An open string analogue of Viterbo functoriality

By M. Abouzaid, P. Seidel. 2007

328 citations recorded by the source corpus.

In this preprint, we look at exact Lagrangian submanifolds with Legendrian boundary inside a Liouville domain. The analogue of symplectic cohomology for such submanifolds is called “wrapped Floer cohomology”. We construct an A1 ‐structure on the underlying wrapped Floer complex, and (under suitable assumptions) an A1 ‐ homomorphism realizing the restriction to a Liouville subdomain. The construction of the A1 ‐structure relies on an implementation of homotopy direct limits, and involves some new moduli spaces which are solutions of generalized continuation map equations. 53D40

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/96e550471a17a31de40d3131b5a5d4c87f8c9707

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Citations in the corpus

Fukaya A∞-structures associated to Lefschetz fibrations. I (2009, neutral): “Similar asymmetries appear in other theories with non-compact target spaces, such as wrapped Floer cohomology [1] or string topology.”
Fukaya categories and bordered Heegaard-Floer homology (2010, methodology): “The objects L,L′, L′′ can be viewed as Lefschetz thimbles for the Lefschetz fibration induced by p on the symmetric product; in the corresponding Fukaya category, the statement that L′′ ≃ Cone(u) follows from a general result of Seidel [15, Proposition 18.23].”
Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry (2013, background): “y on [ω] +iB ∈H2(X,C/2πiZ). Under appropriate conditions one can deﬁne a version of F(X,ω,B) called wrapped Fukaya category in case when Xis non-compact and endowed with a proper map H: X→[0,+∞) (see [AbSe]). In that case Lagrangian submanifolds supporting objects are non-compact, having “good” beha”
Effect of Legendrian surgery (2009, methodology): “In the Appendix, written by M. Maydanskiy and S. Ganatra, this description is used to deduce P. Seidel’s conjectur [42] from the results of the current paper.”
Symplectic cohomology and Viterbo's theorem (2013, methodology): “For completeness, we mention the two other approaches: the first is called wrapped Floer cohomology, and can be obtained analogously to symplectic cohomology by taking a direct limit over Lagrangian Floer cohomology groups defined using Hamiltonians of increasing slope [11].”
Topological quantum field theory structure on symplectic cohomology (2010, background): “To prove compactness results, the following lemma will be useful (it is analogous to Abouzaid-Seidel [3, Lemma 7.2], adapted to our setup).”
Mirror symmetry for log Calabi-Yau surfaces I (2011, background): “instance of homological mirror symmetry, we expect: Conjecture 0.9. There is an equivalence of triangulated categories DπF wr(U) ≃D(X). 14 MARKGROSS,PAULHACKING,ANDSEANKEEL Remark 0.10. In [S08a] and [AS10] symplectic cohomology and the wrapped Fukaya category are deﬁned only in the exact case. Also”
A geometric criterion for generating the Fukaya category (2010, methodology): “The Lagrangians we allow are exact and are modelled after a Legendrian at infinity (see [ 3 ] and Section 2). In this paper, we define a map”
The wall-crossing formula and Lagrangian mutations (2017, methodology): “P jx ij2 denotes the standard Kahler potential on Cnso that ddc 0 = ! std. Fix a neighbourhood V of T[D. Consider an linearly interpolating family of symplectic forms on V, ! t= tddc + (1 t)ddc 0; t2[0;1]: We wish to apply Moser’s trick to this family intended to provide a symplectomorphism : (V;ddc”
Symplectic homology and the Eilenberg-Steenrod axioms (2015, methodology): “Then the tautological sequence yields the isomorphism SH>0∗ (L) ∼= SH ≤0 ∗−1(L) ∼= Hn−∗+1(L), which was previously conjectured by Seidel, see [36, Conjecture 1.2], and proved from a Legendrian contact homology perspective by Dimitroglou Rizell [31, Theorem 2.5].”
Bilinearised Legendrian contact homology and the augmentation category (2012, background): “This construction is related to the wrapped Fukaya category of [1].”
The monotone wrapped Fukaya category and the open-closed string map (2012, methodology): “A very careful treatment of signs in Floer theory is carried out in Seidel [41], Abouzaid-Seidel [8], Abouzaid [2], Ganatra [25] and Ritter [34].”
On the wrapped Fukaya category and based loops (2009, methodology): “clusion of the right hand side in Tw(W b(T Q)) and the functor F, we obtain the desired result. The reader may want to compare this circuitous construction of a restriction map with the one dened in [8] for an inclusion MinˆMof a Liouville subdomain. The key dierences 4 M. ABOUZAID are that the cons”
Duality between Lagrangian and Legendrian invariants (2017, background): “From this perspective, the case Γ + 6= ∅ is a new construction, that generalizes the theory from [11] in a way analogous to how partially wrapped Fukaya categories [63] generalizes wrapped Fukaya categories [3].”