Organizational Meeting

This quarter, we will devote a large portion of the Geometry, Topology, and Physics seminar to the study of so-called ''wall-crossing 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 4-manifolds, 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 Calabi--Yau 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 wall-crossing formulas (in some cases), involving a study of the 4-dimensional physical theory on R³ x S¹.

There has been some dramatic recent progress in understanding this kind of formula in other contexts, including cases where the curves on a Calabi--Yau 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 wall-crossing formulas (in some cases), involving a study of the 4-dimensional physical theory on R³ x S¹.

We will spend a number of weeks exploring some of these background topics before arriving at the wall-crossing formulae themselves.
Here is a reading list to get us started:
http://arxiv.org/abs/hep-th/0607039 -- D. Joyce: Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds.
http://arxiv.org/abs/0801.3974 -- T. Bridgeland, V. Toledano-Laredo: Stability conditions and Stokes factors.
http://arxiv.org/abs/0807.4723 -- D. Gaiotto, G. Moore, A. Neitzke: Four-dimensional wall-crossing via three-dimensional field theory.
M. Kontsevich, K. Soibelman: to appear.

David Morrison (UCSB)

Introduction to Wall-Crossing Formulas

Abstract: Wall-crossing 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 4-manifolds, 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 Calabi--Yau 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 wall-crossing formulas (in some cases), involving a study of the 4-dimensional physical theory on R³ x S¹.

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.

Andreas Malmendier (UCSB)

The geometry of the Seiberg-Witten curve and BPS states

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 rank-two 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 semi-classical 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 semi-classical.

Audio [ mp3,wma ]; Lecture Notes.

Richard Eager (UCSB)

3D N=4 Supersymmetric Gauge Theories and Hyperkaehler Metrics

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 Kontsevich-Soibelman wall-crossing formula for BPS degeneracies.

Lecture Notes.

No Meeting

David Morrison (UCSB)

Stability in Algebraic Geometry and Wall-Crossing Formulas

Abstract: As observed by Mumford in the early 1960âs, moduli problems in algebraic geometry generally require a notion of stability in order to be well-posed. 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 well-understood behaviors when moving from region to region. We will review this general topic, with an eye towards the relationship with the wall-crossing 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.

Andreas Malmendier (UCSB)

N=2 Supersymmetric Gauge Theory and Mock Theta Functions

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 u-plane 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 u-plane integral are in fact the Donaldson invariants.

Audio [ mp3,wma ]; Lecture Notes.

David Morrison (UCSB)

The Moduli of Sheaves and "Counting Invariants'' for Calabi-Yau Threefolds

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 Calabi--Yau 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 Calabi-Yau 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 "wall-crossing formula.''

Audio [ mp3,wma ]; Lecture Notes.

Richard Eager (UCSB)

Quiver Representations and D0/D2/D6 Bound States

Abstract: Remarkably the geometry of a singular Calabi-Yau threefold is intricately related to the moduli space of representations of an associated quiver with relations. While the relations associated to an arbitrary Calabi-Yau threefold are somewhat mysterious from a mathematical perspective, they are naturally encoded in what physicists call a superpotential. Recently Szendroi and Nagao-Nakajima analyzed fixed points of torus actions on the moduli space of framed A-modules. 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.

Thanksgiving - No Meeting