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

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1 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

2 Plan Reports on joint work with Aganagic, Ng, and Vafa, arxiv: Knot Contact Homology Legendrian DGAs Augmentation varieties Physics Results Relations Chern-Simons theory and topological string The conifold and large N transition GW disk potentials and augmentations Legendrian SFT and recursion relations for the colored HOMFLY

3 Knot contact homology Let K S 3 be a knot. The Lagrangian conormal of K is L K = {p T K S 3 : p TK = 0} S 1 R 2.

4 Knot contact homology Let K S 3 be a knot. The Lagrangian conormal of K is L K = {p T K S 3 : p TK = 0} S 1 R 2. The Legendrian conormal is the ideal boundary of L K : Λ K = L K U S 3, where U S 3 is the unit conormal {(q, p) T S 3 : p = 1} S 3 S 2, with the contact form α = pdq, Λ K Sx 1 Sp 1 T 2, where Sx 1 is the longitude and Sp 1 the meridian.

5 Knot contact homology Knot contact homology is the Legendrian contact homology of the Legendrian conormal of a knot (or link), which we describe next.

6 Knot contact homology Knot contact homology is the Legendrian contact homology of the Legendrian conormal of a knot (or link), which we describe 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 line of R beginning and ending on a Legendrian submanifold is a Reeb chord of Λ. For Λ K Reeb chords correspond to binormal geodesic chords on K, and are the critical points of the action functional γ γ α for curves γ with endpoints on Λ K.

7 Knot contact homology Knot contact homology is the Legendrian contact homology of the Legendrian conormal of a knot (or link), which we describe 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 line of R beginning and ending on a Legendrian submanifold is a Reeb chord of Λ. For Λ K Reeb chords correspond to binormal geodesic chords on K, and are the critical points of the action functional γ γ α for curves γ with endpoints on Λ K. The grading c of a Reeb chord c is defined by a Maslov index. For binormal geodesics in knot contact homology the grading is the Morse index: min = 0, sad = 1, max = 2.

8 Knot contact homology Legendrian contact homology is in a sense the Floer homology of the action functional γ α. Because of certain bubbling phenomena this theory becomes more non-linear than in the ordinary case.

9 Knot contact homology Legendrian contact homology is in a sense the Floer homology of the action functional γ α. Because of certain bubbling phenomena this theory becomes more non-linear than in the ordinary case. The symplectization of U S 3 is U S 3 R with symplectic form d(e t α) where t R. Fix an almost complex structure J such that J( t ) = R, J(ker α) = ker α, and dα(v, Jv) > 0.

10 Knot contact homology Legendrian contact homology is in a sense the Floer homology of the action functional γ α. Because of certain bubbling phenomena this theory becomes more non-linear than in the ordinary case. The symplectization of U S 3 is U S 3 R with symplectic form d(e t α) where t R. Fix an almost complex structure J such that J( t ) = R, J(ker α) = ker α, and dα(v, Jv) > 0. If c is a Reeb chord of Λ K then c R is a J-holomorphic strip with boundary on the Lagrangian submanifold Λ K R.

11 Knot contact homology The Legendrian contact homology algbera of Λ K is the free unital (non-commutative) algebra A(Λ K ) = C[H 2 (U S 3, Λ K )] Reeb chords = C[e ±x, e ±p, Q ±1 ] Reeb chords

12 Knot contact homology The Legendrian contact homology algbera of Λ K is the free unital (non-commutative) algebra A(Λ K ) = C[H 2 (U S 3, Λ K )] Reeb chords = C[e ±x, e ±p, Q ±1 ] Reeb chords A(Λ K ) is a DGA. The differential : A(Λ K ) A(Λ K ) is linear, satisfies Leibniz rule, decreases grading by 1, and is defined on generators through a holomorphic curve count. The DGA (A(Λ K ), ) is invariant under deformations up to homotopy and in particular up to quasi-isomorphism.

