Calabi-Yau

• Tags
  • #AG #AG/deformation-theory
• Refs:
  • #todo/add-references
• Links:
  • Enriques-Kodaira Classification
  • canonical bundle

Calabi-Yau

• Various definitions:
  • A compact Kahler manifold with vanishing first Chern class \(c_1 = 0\) which is Ricci-flat.
  • A smooth proper variety $X\in {\mathsf{Var}}_{/ {k}} $ with trivial canonical bundle, so \(\omega_X \coloneqq\bigwedge\nolimits^{\dim X}\Omega^1_{X_{/ {k}} } \cong{\mathcal{O}}_X\). When \(k={\mathbf{C}}\), the trivialization must be holomorphic and not just topological!
  • A riemannian manifold \(X\) of even real dimension \(\dim_{\mathbf{R}}(X) = 2n\) with Holonomy \({\mathrm{holon}}(X) \subseteq {\operatorname{SU}}_n \subset {\operatorname{O}}_{2n}({\mathbf{R}})\).

Motivations

• Setting for Unsorted/mirror symmetry : the symplectic geometry of a Calabi-Yau is “the same” as the complex geometry of its mirror.

• Applications: Physicists want to study \(G_2\) manifolds (an exceptional Lie group, automorphisms of octonions), part of \(M{\hbox{-}}\)theory uniting several superstring theories, but no smooth or complex structures.

  • Indirect approach: compactify add one small \(S^1\) dimension to a 10 dimensional space to compactify, yielding an 11 dimensional space.
    • Why 10 dimensions: 4 from spacetime and 6 from a “small” Calabi-Yau threefold.

• Yau, Fields Medal 1982: There are Ricci-flat but non-flat (nontrivial holonomy) projective complex manifolds of dimensions \(\geq 2\).

Examples/Classification

• Examples:
  • \(\dim X = 1\): Elliptic Curves.
  • \(\dim X = 2\): K3 surfaces.
• Compact classification for \({\mathbf{C}}{\hbox{-}}\)dimension:
  • Dimension 1: 1 type, all elliptic curves (up to homeomorphism)
  • Dimension 2: 1 type, K3 Surface
  • Dimension 3: (threefolds) #open/conjectures that there is a bounded number, but unknown. At least 473,800,776!

Appearance in mirror symmetry: