Mirror symmetry of abelian varieties and multi-theta functions

By K. Fukaya. 2002

131 citations recorded by the source corpus.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/50e2531833f841bebf51cae6b4e0ac192779de6a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

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 [16, 23, 28], it has so far only been proved for elliptic curves [34], the quartic K3 surface [37], and their products [7].”
Homological mirror symmetry for the 4-torus (2009, unknown): “Amusingly, the argument relies only on the proof of Homological Mirror Symmetry for the elliptic curve; we need not appeal to the deeper work of Fukaya [ Fu02 ] or of Kontsevich-Soibelman [KS01], nor perform any (serious) Floer theoretic computation in T4.”
Homological mirror symmetry for the quartic surface (2003, methodology): “ubmanifolds are supposed to be transverse to the ﬁbres. Both restrictions are serious ones from the point of symplectic topology, and there has been some work on how to remove them, see in particular [15]. Actually, this entire circle of ideas is of little importance for our proof (although the SYZ ”
A∞-Structures on an Elliptic Curve (2000, unknown): “In general the equality of double products and triple Massey products in A∞-categories corresponding to a mirror dual pair (symplectic torus, abelian variety) was established by Fukaya in [3].”
The Symplectic Geometry of Cotangent Bundles from a Categorical Viewpoint (2007, background): “From the description we have just given, one can already infer one basic philosophical point, namely the interpretation of Lagrangian submanifolds in M as (some kind of) sheaves on the base Z. This can be viewed as a limit of standard ideas about Lagrangian torus fibrations in mirror symmetry [ 12 ,”
On the homological mirror symmetry conjecture for pairs of pants (2010, background): “Complete or partial results in this case are known for elliptic curves [43, 42], abelian varieties [20] (see [5] for the case of the four-torus), Strominger-Yau-Zaslow dual torus fibrations [32], and K3 surfaces [46].”
Yukawa couplings in intersecting D-brane models (2003, background): “The precise relation between the moduli space of these two set of objects is still an open problem, but in some simple cases as Lagrangian tori in M = T 2n they can be shown to be equal [56].”
Homological mirror symmetry for the genus two curve (2008, result): “There, we now have complete proofs of some cases [44, 49] and partial results for many more [32, 24].”
Homological mirror symmetry for Calabi–Yau hypersurfaces in projective space (2011, unknown): “Complete or partial proofs of homological mirror symmetry for closed CalabiYau varieties are known for elliptic curves [42, 40], abelian varieties [15] (see [2] for the case of the four-torus), Strominger-Yau-Zaslow dual torus fibrations [26], and the quartic K3 surface [44].”
Cyclic symmetry and adic convergence in Lagrangian Floer theory (2009, background): “(This project was started around 1998 in [5].”