I started writing this post this past weekend, but got stuck really quickly and then kept putting it off. I don’t want to leave anyone following this hanging with no idea what the A-model is. This is harder for me to describe than the A-model for some reason. Mostly I’m running into the problem of either just saying what the A-side is without explanation or I’m getting too bogged down in details. Both seem bad. In conclusion, I think I’ll err on the side of too few details, and then hopefully make sense of what is going on by completely describing mirror symmetry in the easiest case possible: the one dimensional case, i.e. for an elliptic curve.

I’m going to semi-cheat right off and refer to posts over a year old. Recall what a symplectic form is on a smooth manifold is. It is just a closed non-degenerate 2-form. A smooth manifold plus symplectic form is called a symplectic manifold. The cotangent bundle always has a canonical symplectic form on it. An example that may be less well-known is that any smooth complex projective variety is symplectic because the Fubini-Study Kähler form on restricts to a symplectic form.

If we just think about vector spaces for a second, then given a symplectic form, we say that a subspace is isotropic if and coisotropic if . The subspace is Lagrangian if it is both isotropic and coisotropic. This extends to manifold language easily by saying an embedded submanifold is Lagrangian if the tangent subspace is Lagrangian for every point of . If you want to get used to these definitions, a quick exercise would be to check that the zero section of the cotangent bundle is Lagrangian with respect to the canonical symplectic structure.

My second semi-cheat is to ask you to recall the definition of an almost complex structure from close to two years ago. The way to think about it is that it is a bundle map that behaves similarly to “multiplication by “. The condition is that , and indeed multiplication by when identifying gives an example of an almost complex structure. In fact, since we’ll always work over , any complex manifold does have multiplication by as a natural almost complex structure.

It is possible that all these things are related by the following. Suppose is a symplectic manifold, an almost complex structre, and a Riemannian metric. These three structures are called compatible if . I am far out of my depth here, but I’m pretty sure such a manifold is called Kähler if this happens, but maybe some slight more conditions are needed (e.g. does this automatically imply that is Hermitian? If so, then this is definitely what people call Kähler).

Now for the definition of the A-model. Let be a Kähler (in the sense of the previous paragraph) manifold. We define the Fukaya category to have as objects the Lagrangian submanifolds. The morphisms require a bit of technicality to define, but essentially are a way to intersect the submanifolds. It involves all the structures above and is called Floer cohomology. Recall that we’re merely sketching an idea here! Somehow this should be an or dg-category if you remember from last time, and this just comes from the fact that the morphisms have to do with cohomology classes of intersections.

If you’ve been following this at all, then you should be in utter amazement. We can state mirror symmetry now as an equivalence of categories where is a Calabi-Yau. Why is this amazing (for those not following along)? Look at the left side of this equivalence. The bounded derived category of coherent sheaves (in the Zariski topology!!) on is something that has to do purely with the algebraic data of . I mean, the Zariski topology is algebraic, the definition of coherent is very algebraic, the construction of the derived category is algebraic, etc.

The right hand side seems to have forgotten all of the algebraic data. You forget that it is a variety and instead think of it as a smooth manifold. You consider a bunch of structure that helps you study the smooth structure. You consider Lagrangian submanifolds. The Fukaya category is almost entirely analytic in nature. But now the conjecture of Kontsevich mirror symmetry is that the two are always equivalent. That’s it for today. There should be one more post in this series in which I try to sketch the conjecture in the case of an elliptic curve.

March 31, 2012 at 6:57 pm

To be Kahler, you need to add that the almost-complex structure is complex. But the Fukaya category is defined for almost-Kahler manifolds (modulo hedging about picking the right almost-complex structure) so the typos cancel out.

By the way… I wonder whether there’s any connection at all between homological mirror symmetry and GAGA. I mean, both are about relating an algebraic thing with an analytic thing, and both HMS and GAGA say something about (in particular) complex projective varieties.

April 2, 2012 at 1:18 pm

Thanks! I’m not sure what is meant by “add that the almost-complex structure is complex.” I hunted around for a definition, but couldn’t find one. Does this just mean X is a complex manifold rather than a manifold with almost-complex structure? It isn’t clear from what I wrote, but I’m assuming we are starting with a complex manifold.

The GAGA thing is a great point. I guess one could try to argue that HMS is not a surprising conjecture from that point of view. GAGA says something like the category of coherent sheaves in Zariski is equivalent to the category of coherent sheaves analytically. So one could re-interpret the “algebraic side” of HMS almost entirely analytically through GAGA and then the statement says that two analytic things are related when you are in this really nice situation where all the structures play nice with each other.

April 2, 2012 at 1:37 pm

Yeah, that’s what I meant, that you need to assume that the manifold M is complex, and that the complex structure J on the tangent bundle actually comes from the complex structure of M. I guess you actually were suggesting that hypothesis in the last sentence of the fourth paragraph.