Roadmap for Mirror Symmetry

Auroux's notes for a course on mirror symmetry at Berkeley.

They look interesting and they cover a lot of material.

Can't really say anything about the physics ... :-)

But for Kontsevich's HMS conjecture, my personal (very biased) all-time-favorite list is:

• Two great survey papers to start with are: "A beginner's introduction to Fukaya categories" and "A symplectic prolegomenon".
• Following that, there are Denis's notes from a graduate class he taught two times. Notes can be found in: 18.969 (MIT)/Math 277 (Berkeley).
• Another great source is Nick Sheridan's lectures at IAS and Jussieu summer school. They are extremely clear and also include some overview of the operadic background.
• James Pascaleff's class M392C (UT Austin) covers a lot of the basic aspects of Lagrangian Floer (co)homology (which is a neccesary component in defining the Fukaya category) beyond the usual 1-hour lecture. He also connects the actual theory with the standard heurestics ("Floer theory is Morse theory for the action functional!").
• Sheel's graduate class Math 257B has a really nice summary of the analysis and algebra needed to define a Fukaya category with an eye toward LG-models
• Seidel's notes from a topics class on equivariant mirror symmetry ("Lectures on Categorical Dynamics and Symplectic Topology") include a wealth of material on almost every subject you can imagine from Derived Picard groups, via Hochschild homology and cohomology, and to the actual setup of a Fukaya category for the surface case ... all in modular, concise (and dense) lectures.
• Hiro Lee Tanaka taught a graduate class on Fukaya categories called: Fukaya Categories, Sheaves, and Cosheaves. It includes some nice examples of HMS, and also discusses the bigger abstract framework (like TFT's, $(\infty,1)$-categories, CY-categories,...)
• For SYZ (besides Auroux's class), Siu Cheung Liu taught a couple of classes on mirror symmetry for an SYZ perspective a couple of years ago. The notes are on his website.
• A really nice book on derived categories is "Fourier-Mukai transforms in Algebraic Geometry". Sheel Gantara's thesis (more generally, the "Preliminaries" section in all the papers of Gantara-Perutz-Sheridan) provide a lot of background and references on $A_\infty$-algebra and noncommutative geometry with a mirror symmetry application in mind.

After that, one should probably start reading research papers (e.g. Paul's early papers like "A long exact sequence for symplectic Floer cohomology" and "Graded Lagrangian submanifolds" are a good source for the nuts and bolts of Floer theory, culminating in the definition of the Fukaya category in the exact setting).

To understand the Fukaya category for a closed, non-monotone, symplectic manfold you will probably need virtual perturbation techniques. But this is a different story altogether ....

For an understanding of the SYZ conjecture it is necessary to understand the framework of D-branes. I believe that a very good introduction, still unmatched in its comprehensiveness, is the book by Hori et al. on "Mirror Symmetry". This book has been made freely available by the Clay Math Institute and can be downloaded from their website http://www.claymath.org/library/monographs/cmim01.pdf. [ or from http://www.worldcat.org/title/mirror-symmetry/oclc/491393219 ] A more recent reference is the book by Joyce on "Riemannian Holonomy Groups and Calibrated Geometry", which contains a discussion of the SYZ conjecture from a more rigorous point of view.