Colored HOMFLYPT counts holomorphic curves
CCOLORED HOMFLYPT COUNTS HOLOMORPHIC CURVES
TOBIAS EKHOLM AND VIVEK SHENDE
Abstract.
We compute the contribution of all multiple covers of an isolated rigid em-bedded holomorphic annulus, stretching between Lagrangians, to the skein-valued count ofopen holomorphic curves in a Calabi-Yau 3-fold. The result agrees with the predictions fromtopological string theory and we use it to prove the Ooguri-Vafa formula [24] that identifiesthe colored HOMFLYPT invariants of a link with a count of holomorphic curves ending onthe conormal Lagrangian of the link in the resolved conifold. This generalizes our previouswork [14] which proved the result for the fundamental color. Introduction
In [24], extending the ideas of [26, 15], Ooguri and Vafa made a remarkable prediction:the colored HOMFLYPT invariant of a link in the 3-sphere is the count of all holomorphiccurves in the resolved conifold with boundary on the shifted Lagrangian conormal of the link.This prediction both implied novel structural features of the knot invariants, and opened thedoor to counting holomorphic curves by manipulating knot invariants [4, 3, 2]; i.e., countingsolutions to the Cauchy-Riemann equation – a nonlinear PDE – by solving a problem incombinatorial representation theory.Much mathematical evidence supports this picture, e.g. [19, 17, 23, 18, 22, 6, 21]. However,the main ingredient – a deformation invariant open Gromov-Witten theory – was missing.Indeed, deformation invariance of Gromov-Witten curve counts amounts, naively, to thefollowing assertion: for a generic 1-parameter family of data, the boundary of the parame-terized solution space sits entirely above the boundary of the family. This is well-known to befalse for moduli of curves with boundary, which have additional codimension one boundarycomponents associated to hyperbolic and elliptic boundary bubbling.In [14] we showed: at these extra boundaries in moduli the boundaries of the curvesthemselves look exactly like the terms in the framed skein relation (Figure 1). Recall thatthe framed skein module Sk( L ) is the free module generated by framed links in the (three-dimensional) Lagrangian L modulo the skein relations. If we count curves by the isotopyclass of their boundary in Sk( L ), then the result is invariant [14]. In the same article,we also proved the Ooguri-Vafa formula in the simplest case, showing that the (uncolored)HOMFLYPT counts curves in the simplest homology class.The full Ooguri-Vafa prediction amounts to one identity per integer partition. Here,the integer partitions index on the one hand the possible windings of the boundary of theholomorphic curve around the longitude in conormal Lagrangian (a solid torus), and on theother hand the possible ‘colors’ of the colored HOMFLYPT polynomial. In this language, thecase treated in [14] corresponds to the partition “1=1”. Here we will treat general partitions. TE is supported by the Knut and Alice Wallenberg Foundation and the Swedish Research Council.VS is partially supported by the NSF grant CAREER DMS-1654545. a r X i v : . [ m a t h . S G ] J a n TOBIAS EKHOLM AND VIVEK SHENDE
To explain what this requires, let us sketch the argument from [14]. Given a link K ⊂ S ,one forms the conormal L K ⊂ T ∗ S and shifts it off the zero section. For an appropriatecomplex structure, any non-constant holomorphic curve in T ∗ S with boundary on S ∪ L K is a cover of the holomorphic annuli with boundaries tracing the path of the original link asthe conormal is shifted off. Focus attention on the lowest degree term, i.e., the embeddedannulus. There is exactly one per component of the link. Counting in the skein, this givesthe class (cid:104) K (cid:105) ⊂ Sk( S ), tensored with some longitude factors in the skeins of the conormalLagrangians recording the fact that the map went only once around each component of thelink. Perform the conifold transition by first stretching around the zero section of T ∗ S .Because S ∗ S has no index zero Reeb orbits, in the stretching all holomorphic curves mustend up in T ∗ S \ S for sufficiently stretched complex structures. (These may later beidentified with curves in the conifold.) We count curves by their boundaries, so these arecounted in the skein as some C K · (cid:104)∅(cid:105) ∈ Sk( S ), where C K just counts the curves whichwind once around each component of K . By invariance, C K · (cid:104)∅(cid:105) = (cid:104) K (cid:105) , so C K is nothingother than the HOMFLYPT polynomial of K . On the other hand, the curves counted by C K persist to the conifold, so the count is valid there as well. - == = a a -1 z = a - a -1 z Figure 1.