13 Knot contact homology

14 Knot contact homology

15 Knot contact homology In general, the knot contact homology can be explicitly computed, via Morse flow trees as an adiabatic limit of holomorphic disks, from a braid presentation of a knot. For a braid on n strands the algebra has n(n 1) generators in degree 0, n(2n 1) in degree 1, and n 2 in degree 2.

16 Knot contact homology In general, the knot contact homology can be explicitly computed, via Morse flow trees as an adiabatic limit of holomorphic disks, from a braid presentation of a knot. For a braid on n strands the algebra has n(n 1) generators in degree 0, n(2n 1) in degree 1, and n 2 in degree 2. The unknot A(Λ U ) = C[e ±x, e ±p, Q ±1 ] c, e, c = 1, e = 2, e = c c = 0, c = 1 e x e p + Qe x e p

17 Knot contact homology The right handed trefoil (differential in degree 1): A(Λ T ) = C[e ±x, e ±p, Q ±1 ] a 12, a 21, b 12, b 21, c ij, e ij i,j {1,2}, a ij = 0, b ij = c ij = 1, e ij = 2, b 12 = e x a 12 a 21, b 21 = e x a 21 a 12, c 11 = e p e x e x (2Q e p )a 12 Qa12a 2 21, c 12 = Q e p + e p a 12 + Qa 12 a 21, c 21 = Q e p + e p e x a 21 + Qa 12 a 21, c 22 = e p 1 Qa 21 + e p a 12 a 21,

18 Augmentation variety Consider A = A(Λ K ) as a family over (C ) 3 of C-algebras, where points in (C ) 3 correspond to values of coefficients (e x, e p, Q).

19 Augmentation variety Consider A = A(Λ K ) as a family over (C ) 3 of C-algebras, where points in (C ) 3 correspond to values of coefficients (e x, e p, Q). An augmentation of A is a chain map ɛ: A C, ɛ = 1, of DGAs, where we consider C as a DGA with trivial differential concentrated in degree 0.

20 Augmentation variety Consider A = A(Λ K ) as a family over (C ) 3 of C-algebras, where points in (C ) 3 correspond to values of coefficients (e x, e p, Q). An augmentation of A is a chain map ɛ: A C, ɛ = 1, of DGAs, where we consider C as a DGA with trivial differential concentrated in degree 0. The augmentation variety V K is the algebraic closure of { (e x, e p, Q) (C ) 3 : A has augmentation }.

21 Augmentation variety The augmentation polynomial A K is the polynomial corresponding to the hypersurface V K. It can be computed from the formulas for the differential by elimination theory.

22 Augmentation variety The augmentation polynomial A K is the polynomial corresponding to the hypersurface V K. It can be computed from the formulas for the differential by elimination theory. The unknot U: A U (e x, e p, Q) = 1 e x e p + Qe x e p.

23 Augmentation variety The augmentation polynomial A K is the polynomial corresponding to the hypersurface V K. It can be computed from the formulas for the differential by elimination theory. The unknot U: A U (e x, e p, Q) = 1 e x e p + Qe x e p. The trefoil T : A T (e x, e p, Q) = (e 4p e 3p )e 2x + (e 4p Qe 3p + 2Q 2 e 2p 2Qe 2p Q 2 e p + Q 2 )e x + ( Q 3 e p + Q 4 ).

24 Physics results We apply Witten s argument, equating partition functions in U(N) Chern-Simons gauge theory on a 3-manifold M and in open topological strings in T M with N Lagrangian branes on the 0-section M, to the case of a knot K in M = S 3.

25 Physics results We apply Witten s argument, equating partition functions in U(N) Chern-Simons gauge theory on a 3-manifold M and in open topological strings in T M with N Lagrangian branes on the 0-section M, to the case of a knot K in M = S 3. Add an additional Lagrangian brane along L K. Integrating the strings in L K S 3 = K out corresponds to inserting det(1 Hol(K)e x ) 1 in the Chern-Simons path integral, where Hol(K) is the holonomy of the U(N)-connection around K and e x the holonomy of the U(1)-connectionon L K.

