Matrix factorizations and mirror symmetry

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\underline{Matrix factorizations and mirror symmetry}
 
The purpose of this talk is to introduce matrix factorizations (``sheaves
with
a potential''). These form categories which have general properties
analogous
to Fukaya categories, and indeed appear as mirrors in the non-Calabi-Yau
case. A good informal reference is [Kapustin-Li, D-branes in LG models and
algebraic geometry]. The most systematic approach 
is in [Orlov, Triangulated categories of singularities and D-branes in 
Landau-Ginzburg models]. A good exposition would cover both definitions,
since
each has its advantages. To read Orlov, one needs to know about
triangulated
categories at the level of [Gelfand-Manin, homological algebra].
 
\underline{Plan}
 
{\it Definition.} Matrix factorizations for a potential $W$. They form
a differential graded category, and the resulting homotopy category has
mapping cones (technically, is triangulated). Older literature has a 
more commutative algebra viewpoint, going back to [Eisenbud, Homological
algebra on a complete intersection, with an application to group
representations]
 
{\it Examples.} The nondegenerate potential $W = x_1^2 + \cdots + x_n^2$
is treated in [Kapustin-Li, op. cit.]. The relevant thing for us that
there
is a single basic matrix factorization, whose endomorphism algebra is
semisimple (a Clifford algebra). There are a lot of other examples in
the older literature, for instance [Knoerrer, 
The Cohen-Macaulay modules over simple hypersurface singularities].
 
{\it More theory.} This would be a good point to introduce Orlov's 
definition. It shows that if $W^{-1}(0)$ is smooth, the category of
matrix factorizations is zero. Note also Orlov's recent preprint
on idempotent completions, which shows that the Karoubi completion
of the category of matrix factorizations only depends on the formal
neighbourhood of the critical points. This is important, since it
shows that up to Karoubi completion, we understand the category fully
if $W^{-1}(0)$ has only ordinary double points (described in a formal
neighbourhood by the nondegenerate potential above).
 
{\it Mirror symmetry.} We should look at the mirror potentials for
${\mathbb C}P^1$ and ${\mathbb C}P^1 \times {\mathbb C}P^1$, and 
try to identify the matrix factorizations mirror to various important
Lagrangian submanifolds. This is much easier to do in Orlov's picture,
actually. For ${\mathbb C}P^1 \times {\mathbb C}P^1$, the main
important point is to identify the standard Lagrangian torus (with
its four different $Spin$ structures) and the anti-diagonal sphere.
Unfortunately, this is unpublished work of Auroux-Katzarkov-Orlov,
but it's not hard to at least guess the result.
 
\underline{Notes} 
 
The important point about $\mathbb{C} P^1 \times \mathbb{C} P^1$ is that
the two critical points lying in the same fibre interact nontrivially.
Of course, if one passes to Karoubi completions, that interaction becomes
trivial. In the mirror picture, this corresponds to the fact that 
various Lagrangian submanifolds will split as direct sums of simpler
objects once one Karoubi-completes the Fukaya category.
 
\underline{Dependencies}
 
This depends on the talk about Lagrangian submanifolds in Fano varieties.
It also uses derived categories of coherent sheaves.
 
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