The HOMFLYPTskein relations. Here we will take z = q / − q − / .To treat the general case, we need to solve twoproblems, one foundational and one calculational.The foundational problem is the usual difficulty ofachieving transversality in settings where multiplecovers may appear (which we avoided in [14] by con-sidering only the curves going once around), compli-cated by our need to not perturb constant curves.This is dealt with in [12]. Here we treat the cal-culational question: how does the multiply coveredholomorphic annulus contribute to the skein?Let us first recall how the answer is predicted by[24]. The boundary of the worldsheet of a string ina Lagrangian brane introduces a line defect into theChern-Simons theory on the brane. When the stringworldsheet is embedded it can be shown to contributethe simplest Wilson line: trace of the holonomy in thefundamental representation [26, Section 4.4]. Thus, going n times around should be the traceof the n th power of the holonomy. In the case under consideration we have n -fold covers ofannuli which come with a 1 /n automorphism factor. Now, exponentiating the sum of thesecontributions (and using a standard identity from symmetric function combinatorics) onefinds that the total contribution of all disconnected curves should be (cid:80) λ W λ ⊗ W λ , where W λ denotes the Wilson line given by taking trace in the representation corresponding to thepartition λ .For various reasons it is hard to make direct mathematical use of the above argument:neither Chern-Simons theory nor topological string theory have proper mathematical founda-tions, which makes it hard to interpret the statement that worldsheets in the latter contributeWilson lines to the former. Instead in [14] we turned the correspondence between embed-ded curves and simple Wilson lines into a definition – count holomorphic curves by theirboundaries in the skein – which we showed was consistent (in full generality, this requires[12]). OLORED HOMFLYPT COUNTS HOLOMORPHIC CURVES 3
This definition still does not allow for a direct import of the argument from [24]: wedemand that all curves have embedded boundary, which means we can not deal directlywith the multiply covered annulus. Furthermore, the skein relation requires us to countall disconnected curves, which means we can only see the exponentiated version of thecount discussed above. Nevertheless, the final answer given by the Ooguri-Vafa argument isintelligible in our treatment and we can try and verify it.Our version of W λ is the element in the skein of the solid torus with the following property:given a framed knot K , the ordinary HOMFLYPT polynomial of the cable W λ ( K ) is thesame as the λ -colored HOMFLYPT polynomial of K . These elements have a skein theoreticcharacterization as follows. The skein of the solid torus has an endomorphism P , givenby encircling by a meridional loop. The W λ are characterized up to scalar multiple as theeigenvectors of this endomorphism with eigenvalues [20](1) P , W λ = ( (cid:13) + a ( q / − q − / ) c λ ( q )) · W λ Here (cid:13) means an unknot (the boundary of an embedded disk with standard framing), or inother words simply the scalar ( a − a − )( q / − q − / ) − . The quantity c λ ( q ) is the ‘contentpolynomial’ of λ ; its value is irrelevant to us here save only for the fact that it determines λ . The scalar multiple is fixed by the quantum dimension formula of [25], which we recallbelow in Equation (2).The W λ span the part of the skein generated by links winding only positively along thelongitude. Our links will always be positive, being small perturbations of positive multiplecovers of the longitude.We prove here the following version of the Ooguri-Vafa local calculation: Theorem 1.1.
Let C be a totally isolated rigid holomorphic annulus, with boundaries K , K .Then the total contribution of C to the skein valued curve count is (cid:88) λ γ | λ | W λ ( K ) ⊗ W λ ( K ) This is a sum over all integer partitions λ , where | λ | is the sum of parts of λ , and γ is somesigned monomial in the framing variables. The notion of totally isolated rigid holomorphic annulus (see Definition 2.2 below) is acondition ensuring that the holomorphic annulus is modeled on the kind we discuss above,and moreover that we may consider its perturbations independently of perturbations neededfor other curves. The condition is certainly satisfied in the Ooguri-Vafa situation, when theannulus is the only holomorphic curve around.Let us sketch the proof of Theorem 1.1. By locality of the perturbation scheme we maywork in some appropriate local model. Perhaps the simplest imaginable is the cotangentbundle of the solid torus, where we take the zero section and a shifted off conormal ofthe longitude. Studying boundaries of one-dimensional moduli of curves with one positivepuncture as in [1, 8, 11, 13], we find a relation: (( P , −(cid:13) ) ⊗ a − a ⊗ ( P , −(cid:13) ))Ψ = 0, whereΨ is the total contribution of all (perturbed) multiple covers of the holomorphic annulus.Here a , a are the ‘ a ’ elements of the skeins of the two solid tori. From the formula (1), itfollows immediately that Ψ is itself ‘diagonal’ in the eigenbasis, i.e., a sum of W λ ⊗ W λ withsome coefficients. To determine the coefficients it suffices to study the unknot, which we didin [13].Combining Theorem 1.1 with the results of [14, 12], we deduce TOBIAS EKHOLM AND VIVEK SHENDE
Theorem 1.2.