26 Physics results Expanding det(1 e x Hol(K)) 1 = k tr Sk Hol(K) e kx then gives Ψ K (e x, e gs, Q) : = Z GW (L K )/Z GW = k tr Sk Hol(K) e kx = k H k (K) e kx, where H k (K) is a HOMFLY-invariant which is polynomial in q = e gs and Q = q N = e gsn. Here the string coupling constant is g s = 2πi N+k, where k is the level in Chern-Simons path integral.

27 Physics results X = T S 3 is a quadric in C 4, a resolution of a cone. There is another resolution Y, total space of O( 1) O( 1) CP 1.

28 Physics results Gopakumar-Vafa show: if area(cp 1 ) = t = Ng s, and Q = e t = q N, then Z GW (X ; N, g s ) = Z GW (Y ; g s, Q), relating open curve counts in X to closed curve counts in Y.

29 Physics results For a knot K S 3, L K can be shifted off the 0-section by a non-exact Lagrangian isotopy. Then L K Y and Gopakumar-Vafa s argument gives Ψ K (e x, e gs, Q) = Z GW (Y ; L K )/Z GW (Y )

30 Physics results Witten s argument relates constant curves on L K to GL(1)-gauge theory on the solid torus which corresponds to ordinary QM in the periods of the connection. This gives pψ K (x) = g s x Ψ K (x), and the usual asymptotics: ( 1 Ψ K (x) = exp g s ) pdx +....

31 Physics results Witten s argument relates constant curves on L K to GL(1)-gauge theory on the solid torus which corresponds to ordinary QM in the periods of the connection. This gives pψ K (x) = g s x Ψ K (x), and the usual asymptotics: ( 1 Ψ K (x) = exp g s ) pdx From the GW -perspective Ψ K (x) = exp ( ) 1 W K (x) +..., g s where W K (x) is the disk potential of L K and where... counts curves with 2g 2 + h > 1.

32 Physics results We conclude that as we vary L K p = W K x.

33 Physics results We conclude that as we vary L K p = W K x. This equation gives a local parameterization of an algebraic curve that in all examples agree with the augmentation variety (and which is important from a mirror symmetry perspective).

34 Augmentations and exact Lagrangian fillings Exact Lagrangian fillings L of Λ K in T S 3 induces augmentations by ɛ(a) = M A (a) A. a =0 The map on coefficients are just the induced map on homology.

35 Augmentations and exact Lagrangian fillings There are two natural exact fillings of Λ K : L K and M K S 3 K. Thus, e p = 1 and e x = 1 belong to V K Q=1 for any K.

36 Augmentations and exact Lagrangian fillings There are two natural exact fillings of Λ K : L K and M K S 3 K. Thus, e p = 1 and e x = 1 belong to V K Q=1 for any K. For the unknot A U (e x, e p, Q = 1) = (1 e x )(1 e p ).

37 Augmentations and exact Lagrangian fillings There are two natural exact fillings of Λ K : L K and M K S 3 K. Thus, e p = 1 and e x = 1 belong to V K Q=1 for any K. For the unknot A U (e x, e p, Q = 1) = (1 e x )(1 e p ). For the trefoil A T (e x, e p, Q = 1) = (1 e x )(1 e p )(e 3p 1).

38 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 compactness still holds: closed disks lie in a compact, and disks at infinity are as for Λ K R.

39 Augmentations and non-exact Lagrangian fillings The previous definition of a chain map does not work because of new boundary phenomena. Compare the family of real curves in C 2, xy = ɛ, ɛ 0.

40 Augmentations and non-exact Lagrangian fillings We resolve this problem by using bounding chains: fix a chain σ D for each rigid disk D that connects its boundary in L K to a multiple of a standard homology generator at infinity.

