arXiv subjects

Tobias Ekholm

Publications and source records attributed to Tobias Ekholm.

At least 37 records · Page 2Linked to original sources

arXiv2020

Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities

The relation between open topological strings and representation theory of symmetric quivers is explored beyond the original setting of the knot-quiver correspondence. Multiple cover generalizations of the skein relation for boundaries of holomorphic disks on a Lagrangian brane are observed to generate dual quiver descriptions of the geometry. Embedding into M-theory, a large class of dualities of 3d $\mathcal{N}=2$ theories associated to quivers is obtained. The multi-cover skein relation admits a compact formulation in terms of quantum torus algebras associated to the quiver and in this language the relations are similar to wall-crossing identities of Kontsevich and Soibelman.

hep-th↗

arXiv2020

Higher genus knot contact homology and recursion for colored HOMFLY-PT polynomials

We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal tori of knots and links. Using large $N$ duality and Witten's connection between open Gromov-Witten invariants and Chern-Simons gauge theory, we relate the SFT of a link conormal to the colored HOMFLY-PT polynomials of the link. We present an argument that the HOMFLY-PT wave function is determined from SFT by induction on Euler characteristic, and also show how to, more directly, extract its recursion relation by elimination theory applied to finitely many noncommutative equations. The latter can be viewed as the higher genus counterpart of the relation between the augmentation variety and Gromov-Witten disk potentials established by Aganagic, Vafa, and the authors, and, from this perspective, our results can be seen as an SFT approach to quantizing the augmentation variety.

math.SG↗

arXiv2019

Holomorphic curves for Legendrian surgery

Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $Λ\subset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $Λ$ we get a Weinstein cobordism $X_Λ$ with a collection of Lagrangian co-core disks $C$ corresponding to $Λ$. In \cite{BEE, EL} it was shown that the wrapped Floer cohomology $CW^{\ast}(C)$ of $C$ in the Weinstein manifold $X'_Λ=X\cup X_Λ$is naturally isomorphic to the Legendrian differential graded algebra $CE^{\ast}(Λ)$ of $Λ$ in $Y$. The argument uses properties of moduli spaces of holomorphic curves, the proofs of which were only sketched. The purpose of this paper is to provide proofs of these properties.

math.SG↗

arXiv2018

Lagrangian fillings and complicated Legendrian unknots

An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $Σ$ can be glued to itself along $Σ$. This gives a Legendrian embedding $Λ(L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $Λ(L,L)$ is determined by formal data: the manifold $L$ together with a trivialization of its complexified tangent bundle. In particular, if $L$ is a disk then $Λ(L,L)$ is the Legendrian unknot.

math.SG↗

arXiv2018

Non-loose Legendrian spheres with trivial Contact Homology DGA

Loose Legendrian n-submanifolds, for n at least 2, were introduced by Murphy and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial.

math.SG↗

arXiv2017

Knot contact homology and open Gromov-Witten theory

Knot contact homology studies symplectic and contact geometric properties of conormals of knots in 3-manifolds using holomorphic curve techniques. It has connections to both mathematical and physical theories. On the mathematical side, we review the theory, show that it gives a complete knot invariant, and discuss its connections to Fukaya categories, string topology, and micro-local sheaves. On the physical side, we describe the connection between the augmentation variety of knot contact homology and Gromov-Witten disk potentials, and discuss the corresponding higher genus relation that quantizes the augmentation variety.

math.SG↗

arXiv2017

Nearby Lagrangian fibers and Whitney sphere links

Let n>3, and let L be a Lagrangian embedding of an n-disk into the cotangent bundle of n-dimensional Euclidean space that agrees with the cotangent fiber over a non-zero point x outside a compact set. Assume that L is disjoint from the cotangent fiber at the origin. The projection of L to the base extends to a map of the n-sphere into the complement of the origin in Euclidean n-space . We show that this map is homotopically trivial, answering a question of Y. Eliashberg. We give a number of generalizations of this result, including homotopical constraints on embedded Lagrangian disks in the complement of another Lagrangian submanifold, and on two-component links of immersed Lagrangian spheres with one double point in 2n-dimensional space, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from the authors' previous work, constructing bounding manifolds from moduli spaces of Floer-holomorphic disks, with symplectic field theory.

