WALL CROSSING
IHES, November 10-12, 2011
Abstract of the minicourse of
Maxim Kontsevich
Wall crossing is a relatively new phenomenon which appears in several
mathematical frameworks:
1) (homological algbera): theory of Donaldson-Thomas invariants, i.e.
of the countiung of stable objects in triangulated 3-dimensional
Calabi-Yau categories (by Joyce and Song, and by Kontsevich and
Soibelman).
2) (combinatorics): quiver mutations by Fomin and Zelevinsky, cluster
algebras, canonical bases of Fock and Goncharov.
3) (complex and differential geometry): explicit formulas for the
hyperkahler metric on the Hitchin moduli space (by Gaiotto, Moore and
Neitzke).
4) (analysis) Stokes phenomenon for the the Bohr-Sommerfeld
asymptotics of spectra and eigenfunctions of polynomial differential
operators depending on small parameters (works of Ecalle and Voros in
80-ies).
Also the wall crossing has "physcial" origins in supersymmetric
gauge/string theory, as marginal stability of BPS particles and
supersymmetric black holes.
The moduli spaces of meromorphic quadratic differentials serves as
the most basic example. In this case one can write the wall crossing
formulae explicitly, and see directly its various algebraic and
geometric interpretations. Moreover, the natural objects of study of
the dynamics of the Teichmuller geodesic flow and interval exchange
maps (i.e. separatrix intervals and closed geodesics) are exactly the
same as the BPS states in physics, stable objects in algebra, cluster
coordinates in combinatorics.
Our goal in this series of lectures is to introduce the community of
specialists in flat surfaces and Teichmuller flow (and everybody in
fact!) to the new exiting developments giving new viewpoints on the
familiar objects, and proposing new venues for generalizations.