Abstract |
In the last three years a new concept - the concept of wall crossing
has emerged. The current situation with wall crossing phenomena, after
papers of Seiberg-Witten, Gaiotto-Moore-Neitzke, Vafa-Cecoti and
seminal works by Donaldson-Thomas, Joyce-Song,
Maulik-Nekrasov-Okounkov-Pandharipande, Douglas, Bridgeland, and
Kontsevich-Soibelman, is very similar to the situation with Higgs
Bundles after the works of Higgs and Hitchin - it is clear that a
general "Hodge type" of theory exists and needs to be developed.
Nonabelian Hodge theory did lead to strong mathematical applications - uniformization,
Langlands program to mention a few. In the wall crossing
it is also clear that some "Hodge type" of theory exists -
Stability Hodge Structure (SHS). This theory needs to be developed
in order to reap some
mathematical benefits - solve long standing problems in algebraic
geometry. In this paper we look at SHS from the perspective of
Landau-Ginzburg models and we look at some applications. We
consider simple examples and explain some conjectures these examples
suggest.
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