'Mirror Symmetry' is the name for an exciting and fast-moving research area at the boundary between Mathematics (Differential and Algebraic Geometry) and Theoretical Physics (String Theory). It concerns Calabi-Yau 3-folds, six-dimensional spaces with a rich geometrical structure, which appear in String Theory as an ingredient of the space-time vacuum — different choices of Calabi-Yau 3-fold would lead to different spectra of particles and different physics in our four-dimensional universe. String Theorists believe that each Calabi-Yau 3-fold has an associated quantum theory, a Super Conformal Field Theory (SCFT). Mirror Symmetry is the idea that two different Calabi-Yau 3-folds M, M* can have 'the same' SCFT after a sign change. This sets up a mysterious, non-classical relationship between M and M*, and leads to mathematically extraordinary conjectures, which have been verified in many cases.
A (partial) geometrical explanation of Mirror Symmetry is Kontsevich's 'Homological Mirror Symmetry Conjecture', which says there should be an equivalence of triangulated categories between the derived category Dbcoh(M) of coherent sheaves on M (this depends on the complex algebraic geometry of M) and the derived Fukaya category DbF(M*) of Lagrangians on M* (this depends on the symplectic geometry of M*). It relates complex algebraic geometry and symplectic geometry, very different subjects, in surprisi ng ways.
Calabi-Yau manifolds and Homological Mirror Symmetry involve many different areas of Mathematics and Physics:
- Differential Geometry: a Calabi-Yau manifold is a smooth manifold M equipped with four compatible differential geometric structures: a complex structure J, a Riemannian metric g, a Kähler form ω, and a holomorphic volume form Ω.
- Algebraic Geometry: forgetting about g and ω, (M,J) is a complex manifold, and can be studied using complex algebraic geometry. The holomorphic volume form Ω, and coherent sheaves and their derived category Dbcoh(M), are also complex algebraic objects.
- Symplectic Geometry: forgetting about J, g and Ω, (M,ω) is a symplectic manifold, and can be studied using symplectic geometry. Lagrangian submanifolds and the derived Fukaya category DbF(M*) are also symplectic objects.
- Calibrated geometry: the calibrated submanifolds of Calabi-Yau manifolds, special Lagrangian submanifolds, are important in Mirror Symmetry, and are believed to represent 'semistable objects' in the derived Fukaya category.
- Analysis: examples of Calabi-Yau manifolds are constructed using Yau's solution of the Calabi Conjecture, an analytic result on solutions of a nonlinear partial differential equation. Special Lagrangian geometry also involves a lot of analysis.
- Category Theory and Homological Algebra: coherent sheaves coh (M) are an abelian category; the Fukaya category F(M*) is an A∞-category. These are very different structures. To make them equivalent one must apply a complicated process called deriving, which turns both into triangulated categories.
- String Theory: is how the whole mess got started in the first place.
This profusion of subjects makes Homological Mirror Symmetry at once a very exciting, and a very challenging area.
Disclaimer by Dominic Joyce: In October 2011, together with Kobi Kremnizer, Balázs Szendröi and Raphaël Rouquier, I started another research group on Geometry and Representation Theory. In case having two different research groups should appear greedy, I should point out that there is a substantial overlap between the two groups: anything I'm involved with that is vaguely to do with Donaldson-Thomas theory and algebraic geometry comes under both headings. Most of my energy will be going in 'Geometry and Representation Theory' directions for the next few years, but I am also collaborating on some projects in Symplectic Geometry wearing only my 'Homological Mirror Symmetry' hat.