Topological quantum field theory structure on symplectic cohomology

By Alexander F. Ritter. 2010

150 citations recorded by the source corpus.

We construct the topological quantum field theory (TQFT) on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and we prove that Viterbo restriction preserves the TQFT. This yields new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary ∂ M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian in ∂ M. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/20aefd552b2239872193833ee06c03a9c3194cdf

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Citations in the corpus

Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “One can embed this into a more general TQFT structure, of which we will now recall a version (related expositions are [36, 40, 32]).”
Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors (2017, background): “Note that this is the negative of the action used in [33].”
$S^1$-equivariant symplectic homology and linearized contact homology (2012, background): “Symplectic homology has the structure of a ring with unit [55, 47, 52], the unit being the image of 1 ∈ H(W ) under the map H(W ) ≃ H2n(W,∂W ) → SHn(W ).”
The monotone wrapped Fukaya category and the open-closed string map (2012, methodology): “We sketch the argument from [34] that the product μ(2) is unital.”
Symplectic cohomology and Viterbo's theorem (2013, methodology): “The construction also appears in [76], where it is observed that the same methods can be used to define operations parametrised by the homology of moduli spaces of curves of arbitrary genus (see also [68]).”
Symplectic homology and the Eilenberg-Steenrod axioms (2015, methodology): “3 in [65], except for the statement that involves SH∗(V, ∂V ), which is a consequence of Poincaré duality.”
The positive equivariant symplectic homology as an invariant for some contact manifolds (2015, background): “5 (Abouzaid, [22]) Let (W ′, ω′ = dλ′) be an exact symplectic manifold with contact type boundary ∂W ′, such that the Liouville vector field points inwards.”
Floer theory for negative line bundles via Gromov–Witten invariants (2011, methodology): “In [21], we constructed c as a count of spiked discs (a −∇H flowline from a critical point of H to the centre of a disc satisfying a Floer continuation equation).”