Knot Contact Homology, Chern-Simons Theory, and ... · Knot contact homology Knot contact...

Knot Contact Homology, Chern-Simons Theory,and Topological Strings

Tobias Ekholm

Uppsala University and Institute Mittag-Leffler, Sweden

Symplectic Topology, Oxford, Fri Oct 3, 2014

Reports on joint work with Aganagic, Ng, and Vafa, arXiv:1304.5778

Knot Contact Homology

Legendrian DGAsAugmentation varieties

Physics Results

Chern-Simons theory and topological stringThe conifold and large N transition

Relations

GW disk potentials and augmentationsLegendrian SFT and recursion relations for the coloredHOMFLY

Knot contact homology

Let K ⊂ S3 be a knot. The Lagrangian conormal of K is

LK = {p ∈ T ∗KS3 : p|TK = 0} ≈ S1 × R2.

The Legendrian conormal is the ideal boundary of LK :

ΛK = LK ∩ U∗S3,

where U∗S3 is the unit conormal

{(q, p) ∈ T ∗S3 : |p| = 1} ≈ S3 × S2,

with the contact form α = pdq,

ΛK ≈ S1x × S1

p ≈ T 2,

where S1x is the longitude and S1

p the meridian.

Let K ⊂ S3 be a knot. The Lagrangian conormal of K is

LK = {p ∈ T ∗KS3 : p|TK = 0} ≈ S1 × R2.

The Legendrian conormal is the ideal boundary of LK :

ΛK = LK ∩ U∗S3,

where U∗S3 is the unit conormal

{(q, p) ∈ T ∗S3 : |p| = 1} ≈ S3 × S2,

with the contact form α = pdq,

ΛK ≈ S1x × S1

p ≈ T 2,

where S1x is the longitude and S1

p the meridian.

Knot contact homology is the Legendrian contact homologyof the Legendrian conormal of a knot (or link), which wedescribe next.

The Reeb vector field R of α: dα(R, ·) = 0 and α(R) = 1.For α = p dq, the flow of R is the geodesic flow. A flow lineof R beginning and ending on a Legendrian submanifold is aReeb chord of Λ. For ΛK Reeb chords correspond to binormalgeodesic chords on K , and are the critical points of the actionfunctional γ 7→

∫γ α for curves γ with endpoints on ΛK .

The grading |c | of a Reeb chord c is defined by a Maslovindex. For binormal geodesics in knot contact homology thegrading is the Morse index: min = 0, sad = 1, max = 2.

Legendrian contact homology is in a sense the “Floerhomology” of the action functional

∫γ α. Because of certain

bubbling phenomena this theory becomes more non-linearthan in the ordinary case.

The symplectization of U∗S3 is U∗S3 × R with symplecticform d(etα) where t ∈ R. Fix an almost complex structure Jsuch that J(∂t) = R, J(kerα) = kerα, and dα(v , Jv) > 0.

If c is a Reeb chord of ΛK then c ×R is a J-holomorphic stripwith boundary on the Lagrangian submanifold ΛK × R.

The Legendrian contact homology algbera of ΛK is the freeunital (non-commutative) algebra

A(ΛK ) = C[H2(U∗S3,ΛK )]⟨

Reeb chords⟩

= C[e±x , e±p,Q±1]⟨

Reeb chords⟩

A(ΛK ) is a DGA. The differential ∂ : A(ΛK )→ A(ΛK ) islinear, satisfies Leibniz rule, decreases grading by 1, and isdefined on generators through a holomorphic curve count.The DGA (A(ΛK ), ∂) is invariant under deformations up tohomotopy and in particular up to quasi-isomorphism.

The Legendrian contact homology algbera of ΛK is the freeunital (non-commutative) algebra

A(ΛK ) = C[H2(U∗S3,ΛK )]⟨

Reeb chords⟩

= C[e±x , e±p,Q±1]⟨

Reeb chords⟩

A(ΛK ) is a DGA. The differential ∂ : A(ΛK )→ A(ΛK ) islinear, satisfies Leibniz rule, decreases grading by 1, and isdefined on generators through a holomorphic curve count.The DGA (A(ΛK ), ∂) is invariant under deformations up tohomotopy and in particular up to quasi-isomorphism.

