• Tags
  • #todo/untagged
• Refs:
  • #todo/add-references
• Links:
  • Fukaya category
  • derived category of coherent sheaves
  • Hodge theory MOC
  • Calabi-Yau, kahler,
  • Gromov-Witten invariants
  • symplectic geometry
  • SYZ
  • triangulated category
  • Gorenstein
  • del Pezzo
  • Log geometry
  • exceptional collection
  • McKay correspondence
  • Fano
  • Landau Ginzberg
  • vanishing cycles
  • del Pezzo
  • exceptional collection
  • invertible polynomial

Mirror Symmetry

Slogan: the symplectic geometry of a Calabi-Yau should have “the same” enmerative invariants as those in the complex-analytic geometry of its mirror. The homological mirror symmetry conjecture predicts a correspondence between the derived category of coherent sheaves of a variety and the symplectic data (packaged in the Fukaya category) of its mirror object.

Lagrangian submanifolds

For \(X\) a Calabi-Yau - A-side: the Fukaya category of \(X\) corresponds to \(A{\hbox{-}}\)branes on \(X\), so roughly Lagrangian submanifolds equipped with a flat bundle. - B-side: DCoh of \(X\) corresponds to \(B{\hbox{-}}\)branes on \(X\).

Other field theories replace \(X\) with a Fano.

For Fanos

Importance of toric structure:

Examples

Showing an congruence of Weil zeta functions for a K3