Symplectic cohomology and duality for the wrapped Fukaya category

By Sheel Ganatra. 2013

148 citations recorded by the source corpus.

Consider the wrapped Fukaya category W of a collection of exact Lagrangians in a Liouville manifold. Under a non-degeneracy condition implying the existence of enough Lagrangians, we show that natural geometric maps from the Hochschild homology of W to symplectic cohomology and from symplectic cohomology to the Hochschild cohomology of W are isomorphisms, in a manner compatible with ring and module structures. This is a consequence of a more general duality for the wrapped Fukaya category, which should be thought of as a non-compact version of a Calabi-Yau structure. The new ingredients are: (1) Fourier-Mukai theory for W via a wrapped version of holomorphic quilts, (2) new geometric operations, coming from discs with two negative punctures and arbitrary many positive punctures, (3) a generalization of the Cardy condition, and (4) the use of homotopy units and A-infinity shuffle products to relate non-degeneracy to a resolution of the diagonal.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7272cec79917691409b96adb86579de22b4fed37

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Citations in the corpus

The monotone wrapped Fukaya category and the open-closed string map (2012, methodology): “Abouzaid [2] proved the generation criterion in the exact setup using quadratic Hamiltonians, and Ganatra uses the same setup in his discussion [25] of the module structure of the string maps.”
Fukaya A∞-structures Associated to Lefschetz Fibrations. II (2014, background): “Open-closed string maps [38, 1, 18] establish a relationship between the two flavours.”
Homological mirror symmetry for punctured spheres (2011, background): “This Fukaya category, whose objects also include non-compact Lagrangian submanifolds, more accurately reflects the symplectic geometry of open symplectic manifolds, and by recent work [2, 19], is known in some generality to be homologically smooth in the sense of Kontsevich [29] (homological smoothn”
On partially wrapped Fukaya categories (2016, methodology): “principle does not apply in the presence of a timedependent perturbing Hamiltonian, but by choosing a symplectization coordinate r= p Hewe nd ourselves in Ganatra’s setup and can apply Theorem A.1 of [19]. Fix J t2JS 1 reg(M;H t). The dierential @on SC(M) is given by @x + = X deg(x ) deg(x+)=1 #Q(x ”
Mirror symmetry: from categories to curve counts (2015, methodology): “We use without detailed exposition the dg category [ C , C ] of A ∞ C C bimodules, for which there are now many references (see e.g., [Sei08a, Tra08, Gan13, She15a]).”
A Beginner’s Introduction to Fukaya Categories (2013, background): “(One guiding principle which might explain why this approach is so successful is that algebraic deformations of Fukaya categories are often geometric: natural “closed-open” maps from the quantum or symplectic cohomology ofM to the Hochschild cohomology of its ordinary or wrapped Fukaya category ofte”
Covariantly functorial wrapped Floer theory on Liouville sectors (2017, methodology): “ Y 0;:::;Y k2C P(X;Y 0) C(Y 0;Y 1)[1] C(Y k 1;Y k)[1] Q(Y k;Z) (3.8) with dierential and bimodule maps given by summing over ways to contract tensor chains by the A 1and bimodule structure maps, see [48,25] for more details. Note that this construction preserves cobrancy (i.e. if the morphism spaces”
On the Fukaya category of a Fano hypersurface in projective space (2013, background): “[14] Sheel Ganatra.”
Towards Mirror Symmetry for Varieties of General Type (2012, result): “and the middle one is the quantum cohomology of S. These are conjectured to be isomorphisms under certain conditions, see [Ko94], [AFOO] and for references with SH in place of QH see [Sei07], [Ab10], [Ga13]. For the following considerations, let us assume 2This notion was introduced by Kontsevich an”