
Geometry, Topology, and Physics Seminar, Fall 2008
Organizers:
Andreas Malmendier
and Dave Morrison.
Meets 4:00  5:30 p.m. Fridays in South Hall 6635.
This quarter, we will devote a large portion of the Geometry, Topology, and Physics seminar to the study of
socalled "wallcrossing formulas."
Other Quarters: [
Fall, 2021;
Winter, 2020;
Fall, 2019;
Spring, 2018;
Winter, 2018;
Fall, 2017;
Spring, 2017;
Wnter, 2017;
Fall, 2016;
Spring, 2016;
Winter, 2016;
Fall, 2015;
Spring, 2015;
Winter, 2014;
Fall, 2013;
Fall, 2012;
Fall, 2011;
Winter, 2011;
Spring, 2010;
Winter, 2010;
Fall, 2009;
Spring, 2009;
Winter, 2009;
Fall, 2008;
Spring, 2008;
Winter, 2008;
Fall, 2007;
Spring, 2007;
Winter, 2007;
Fall, 2006
]
Sept. 26 
This quarter, we will devote a large portion of the Geometry, Topology, and Physics seminar to the study of
socalled ''wallcrossing formulas.'' Such formulas first arose in the celebrated work of Seiberg & Witten, where
they related the count of ''BPS quantum states'' in one part of the moduli space (of certain quantum field theories)
to the count in another part of the moduli space: this led directly to a new method for computing Donaldson
invariants of 4manifolds, and the rest is history.
There has been some dramatic recent progress in understanding this kind of formula in other contexts, including
cases where the curves on a CalabiYau manifold are being counted. The ingredients for this progress include
considerations of symplectic geometry and a study of ''Stokes factors'' in differential equations. There is also a
beautiful physics argument for the wallcrossing formulas (in some cases), involving a study of the 4dimensional
physical theory on R^{3} x S^{1}.
There has been some dramatic recent progress in understanding this kind of formula in other contexts, including
cases where the curves on a CalabiYau manifold are being counted. The ingredients for this progress include
considerations of symplectic geometry and a study of ''Stokes factors'' in differential equations. There is also a
beautiful physics argument for the wallcrossing formulas (in some cases), involving a study of the 4dimensional
physical theory on R^{3} x S^{1}.
We will spend a number of weeks exploring some of these background topics before arriving at the wallcrossing
formulae themselves.
Here is a reading list to get us started:
http://arxiv.org/abs/hepth/0607039  D. Joyce: Holomorphic generating functions for invariants counting coherent sheaves on CalabiYau 3folds.
http://arxiv.org/abs/0801.3974  T. Bridgeland, V. ToledanoLaredo: Stability conditions and Stokes factors.
http://arxiv.org/abs/0807.4723  D. Gaiotto, G. Moore, A. Neitzke: Fourdimensional wallcrossing via threedimensional field theory.
M. Kontsevich, K. Soibelman: to appear.

Oct. 3 
David Morrison (UCSB)
Abstract:
Wallcrossing formulas first arose in the celebrated work of Seiberg and
Witten, where they related the count of ''BPS quantum states'' in one part
of the moduli space (of certain quantum field theories) to the count in
another part of the moduli space: this led directly to a new method for
computing Donaldson invariants of 4manifolds, and the rest is history.
We will introduce this area of study, aiming at the dramatic recent progress
in understanding this kind of formula in other contexts, including cases
where the curves on a CalabiYau manifold are being counted. The
ingredients for this progress include considerations of symplectic geometry
and a study of ''Stokes factors'' in differential equations. There is also
a beautiful physics argument for the wallcrossing formulas (in some cases),
involving a study of the 4dimensional physical theory on R^{3} x S^{1}.
Our study will also be related to an interesting theme in algebraic geometry
during the past several years: how a moduli space changes when the
''stability condition'' used to define that moduli space changes.
Audio [ mp3,wma ];
Lecture Notes.

Oct. 10 
Andreas Malmendier (UCSB)
Abstract:
The moduli spaces of vacua for the topological N=2 supersymmetric SU(2)
gauge theories on CP^2 with (doublet) hypermultiplets are Jacobian
rational elliptic surfaces over CP^1 (with an analytical marking). We
review how the number and type of the singular fibers of the moduli spaces
vary with the number and masses of the additional matter fields.
The period bundle of the elliptic surface defines a ranktwo SL(2,Z)
bundle equipped with a special Kaehler connection. The bundle contains a
flat submanifold which intersect each fiber in a full integer lattice;
this is the charge lattice of the BPS states. The spectrum of the stable
semiclassical BPS states defines a unique flat holomorphic line bundle on
the rational elliptic surface. Finally, we will review the construction of
the line of marginal stability separating the strong coupling BPS spectrum
from the semiclassical.
Audio [ mp3,wma ];
Lecture Notes.

