Homological mirror symmetry for the 4-torus

By M. Abouzaid, I. Smith. 2009

58 citations recorded by the source corpus.

We use the quilt formalism of Mau-Wehrheim-Woodward to give a sufficient condition for a finite collection of Lagrangian submanifolds to split-generate the Fukaya category, and deduce homological mirror symmetry for the standard 4-torus. As an application, we study Lagrangian genus two surfaces of Maslov class zero, deriving numerical restrictions on the intersections of such a surface with linear Lagrangian 2-tori in in the 4-torus.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/219dbb7aa58b95fb5d76f7d3c27c510c97dd7ced

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. D. Auroux
2. Nick Sheridan
3. L. Katzarkov
4. Yankı Lekili
5. M. Abouzaid
6. I. Smith
7. Alexander I. Efimov
8. Dmitri Orlov

In Geometry and Topology

1. D. Auroux
2. Nick Sheridan
3. I. Smith
4. M. Abouzaid
5. L. Katzarkov
6. P. Seidel
7. Yankı Lekili
8. J. Evans

In Mathematical Physics

1. D. Auroux
2. Nick Sheridan
3. I. Smith
4. M. Abouzaid
5. L. Katzarkov
6. Yankı Lekili
7. P. Seidel
8. J. Evans

Citations in the corpus

• Floer cohomology and pencils of quadrics (2010, methodology): “We will combine this general theory with an argument from [2], which uses properties of Yoneda embeddings to deduce that, in special cases, the Mau-Wehrheim-Woodward functor is actually fully faithful.”
• Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, background): “of coherent sheaves on Y (on the derived level). Complete or partial proofs of homological mirror symmetry for Calabi-Yau varieties are known for elliptic curves [10, 11], abelian varieties [12] (see [13] for the case of the four-torus), Strominger-Yau-Zaslow dual torus fibrations [14], and the quart”
• Generating the Fukaya categories of Hamiltonian 𝐺-manifolds (2015, background): “Finding a split-generator is also the first step in many modern proofs of homological mirror symmetry (for example [7, 57, 59]).”
• Homological mirror symmetry for punctured spheres (2011, background): “This conjecture has been studied extensively, and while evidence has been gathered in a number of examples including abelian varieties [15, 27, 22], it has so far only been proved for elliptic curves [33], the quartic K3 surface [36], and their products [7].”