Research Seminar in Mathematics (MTH G450 - 30757)
Organized by Professor Jonathan Weitsman
Guest Speaker: Andy Neitzke
Harvard University
Title: Interface between geometry and string theory
Date: Tuesday, December 2, 2008
Time: 1:00 p.m.
Location: 509 Lake Hall
Pre-talk (1-2): "Introduction to BPS wall-crossing"
I will discuss the integer invariants that come from quantum field
theories. These field theories have continuous moduli, often
identified with the moduli of some geometric object, and one wants to
know a "wall-crossing formula" describing how the integer invariants
jump when one crosses special codimension-1 walls in the moduli
space. I will describe two broad classes of example where this problem has
been solved. One class was studied by Cecotti-Vafa in the early 1990's and
mathematically has to do with singularity theory. The other class was
studied by Kontsevich-Soibelman very recently and has to do with the
geometry of Calabi-Yau threefolds (or categories).
Break (2:00 - 2:30)
Research Talk (2:30 - 3:00): "Hyperkahler geometry and BPS wall-crossing" - Kontsevich-Soibelman's wall-crossing formula looks rather mysterious when first encountered. I will explain how to understand this wall-crossing formula geometrically: it is a consistency condition for a new construction of hyperkahler spaces. This is work done in collaboration with Davide Gaiotto and Greg Moore.
Department tea (3:30 - 4:00)
Research Talk 2(4:00 - 5:00):
"BPS wall-crossing in moduli spaces of Higgs bundles"
I will describe a concrete example of the abstract setup of the previous talk, where
the hyperkahler space in question is a moduli space of ramified Higgs
bundles. These spaces have been much studied recently in connection with the
geometric Langlands program. In this case the integer invariants and their
wall-crossing turn out to have a rather concrete meaning in terms of
foliations on a punctured curve. We obtain a new description of the geometry
of the moduli spaces. This is work in progress with Davide Gaiotto
and Greg Moore.
Discussion at 5:00 p.m. followed by dinner with the speaker.
Previous Talks:
Fall 2008:9/16, 9/23, 10/07,10/28, 11/18
Started: 23 September 2008 Last modified: 11 February 2009
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