Oct. 17 
Richard Eager (UCSB)
Abstract:
Seiberg and Witten famously determined the dynamics of four
dimensional gauge theories with N=2 supersymmetry using constraints from
low energy effective field theory. Compactification to three dimensions
yields a sigma model with N=4 supersymmetry and a hyperkaehler manifold as
its target space. We describe how the spectrum of BPS instantons is
encoded in the hyperkaehler metric and Hitchin's construction of hyperkaehler
metrics using twistor methods. These techniques provide the foundation for
Gaiotto, Moore, and Neitzke's recent work on the KontsevichSoibelman
wallcrossing formula for BPS degeneracies.
Lecture Notes.

Oct. 24 
No Meeting

Oct. 31 
David Morrison (UCSB)
Abstract:
As observed by Mumford in the early 1960âs, moduli problems in algebraic
geometry generally require a notion of stability in order to be wellposed. Although the
possibility of different choices of stability condition was implicit in Mumfordâs original
work, it is only in the last 15 years that systematic investigations have been made
into stability choices and their effects on moduli spaces. These days, one regonizes
that there is generally a parameter space for the stability choices, with the algebrogeometric
structure on the corresponding moduli space constant within regions of this
parameter space but changing from region to region. Often, one can put a natural
metric on these moduli spaces, which varies continuously with the stability parameters
within a region, and has certain wellunderstood behaviors when moving from region
to region.
We will review this general topic, with an eye towards the relationship with the
wallcrossing formulas for BPS counting functions which have been the underlying
topic of this seminar for the fall quarter. Our primary example will be the moduli
spaces of representations of the conifold superpotential algebra, as studied in recent
work of Nagao and Nakajima.
Audio [ mp3,wma ];
Lecture Notes.

Nov. 7 
Andreas Malmendier (UCSB)
Abstract:
The moduli spaces of vacua for the topological N=2 supersymmetric
gauge theories with gauge group SU(2) or SO(3) on CP^2 with massless hypermultiplets are Jacobian
rational elliptic surfaces over CP^1. The uplane integrals for these moduli spaces
compute interesting topological invariants. To evaluate the integrals one
needs to integrate by parts using nonholomorphic modular forms or Mock theta functions.
We explain what these Mock theta functions are for the N=2 gauge theory with gauge group SU(2) and SO(3) and how
their modular properties relate to the BPS spectrum. Time permitting, I will also sketch
how  in the case of the SO(3)gauge theory on CP^2  one can prove using Mock theta functions that the invariants from the uplane integral are in fact
the Donaldson invariants.
Audio [ mp3,wma ];
Lecture Notes.

Nov. 14 
David Morrison (UCSB)
Abstract:
We will continue our discussion of stability and moduli spaces in algebraic
geometry, focussing on a particular problem: the moduli of sheaves on a
certain CalabiYau threefold known as the "resolved conifold''. We will
explain the recent results of Nagao and Nakajima on this problem, as well as
describing the
connection with certain kinds of "counting invariants'' on this CalabiYau
threefold (related to the counting of BPS states in physics). There are
many different ways to evaluate these counting invaraints, which (as we
shall explain) are related by a "wallcrossing formula.''
Audio [ mp3,wma ];
Lecture Notes.

Nov. 21 
Richard Eager (UCSB)
Abstract:
Remarkably the geometry of a singular CalabiYau threefold is intricately
related to the moduli space of representations of an associated quiver with
relations. While the relations associated to an arbitrary CalabiYau
threefold are somewhat mysterious from a mathematical perspective, they are
naturally encoded in what physicists call a superpotential. Recently
Szendroi and NagaoNakajima analyzed fixed points of torus actions on the
moduli space of framed Amodules. In this talk we will relate their work to
the physical interpretation given by Chuang and Jafferis in terms of
D0/D2/D6 BPS bound states.
Audio [ mp3,wma ];
Lecture Notes.

Nov. 28 
Thanksgiving  No Meeting


