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| 65-135: |
Seminar on Mathematical Physics: Supergeometry
(
Deutsche Version)
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| Lecturer: |
Christoph
Sachse
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| Office hours: |
flexible (upon request)
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| Content: |
1. Linear and commutative superalgebra
2. Supermanifolds
3. Functors of points, categorical description of supermanifolds
4. Supergroups
5. Mapping spaces of supermanifolds, (possibly in the context of sigma models)
6. Supergeometry in physics (several possible topics, to be chosen regarding the interests of participants)
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| Ziel: |
The goal of this seminar is to give an introduction to the field of supergeometry. Supergeometry is an extension of
commutative geometry which allows anticommuting ``functions'' and coordinates on the same footing as commuting ones.
The initial motivation for its introduction comes from physics which also gave rise to its most interesting examples
and problems to this day. Although much of the classical differential geometry can be carried over to this setting,
the theory has a considerably more algebraic flavor and new phenomena arise.
We will start by introducing the algebraic foundations of supergeometry: supercommutative algebras, Lie superalgebras and
their modules. We will then move on towards geometry, introducing the necessary tools along the way, such as sheaves
and ringed spaces. Our goal is to be able to give overviews of some topics relevant for current research. This will
include supergroups and their representations and the categorical description of supermanifolds. Depending on the
interests of participants we can have a look at infinite-dimensional supergeometry and the formalism of superfields in physics,
e.g., in supergravity and supersymmetric sigma models.
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| Vorkenntnisse: |
This class adresses students of all mathematical curricula as well as master students of physics or
mathematical physics. Required is only a background in linear algebra as well as some basic idea of geometry, e.g., from
a one-semester course in differential geometry or general relativity. It could be useful, though not required, to have
gained some familiarity with basic algebraic concepts such as rings, local rings and modules. Necessary background can
be acquired from textbooks during the seminar.
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| Literatur: |
P. Deligne, J. Morgan: Notes on Supersymmetry (following Joseph Bernstein), AMS 1999
V.S. Varadarajan: Supersymmetry for Mathematicians: An Introduction, AMS 2004
J. Bagger, J. Wess: Supersymmetry and Supergravity, Princeton University Press, 1992
C. Baer:
Nichtkommutative Geometrie (Vorlesungsskript, in German)
F. Constantinescu, H.F. de Groote: Geometrische und algebraische Methoden der Physik: Supermannigfaltigkeiten und Virasoro-Algebren,
Teubner Studienbuecher, 1994
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| Zeit und Ort: |
Wednesday 2-4 pm, room 432 (Geomatikum), Preliminary meeting: Thursday, Feb. 3, 2011, 1-2 pmm room 327 (Geomatikum)
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| Necessary contribution for credit: |
Persistant participation in the seminar and one oral presentation.
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