Fukaya categories and deformations

By P. Seidel. 2002

130 citations recorded by the source corpus.

This is an informal (and mostly conjectural) discussion of some aspects of Fukaya categories. We start by looking at exact symplectic manifolds which are obtained from a closed Calabi-Yau by removing a hyperplane section. We look at the possible geometric significance of Hochschild cohomology in this situation, and how one can try to get from the Fukaya category of the exact manifold to that of the closed Calabi-Yau. Also included is a brief discussion of the role of Lefschetz pencils, and a bit of general deformation theory. To appear in the Proceedings of the Beijing ICM.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f63d9cb42afa15d9889500a8d937bfba5d42cc1

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Citations in the corpus

The monotone wrapped Fukaya category and the open-closed string map (2012, unknown): “he cohomological units are eL= CO 0([E])(1) ∈CW∗(L,L) where CO0(·)(1) : QC ∗(E) →CW∗(L,L). Moreover CO([E]) ∈HH (W(E)) is a unit for the cup product. Remark 1.7. String maps were introduced by Seidel [39]. In the closed monotone case, Albers [11] deﬁned maps τ: HF∗(H) →HF∗(L,ϕ1 H(L)) and χ: HF∗(L,ϕ1”
Symplectic Cohomology and q-Intersection Numbers (2010, methodology): “There is a natural map from SH (M) to the Hochschild cohomology of the Fukaya A∞-category [34, 2].”
A biased view of symplectic cohomology (2007, methodology): “One way of formulating this connection is to say that there is a natural map (of commutative rings; actually, even of BV algebras) from symplectic cohomology to the Hochschild cohomology of the Fukaya category [47].”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “These results owe a great deal to him, both because his previous work [43, 44, 45, 46] laid the foundations for them, and because he gave me a lot of guidance and made many useful suggestions along the way.”
A survey of Heegaard Floer homology (2013, unknown): “ U, and deform the Floer complex according to the intersection number with Vz. There are several diﬀerent ways of doing this, giving rise to the diﬀerent ﬂavors of Heegaard Floer homology. Cf. Seidel [90], who ﬁrst introduced such an idea in a diﬀerent context. Without ﬁxing the basepoint, the invar”
Symplectic cohomology and Viterbo's theorem (2013, background): “ holomorphic discs with an interior puncture. This idea has appeared in various incarnations so many times in the literature that we shall necessarily sin by omission in listing potential references: [8,12,39,75]. There are not, as of yet, generalisations of the map Fbeyond cotangent bundles, so it ”
Homological mirror symmetry for the quartic surface (2003, background): “the deformation parameter qis set to zero, while tensoring with ΛQ gives a full subcategory of F(X) (for the deﬁnition of F(X) in higher dimensions, see [18]; and for its presumed relation with F(M), [54]). We emphasize that all properties of Fukaya categories which we use are quite simple ones, and”
Mirror symmetry for log Calabi-Yau surfaces I (2011, background): “from the symplectic cohomology of U to the Hochschild cohomology of its wrapped Fukaya category, which is conjectured to be an isomorphism (under slightly more restrictive conditions than considered here), see [S02], Conjecture 4.”
Topological conformal field theories and Calabi–Yau categories (2004, background): “A CY category C of dimension d (over our base field K) is a linear category with a trace map TrA : Hom(A,A)→ K[d] for each object A of C.”
A-infinity algebras, modules and functor categories (2005, background): “nce of A-inﬁnity structures in geometry and physics became apparent through the work of Getzler-Jones [12], Stasheﬀ [39], Fukaya [9], Kontsevich [21], ..., and later Kontsevich-Soibelman [24], Seidel [36], .... We refer to [19], [20] for a more detailed introduction with numerous references. The pre”
Bordered Heegaard Floer homology (2008, background): “Note that uivi+1−ui+1vi can be interpreted as the determinant of a 2×2 submatrix of the matrix ( u1 u2 ··· un v1 v2 ··· vn ) .”
A Beginner’s Introduction to Fukaya Categories (2013, methodology): “ M0 of the given symplectic manifold M obtained by deleting some complex hypersurface, and then use abstract deformation theory to view the Fukaya category of Mas an A∞-deformation of that of M0 (cf. [41]). The Hochschild cohomology class that determines the deformation is then often determined by s”