen from above.
where we keep additional cycles in order to keep track of obstruction chains.
The continuity argument from Section 6.7 requires that we remove x 1 , where different obstruction cycles meet by passing to the critical point. Using the symmetry of the unknot potential under exchange of x and p, we find the critical point condition Re-substituting this, we find that a "potential" for K is given by which, in the light of our knowledge of augmentations for split links, gives correct augmentations only if p 1 = · · · = p n .
Looking at the Lagrangian filling, we can see these phenomena from a geometric perspective. Applying [39], we get a description of holomorphic disks in terms of flow trees and in this case this is straightforward. The Lagrangian M K/K 0 obtained by anti-brane addition has Bott circles of selfintersection points. Perturbing to a Morse situation leaves two intersection points on each circle, and the rigid disks have a puncture at the double point with the smallest action. By [39], these rigid disks correspond to flow lines starting at saddle points and hitting the cusp edge as shown in Figure 18.
These rigid disks are indeed obstructions, with difference classes p 1 , p 1 + p 2 , Figure 18: The flow trees that correspond to rigid disks, and the boundaries of the corresponding rigid disks lying in a fiber punctured sphere of N K/K 1 .
. . . , p 1 + · · · + p n−1 as indicated there. In the exact case, we find that the obstructions then vanish only if p 1 = · · · = p n = 0. In the general case, one would expect these equations to be quantum corrected to which is indeed the case if we use a model where the rigid disks lie close to each conormal piece and look like the disks on the conormal of the corresponding unknot, i.e. the disks lie near the cones in Figure 17. This is in agreement with the above algebraic analysis.

Legendre transform of superpotentials from a geometric perspective
In this section we consider the duality between the potentials of different Lagrangian fillings mentioned in Section 3.2 from a geometric perspective.
Let K be a knot, let L K be its conormal filling, and let M K be a filling that fills in the longitude, e.g. the knot complement if the knot is fibered or the filling N K∪ constrained by (6.9). Consider a branch of the augmentation variety defined by L K and parametrized by x: x → x, ∂W ∂x .
Parametrizing the curve by p instead, we get a function p → x(p), and the corresponding potential is given by This indicates that the GW-potential of M K should be Legendre dual to the GW-potential of L K . We next discuss this duality heuristically from a geometric viewpoint.
First consider holomorphic disks with boundary on L K . We fix a perturbation scheme so that no holomorphic disk intersects the central S 1 in the conormal. Note that the construction of M K involves gluing L K to the 0-section. We assume that the disks on M K are exactly those that come from L K . In order to shift M K off the 0-section, we glue with a twist, taking a curve in the class x + p in Λ K ⊂ M K to the curve in class x in Λ K ⊂ L K .
Looking at the holomorphic disks of M K in a neighborhood of the knot, we see that they look like the disks in L K except they all link with linking number 1. Consider now a 1-parameter deformation which changes this, energy level by energy level. First we let the disk on the second energy level where x(p) is the function discussed above. See Figure 19.
Differentiating with respect to p gives and U (p) is the Legendre transform of W (x).
6.10 Augmentations, the cord algebra, and flat connections on exact fillings In order to find other geometric sources for augmentations, we consider the exact Lagrangian filling M K with topology of the link complement S 3 − K from a different perspective, in line with the discussion at the end of Section 4.2.
Let Zπ denote the group ring of the fundamental group π := π 1 (S 3 −K) = π 1 (M K ). We define a chain map Φ : is the element in Zπ corresponding to γ ∈ π. The usual argument using the description of the boundary of the 1-dimensional moduli space as two-level disks gives the chain map property Φ • ∂ = 0.
Recall the concrete construction of M K (see Figure 7), where the conormal of the link was added to the 0-section via Lagrange surgery. Compactness properties of the space of holomorphic disks with boundary on L K ∪ S 3 ⊂ T * S 3 were studied in [40], where it was shown that for generic data there is a constant C > 0 such that the number of corners (where the disk boundary switches between L K and S 3 ) for any disk is at most C. The corners of disks with boundary on L K ∪ S 3 can be smoothed to give a disk on M K , and conversely, disks with boundary on M K limit to disks with corners on L K ∪ S 3 as the surgery degenerates. For isolated double points this relation was studied in [36].
In order to describe the relation between holomorphic disks with boundary on L K ∪ S 3 and disks with boundary on M K , we first characterize the corners of disks. At any corner the boundary orientation of the disk gives one incoming and one outgoing arc. We say that a corner with incoming arc along S 3 and outgoing along L K is positive and that a corner with the opposite behavior is negative. Generically, rigid disks have only non-degenerate corners. In 1-parameter families of disks with boundary on L K ∪ S 3 , there are also degenerate corners where the disk has a vanishing derivative in the fiber direction and in a neighborhood of which two nondegenerate corners collide.
The relation between disks on L K ∪S 3 and disks on the "surgered" version M K were studied in [36]. Using this result, or alternatively the correspondence between flow trees and holomorphic disks established in [39] together with standard gluing theorems, one establishes the following results, beginning with the case of rigid disks.