Let K ⊂ S be a link, and L K ⊂ X the associated Lagrangian in theresolved conifold. Let P K,λ ( a, q ) be the λ -colored HOMFLYPT polynomial of K . Let Z ∈ Sk( L K )(( Q / )) be the skein-valued open Gromov-Witten invariant. For appropriate choiceof 4-chain, Z | z = q / − q − / = (cid:88) λ P K,λ ( Q / , q ) · W λ Totally isolated annuli
Let us fix some terminology. Recall that for the perturbation scheme in [12] we alwaystake almost complex structures which are standard in some neighborhood of the Lagrangians,and moreover only perturb the Cauchy-Riemann equation in the complement of some smallerneighborhood of the Lagrangians.
Definition 2.1.
Let T be the solid torus, which we identify as S × R . Consider itscotangent bundle, T ∗ T = T ∗ S × T ∗ R . Inside T ∗ R , consider the zero section Z and thethe conormal at zero, N . Inside T ∗ S = S × R , we write S (cid:15) for the circle over (cid:15) ∈ R ; theseare all Lagrangian circles but only S is exact.Consider L = Z × S and L = N × S (cid:15) . For a standard complex structure, there will be aunique holomorphic annulus stretching between them, namely S × [0 , (cid:15) ] × ⊂ S × R × T ∗ R .We term this the model annulus . Definition 2.2.
Suppose given some (possibly disconnected) Lagrangian L . Recall from[14, 12] that we always take complex structures standard within some (cid:15) neighborhood of theLagrangian. We say a holomorphic annulus ending on L is standard if it is contained withinthis (cid:15) neighborhood, and some neighborhood of the annulus can be identified with the modelannulus. Remark . Note that a small deformation of any two Lagrangians that meet cleanly alonga non null-homologous knot bounds a standard annulus after a small shift.
Definition 2.4.
We say a standard annulus is totally isolated if in addition no other holo-morphic curves with boundary on L enter its standard neighborhood (aside from multiplecovers of the annulus).Here we will restrict ourselves to the totally isolated case, but we expect more sophisticatedarguments (such as in [16]) will allow us to treat more general annuli. The ‘totally isolated’restriction allows us to avoid discussing any properties of the perturbation scheme, beyondlocality. In particular we have: Lemma 2.5.
There is some element Ψ ∈ Sk( T ) ⊗ Sk( T ) such that the total contributionof all multiple covers of a totally isolated annulus to the Gromov-Witten invariant is alwaysgiven by the cabling of its boundary by Ψ (up to monomial change of framing variables).Proof. The perturbation scheme of [12] is local in the sense that we are free to independentlyperturb maps with disjoint images. In particular we may always transplant some givenperturbation of the model annulus and all its multiple covers to any occurence of a totallyisolated annulus. (cid:3)
OLORED HOMFLYPT COUNTS HOLOMORPHIC CURVES 5 Constraints from the cotangent bundle of the solid torus
Here we study the model geometry of Definition 2.1. We regard T ∗ S × T ∗ R as a Liouvillemanifold by taking the radial Liouville form on T ∗ R , i.e., rounding corners in the T ∗ R -factor. We write Λ and Λ for the Legendrians at infinity of the Lagrangians L , L .We are interested in boundaries of moduli of 1-parameter families of curves asymptoticto chords on Λ ∪ Λ . Boundaries coming from interior breaking we cancel using the skeinrelations as in [14, 12, 13]; the remainder are the SFT breakings which correspond to two-levelcurves [5]. These are pairs of: a rigid curve in the symplectization of the contact boundary,and a rigid curve in the original manifold, with matching asymptotic Reeb chords. We studysuch curves with a single positive puncture at Reeb chord connecting a component of Λ toitself, and we will characterize which such curves may appear in the symplectization.For each Reeb chord of the Legendrian Hopf link, the Legendrian Λ ∪ Λ has a Bottfamily, canonically parameterized by the zero section S ⊂ T ∗ S . We resolve this degener-acy perturbing the contact form using a Morse function on S with one minimum. Afterperturbation, rigid holomorphic curves on Λ ∪ Λ sit near this point, and understandingthem reduces to the corresponding problem for the Legendrian Hopf link. More precisely: Lemma 3.1.