In general, the knot contact homology can be explicitlycomputed, via Morse flow trees as an adiabatic limit ofholomorphic disks, from a braid presentation of a knot. For abraid on n strands the algebra has n(n − 1) generators indegree 0, n(2n − 1) in degree 1, and n2 in degree 2.

The unknot

A(ΛU) = C[e±x , e±p,Q±1]〈c, e〉, |c | = 1, |e| = 2,

∂e = c − c = 0, ∂c = 1− ex − ep + Qexep

In general, the knot contact homology can be explicitlycomputed, via Morse flow trees as an adiabatic limit ofholomorphic disks, from a braid presentation of a knot. For abraid on n strands the algebra has n(n − 1) generators indegree 0, n(2n − 1) in degree 1, and n2 in degree 2.

The unknot

A(ΛU) = C[e±x , e±p,Q±1]〈c, e〉, |c | = 1, |e| = 2,

∂e = c − c = 0, ∂c = 1− ex − ep + Qexep

The right handed trefoil (differential in degree 1):

A(ΛT ) = C[e±x , e±p,Q±1]〈a12, a21, b12, b21, cij , eij〉i ,j∈{1,2},|aij | = 0, |bij | = |cij | = 1, |eij | = 2,

∂b12 = e−xa12 − a21,

∂b21 = exa21 − a12,

∂c11 = epex − ex − (2Q − ep)a12 − Qa212a21,

∂c12 = Q − ep + epa12 + Qa12a21,

∂c21 = Q − ep + epexa21 + Qa12a21,

∂c22 = ep − 1− Qa21 + epa12a21,

Augmentation variety

Consider A = A(ΛK ) as a family over (C∗)3 of C-algebras,where points in (C∗)3 correspond to values of coefficients(ex , ep,Q).

An augmentation of A is a chain map ε : A → C, ε ◦ ∂ = 1, ofDGAs, where we consider C as a DGA with trivial differentialconcentrated in degree 0.

The augmentation variety VK is the algebraic closure of{(ex , ep,Q) ∈ (C∗)3 : A has augmentation

The augmentation polynomial AK is the polynomialcorresponding to the hypersurface VK . It can be computedfrom the formulas for the differential by elimination theory.

The unknot U:

AU(ex , ep,Q) = 1− ex − ep + Qexep.

The trefoil T :

AT (ex , ep,Q) = (e4p − e3p)e2x

+ (e4p − Qe3p + 2Q2e2p − 2Qe2p − Q2ep + Q2)ex

+ (−Q3ep + Q4).

The unknot U:

The trefoil T :

+ (−Q3ep + Q4).

The unknot U:

The trefoil T :

+ (−Q3ep + Q4).

Physics results

We apply Witten’s argument, equating partition functions inU(N) Chern-Simons gauge theory on a 3-manifold M and inopen topological strings in T ∗M with N Lagrangian branes onthe 0-section M, to the case of a knot K in M = S3.

Add an additional Lagrangian brane along LK . Integrating thestrings in LK ∩ S3 = K out corresponds to inserting

det(1− Hol(K )e−x)−1

in the Chern-Simons path integral, where Hol(K ) is theholonomy of the U(N)-connection around K and e−x theholonomy of the U(1)-connectionon LK .

Physics results

We apply Witten’s argument, equating partition functions inU(N) Chern-Simons gauge theory on a 3-manifold M and inopen topological strings in T ∗M with N Lagrangian branes onthe 0-section M, to the case of a knot K in M = S3.

Add an additional Lagrangian brane along LK . Integrating thestrings in LK ∩ S3 = K out corresponds to inserting

det(1− Hol(K )e−x)−1

in the Chern-Simons path integral, where Hol(K ) is theholonomy of the U(N)-connection around K and e−x theholonomy of the U(1)-connectionon LK .