Theorem 6.15. For all sufficiently small surgery parameters, there is the following many-to-one correspondence between rigid holomorphic disks with boundary on M K and disks with boundary on L K ∪ S 3 : To each disk with 2k corners corresponds 2 k disks obtained by smoothing the corners of the disk. At a positive corner the smoothing is unique, see Figure 20, and at each negative corner there are two smoothings and the homotopy class of the corresponding boundaries differ by the meridian class in Λ K , see Figure 21.
We next consider 1-parameter families of disks. Consider such a 1-  It thus remains to describe the smoothed family for parameters in a small neighborhood of a disk with a degenerate corner. Consider such a family parametrized by t ∈ [−δ, δ], with two nondegenerate corners for t ∈ [−δ, 0), a degenerate corner at t = 0, and no corner at t ∈ (0, δ]. Note that the disk with degenerate corner also lies in a moduli space of disks without corners that simply cross through K, see Figure 22. Choose smoothings at all cor- Figure 22: Holomorphic disks with boundary on L K ∪ S 3 near a disk intersecting the knot. ners except for the corners near the degenerate corner. Write M ± for the two branches of the moduli space corresponding to the two smoothings of corners of disks (−δ, 0), see Figure 23. Also write N ± for the two components of the moduli space without corners in the complement of the disk that intersects K, see Figure 24.   There is a topological interpretation for knot contact homology in homological degree 0 (and Q = 1) known as the cord algebra [30]. Theorems 6.15 and 6.16 precisely match with the generators and relations of the cord algebra. Indeed, work in progress [41] aims at establishing that the map counting holomorphic disks with boundary on L K ∪ S 3 gives an isomorphism between the degree zero contact homology of Λ K and the cord algebra using a version of string topology. Together with Theorems 6.15 and 6.16, this would lead to a proof of the following result that we state as a conjecture: Conjecture 6.17. The chain map Φ induces an isomorphism between the degree 0 knot contact homology (with Q = 1) and the cord algebra.
The fact that these two algebras are isomorphic is already known [30], but this conjecture would give a geometric reason for the isomorphism.
As an aside, it is straightforward to show that the chain map Φ is surjec-

The cord algebra and small non-exact deformations
In this section we discuss a possible generalization of the material in Section 6.10 and consider maps from the DGA of Λ K into the group ring of the fundamental group of a knot complement in presence of quantum corrections.
The picture studied here applies for small shifts (i.e. for Lagrangian fillings that are sufficiently close to being exact).
Let K = K 1 ∪· · ·∪K n be an n-component link with complement M K that can be shifted off of the 0-section; for example, K could be a link with an adjoined braid axis, as discussed in Section 6.7. In the limit where periods are small, it is reasonable to expect that the boundaries of all rigid holomorphic disks lie in a small neighborhood of the link. Here the disks look basically like the corresponding disks in the conormals of the link components that are glued in with a twist. We restrict attention to this case and write x = (x 1 , . . . , x n ) and p = (p 1 , . . . , p n ) for the longitude and meridian homology generators in Λ K , where H 1 (Λ K ) = H 1 (Λ K 1 ) ⊕ · · · ⊕ H 1 (Λ Kn ) and x j , p j ∈ H 1 (Λ K j ). Since boundaries of holomorphic disks lie in a (punctured) tubular neighborhood of K we fix obstruction chains from the disks' boundaries to linear combinations of fixed homology generators x j , p j at infinity. As above tree configurations then define a generating function (GW-potential) where k = (k 1 , . . . , k n ) and l = (l 1 , . . . , l n ) both range over all vectors in Z n and r ranges over Z.
In analogy with the above, we change variables in the coefficient ring of the DGA of Λ K by and write A ml for the DGA with transformed coefficient variables.
Let C Q π denote the group ring of the fundamental group of the link complement π = π 1 (S 3 − K) over C, with the commutative variable Q ±1 adjoined. Define the algebra map Φ : where γ ∈ π is the homotopy class of the boundary of the configuration, σ is an obstruction chain as above, and M γ (a; σ) is the moduli space of quantum corrected holomorphic disks with boundary on M K and one positive puncture at the Reeb chord a. Note that this map makes sense as a map into C Q π: since this is a module over the commutative ring C[π 1 (Λ K )], we need not keep track of the order of intersections of the reference chains in the boundary when all holomorphic disks lie near the link components.
The usual argument analyzing the boundaries of 1-dimensional moduli spaces shows that the map Φ : A ml → C Q π is a chain map. Thus if R Q π denotes the subalgebra of C Q π that is generated by the image of Φ, then any algebra map R Q π → C induces an augmentation. We next argue that the parts of the augmentation variety which come from representations of R Q π are Lagrangian with respect to the symplectic form n j=1 dx j ∧ dp j . To see this, write X j and P j for the "logarithmic coefficient variables" in the algebra A ml , see (6.11). Fix a link component K j of K and write the function lk j , defined on generators γ ∈ π of Cπ, for the linking number with K j : lk j (γ) = lk(K j , γ). Write lk −1 (0) for the set of generators with vanishing linking number with each K j . Since any generator γ in lk −1 (0) is zero in homology, it is homotopic to a product of commutators: Consider now an algebra morphism parametrized by P j and note that We have where ξ k ∈ lk −1 (0). Thus since by symmetry of linking numbers, lk(K j , K k ) = lk(K k , K j ). Hence X = X(P ) is Lagrangian with respect to the symplectic form n j=1 dX j ∧ dP j , and since the change of coordinates (X, P ) to (x, p) takes n j=1 dX j ∧ dP j to n j=1 dx j ∧ dp j , we find that the original augmentation variety is Lagrangian as well.

