Homological mirror symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
In an address to the 1994 International Congress of Mathematicians in Zürich, Kontsevich (1994) speculated that mirror symmetry for a pair of Calabi–Yau manifolds X and Y could be explained as an equivalence of a triangulated category constructed from the algebraic geometry of X (the derived category of coherent sheaves on X) and another triangulated category constructed from the symplectic geometry of Y (the derived Fukaya category).
Edward Witten originally described the topological twisting of the N=(2,2) supersymmetric field theory into what he called the A and B model topological string theories. These models concern maps from Riemann surfaces into a fixed target—usually a Calabi–Yau manifold. Most of the mathematical predictions of mirror symmetry are embedded in the physical equivalence of the A-model on Y with the B-model on its mirror X. When the Riemann surfaces have empty boundary, they represent the worldsheets of closed strings. To cover the case of open strings, one must introduce boundary conditions to preserve the supersymmetry. In the A-model, these boundary conditions come in the form of Lagrangian submanifolds of Y with some additional structure (often called a brane structure). In the B-model, the boundary conditions come in the form of holomorphic (or algebraic) submanifolds of X with holomorphic (or algebraic) vector bundles on them. These are the objects one uses to build the relevant categories. They are often called A and B branes respectively. Morphisms in the categories are given by the massless spectrum of open strings stretching between two branes.
The closed string A and B models only capture the so-called topological sector—a small portion of the full string theory. Similarly, the branes in these models are only topological approximations to the full dynamical objects that are D-branes. Even so, the mathematics resulting from this small piece of string theory has been both deep and difficult.
Only in a few examples have mathematicians been able to verify the conjecture. In his seminal address, Kontsevich commented that the conjecture could be proved in the case of elliptic curves using theta functions. Following this route, Alexander Polishchuk and Eric Zaslow provided a proof of a version of the conjecture for elliptic curves. Kenji Fukaya was able to establish elements of the conjecture for abelian varieties. Later, Kontsevich and Yan Soibelman provided a proof of the majority of the conjecture for nonsingular torus bundles over affine manifolds using ideas from the SYZ conjecture. In 2003, Paul Seidel proved the conjecture in the case of the quartic surface. In 2002 Hausel & Thaddeus (2002) explained SYZ conjecture in the context of Hitchin system and Langlands duality.
The dimensions hp,q of spaces of harmonic (p,q)-differential forms (equivalently, the cohomology, i.e., closed forms modulo exact forms) are conventionally arranged in a diamond shape called the Hodge Diamond. For a three-dimensional manifold, for example, the Hodge diamond has p and q ranging from 0 to 3:
Mirror symmetry translates the dimension number of the (p, q)-th differential form hp,q for the original manifold into hn-p,q of that for the counter pair manifold. Namely, for any Calabi–Yau manifold the Hodge diamond is unchanged by a rotation by π radian and the Hodge diamonds of mirror Calabi–Yau manifolds are related by a rotation by π/2 radian.
In the case of an elliptic curve, which is viewed as a 1-dimensional Calabi–Yau manifold, the Hodge diamond is especially simple: it is the following figure.
In the 3-dimensional case, in usual called Calabi–Yau manifold, a very interesting thing happens. There are sometimes mirror pairs, say M and W, that have symmetric Hodge diamonds each other along diagonal straight line.
M and W correspond to A- and B-model in string theory. Mirror symmetry does not only replace the homological dimensions but also symplectic structure and complex structure on the mirror pairs. That is the origin of homological mirror symmetry.
In 1990-1991, Philip Candelas, Xenia C. de la Ossa, and Paul S. Green et al. (1991) had a major impact not only on enumetive algebraic geometry but on the whole mathematics and motivated Kontsevich (1994). The mirror pair of two quintic threefolds in this paper have the following Hodge diamonds.
- Topological quantum field theory
- Category theory
- Floer homology
- Fukaya category
- Derived category
- Quintic threefold
- Candelas, Philip; de la Ossa, Xenia C.; Green, Paul S.; Parkes, Linda (1991). "A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory". Nuclear Physics B 359 (1): 21–74. Bibcode:1991NuPhB.359...21C. doi:10.1016/0550-3213(91)90292-6. MR 1115626.
- Kontsevich, Maxim (1994). "Homological algebra of mirror symmetry". arXiv:alg-geom/9411018.
- Kontsevich, Maxim; Soibelman, Yan (2000). "Homological Mirror Symmetry and torus fibrations". arXiv:math.SG/0011041.
- Seidel, Paul (2003). "Homological mirror symmetry for the quartic surface". arXiv:math.SG/0310414.
- Hausel, Tamas; Thaddeus, Michael (2002). "Mirror symmetry, Langlands duality, and the Hitchin system". arXiv:math.DG/0205236.