Physics results

Expanding

det(1− e−xHol(K ))−1 =∑k

trSk Hol(K ) e−kx

then gives

ΨK (ex , egs ,Q) : = ZGW (LK )/ZGW

〈trSk Hol(K )〉e−kx =∑k

Hk(K ) e−kx ,

where Hk(K ) is a HOMFLY-invariant which is polynomial inq = egs and Q = qN = egsN . Here the string couplingconstant is gs = 2πi

N+k , where k is the level in Chern-Simonspath integral.

Physics results

X = T ∗S3 is a quadric in C4, a resolution of a cone. There isanother resolution Y , total space of O(−1)⊕O(−1)→ CP1.

Physics results

Gopakumar-Vafa show: if area(CP1) = t = Ngs , andQ = et = qN , then

ZGW (X ;N, gs) = ZGW (Y ; gs ,Q),

relating open curve counts in X to closed curve counts in Y .

Physics results

For a knot K ⊂ S3, LK can be shifted off the 0-section by anon-exact Lagrangian isotopy. Then LK ⊂ Y andGopakumar-Vafa’s argument gives

ΨK (ex , egs ,Q) = ZGW (Y ; LK )/ZGW (Y )

Physics results

Witten’s argument relates constant curves on LK toGL(1)-gauge theory on the solid torus which corresponds toordinary QM in the periods of the connection. This gives

pΨK (x) = gs∂

∂xΨK (x),

and the usual asymptotics:

ΨK (x) = exp

∫pdx + . . .

From the GW -perspective

ΨK (x) = exp

gsWK (x) + . . .

where WK (x) is the disk potential of LK and where . . .counts curves with 2g − 2 + h > −1.

Physics results

Witten’s argument relates constant curves on LK toGL(1)-gauge theory on the solid torus which corresponds toordinary QM in the periods of the connection. This gives

pΨK (x) = gs∂

∂xΨK (x),

and the usual asymptotics:

ΨK (x) = exp

∫pdx + . . .

From the GW -perspective

ΨK (x) = exp

gsWK (x) + . . .

where WK (x) is the disk potential of LK and where . . .counts curves with 2g − 2 + h > −1.

Physics results

We conclude that as we vary LK

p =∂WK

This equation gives a local parameterization of an algebraiccurve that in all examples agree with the augmentation variety(and which is important from a mirror symmetry perspective).

Physics results

We conclude that as we vary LK

p =∂WK

This equation gives a local parameterization of an algebraiccurve that in all examples agree with the augmentation variety(and which is important from a mirror symmetry perspective).

Augmentations and exact Lagrangian fillings

Exact Lagrangian fillings L of ΛK in T ∗S3 inducesaugmentations by

ε(a) =∑|a|=0

|MA(a)|A.

The map on coefficients are just the induced map onhomology.

There are two natural exact fillings of ΛK : LK andMK ≈ S3 − K . Thus, ep = 1 and ex = 1 belong to VK |Q=1

for any K .

For the unknot AU(ex , ep,Q = 1) = (1− ex)(1− ep).

For the trefoil AT (ex , ep,Q = 1) = (1− ex)(1− ep)(e3p − 1).

for any K .

Augmentations and non-exact Lagrangian fillings

The Lagrangian conormal can be shifted off of the 0-section.It becomes non-exact but a similar version of compactnessstill holds: closed disks lie in a compact, and disks at infinityare as for ΛK × R.

The previous definition of a chain map does not work becauseof new boundary phenomena.

Compare the family of real curves in C2, xy = ε, ε→ 0.

We resolve this problem by using bounding chains: fix a chainσD for each rigid disk D that connects its boundary in LK to amultiple of a standard homology generator at infinity.

We introduce quantum corrected holomorphic disks withpunctures: these are ordinary holomorphic disks with allpossible insertions of σ along the boundary. (CompareFukaya’s unforgettable marked points.) In the moduli spaceMA(a;σ) of quantum corrected disks, boundary bubblingbecome interior points.