Examples
Here we provide a few computations of augmentation varieties for links. The augmentation variety and the mirror-symmetry variety coincide in all cases where they have both been computed, and we denote both by V K . In addition, we present evidence for the conjectures presented in sections 4 and

5.
Let us summarize what we expect, based on the discussion of the previous sections. Let K be an n-component link. The mirror/augmentation variety V K has a Zariski-closed subset (union of irreducible components) V K (P ) for each primitive partition P of {1, . . . , n}, where we recall that primitive partitions divide K into non-split sublinks K α . For fixed Q, we have , where K α corresponds to the subset P α ⊂ {1, . . . , n}. Moreover, for fixed Q, V Kα ⊂ (C * ) 2nα is expected to be a complex-Lagrangian variety of dimension n α , where n α is the cardinality of P α , and the corresponding potential defining V Kα is associated to the Lagrangian filling M Kα as follows. Let e x i , e p i denote the coordinates of (C * ) 2nα ; then and W (p 1 , ..., p nα ) = lim λ→0 (λ log H p 1 /λ,...,pn α /λ (K(P ))), where H p 1 /λ,...,pn α /λ (K(P )) are the HOMFLY invariants of the link K(P ) colored by symmetric representations of length p n i /λ.
Furthermore, let P 1 , P 2 be primitive partitions such that P 2 is a refinement of P 1 . If there is no primitive P such that P is a refinement of P 1 and P 2 is a refinement of P , then we expect codim(V K (P 1 ) ∩ V K (P 2 )) = 1.
More generally, if we have a sequence P 1 , . . . , P r of primitive partitions such that each is a refinement of the previous one and the sequence cannot be lengthened by inserting another partition in the middle, then we expect codim(V K (P 1 ) ∩ V K (P r )) ≤ r.
In particular, we expect V K (n) = V K (1 · · · n) and to intersect at least in a curve.

Torus links
Here we give the augmentation varieties for various torus links, obtained from knot contact homology. In all cases, each link component is an unknot, and so V K (1 n ) is the vanishing set of where n is the number of components.
One can show that V K (3) intersects V K ((12)(3)) in a surface, which projects in (λ 1 , µ 1 , λ 2 , µ 2 ) space to a curve containing V Hopf (2) The graph Γ K for the (3,3) torus link is shown in Figure 25, and the intersection dimensions are in accord with Conjecture 5.3.

Torus links from Chern-Simons theory
Before presenting other examples, in this subsection we will study torus links from the perspective of Chern-Simons theory. We will show that for the (2, 2f ) torus links considered in the previous subsection, the mirror variety of the link, obtained from Chern-Simons theory and large N duality, agrees with the augmentation variety. Moreover, we will see that the geometry of the mirror variety is indeed captured by the link graph Γ K . We expand our discussion to (n, nf ) torus links, which provide a good testing ground of the general relation we spelled out earlier between a linkK that is an n-parallel For the unknot K in framing f , colored by an arbitrary representation R, the HOMFLY polynomial is given by a well-known expression: where S RR and T RR = T R δ R R are the SU (N ) WZW S and T matrices, as before. In particular, S 0R /S 00 is just the quantum dimension of the SU (N ) representation R. From this, the partition function of n branes on K is easy to compute: In the above, ∆(z 1 , . . . , z n ) is the usual Vandermonde determinant 1≤i<j≤n (z i − z j ), and B f (m) is the HOMFLY polynomial for an unknot in framing f , colored by a totally symmetric representation of rank m: By ∝ we mean equality up to overall normalization. We also drop factors that ultimately only shift what we mean by holonomy.
In the classical limit, the Vandermonde determinant does not contribute, correspondingly, up to terms suppressed by g s relative to the leading order: This is simply a product of n copies of a single brane partition function It is easy to show that (µ, λ) = (e p , e x ) with p defined by lies on a Riemann surface that is mirror to the unknot with framing f (the unusual framing replaces λ by λµ f ): Thus, with rank n representations coloring the knot K, Geometrically, this comes from n Lagrangians conormal to the unknot, which corresponds to the partition The corresponding mirror variety is just n copies of V K : This is consistent with the analysis of higher representations based on knot contact homology which we gave in Section 5.6. We will consider the case n = 2 in some detail. The partition function has two saddle points. At one, corresponding to P = (1)(2), we have where W K is the potential of a single knot, and This agrees with our statements in Sections 4.6 and 5.6 that rank n representations of a knot are a subset of the augmentations of the link which is the n-copy of the knot. In this filling, both meridians are filled, µ 1,2 = e p 1,2 ∼ 1, and we get two disconnected Lagrangians, as befits the fact that the corresponding partition is P = (1)(2).
There is one more saddle point, which as we will see corresponds to intersection of and a variety obtained by eliminating z = e q from the equations We can solve for λ 1,2 and z in these equations for any µ 1,2 , consistent with our assumption that treating µ 1,2 as a continuous variable was justified. Thus we have shown the saddle point corresponding to V (2) indeed exists.
