Symplectic Field Theory Research Articles

Coherent orientations in symplectic field theory revisited

Coherent orientations in symplectic field theory revisited

On doubly symmetric periodic orbits

In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.e., those which are symmetric with respect to two (distinct) commuting antisymplectic involutions. These are ubiquitous in several problems of interest in mechanics. We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) All covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory (Eliashberg et al. Geom Funct Anal Special Volume Part II:560–673, 2000); (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined in Frauenfelder et al. in Symplectic methods in the numerical search of orbits in real-life planetary systems. Preprint arXiv:2206.00627). The above results follow from: (5) A symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in Frauenfelder and Moreno 2021) differ.

An elementary alternative to ECH capacities

The embedded contact homology (ECH) capacities are a sequence of numerical invariants of symplectic four-manifolds that give (sometimes sharp) obstructions to symplectic embeddings. These capacities are defined using embedded contact homology, and establishing their basic properties currently requires Seiberg-Witten theory. In this paper we define a sequence of symplectic capacities in four dimensions using only basic notions of holomorphic curves. The capacities satisfy the same basic properties as ECH capacities and agree with the ECH capacities for the main examples for which the latter have been computed, namely convex and concave toric domains. The capacities are also useful for obstructing symplectic embeddings into closed symplectic four-manifolds. This work is inspired by a recent preprint of McDuff and Siegel [D. McDuff, K. Siegel, arXiv [Preprint] (2021)], giving a similar elementary alternative to symplectic capacities from rational symplectic field theory (SFT).

Cylindrical contact homology of 3–dimensional Brieskorn manifolds

Cylindrical contact homology is a comparatively simple incarnation of symplectic field theory whose existence and invariance under suitable hypotheses was recently established by Hutchings and Nelson. We study this invariant for a general Brieskorn 3-manifold $\Sigma(a_1,\ldots, a_n)$, and give a complete description of the cylindrical contact homology for this 3-manifold equipped with its natural contact structure, for any $a_j$ satisfying $\frac{1}{a_1} + \cdots + \frac{1}{a_n} < n-2$.

A Symplectic Dynamics Proof of the Degree–Genus Formula

We classify global surfaces of section for the Reeb flow of the standard contact form on the 3-sphere (defining the Hopf fibration), with boundaries oriented positively by the flow. As an application, we prove the degree-genus formula for complex projective curves, using an elementary degeneration process inspired by the language of holomorphic buildings in symplectic field theory.

A polyfold proof of the Arnold conjecture

We give a detailed proof of the homological Arnold conjecture for nondegenerate periodic Hamiltonians on general closed symplectic manifolds M via a direct Piunikhin–Salamon–Schwarz morphism. Our constructions are based on a coherent polyfold description for moduli spaces of pseudoholomorphic curves in a family of symplectic manifolds degenerating from {{mathbb {C}}{mathbb {P}}}^1times M to {{mathbb {C}}}^+ times M and {{mathbb {C}}}^-times M, as developed by Fish–Hofer–Wysocki–Zehnder as part of the Symplectic Field Theory package. To make the paper self-contained we include all polyfold assumptions, describe the coherent perturbation iteration in detail, and prove an abstract regularization theorem for moduli spaces with evaluation maps relative to a countable collection of submanifolds. The 2011 sketch of this proof was joint work with Peter Albers, Joel Fish.

Smooth Kuranishi structure of the space of Morse trajectories

Our aim here is to explain a new technique for the construction of a smooth Kuranishi structure, which we used in our previous article about the construction of symplectic field theory. We explain this technique in the case of a Morse chain complex.

SFT COMPUTATIONS AND INTERSECTION THEORY IN HIGHER-DIMENSIONAL CONTACT MANIFOLDS

AbstractI construct infinitely many nondiffeomorphic examples of $5$ -dimensional contact manifolds which are tight, admit no strong fillings and do not have Giroux torsion. I obtain obstruction results for symplectic cobordisms, for which I give a proof not relying on the polyfold abstract perturbation scheme for Symplectic Field Theory (SFT). These results are part of my PhD thesis [23], and are the first applications of higher-dimensional Siefring intersection theory for holomorphic curves and hypersurfaces, as outlined in [23, 24], as a prequel to [30].

Homological algebra related to surfaces with boundary

In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus.

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.