41 Augmentations and non-exact Lagrangian fillings We introduce quantum corrected holomorphic disks with punctures: these are ordinary holomorphic disks with all possible insertions of σ along the boundary. (Compare Fukaya s unforgettable marked points.) In the moduli space M A (a; σ) of quantum corrected disks, boundary bubbling become interior points.

42 Augmentations and non-exact Lagrangian fillings Analyzing the boundary then shows that ɛ(a) = M A (a; σ)e A a =1 is a chain map provided p = W K x.

43 Augmentations and non-exact Lagrangian fillings We find that p = W K x parameterizes a branch of the augmentation variety.

44 Augmentations and non-exact Lagrangian fillings We find that p = W K x parameterizes a branch of the augmentation variety. Here W K is the Gromov-Witten disk potential which is a sum over weighted trees, where there is a holomorphic disk at each vertex of the tree and where each edge is weighted by the linking number of the boundaries of the disks at the vertices of its endpoints, as measured by the bounding chains.

45 Many component links For an m-component link, the GW-disk potential of the conormal filling sees only the individual knot components. We thus need other Lagrangian fillings of the conormal tori. Such fillings indeed exist in special cases. In the general case there are candidates. The GW-disk potential W K of such a filling gives a Lagrangian (locally x j = W p j ) augmentation variety V K in (C ) m (C ) m.

46 Physical quantization of V K To quantize V K, consider the topological A-model on T T 2m with a co-isotropic A-brane B on the whole space and a Lagrangian brane L on V K. The algebra of open string states from B to B gives the Weyl algebra. Strings with one end on B and the other on L has a unique ground state which gives a fermion ψ K (x) on V K. The theory of this fermion is the D-model on V K. Conjecture: the expectation value of this fermion living on V K is the (HOMFLY) function, ψ K (x) = Ψ K (x) = Ψ K (x 1,..., x m ). The action of B B strings on B V K strings gives a D-module with characteristic variety V K, this is in line with the intersection behavior of various components of V K.

47 Physical quantization of V K To quantize V K, consider the topological A-model on T T 2m with a co-isotropic A-brane B on the whole space and a Lagrangian brane L on V K. The algebra of open string states from B to B gives the Weyl algebra. Strings with one end on B and the other on L has a unique ground state which gives a fermion ψ K (x) on V K. The theory of this fermion is the D-model on V K. Conjecture: the expectation value of this fermion living on V K is the (HOMFLY) function, ψ K (x) = Ψ K (x) = Ψ K (x 1,..., x m ). The action of B B strings on B V K strings gives a D-module with characteristic variety V K, this is in line with the intersection behavior of various components of V K. This is in line with Garoufalidis result that the colored HOMFLY polynomial is q-holonomic.

48 Mathematical quantization of V K Legendrian SFT It is possible to define a version of Legendrian SFT in the current set up which we conjecture gives the D-module defined by the colored HOMFLY polynomial.

49 Mathematical quantization of V K Legendrian SFT It is possible to define a version of Legendrian SFT in the current set up which we conjecture gives the D-module defined by the colored HOMFLY polynomial. Let the degree 1 Reeb chords be denoted b 1,... b m and the degree 0 Reeb chords a 1,..., a n

50 Mathematical quantization of V K Legendrian SFT It is possible to define a version of Legendrian SFT in the current set up which we conjecture gives the D-module defined by the colored HOMFLY polynomial. Let the degree 1 Reeb chords be denoted b 1,... b m and the degree 0 Reeb chords a 1,..., a n Additional data: a Morse function f on L K which gives obstruction chains. A 4-chain C K for L K with C K = 2[L K ] which looks like ±J f near the boundary.

