Quantum Gravity Seminar - Fall 2004
Gauge Theory and Topology
John Baez and Derek Wise
In the 2004-2005 academic year, our seminar is about
gauge theory and topology. To lay the background,
we started with a short history of n-categories in physics,
following a course John taught this summer at Cambridge University.
Together with
Aaron Lauda,
he is turning this course into a paper, and you can see
a draft here:
You might also like to review some definitions leading up to
the concept of "topological quantum field theory":
Derek Wise is
writing notes for the seminar:
-
Week 0
(Sept. 23) - A history of n-categorical
physics from Maxwell's thoughts on relativity to Weyl's introduction of
"gauge invariance" to Heisenberg's matrix mechanics.
-
Week 1 (Sept. 28, 30)
- A history of n-categorical physics
from Born's probability interpretation of quantum mechanics,
to Feynman path integrals, to Mac Lane's introduction of monoidal
and symmetric monoidal categories.
-
Week 2 (Oct. 5, 7) - A
history of n-categorical physics from Bénabou's introduction
of bicategories to Penrose's spin networks.
-
Week 3 (Oct. 12, 14) - A
history of n-categorical physics from the Ponzano-Regge model
of 3d quantum gravity, to Grothendieck's dreams about
ω-categories, to string theory.
-
Week 4 (Oct. 19, 21) - A
history of n-categorical physics from Segal and Atiyah's
definitions of conformal and topological quantum field theories
to Joyal and Street's definition of braided monoidal categories.
-
Week 5
(Oct. 26) - Conclusion of our history of n-categorical
physics: from TQFTs to the periodic table of n-categories.
-
Week 6
(Nov. 2, 4) - Constructing 2d TQFTs from semisimple algebras:
an exposition of the work of
Fukuma,
Hosono and Kawai.
-
Week 7
(Nov. 9) - Constructing 2d TQFTs from semisimple algebras,
continued.
-
Week 8
(Nov. 16, 18) - Constructing 2d TQFTs from semisimple algebras,
concluded.
-
Week 9
(Nov. 23) - Computing the vector space for a circle in a 2d
TQFT: it's the center of the semisimple algebra we started with!
-
Week 10
(Nov. 20, Dec. 2) - A final description of the 2d TQFT obtained
from a semsimple algebra. A key example of a semisimple algebra,
leading ultimately to gauge theory: the group algebra of a finite group.
A sneak preview of next quarter, in which we'll build 3d TQFTs by
categorifying all these ideas.
For more, go on to the Winter 2005
notes!
If you discover any errors in the course notes
please email John, and we'll try to correct them.
We'll keep a list of errors that
haven't been fixed yet.
You can also download
TeX or LaTeX files of the
category theory definitions, homework problems and
solutions, if for some bizarre reason you want
them. However, the authors keep all rights to this work, except
when stated otherwise.
baez@math.removethis.ucr.andthis.edu
© 2004 John Baez and Derek Wise