Analyzing the boundary then shows that

ε(a) =∑|a|=1

MA(a;σ)eA

is a chain map provided p = ∂WK∂x .

We find that p = ∂WK∂x parameterizes a branch of the

augmentation variety.

Here WK is the Gromov-Witten disk potential which is a sumover weighted trees, where there is a holomorphic disk at eachvertex of the tree and where each edge is weighted by thelinking number of the boundaries of the disks at the verticesof its endpoints, as measured by the bounding chains.

We find that p = ∂WK∂x parameterizes a branch of the

augmentation variety.

Here WK is the Gromov-Witten disk potential which is a sumover weighted trees, where there is a holomorphic disk at eachvertex of the tree and where each edge is weighted by thelinking number of the boundaries of the disks at the verticesof its endpoints, as measured by the bounding chains.

Many component links

For an m-component link, the GW-disk potential of theconormal filling sees only the individual knot components. Wethus need other Lagrangian fillings of the conormal tori. Suchfillings indeed exist in special cases. In the general case thereare candidates. The GW-disk potential WK of such a fillinggives a Lagrangian (locally xj = ∂W

∂pj) augmentation variety

VK in (C∗)m × (C∗)m.

Physical quantization of VK

To quantize VK , consider the topological A-model on T ∗T 2m

with a co-isotropic A-brane B on the whole space and aLagrangian brane L on VK . The algebra of open string statesfrom B to B gives the Weyl algebra. Strings with one end onB and the other on L has a unique ground state which gives afermion ψK (x) on VK . The theory of this fermion is theD-model on VK . Conjecture: the expectation value of thisfermion living on VK is the (HOMFLY) function,

〈ψK (x)〉 = ΨK (x) = ΨK (x1, . . . , xm).

The action of B − B strings on B − VK strings gives aD-module with characteristic variety VK , this is in line withthe intersection behavior of various components of VK .

This is in line with Garoufalidis’ result that the coloredHOMFLY polynomial is q-holonomic.

Physical quantization of VK

To quantize VK , consider the topological A-model on T ∗T 2m

with a co-isotropic A-brane B on the whole space and aLagrangian brane L on VK . The algebra of open string statesfrom B to B gives the Weyl algebra. Strings with one end onB and the other on L has a unique ground state which gives afermion ψK (x) on VK . The theory of this fermion is theD-model on VK . Conjecture: the expectation value of thisfermion living on VK is the (HOMFLY) function,

〈ψK (x)〉 = ΨK (x) = ΨK (x1, . . . , xm).

The action of B − B strings on B − VK strings gives aD-module with characteristic variety VK , this is in line withthe intersection behavior of various components of VK .

This is in line with Garoufalidis’ result that the coloredHOMFLY polynomial is q-holonomic.

Mathematical quantization of VK – Legendrian SFT

It is possible to define a version of Legendrian SFT in thecurrent set up which we conjecture gives the D-moduledefined by the colored HOMFLY polynomial.

Let the degree 1 Reeb chords be denoted b1, . . . bm and thedegree 0 Reeb chords a1, . . . , an

Additional data: a Morse function f on LK which givesobstruction chains. A 4-chain CK for LK with ∂CK = 2[LK ]which looks like ±J∇f near the boundary.

Legendrian SFT

We define the GW-potential eF as the generating function oforiented graphs with holomorphic curves at vertices, andintersections with chains at the edges weighted by ±1

F =∑

Cχ,m,I g−χs emxai1 . . . air

Legendrian SFT

Let H(bj) denote the count of rigid holomorphic curves in thesymplectization with a positive puncture at bj :

H(bj) =∑

Cχ,m,n,I ,J g−χs emxenp ai1 . . . air g

∂aj1. . .

∂ajl.

Then if p = gs∂∂x , e−FH(bj)e

F counts ends of a1-dimensional moduli space and in particular:

H(bj)eF = 0.

We conjecture that the above system give a generator AK

with AKeF0 = 0 for the D-module ideal corresponding to the

colored HOMFLY of K (generators AjK if K is a link).

