Differential Geometry
Mirror Symmetry
Mirror symmetry united string physics and algebraic geometry into one of the greatest achievements of mathematical physics.
- Mirror symmetry predicts curve counts, physics helping pure mathematics.
- Applied in topological field theories and quantum gravity.
Предварительные знания
Calabi-Yau Mirror Symmetry
Mirror symmetry was first observed numerically in 1990 by Candelas, de la Ossa, Green, and Parkes when they predicted the count of rational curves on the quintic threefold (a 9-digit prediction confirmed by Givental in 1996). Kontsevich's homological mirror symmetry conjecture (1994) reframed the duality as an equivalence of derived categories, generating thirty years of research and an entire ICM plenary in 2018.
Mirror symmetry is a duality between pairs of Calabi-Yau threefolds M and W that swaps their Hodge numbers: h^{1,1}(M) = h^{2,1}(W). Its discovery transformed both mathematics and string theory.
What happens to the Hodge numbers when passing to the mirror manifold?
Gromov-Witten Invariants and the Quintic Prediction
Gromov-Witten invariants count the 'virtual number' of rational curves on a manifold. Mirror symmetry allowed Candelas-de la Ossa-Parks-Vafa to compute all n_d for the quintic using classical geometry of the mirror.
How did mirror symmetry enable the computation of Gromov-Witten invariants for the quintic?
Key Ideas
- A mirror pair (M,W) exchanges h^{1,1} and h^{2,1}.
- CDGP computed GW invariants of the quintic via its mirror (1991).
- SYZ conjecture explains mirror symmetry via T-duality.
Further Directions
The constructions studied here open paths to related areas of geometry.
- dg-28 — extends
Вопросы для размышления
- Why is mirror symmetry a genuine symmetry and not just a numerical coincidence?
- How does the SYZ conjecture geometrically explain the mirror exchange?