Minimal index Reeb chords for Λ ∪ Λ are in natural bijection with minimalindex chords for the Legendrian Hopf link. Rigid (up to translation) curves with one positivepuncture are arbitrarily close to configurations of the following form: a curve of the Hopf linkover the minimum in the S -family of self chords, and in case there are negative punctures,continue as flow lines to the minimum over trivial strips in the S -families of these.Proof. When resolving the Bott degeneracy we must glue holomorphic curves and Morseflow lines to the minima in the S Bott-families. This is straightforward in the case underconsideration, compare e.g., [9] for similar gluing results. Another approach is to work withflow trees throughout, and refer to [7] for the relation to holomorphic curves. This wascarried out for Legendrian tori constructed from general Legendrian isotopies of Legendrianlinks in [10, Theorem 1.1]. In that setting, the case under consideration here, correspondsto the trivial isotopy of the standard Legendrian Hopf link. (cid:3)
We now recall the chords and curves for the Hopf link. c m m c Figure 2.
The Lagrangian projection of the Legendrian Hopf link
Lemma 3.2.
The rigid curves in the symplectization with one puncture at c are: no-negative-puncture disks D going to the left and D (cid:48) going to the right in Figure 2; and thereis a two-negative-puncture disk T which follows along D (cid:48) until arriving at m , then changesto the other component, travels the short arc to m , then returns to again follow D (cid:48) .Similarly, for c there is D going to the right, D (cid:48) going to the left, and a two-negative-puncture T which again changes components at m and m . TOBIAS EKHOLM AND VIVEK SHENDE
The torus Λ comes with 4 points where it meets the Reeb chords; similarly for Λ . Wefix a capping path for the self-chord, but do not fix any for the mixed chord. Lemma 3.3.
We use capping paths given by the boundaries ∂D (cid:48) , ∂D (cid:48) . With this choice, ∂D , ∂D become the meridians of their respective tori, ∂D (cid:48) , ∂D (cid:48) become trivial, and ∂T and ∂T are isotopic (rel boundary).Proof. Clear from Lemmas 3.1 and 3.2. (cid:3)
Proposition 3.4.
The count Ψ of all bounded curves in the interior is annihilated by ( P , − (cid:13) ) ⊗ a − a ⊗ ( P , − (cid:13) ) Proof.
Let us write M ( c ) and M ( c ) for the moduli spaces of holomorphic curves in thecotangent bundle of the solid torus with boundaries on L ∪ L and with one positive puncture,asymptotic, respectively, to c or c . These moduli spaces are one dimensional. They haveboundaries of two kinds, coming from boundary degenerations in the interior, and SFTdegenerations at infinity. Degenerations of the first kind are cancelled by working in theskein (and appropriately weighting curves by Euler characteristic of the domain and 4-chainintersections as explained in [14]). The sum of all SFT degenerations must therefore vanishwhen evaluated in the skein. Such degenerations are two-level rigid curves. There are twopossibilities: either a disk D i or D (cid:48) i at infinity, plus bounded curves in the interior; or T or T , plus some curves in the interior with two positive punctures asymptotic to m and m .Let Ψ denote the count of all bounded curves in the interior. It takes value in Sk + ( L ∪ L ).We write L (cid:48)(cid:48) and L (cid:48)(cid:48) for these Lagrangians with the two points where the mixed Reeb chordsenter and leave. We write Ψ (cid:48)(cid:48) ∈ Sk( L (cid:48)(cid:48) ∪ L (cid:48)(cid:48) ) for the count of all holomorphic curves withtwo punctures, asymptotic to these two chords.We obtain two equations in Sk( L ∪ L ) from the boundaries of M ( c ) and M ( c ). Up tounknown framing factors (signed monomials in framing variables on every term) these are:( ∂D + ∂D (cid:48) ) ◦ Ψ + ( ∂T ) ◦ (cid:48)(cid:48) Ψ (cid:48)(cid:48) = 0( ∂D + ∂D (cid:48) ) ◦ Ψ + ( ∂T ) ◦ (cid:48)(cid:48) Ψ (cid:48)(cid:48) = 0Here the ◦ (cid:48)(cid:48) is gluing of a T × [0 ,
1] with two marked points on T × ∂T ∼ ∂T , so subtracting (multiplesby some framing factor) we may cancel the Ψ (cid:48)(cid:48) term and obtain:(( γ P , + γ (cid:48) (cid:13) ) ⊗ ⊗ ( γ P , + γ (cid:48) (cid:13) ))Ψ = 0Here the γ i are framing factors we now write explicitly. This operator preserves the naturalgrading by number of boxes in each factor of Sk + ( T ). We determine the framing factorsfrom the first terms of Ψ = 1 + γ · W (cid:3) ⊗ W (cid:3) + · · · . From the zeroeth term, we see that γ = − γ (cid:48) and γ = − γ (cid:48) . Now we study the first term: (cid:18) γ ( P , − (cid:13) ) ⊗ ⊗ γ ( P , − (cid:13) ) (cid:19) ( W (cid:3) ⊗ W (cid:3) ) = 0Using Equation (1), we see a γ + a γ = 0. (cid:3) Corollary 3.5.