Automatic transversality in contact homology II: filtrations and computations

This paper is the sequel to the previous paper [Nelson, Abh. Math. Semin. Univ. Hambg. 85 (2015) 125–179], which showed that sufficient regularity exists to define cylindrical contact homology in dimension three for nondegenerate dynamically separated contact forms, a subclass of dynamically convex contact forms. The Reeb orbits of these so-called dynamically separated contact forms satisfy a uniform growth condition on their Conley–Zehnder indices with respect to a free homotopy class. Given a contact form which is dynamically separated up to large action, we demonstrate a filtration by action on the chain complex and show how to obtain the desired cylindrical contact homology by taking direct limits. We give a direct proof of invariance of cylindrical contact homology within the class of dynamically separated contact forms, and elucidate the independence of the filtered cylindrical contact homology with respect to the choice of the dynamically separated contact form and almost complex structure. We also show that these regularity results are compatible with geometric methods of computing cylindrical contact homology of prequantization bundles, proving a conjecture of Eliashberg [Symplectic field theory and its applications, International Congress of Mathematicians I (European Mathematical Society, Zürich, 2007) 217–246] in dimension three.

Symplectic Field Theory of the Galilean Covariant Scalar and Spinor Representations

We explore the concept of the extended Galilei group, a representation for the symplectic quantum mechanics in the manifold G, written in the light-cone of a five-dimensional de Sitter space-time in the phase space. The Hilbert space is constructed endowed with a symplectic structure. We study the unitary operators describing rotations and translations, whose generators satisfy the Lie algebra of G. This representation gives rise to the Schr¨odinger (Klein–Gordon-like) equation for the wave function in the phase space such that the dependent variables have the position and linear momentum contents. The wave functions are associated to the Wigner function through the Moyal product such that the wave functions represent a quasiamplitude of probability. We construct the Pauli–Schr¨odinger (Dirac-like) equation in the phase space in its explicitly covariant form. Finally, we show the equivalence between the five-dimensional formalism of the phase space with the usual formalism, proposing a solution that recovers the non-covariant form of the Pauli–Schr¨odinger equation in the phase space.

Nearby Lagrangian fibers and Whitney sphere links

Let $n&gt;3$, and let $L$ be a Lagrangian embedding of $\mathbb{R}^{n}$ into the cotangent bundle $T^{\ast }\mathbb{R}^{n}$ of $\mathbb{R}^{n}$ that agrees with the cotangent fiber $T_{x}^{\ast }\mathbb{R}^{n}$ over a point $x\neq 0$ 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 $S^{n}$ into $\mathbb{R}^{n}\setminus \{0\}$. We show that this map is homotopically trivial, answering a question of 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 $T^{\ast }\mathbb{R}^{n}$, under suitable dimension and Maslov index hypotheses. The proofs combine techniques from Ekholm and Smith [Exact Lagrangian immersions with a single double point, J. Amer. Math. Soc. 29 (2016), 1–59] and Ekholm and Smith [Exact Lagrangian immersions with one double point revisited, Math. Ann. 358 (2014), 195–240] with symplectic field theory.

Applications of Polyfold Theory I: The Polyfolds of Gromov–Witten Theory

In this paper we start with the applications of polyfold theory to symplectic field theory.

ANALYSIS OF CONTACT CAUCHY–RIEMANN MAPS II: CANONICAL NEIGHBORHOODS AND EXPONENTIAL CONVERGENCE FOR THE MORSE–BOTT CASE

This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone $S^2 \times S^2$. The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for $T^*\mathbb{T}^2$, i.e.~it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

Infinitely many exotic monotone Lagrangian tori in ℂℙ2

Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple - positive integers satisfying a^2 + b^2 + c^2 = 3abc - there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.

Legendrian knots and exact Lagrangian cobordisms

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair (X,L) consisting of an exact symplectic manifold X and an exact Lagrangian cobordism L\subset X which agrees with cylinders over Legendrian links \Lambda_+ and \Lambda_- at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of \Lambda_+ to that of \Lambda_- . We give a gradient flow tree description of the DGA maps for certain pairs (X,L) , which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.

Symplectic Field Theory of a Disk, Quantum Integrable Systems, and Schur Polynomials

We consider commuting operators obtained by quantization of Hamiltonians of the Hopf (aka dispersionless KdV) hierarchy. Such operators naturally arise in the setting of Symplectic Field Theory (SFT). A complete set of common eigenvectors of these operators is given by Schur polynomials. We use this result for computing the SFT potential of a disk.

OnJ-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends

Symplectic Field Theory studies J-holomorphic curves in almost complex manifolds with cylindrical ends. One natural generalization is to replace 'cylindrical' by 'asymptotically cylindrical'. In this article, we generalize the asymptotic results about the behavior of J-holomorphic curves near infinity to the asymptotically cylindrical setting. We also sketch how these asymptotic results allow the main compactness theorems in Symplectic Field Theory proved by Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder to be extended to the asymptotically cylindrical case.