51 Legendrian SFT

52 Legendrian SFT We define the GW-potential e F as the generating function of oriented graphs with holomorphic curves at vertices, and intersections with chains at the edges weighted by ± 1 2 : F = C χ,m,i gs χ e mx a i1... a ir

53 Legendrian SFT Let H(b j ) denote the count of rigid holomorphic curves in the symplectization with a positive puncture at b j : H(b j ) = C χ,m,n,i,j g χ s e mx e np a i1... a ir gs l... a j1 Then if p = g s x, e F H(b j )e F counts ends of a 1-dimensional moduli space and in particular: H(b j )e F = 0. a jl. We conjecture that the above system give a generator ÂK with ÂK e F 0 = 0 for the D-module ideal corresponding to the colored HOMFLY of K (generators Âj K if K is a link).

54 Legendrian SFT and HOMFLY For the unknot there are no (formal) higher genus curves and the operator equation is Â U (e x, e p, Q) = (1 e x e p Qe x e p )Ψ U = 0.

55 Legendrian SFT and HOMFLY For the unknot there are no (formal) higher genus curves and the operator equation is Â U (e x, e p, Q) = (1 e x e p Qe x e p )Ψ U = 0. For the Hopf link L Reeb chord generators are as for the trefoil. The relevant parts for the operator H is as follows: H(c 11 ) = (1 e x 1 e p 1 + Qe x 1 e p 1 ) + g 2 s a12 a21 + O(a), H(c 22 ) = (1 e x 2 e p 2 + Qe x 2 e p 2 ) + Qe x 2 e p 2 g 2 s a12 a21 + O(a), H(c 12 ) = (e p 2 e p 1 Qe x 2 e p 2 )g s a12 + gs 1 (e gs 1)(1 e x 2 )a 21 + O(a 2 ), H(c 21 ) = (Qe x 1 e p 1 e gs e p 1 e p 2 )g s a21 + gs 1 ( (e g s (e gs 1) e 2gs (e gs 1)e x 1 )e p 1 e p 2 +(e gs 1)Qe x 1 e p 1 gs 2 ) a12 a21 a12 + O(a 2 ).

56 Legendrian SFT and HOMFLY After the change of variables, e x 1 = e g s e x 1, e p 1 = e g s e p 1 ; e x 2 = Q 1 e x 2, e p 2 = e g s Q 1 e p 2 ; Q = e gs Q, g s = g s, we find the D-module ideal generators Â 1 L = (ex 1 e x 2 ) + (e p 1 e p 2 ) Q(e x 1 e p 1 e x 2 e p 2 ) Â 2 L = (1 e gs e x 1 e p 1 + Qe x 1 e p 1 )(e x 1 e p 2 ) Â 3 L = (1 e gs e x 2 e p 2 + Qe x 2 e p 2 )(e x 2 e p 1 ), in agreement with HOMFLY.

57 Legendrian SFT and HOMFLY Similarly, for the trefoil we get the D-module ideal generator Â T = (Q 2 Qe p )e gs (Q e gs e 2p )Q 2 + ( (Q 2 Qe p )e 5gs (Q e gs e 2p )e 2p) e x + ( (qe 2p e p )e 3gs (Q e 3gs e 2p )Q 2) e x + ( e gs (Q e gs e 2p )(Q e 2gs e 2p )(Q e 3gs e 2p ) ) e x + ( (e gs e 2p e p )e 7gs (Q e 3gs e 2p )e 2p) e 2x. which after a change of variables is in agreement with HOMFLY.

58 Legendrian SFT and HOMFLY Similarly, for the trefoil we get the D-module ideal generator Â T = (Q 2 Qe p )e gs (Q e gs e 2p )Q 2 + ( (Q 2 Qe p )e 5gs (Q e gs e 2p )e 2p) e x + ( (qe 2p e p )e 3gs (Q e 3gs e 2p )Q 2) e x + ( e gs (Q e gs e 2p )(Q e 2gs e 2p )(Q e 3gs e 2p ) ) e x + ( (e gs e 2p e p )e 7gs (Q e 3gs e 2p )e 2p) e 2x. which after a change of variables is in agreement with HOMFLY. Note that the classical limit g s 0 of this is (Q e 2p )A T (e x, e p, Q).

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