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Electric-Magnetic Duality and the Geometric Langlands Program
Tuesdays 4:00-6:00 pm in Room 4-237, MIT
The goal of this seminar is to understand the recent paper by A. Kapustin
and E. Witten
of this title (hep-th/0604151),
which relates the geometric Langlands
conjecture
over the complex numbers to electric-magnetic duality in supersymmetric
gauge theory.
In their words:
The geometric Langlands program can be described in a natural way by
compactifying on
a Riemann surface C a twisted version of N=4 super Yang-Mills theory in
four dimensions.
The key ingredients are electric-magnetic duality of gauge theory, mirror
symmetry of sigma-models,
branes, Wilson and 't Hooft operators, and topological field theory.
Seemingly esoteric notions of the
geometric Langlands program, such as Hecke eigensheaves and D-modules,
arise naturally
from the physics.
An introduction to quantum field theory for mathematicians
may be found in Quantum fields and Strings: a course for mathematicians,
by P. Deligne et al. The following words from physics (which were used in
the October 17 talk) can be found in the index and/or glossary of this
2-volume textbook:
N=1,2,4 supersymmetry;
dimensional reduction;
scalars, spinors, vectors;
particle in QFT, mass of a particle;
symmetry breaking;
Goldstone theorem, Goldstone bosons;
Higgs mechanism;
low energy effective theory;
(the moduli space of) vacua of a QFT;
solitons, BPS states
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February 13 (Seminar Poster)
Speaker: Sergei Gukov (UCSB)
Title: Surface Operators in Gauge Theory and the Ramified Geometric Langlands
Abstract:
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February 20 (Seminar Poster)
Speaker: Roman Bezrukavnikov (MIT)
Title: Some ingredients of tamely ramified geometric Langlands duality
Abstract:
I will be aiming at giving a mathematical introduction to (some
aspects of) the paper hep-th/0612073 by Gukov and Witten. I will describe
the key ingredients in the algebraic approach to tamely ramified
Langlands duality -- local systems with regular singularity, bundles with
parabolic structures and categorification of affine Hecke algebra --
trying to link them to their "physical" or differential geometric
counterparts -- singular solutions of Hitchin equations and surface and
line operators in gauge theory.
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February 27 (Seminar Poster)
Speaker: I.M. Singer (MIT)
Title: HOW WILSON AND THOOFT OPERATORS ACT ON BRANES
Abstract:
I’ll first describe how the Wilson loop operator acts on Branes. The tHooft operator will
be introduced as the S-dual of the Wilson operator. Some needed facts about the Hitchin
module space MH will be reviewed. If time permits, I’ll give some of the physics
background to Wilson & tHooft operators. [The confinement problem; the original
introduction of LG into gauge theory]
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March 6 (Seminar Poster)
Speaker: I.M. Singer (MIT)
Title: HOW WILSON AND THOOFT OPERATORS ACT ON BRANES
Abstract:
Continued
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March 13 (Seminar Poster)
Speaker: Christopher Beasley (Harvard)
Title: Abelian Duality for Wilson and THooft Loops
Abstract:
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March 20 (Seminar
Poster)
Speaker: Marco Gualtieri (MIT)
Title: D-BRANES ON POISSON VARIETIES
Abstract:
I will describe the notion of D-brane on a
holomorphic Poisson manifold, a particular case of which is the
space-filling brane on a hyper-Kahler manifold, used by
Kapustin-Witten in the relation to D-modules. I will then describe how
to construct such branes and what their relation is to generalize
Kahler geometry.
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April 3 (Seminar
Poster)
Speaker: D. Jeremy Copeland (MIT)
Title:SPECTRAL COVERS AND T'HOOFT OPERATORS
Abstract:
We begin this talk by discussing how one describes explicitly (the general locus of)
Hitchin's moduli space. We will discuss spectral covers and the way in which one
constructs the moduli space as a torus fibration over the characteristic variety. We will
use the examples of SUn and PSUN to illustrate the roles played by Langlands dual
groups, and possibly discuss how this should appear in general. In the second part, we
will explicitly construct t'Hooft operators. I expect the discussion to be very concrete
and not to rely too heavily on the physical picture.
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April 10 (Seminar
Poster)
Speaker: Paul Norbury (Boston University)
Title: BOGOMOLNY EQUATIONS AND THE SPACE OF HECKE MODIFICATIONS
Abstract:
The moduli space of monopoles, i.e. solutions to the Bogomolny equations, over a surface
times an interval produces the space of Hecke modifications with added structure. The
extra structure arises because monopoles use the explicit metric on the surface while
Hecke modifications depend only on the conformal structure on the surface. I will
describe the moduli spaces of monopoles and the new perspectives they bring to the space
of Hecke modifications. Much of what I will say is contained in chapter 10 of Kapustin
and Witten.
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April 17 (Seminar
Poster)
Speaker: Allan Adams (M.I.T.) and I.M. Singer (M.I.T.)
Title: A & B Sigma models for 
Abstract:
Singer will review (1) twisted N = 2SSYM giving The Donaldson invariants as
observables. (2) The Topological Sigma models of type A & B.
Adams will review how S - duality of 4 d N = 4 SSYM induces mirror symmetry of the
various sigma models on .
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May 1 (Seminar Poster)
Speaker: Philip Boalch (ENS at CNRS)
Title: Geometry of moduli spaces of
meromorphic connections on curves
Abstract:
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May 8 (Seminar Poster)
Speaker: Allan W. Adams (M.I.T.)
Title: A-Branes and D-Modules
Abstract:
We will review the identification of strings on A-branes as D-modules, focusing on
explicit examples of A-branes in the Hitchin fibration and the D-modules they provide.
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Organizing Committee:
Aliaa Barakat
Roman Bezrukavnikov
Pavel Etingof
Marco Gualtieri
Is Singer
Fall 2006 Schedule
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