Physics Around Mirror Symmetry
G_2 manifolds play
the analogous role in M-theory that Calabi-Yau manifolds play in string
theory. There has been work in the physics community on conjectural
"mirror symmetry" in this context, and it has also been observed that
singularities are necessary for a satisfactory theory. After a very
brief review of these physical developments (by a mathematician who
doesn't necessarily understand the physics), I will give a mathematical
introduction to G_2 conifolds. I will then proceed to give a detailed
Chiral gauge theories in two dimensions with (0,2) supersymmetry admit a
much broader, and more interesting, class of vacuum solutions than
their better studied (2,2) counterparts. In this talk, we will explore
some of the possibilities that are offered by this additional freedom by
including field-dependent theta-angles and FI parameters. The moduli
spaces that will result from this procedure correspond to heterotic
string backgrounds with non-trivial H-flux and NS-brane sources. Along
I
will describe a new method for understanding a large class of
generalized complex manifolds, in which we view them as usual
symplectic structures on a manifold with a kind of log structure. I
will explain this structure in detail and explain how it can be used
to prove a Tian-Todorov unobstructedness theorem as well as
topological obstructions for existence of nondegenerate generalized
complex structures.
The concept of wall-crossing structure (WCS for short) was introduced
recently in my joint work with Maxim Kontsevich. WCS appear in different
disguises in the theory of Donaldson-Thomas invariants of Calabi-Yau
3-folds, quiver representations,integrable systems of Hitchin type,
cluster algebras, Mirror Symmetry, etc.
I plan to discuss the
definition of WCS and illustrate it in several well-known examples. If
time permits I will speak about a special class of WCS called rational