Legendrian SFT and HOMFLY

For the unknot there are no (formal) higher genus curves andthe operator equation is

AU(ex , ep,Q) = (1− ex − ep − Qexep)ΨU = 0.

For the Hopf link L Reeb chord generators are as for thetrefoil. The relevant parts for the operator H is as follows:

H(c11) = (1− ex1 − ep1 + Qex1ep1) + g2s ∂a12∂a21 +O(a),

H(c22) = (1− ex2 − ep2 + Qex2ep2) + Qex2ep2g2s ∂a12∂a21 +O(a),

H(c12) = (ep2e−p1 − Qex2ep2)gs∂a12

+ g−1s (e−gs − 1)(1− ex2)a21 +O(a2),

H(c21) = (Qex1ep1 − egs ep1e−p2)gs∂a21

+ g−1s

((egs (egs − 1)− e2gs (egs − 1)ex1)ep1e−p2

+(egs − 1)Qex1ep1g2s ∂a12∂a21

)a12 +O(a2).

For the unknot there are no (formal) higher genus curves andthe operator equation is

AU(ex , ep,Q) = (1− ex − ep − Qexep)ΨU = 0.

For the Hopf link L Reeb chord generators are as for thetrefoil. The relevant parts for the operator H is as follows:

H(c11) = (1− ex1 − ep1 + Qex1ep1) + g2s ∂a12∂a21 +O(a),

H(c22) = (1− ex2 − ep2 + Qex2ep2) + Qex2ep2g2s ∂a12∂a21 +O(a),

H(c12) = (ep2e−p1 − Qex2ep2)gs∂a12

+ g−1s (e−gs − 1)(1− ex2)a21 +O(a2),

H(c21) = (Qex1ep1 − egs ep1e−p2)gs∂a21

+ g−1s

((egs (egs − 1)− e2gs (egs − 1)ex1)ep1e−p2

+(egs − 1)Qex1ep1g2s ∂a12∂a21

)a12 +O(a2).

After the change of variables,

ex′1 = egs ex1 , ep

′1 = egs ep1 ;

ex′2 = Q−1e−x2 , ep

′2 = e−gsQ−1e−p2 ;

Q ′ = egsQ, g ′s = −gs ,

we find the D-module ideal generators

A1L = (ex1 − ex2) + (ep1 − ep2)− Q(ex1ep1 − ex2ep2)

A2L = (1− e−gs ex1 − ep1 + Qex1ep1)(ex1 − ep2)

A3L = (1− e−gs ex2 − ep2 + Qex2ep2)(ex2 − ep1),

in agreement with HOMFLY.

Similarly, for the trefoil we get the D-module ideal generator

AT = (Q2 − Qep)egs (Q − egs e2p)Q2

+((Q2 − Qep)e5gs (Q − egs e2p)e2p

+((qe2p − ep)e3gs (Q − e3gs e2p)Q2

+(egs (Q − egs e2p)(Q − e2gse2p)(Q − e3gs e2p)

+((egs e2p − ep)e7gs (Q − e3gs e2p)e2p

)e2x .

which after a change of variables is in agreement withHOMFLY.

Note that the classical limit gs → 0 of this is

(Q − e2p)AT (ex , ep,Q).

Similarly, for the trefoil we get the D-module ideal generator

AT = (Q2 − Qep)egs (Q − egs e2p)Q2

+((Q2 − Qep)e5gs (Q − egs e2p)e2p

+((qe2p − ep)e3gs (Q − e3gs e2p)Q2

+(egs (Q − egs e2p)(Q − e2gse2p)(Q − e3gs e2p)

+((egs e2p − ep)e7gs (Q − e3gs e2p)e2p

)e2x .

which after a change of variables is in agreement withHOMFLY.

Note that the classical limit gs → 0 of this is

(Q − e2p)AT (ex , ep,Q).

Knot Contact Homology, Chern-Simons Theory, and ... · Knot contact homology Knot contact...

Documents

Transcript of Knot Contact Homology, Chern-Simons Theory, and ... · Knot contact homology Knot contact...