Department of Mathematics - UC Santa Barbara

Geometry, Topology, and Physics Seminar, Fall 2009

Organizers: Andreas Malmendier and Dave Morrison.
Meets 4:00 - 5:30 p.m. Fridays in South Hall 6635.

Other Quarters: [ 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 ]

Oct. 2

Dave Morrison (UCSB)

N=2 dualities and Riemann surfaces

Abstract: N=2 supersymmetric field theories in four dimensions have been studied from many points of view, notably by Seiberg and Witten in the mid-1990's who introduced an associated Riemann surface and used its properties to derive remarkable results about the physics, and remarkable consequences for mathematics. In the past six months, work of Gaiotto and collaborators has shown that these theories can be studied by means of *another* Riemman surface, this time used to compactify the six-dimensional N=(2,0) field theories to obtain a four-dimensional theory. Moreover, the two-dimesional field theory on this "other" Riemann surface is related in many interesting ways to the N=2 four-dimensional field theory.

This lecture will give an overview of these developments, which will be described in more detail in future lectures of this seminar.

Lecture notes.

Oct. 9

Andreas Malmendier (UCSB)

Kummer surfaces from Seiberg-Witten curves

Abstract: Jacobian elliptic surfaces are elliptic surfaces with sections. They play a key role in gauge theory as well as in string theory. In gauge theory, the Seiberg-Witten curve of SU(2) gauge theory arises as a pencil generated by two cubics in the plane forming a rational elliptic surface. In F-theory, K3 surfaces constructed as elliptic surfaces with sections are also of special importance.
In my talk, I will start with the Weierstrass normal form of the Seiberg-Witten curve for pure SU(2) gauge theory. By carrying out a base transformation, one obtains a 3-parameter family of elliptically fibered K3 surfaces. The gauge theoretic relation between the Seiberg-Witten curves with N_f=2 and N_f=0 hypermultiplets in turn defines a Shioda-Inose structure on each K3 surface in the family with quotient birational to a Kummer surface of the Jacobian of a genus-two curve.

Audio [ mp3, wma ]; Lecture notes.

Oct. 16

Dave Morrison (UCSB)

Much ado about N=2

Abstract: I will continue to review the recent progress in four-dimensional N=2 supersymmetric field theories. In this lecture, I will concentrate on the conformal building blocks and associated dualities introduced by Gaiotto in arXiv:0904.2715. If time permits, I will also discuss how these constructions are related to the study of wall-crossing formulas made by Gaiotto, Moore, and Neitzke.

Audio [ mp3, wma ]; Lecture notes.

Oct. 23

Francesco Benini (Princeton)

M5-branes wrapped on Riemann surfaces

Abstract: I will fastly review Gaiotto's construction of N=2 SCFTs via M5-branes wrapped on Riemann surfaces, including an alternative type IIB construction suitable to study their flavor symmetries. Then I will focus on N=1 constructions, and the computation of central charges from supergravity. The method also gives some information on the recently discovered 2d-4d correspondence with Toda theories.

Audio [ mp3, wma ]; Slides.

Oct. 30

No meeting this week

Nov. 6

Herman Verlinde (Princeton)

More about S-dual probes of supersymmetric gauge theory

Abstract: I will give further details about arXiv:0909.0945 and related work.

Audio [ mp3, wma ]; Lecture notes.

Nov. 13, 20

No meeting this week

Nov. 27

Thanksgiving - No meeting

Dec. 15

The High Energy Seminar will meet in KITP SSR at 12:30 pm.

Paul Aspinwall (Duke)

Decompactifications vs Massless D-Branes in Moduli Space Limits