It is easy to show that for integer values of f , this reproduces the V (2) varieties of (2, 2f ) links obtained for integer values of f . Moreover, we can prove that, for any f = 0, the two varieties intersect on a curve.
For example, for f = 2, corresponding to the (2, 4) link, the variety V (2) is given by a pair of equations consisting of (7.1), which is the same for all f and from solving (7.2). After a shift of framing followed by a change of orientation λ 1,2 , µ 1,2 → λ −1 1,2 , µ −1 1,2 , this agrees with the V (2) component of the augmentation variety of the (2, 4) link in the previous subsection.
Finally, for f = 4, or the (2, 8) link, we get (7.1) and The intersection of the two varieties VK(2) and VK(1 2 ) corresponds to asking that z = z(µ i , λ i ), which solves the equations (7.2) provided µ i , λ i live on V (2), in fact takes a specific value: z = 1. This is because along the locus z = 1, V (2) and V (1 2 ) coincide, as is easily seen from the two equations. Since a priori z is a function of µ i , λ i , this puts one additional constraint on these variables, which reduces V (2) to a curve-this is the curve of its intersection with VK(1 2 ). Moreover, the curve itself is easily seen to be nothing but a copy of V K (1):

Borromean link
The Borromean link, the closure of the 3-braid σ 1 σ −1 2 σ 1 σ −1 2 σ 1 σ −1 2 , consists of three unknot components, any two of which are unlinked. Here we have (2), or (23)(1), since any two-component sublink is split) with where P is a certain polynomial of degree 6 in λ 1 (see the Mathematica notebook for the precise polynomial). It would be interesting to compare this with the direct computation of V K based on HOMFLY coming from symmetric representations on the Borromean link.
For the Borromean link, V K (3) and V K (1 3 ) intersect on a surface, i.e. in codimension 1, as expected from the D-module picture discussed in Section 4. For generic Q, one can calculate that the following surface in (C * ) 6

Other three-component links
Here we briefly consider two more 3-component links. The first is the link K given by the connected sum of two Hopf links. In other words, K consists of three unknots, with two of the unknots, K 1 and K 2 , being meridians of the third, K 3 . Thus K 1 , K 3 are Hopf linked, as are K 2 , K 3 , and K 1 , K 2 are unlinked. This is a special case of the link considered in Section 6.8 (where Here there is no component of V K corresponding to the partition (12)(3), since K 1 ∪ K 2 is split, but all other partitions are primitive, and we have: The graph for this link is shown in Figure 25, and intersections are in accordance with Conjecture 5.3.
Another illustrative example is the 3-component link L8n5, also shown in We start in the case when K is a knot. Then we have a single D-brane probing the conifold Y . The mirror of Y as probed by the brane was conjectured in [1] to be the topological B-model string on the Calabi-Yau threefold We consider a single B-brane wrapping the subvariety The topological B-model on Calabi-Yau manifolds of type (8.1) was shown in [52] to be a theory of a single fermion ψ(x) on the corresponding Riemann surface V K : A K (e x , e p , Q) = 0.
The fermion ψ(x) is the operator that inserts the D-brane at a point x on the Riemann surface 11 , so that The expectation value of the fermion is determined by the geometry of the Riemann surface. In particular, near the classical limit, 11 The Calabi-Yau manifolds that arise from mirror symmetry are a fair bit subtler, as the underlying Riemann surfaces are singular, and one has to describe how the singularities get resolved as a part of the data of quantization. Note that the fermion number conservation requires that a conjugate fermion ψ * (∞) be placed at infinity on V K , which we will often include only implicitly. which corresponds to (p, x) lying on the Lagrangian submanifold V K In quantum theory, as shown in [52], x and p do not commute, but instead and we can view them as operators arising by quantization of V K viewed as a Lagrangian submanifold of with symplectic form Ω = dx ∧ dp, which has its origin in the holomorphic three-form du u ∧ dx ∧ dp on Y K . From the present perspective, the quantization (8.2) expresses the fact that x and p are mirror to the holonomies of the brane on Y around the longitude or (1,0) and the meridian or (0, 1) cycles, respectively, of the torus Λ K at infinity.
Indeed, as discussed in Section 2.4, the theory on the brane is Chern-Simons theory, and the commutation relation between the monodromies along the dual cycles in quantum Chern-Simons theory is well known.
The existence of a fermion and the above non-commutative structure for the space that the Riemann surface lives on was rederived in [53] (see also [54,55]) using string dualities, mapping topological string amplitudes to the partition function of type IIA strings in flat space in a background with D4 and D6 branes intersecting on a Riemann surface, leading to a fermion living on the intersection. This picture was later simplified [56] and reformulated in terms of yet another topological string theory on M 1 . This reformulation involves coisotropic branes for A-models introduced in [57] and studied in a similar context in [58,59]. In that context, one studies only the conformal theory. However, in the context of topological strings, we consider the same worldsheet theory, but couple it to 2d gravity, i.e. we view it as a string theory and in the usual way couple topological strings to the worldsheet metric. The usual topological string theory amplitudes can be reformulated as amplitudes of this simpler topological string theory, where one is studying The D-mirror variety of a n-component link K, This means that quantization of the theory must be quantization of M n , viewed as a phase space with symplectic form Moreover, we must associate to the Lagrangian submanifold V K a wave function that corresponds to Ψ K (x 1 , . . . , x n ) = whose various semi-classical expansions lead to the Lagrangians V K (P ), Generalizing what we already discussed in the context of mirror symmetry and knots, we propose that the mirror is based on the D-model on V K . To define it, we start with the topological A-model on M n studied in [58,59].
