IX Lisbon Summer Lectures in Geometry
The
2008 Lisbon Summer Lectures in Geometry will be held at the Department of Mathematics
of Instituto
Superior Técnico, Lisbon, Portugal,
September 8 to 12, 2008.
Lectures by
Anton Kapustin (California Institute of Technology)
Course: Gauge Theory and the Geometric
Langlands Program
Abstract: Recently, it has been
shown that Geometric Langlands duality is closely
related to properties of quantized supersymmetric
Yang-Mills theory in four dimensions, as well as to Mirror Symmetry and
Topological Field Theory. I will provide an introduction to these developments.
Literature:
·
A. Kapustin and E.
Witten, Electric-Magnetic Duality and the Geometric Langlands
Program, hep-th/0604151
·
S.Gukov and E. Witten, Gauge Theory, Ramification, and the
Geometric Langlands Program, hep-th/0612173
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A. Kapustin, Holomorphic reduction of N=2 gauge theories, Wilson-'t Hooft operators, and S-duality, hep-th/0612119.
For some background, the following papers
are very useful:
·
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
·
E. Witten, Mirror
manifolds and Topological Field Theory, hep-th/9112056.
Colloquium:
Electric-magnetic duality and Langlands duality
Abstract: It is a well-known
fact that Maxwell electrodynamics without sources has a symmetry which
exchanges electric and magnetic fields. This is known as electric-magnetic
duality, and it holds both on the classical and quantum levels. Remarkably,
there is strong evidence that electric-magnetic duality also holds in quantum
Yang-Mills theory, where it relates theories with different (dual) gauge
groups. I will discuss how this phenomenon is related to Langlands
phenomena in representation theory.
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James Sparks
(Mathematical Institute, Oxford),
Course:
Sasaki-Einstein Geometry and the AdS/CFT
Correspondence
Abstract: The aim of these
lectures will be to describe some recent developments in Sasaki-Einstein
geometry, and also to explain, in a way that is hopefully accessible to
geometers, how these results are related to the AdS/CFT
correspondence in string theory. I will begin with a general introduction to Sasakian geometry, which is the odd-dimensional cousin of
Kahler geometry. I will then introduce
Sasaki-Einstein geometry, and describe a number of different constructions of
Sasaki-Einstein manifolds. In particular, I will develop the theory of toric Sasakian manifolds,
culminating with the recent theorem of Futaki-Ono-Wang
on the existence of toric Sasaki-Einstein metrics.
Next I will describe a number of different obstructions to the existence of
Sasaki-Einstein metrics, together with some simple examples. Finally, I will
outline how Sasaki-Einstein manifolds arise as solutions to supergravity, and describe their role in the AdS/CFT correspondence. The latter conjectures that for
each Sasaki-Einstein 5-manifold there exists a corresponding conformal field
theory on R^4. This map is only really understood in certain examples, and
for concreteness I will focus mainly on the toric
case. The conformal field theory is then (conjecturally) described by a gauge
theory on R^4 that is determined from the algebraic geometry of the cone over
the Sasaki-Einstein manifold. Mathematically, this data is encoded by a
bipartite graph on a two-torus. I will conclude by explaining how AdS/CFT relates some of the properties of Sasaki-Einstein
manifolds described earlier to this combinatorial structure.
References: The article
arXiv:math/0701518 [math.DG] reviews much of the
material that I will cover, and also contains references to the original
literature.
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September
9-12, Department of Mathematics, IST
The
Courses will take place in Room PA1 (Floor -1
in Pavilhão de Matemática)
The
Colloquium will take place in Room P12 (Floor 2
in Pavilhão de Matemática)
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SCHEDULE
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Tues, 9
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Wed, 10
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Thu, 11
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Fri, 12
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11:00-11:30
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Kapustin
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Sparks
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11:30-12:00
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12:00-12:30
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14:00-14:30
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Kapustin
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14:30-15:00
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15:00-15:30
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Kapustin
Colloquium
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Kapustin
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15:30-16:00
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coffee
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16:00-16:30
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Sparks
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coffee
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Sparks
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16:30-17:00
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coffee
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17:00-17:30
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17:30-18:00
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coffee
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This is the
ninth series of summer lectures dedicated to Geometry and related subjects held
at IST (see past lectures).
Organizing committee for the current series:
José Mourão, João P. Nunes and
Ricardo Schiappa
Please feel free to contact any of the organizers for further details.
Sponsors:
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Center for Mathematical Analysis, Geometry and Dynamical
Systems - IST (Lisbon)
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Fundação para a Ciência e
a Tecnologia
About the photo.