Ψ = (cid:80) λ n λ · W λ ⊗ W λ , for some n λ ( a , a , q ) . OLORED HOMFLYPT COUNTS HOLOMORPHIC CURVES 7
Proof.
Expand Ψ = (cid:80) λ,µ n λ,µ ( a , a , q ) · W λ ⊗ W µ . Applying the operator of Proposition 3.4and using Equation (1), we find0 = a a ( q / − q − / ) (cid:88) λ,µ n λ,µ ( a , a , q ) · ( c λ ( q ) − c µ ( q )) · W λ ⊗ W µ Since c λ ( q ) − c µ ( q ) vanishes only for λ = µ , the n λ,µ must vanish whenever λ (cid:54) = µ . (cid:3) Remark . Given the value of Ψ we ultimately compute, in fact ( ∂T ) ◦ (cid:48)(cid:48) Ψ (cid:48)(cid:48) must be arather nontrivial quantity.4. Proof of Theorems 1.1 and 1.2
Proof of Theorem 1.1.
We must show that the coefficients n λ ( a , a , q ) from Corollary 3.5are in fact all just γ | λ | , where γ is some signed monomial in the framing variables a , a . Todo so we study in T ∗ S the union of the zero section and the conormal to an unknot, shiftedoff as in [14]. Let us take the zero section as L and the conormal as L .We write (cid:104)·(cid:105) S for the isomorphism from Sk( S ) to an appropriately localized polynomialring in a , q . By Corollary 3.5 and Lemma 2.5, the total count of bounded curves is: (cid:88) λ n λ ( a , a , q ) (cid:104) W λ (cid:105) S · W λ On the other hand, we computed this total count in [13], and showed it was one of thefollowing two things, depending on the orientation of S : (cid:88) λ γ | λ | (cid:104) W λ (cid:105) S · W λ (cid:88) λ γ | λ | (cid:104) W λ (cid:48) (cid:105) S · W λ Here λ (cid:48) is the conjugate partition to λ . In [13] these results appeared via the formulae:(2) (cid:104) W λ (cid:105) S = (cid:89) (cid:3) ∈ λ aq c ( (cid:3) ) / − a − q − c ( (cid:3) ) / q h ( (cid:3) ) / − q − h ( (cid:3) ) / (cid:104) W λ (cid:48) (cid:105) S = (cid:89) (cid:3) ∈ λ aq − c ( (cid:3) ) / − a − q c ( (cid:3) ) / q h ( (cid:3) ) / − q − h ( (cid:3) ) / Thus we see that either n λ ( a , a , q ) = γ | λ | or n λ ( a , a , q ) = γ | λ | (cid:104) W λ (cid:48) (cid:105) S / (cid:104) W λ (cid:105) S . In thesecond case, we would have n λ ( a , a , q ) = a m f λ ( a , q ), where f λ is not a monomial. Butwe also could have closed off L to an S , leaving L alone. From this we would see either n λ ( a , a , q ) = γ | λ | or that n λ ( a , a , q ) = a m (cid:48) g λ ( a , q ), where g λ is not a monomial. The onlyconsistent possibility is n λ ( a , a , q ) = γ | λ | . (cid:3) Remark . In [13], we found two possibilities because we used only the recursion relationcoming from the knot conormal, which cannot know the orientation of S . Here however,both components of the Lagrangian are oriented: we orient their longitudes along the di-rection traversed by the boundary of the annulus, and their meridians along ∂D or ∂D .Having already oriented both components, there is no choice remaining when we close oneoff into a sphere. In principle one could follow carefully the 4-chain conventions to see whichpossibility from [13] arises; but in the above argument we found a trick to avoid doing so. Proof of Theorem 1.2.
This follows immediately from combining Theorem 1.1 (via Lemma2.5) with our previous results [14, Theorem 6.6, 6.7, 7.3; Corollary 7.4], where the “appropri-ate choice of 4-chain” is given. To appeal to the results of [14] in the present context, wheremultiple covers are not excluded for topological reasons, requires an adequate perturbationscheme. The perturbations must also have the locality properties demanded in Lemma 2.5.We construct such a scheme in [12]. (cid:3)
TOBIAS EKHOLM AND VIVEK SHENDE
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