This leads to quantization of the phase space, by putting a single coisotropic Adding an A-brane on V K , the bifundamental strings with one boundary on V K and one on the coisotropic brane have a unique ground state which leads to a fermion ψ(x 1 , . . . , x n ) living on V K . The theory of this fermion is the D-model on V K . In particular, we conjecture that computing the expectation value of this fermion living on V K leads to Ψ K : This conjecture is naturally motivated by extending the case of knots, which we already know works, to the case of links.
Note that the D-model is exact at the annulus level. In this context we in- Our proposal for quantization of V K in terms of the A-model topological string on M n naturally leads to D-modules [58], and provides an a priori way to associate a quantum D-module to the classical Lagrangian V K . Namely, the fermion ψ is a bifundamental string with one boundary on V K and the other on the coisotropic brane on M n . As such, ψ, and hence Ψ K as well, provides a module for strings with two endpoints on the coisotropic brane.
The latter simply correspond to functions on M n , whose algebra is deformed by (8.3). Thus, Ψ K (x 1 , . . . , x n ) generates a D-module for the algebra (8.3).
A strong piece of evidence for our conjecture is the fact, recently proven in [21], that the HOMFLY polynomial of a link H(K) m 1 ,.. whose characteristic variety is a Lagrangian submanifold. This is equivalent, by the discrete Fourier transform in (8.4), to the action of x i and p i on Ψ K (x 1 , . . . , x n ). 12 This fact is predicted by our conjecture for quantization of V K . As an aside, note that the present perspective provides a prediction that the HOMFLY polynomial of the link K colored by representations of n rows is q-holonomic as well.
In the rest of this section we will review the work of [58] and show how this applies to our context.

Review of A-model on M n
To begin with, notice that the torus is in fact hyper-Kähler. In the complex structure we have used so far on M n , which we call I, there is a holomorphic (2, 0) form Ω = i dx i ∧ dp i and a (1, 1) form With this choice, the torus T 2n is Lagrangian, and the variety V K is holomorphic. However, if we let then the triplet of forms ω I , ω J , ω K are the three Kähler forms of M n , viewed as a hyper-Kähler manifold. Note that, in the other two complex structures, J and K, V K becomes Lagrangian rather than holomorphic as it is for I. This fact will be important later.
We consider A-model defined on M n with real symplectic structure ω = Im(Ω) = ω K .
The A-model admits, in addition to the usual Lagrangian branes, which are half the dimension of the symplectic manifold, coisotropic branes [57] that can have lower codimension. On M n , which is hyper-Kähler, there is a canonical example of such a coisotropic brane of rank one that in fact wraps all of M n .
We will denote the canonical coisotropic brane by B c.c . The discussion in this subsection borrows from [58].
On any coisotropic brane there is a line bundle whose curvature F satisfies where we view ω −1 F as a map that takes T M n to itself. In the present case, on the canonical coisotropic brane, F is simply and the coisotropic condition is satisfied since M n is hyper-Kähler, with triplet of Kähler forms ω I , ω J , ω K that satisfy Here brane. We have already seen it is an A-brane in complex structure I. It is also an A-brane in complex structure K, where we take ω I as the symplectic form, simply because ω −1 I F = ω −1 I ω J = K. Finally, in complex structure J it is a B-brane, since in that complex structure F = ω J is the (1, 1) form.
Consider now topological open string with two boundaries on the B c.c brane. The action of the A-model on a disk can be written up to Q-exact terms as where we used Ω = F + iω. This reflects the fact that the instantons of the A-model are holomorphic in complex structure K, ω = ω K . It is easy to show that only degenerate maps contribute [58].
The algebra of open strings is associative, as usual, but due to the presence of the flux F , it becomes non-commutative. Since Ω = i dx i ∧ dp i is exact, the action above introduces a boundary coupling so that p i and x i are canonically conjugate and thus satisfy the standard commutation relations Note that g s corrections in this duality frame are not string loop corrections, but α corrections instead. Locally, this simply implies that The question about the spectrum of the bifundamental strings is local, so we should always get a single fermion ψ(x) at the intersection.

Second-quantizing D-modules
We consider now the A-model string theory with a B c.c brane on M n and a Lagrangian B brane wrapping V . Since M n is hyper-Kähler, there are no higher string loop corrections. The theory is exact at the semi-classical level, and the fermions are free. The only corrections come from the sigma model loops, and these are controlled by g s . Thus, we expect to get a theory of a single free fermion living on V , yet one that is a section of the D n -module.