math.SG↗

arXiv2017

A complete knot invariant from contact homology

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphic-curve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.

math.SG↗

arXiv2017

Knot contact homology, string topology, and the cord algebra

The conormal Lagrangian $L_K$ of a knot $K$ in $\mathbb{R}^3$ is the submanifold of the cotangent bundle $T^* \mathbb{R}^3$ consisting of covectors along $K$ that annihilate tangent vectors to $K$. By intersecting with the unit cotangent bundle $S^* \mathbb{R}^3$, one obtains the unit conormal $Λ_K$, and the Legendrian contact homology of $Λ_K$ is a knot invariant of $K$, known as knot contact homology. We define a version of string topology for strings in $\mathbb{R}^3 \cup L_K$ and prove that this is isomorphic in degree 0 to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in $T^* \mathbb{R}^3$ with boundary on $\mathbb{R}^3 \cup L_K$.

math.SG↗

arXiv2015

Lagrangian exotic spheres

Let k>2. We prove that the cotangent bundles of oriented homotopy (2k-1)-spheres S and S' are symplectomorphic only if the classes defined by S and S' agree up to sign in the quotient group of oriented homotopy spheres modulo those which bound parallelizable manifolds. We also show that if the connect sum of real projective space of dimension (4k-1) and a homotopy (4k-1)-sphere admits a Lagrangian embedding in complex projective space, then twice the homotopy sphere framed bounds. The proofs build on previous work of Abouzaid and the authors, in combination with a new cut-and-paste argument, which also gives rise to some interesting explicit exact Lagrangian embeddings into plumbings. As another application, we show that there are re-parameterizations of the zero-section in the cotangent bundle of a sphere which are not Hamiltonian isotopic (as maps, rather than as submanifolds) to the original zero-section.

math.SG↗

arXiv2015

Symplectic and contact differential graded algebras

We define Hamiltonian simplex differential graded algebras (DGA) with differentials that deform the high energy symplectic homology differential and wrapped Floer homology differential in the cases of closed and open strings in a Weinstein manifold, respectively. The order $m$ term in the differential is induced by varying natural degree $m$ co-products over an $(m-1)$-simplex, where the operations near the boundary of the simplex are trivial. We show that the Hamiltonian simplex DGA is quasi-isomorphic to the (non-equivariant) contact homology algebra and to the Legendrian homology algebra of the ideal boundary in the closed and open string cases, respectively.

math.SG↗

arXiv2015

Symplectic homology product via Legendrian surgery

This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken in [BEE1] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the attaching spheres of critical handles. Here we express the product and $\BV$-operator of symplectic homology in that context.

math.SG↗

arXiv2014

Exact Lagrangian immersions with a single double point

Let k > 2. We show that if a closed orientable 2k-manifold K, with Euler characteristic not equal to -2, admits an exact Lagrangian immersion into complex Euclidean 2k-space with one transverse double point and no other self-intersections, then K is diffeomorphic to the standard sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with boundary in a monotone Lagrangian submanifold obtained by Lagrange surgery on K.

math.SG↗

arXiv2014

Legendrian contact homology in the boundary of a subcritical Weinstein 4-manifold

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplectic homology, this gives a combinatorial description of the symplectic homology of any 4-dimensional Weinstein manifold (and of the linearized contact homology of its boundary). We also study examples and discuss the invariance of the Legendrian homology algebra under deformations, from both the combinatorial and the analytical perspectives.

math.SG↗

arXiv2013

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.

hep-th↗

arXiv2013

Notes on topological strings and knot contact homology

We give an introduction to the physics and mathematics involved in the recently observed relation between topological string theory and knot contact homology and then discuss this relation. The note is based on two lectures given at the Gokova Geometry and Topology Conference, 2013 and reports on joint work by Aganagic, Ng, Vafa, and the author.

math.SG↗

arXiv2013

Exact Lagrangian immersions with one double point revisited

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling of K obtained from solutions to perturbed Cauchy-Riemann equations with boundary on the image f(K) of the immersion. This leads to a new and simplified proof of some of the main results of arXiv:1111.5932, which treated Lagrangian immersions in the case n=2k by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.

math.SG↗

arXiv2013

Constructing exact Lagrangian immersions with few double points

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space $\R^6_\st$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to\R^6_\st$ with vanishing Maslov class.

math.SG↗