Consider the one point function of the fermion, viewed as the string field, inserted at a point x on V , in the background of a single anti-fermion, which we insert at infinity, As it stands, this is not gauge invariant, since ψ is charged under the gauge fields on both the B c.c and the B brane. To fix this, we can modify the correlator by inserting where A is the connection on the B c.c brane and A the connection on the Lagrangian brane B . The coupling to gauge fields is natural from the worldsheet perspective where the correlator is computed by the disk amplitude of the topological A model, and the coupling to gauge fields comes from the standard way the target space gauge fields couple to the worldsheet of the string. 14 If the B brane were the flat brane at p i = 0, in the flat target space M n = (C * ) 2n , then we would find that Ψ(x) = 1, corresponding to expectation value of a fermion in a free field theory. Here however the correlator is modified, and to leading order where W (x) is determined by the local equation of the B -brane in M n , as the set of points with coordinates (x, p) where Existence of such a function W (x) is equivalent to vanishing of i dp i ∧ dx i on V , which in turn follows by the definition of V being a Lagrangian submanifold of M n in complex structure K.
We next discuss how this relates with (8.6). Recall that the connection on the coisotropic brane B c.c does not vanish identically. Rather, the presence of curvature on the coisotropic brane, 14 More precisely, we consider a worldsheet which is a ribbon with one boundary on the B c.c brane and the other on the B brane, and we integrate over all paths that start at the point with coordinates x at the worldsheet time τ → −∞ and go off to x = ∞ at τ → +∞. The propagation in infinite time automatically projects to the vacuum states, and these correspond to fermion insertions. and is the local equation of V as embedded in M n . In the topological A-model on any manifold, holomorphy of the amplitudes implies that the gauge field on the brane is always paired with the symplectic form ω, so it is only the combination F + iω = i dp i ∧ dx i that appears. Consequently, the dependence of the amplitude (8.6) on A implies it depends on i p i dx i . This does not quite lead us to conclude that (8.6) takes the value (8.8) for the following reason: the expectation value (8.6) depends on not only the gauge field A on the coisotropic brane, but also the gauge field A on the Lagrangian brane. However, the condition F = dA does not uniquely determine A to satisfy (8.9), but only determines it up to a gauge transformation. As we shall see, the two deficiencies cancel. Like on the coisotropic brane, on the Lagrangian brane wrapping V one finds two sets of fields that are naturally paired: one set is the components A i of the flat gauge field, and the other is scalar fields φ i that are sections of the normal bundle to V and describe deformations of the Lagrangian. Since V is a Lagrangian and M n symplectic, the normal bundle is just the cotangent space, T * V . In the A-model in complex structure K, the connection on V and the scalars φ i always appear in the combination where we defined p j (x) = A j (x)+iφ j (x) which in effect parametrizes a choice of GL(1) connection on V . This gives us two different yet equivalent ways of deforming the Lagrangian: one by varying the local defining equation p i = p i (x), and the other by turning on the GL(1) connection p j (x). We can use the fact that that p j itself is not gauge invariant to soak up the gauge dependence of the statement that A i = p i . We see that the only invariant combination is the difference p i − p i = ∂W ∂x i (x), and that describes the embedding of the branes. Thus, gauge invariance of the amplitude (8.6) and holomorphy of the topological A-model imply that the one point function takes the value (8.8).
We conclude that the A-model on M n with the two sets of A-branes, the canonically coisotropic brane B c.c and the Lagrangian brane on B , leads to a holonomic D-module for the Weyl algebra D n . Among all the D-modules, the holonomic ones are special: they arise from quantization of Lagrangian (rather than coisotropic) submanifolds, and hence precisely correspond to B branes that are Lagrangian. In particular, it is well known that holonomic D-module are cyclic, i.e. generated by a single element. This corresponds to the fact that among all the (B c.c , B ) strings there is a unique special one, the ground state, which is our fermion. The fact that V is generated by a single element Ψ means that any element Θ ∈ V can be obtained by acting on Ψ with elements of D n , Θ = AΨ for some A ∈ D. The ideal I is typically generated by a finite set of m generators A α satisfying A α (e x i , e gs∂x j )Ψ = 0, (8.10) so that I = D n · A 1 , . . . , A m .
Recall that any element of D n can be written in the form A = k 1 ,...,kn a k 1 ,...,kn (e x 1 , . . . , e xn ) e i k i gs∂x i .
From the semi-classical limit of Ψ in (8.8), it follows that the classical Lagrangian V that we started with is the set of points in M n satisfying A α (e x i , e p j ) = 0, α = 1, . . . m, which in the language of D-modules means that V is the characteristic variety of the D-module V.
There is another, equivalent perspective on the problem. Every A-brane wrapping an arbitrary submanifold V of M n = (C * ) 2n can be viewed as r branes on N = (C * ) n for some r. Such a brane is equipped with a flat GL(r) connection A , which complexifies the U (r) gauge group on the branes, with There is a set of r fermions Ψ a , a = 0, . . . , r − 1, transforming in the fundamental representation of GL(n), that give rise to a covariantly constant section with respect to this connection: The flatness of the GL(n) connection guarantees that there are r linearly independent solutions to this equation. The r linearly independent solutions naturally correspond to r fermions living on the r sheets of V . The fact that these two different descriptions exist is well known in the theory of Dmodules: the fact that they get related to flat connections is one of the key aspects of the theory. Moreover, the relation between the two pictures is also well known. One can prove that there is a basis and a set of r operators P 0 , P 1 , . . . , P r−1 , which are generators of the module V, such that Ψ 0 = P 0 Ψ, Ψ 1 = P 1 Ψ, . . . , Ψ r−1 = P r−1 Ψ.
The operators P 0 , . . . , P r−1 form a basis of the module V viewed as a free module over functions on (C * ) n (that is, with coefficients given by functions of e x i ). Different choices of basis are related by GL(n) gauge transformations.
In particular, we can always choose 1 to be one of the generators, so up to GL(n) transformations, we can set P 0 = 1. In this description, we effectively flatten out the Lagrangian brane, and make it manifest that the theory on the branes is a free theory.

Knots and D-modules
We next consider the general theory developed in Section 8.2 in the special case of Lagrangian submanifolds V = V K of M n where V K is the variety associated to a link K. Above we treated V as smooth, whereas, as we saw in Sections 4, 5, and 6, V K is a union of Lagrangians intersecting according to the graph Γ K . At the outset, classically, the fact that V K is reducible implies that the connection A on the V -brane is reducible, and we get one fermion ψ P for each P . From the physical perspective, when the link is non-split, we expect to get a single theory that unifies all these different V K (P ) into a single object, quantum mechanically.
Thus one would expect all the different components of V K to play a role in the quantization. The fact that the characteristic varieties of irreducible D-modules are in general reducible is a general feature of D-modules, and indeed, here it is naturally forced on us.
Now, recall that we observed that in all computed cases, the graph Γ K encodes the geometry of the intersections of the components in V K . The vertices of the graph correspond to components V K (P ) and for a pair of vertices P , P connected by an edge of the graph, V K (P ) and V K (P ) intersect in codimension 1. For every intersection between V K (P ) and V K (P ) we get additional states at the intersection which are bifundamental (P, P ) strings.
If the intersection is in codimension 1, then by giving expectation value to these bifundamental matter fields, one can fuse the two Lagrangians. 15 One can think of these as corresponding to the off-diagonal components of the connection A on (C * ) n , connecting blocks corresponding to P and P .
This structure matches the mathematical theory very well: it is known [22] that every D-module whose characteristic variety is a union of two components that intersect in codimension greater than 1 is in fact reducible.
Thus, both physically and mathematically, the interaction between the component V K (P ) for different P comes from intersections of codimension 1.

Examples
The simplest case to consider is when the link is totally split: each component is separated from all the rest. In this case, as already noted in Section 5, the augmentation variety has a single component, with P = 1 n . This is the product of V (1) s of the individual knots, and all the other V K (P )'s are empty. This also implies that and that B c.c is the product of the individual B c.c 's for each link component.
Thus the partition function of the topological string should be the product of the partition function for each knot component. This is consistent with the factorization property of HOMFLY for totally split links: where H K i is the HOMFLY invariant associated with the i-th knot component To see how this may come about, it is instructive to consider the simple example of the Hopf link. As described previously, the two components of V K are given by: If the two unknot components were unlinked, we would only have V K (1 2 ), and we would get the product of the partition function of two unknots.
However, the fact that V K (2) is also there changes the story. From the HOMFLY polynomial, we can read off the exact answer for the ideal I of operators that annihilate Ψ. We can write this in two different ways. One is from the perspective of the coisotropic brane, describing which of the B c.c − B c.c strings act trivially on the fermion. This gives the ideal I determined by three quantum difference equations: In the above, X 1,2 = e x 1,2 and P 1,2 = e gs∂x 1,2 are operators that do not commute, due to the flux on the coisotropic brane. Viewed as a set of linear differential equations, this has two solutions, as discussed in Section 4, associated to V (1 2 ) and V (2), so that the general solution is with two a priori arbitrary constants c. As previously discussed, Ψ V (2) simply solves (P 1 − X 2 )Ψ V (2) = 0, (P 2 − X 1 )Ψ V (2) = 0, but Ψ V (1 2 ) is more complicated.
Another way to characterize this is from the perspective of the Lagrangian brane. From this perspective, the fact that the brane has two components V K (1 2 ) and V K (2), each of which covers the X 1 , X 2 plane once, means that we can describe the vacuum of the theory in terms of a flat rank 2 connection on N = (C * ) 2 . Indeed, one can show that the above system of equations is equivalent to a set of two linear equations: where Ψ 0 is what we previously called Ψ.
In this language, we can see how the solution corresponding to V K (1 2 ) is corrected by the presence of V K (2). To get Ψ V K (1 2 ) , consider the projection sending (Ψ 0 , Ψ 1 ) to Ψ 1 . This gives the V K (1 2 ) brane embedded in M 2 in isolation, associated to two unlinked unknots. This is not the exact solution, since the off-diagonal components of the connection do not vanish. But once we have picked a specific solution for Ψ 1 , it determines Ψ 0 in perturbation theory for small g s . In particular, one can easily show that classically, , where the subleading corrections are determined from Ψ 1 by off-diagonal components of A . Finally, to this solution, we can add any multiple of a solution associated to V K (2).
Finally, note that in the case of the Hopf link, we can recover all the data of the link from studying more branes-or more fermions, on the Riemann surface V mirror to the unknot. The exact answer for the HOMFLY for the case of the Hopf link is known to be obtainable by putting two branes on the curve associated to the conifold, one along the x-leg and one along the p-leg [52]. We have seen examples of this in Section 7 as well, when we studied torus links. This should be an example of a more general phenomenon, where the topological string on M k for any k provides a way to get a hierarchy of D-modules for free, simply by inserting more fermions. For any number k of fermions inserted, we get a module for the kn-th Weyl algebra, and moreover all the D-modules that arise are holonomic, with the n-fermion correlation function being the cyclic generator of the D-module.
As an example, consider the linkK given by n parallel copies of a given knot K. In this case, we have two descriptions of the theory. One is the standard one in terms of M n = T * (T 2n ) with a B c.c brane and a single Lagrangian brane wrapping VK. In this theory, the partition function inserted at a point in VK. One of the components of VK is VK(1 n ); this simply equals n copies of V K (1). For most framed knots K,K is a nontrivial link, and when this is the case, VK has additional components. 16 Correspondingly, the partition function (8.12) does not factor. Instead, the S n symmetry of the link translates into the permutation symmetry of VK and the fermion one point function is totally symmetric in its n variables.
The second description is based on the Riemann surface V K in M 1 = T * (T 2 ) which is the mirror for the knot K. We insert n fermions ψ(x 1 ) . . . ψ(x n ) V K at n points on the Riemann surface VK (with n fermions at infinity to cancel the charge). Since the n fermions are totally identical particles that anticommute, the partition function is totally antisymmetric. As we discussed in Section 4, the correspondence between the two pictures is ψ(x 1 , . . . , x n ) VK = ψ(x 1 ) . . . ψ(x n ) V K /∆(x 1 , . . . , x n ).
Dividing by the Vandermonde restores the invariance of the partition function under all permutations of n strands. We saw an example of this in the case of 16 Only when K is the unknot in trivial framing isK a split link.
(n, nf ) links we studied in Section 7.2. There we showed that the partition function (8.11) of the link has the form: 17 ψ(x 1 , . . . , x n ) VK = det(w i−1 (x j ))/∆(x 1 , . . . , x n ). (8.13) The simplicity of this expression has a natural explanation in the second viewpoint: the right hand side is the wave function of n free fermions on V K , provided we treat the fermions as sections of the D-module. Here w j (x) is a basis of fermion wave functions that arises from V K . These shift the vacuum of the theory on V K from the canonical one, where the single particle wave functions are z j (x) = e jx , to w j , so that ψ(x 1 ) · · · ψ(x n ) V K = det(w i−1 (x j )).
where the missing terms are suppressed by powers of g s . The saddle point potential is the sum of two unknot potentials W (x i , Q), with longitude x i . Denoting then λ i , µ i live on a 2-complex-dimensional variety which is Lagrangian and a direct product of two copies of the Riemann surface mirror to the unknot: The potential W Hopf (1 2 )(x 1 , x 2 ) dominates the saddle point as long as the factor 1 1−e −p 2 e x 1 is analytic there. To study the opposite case, consider the Fourier transformΨ Hopf of Ψ Hopf : Ψ Hopf (p 1 , p 2 ) = dx 1 dx 2 e p 1 x 1 /gs+p 2 x 2 /gs Ψ Hopf (x 1 , x 2 ).
The integral over x 2 at fixed x 1 is a Fourier transform Ψ Hopf (p 1 , p 2 ) = dx 1 1 1 − e −p 2 e x 1 exp(−p 1 x 1 /g s )Ψ (x 1 )Ψ −1 (p 2 ), where we used the fact that the Fourier transform of the unknot inverts the wave function. Now, due to the pole in the integrand, the integral will depend on the contour. The saddle point corresponding to V Hopf (2) corresponds to choosing a contour which makes a small circle around the pole at e p 2 = e x 1 .
We could have chosen a different contour, which would have led to the ordinary Fourier transform, and we would have gotten the saddle point corresponding to P = 1 2 instead. The two distinct fillings of Λ K at infinity, associated with the partitions 1 2 and 2, thus correspond to two distinct contours of integration that pick out different saddle point contributions toΨ Hopf (p 1 , p 2 ).

B Chern-Simons and torus link invariants
Here we consider the case of a (n, nf ) torus linkK, for f ∈ Z. This link consists of n parallel copies of an unknot K in framing f , i.e., a (1, f ) torus knot. The HOMFLY polynomial for the unknot in this framing colored by an arbitrary representation R is where S RR and T RR = T R δ R R are the SU (N ) WZW S and T matrices as before. The explicit expressions for these are well known. Using the fact that S 0R /S 00 is the dimension of the corresponding quantum group representation, here the product runs over all boxes at position s = (i, j) in the Young diagram corresponding to R, and a(s) = R i − j, (s) = R T j − i, a (s) = j − 1, l (s) = i − 1. Furthermore, T R = q 1 2 (λ R ,λ R +2ρ) , where λ R is the highest weight vector of an SU (N ) representation R, and ρ is the Weyl vector. This gives There is a nice way to rewrite this by considering the representations R n whose Young diagrams have at most n of rows of length R i for i = 1, . . . , n. Then