Quantum Gravity Seminar - Winter 2005
Gauge Theory and Topology
John Baez and Derek Wise
In the 2004-2005 academic year, our seminar is about
gauge theory and topology. In the Fall
we showed how to construct
2d topological quantum field theories (or TQFTs) from semisimple algebras.
In the Winter,
we categorified all this and saw how to construct 3d TQFTs
from "semisimple 2-algebras". The
fun part was seeing how categorification - the process of replacing
equations by isomorphisms - naturally boosts the dimension by one!
Then we turned to examples: the "Dijkgraaf-Witten models",
which are gauge theories with finite gauge group. We discussed how
these models could be "twisted", and used this as an excuse
to learn about the cohomology of groups.
A bunch of this material will eventually be incorporated in this
You may already enjoy looking at the draft version.
You may also want to review some definitions leading up to
the concept of "topological quantum field theory":
As usual, Derek Wise is
writing notes for the seminar:
(Jan. 4, 6) - Outline of this quarter's course.
2Vect: the 2-category of 2-vector spaces. A sketch of how
we will categorify last quarter's construction of 2d TQFTs to get 3d TQFTs.
(Jan. 11, 13) -
Iterated index notation for categorified tensors; spin foams as
diagrams for categorified tensors.
(Jan. 18, 20) - The concept of "extended TQFT", and how
to construct 3d extended TQFTs from semisimple 2-algebras. Deriving
the 2-3 Pachner move from the pentagon identity.
Two equivalent forms of the bubble move.
(Feb. 1, 3) - Gauge theory on a triangulated manifold.
Connections and gauge transformations
when spacetime is a graph; flat connections when spacetime is a
(Feb. 8, 10) - Computing the partition function in the 2d
Dijkgraaf-Witten model, a TQFT built from the semisimple algebra C[G]
(the group algebra of a finite group G). In this model,
the partition function is a path integral
over the space of flat connections mod gauge transformations.
(Feb. 15, 17) - Description of the space of flat connections mod
gauge transformations in terms of the fundamental group
of our manifold. Similar descriptions of other
"moduli spaces". A beautiful formula for the partition
function of the Dijkgraaf-Witten model in terms of the "groupoid
cardinality" of the "moduli stack" of flat connections.
(For an explanation of groupoid cardinality, see the
Winter 2004 notes, especially starting
on page 53 of week 6.)
(Feb. 22, 24) -
Twisting the multiplication in C[G] by a 2-cocycle;
twisting the associator in Vect[G] by a 3-cocycle.
Twisted Dijkgraaf-Witten models. Group cohomology and Pachner moves.
(Mar. 1, 3) - Group cohomology and TQFTs. The cohomology of the
group G as the cohomology of its classifying space BG.
Computing group cohomology.
(Mar. 8, 10) - Computing group cohomology, continued. Computing
the cohomology of Z/2 using the fact that its classifying
space is the infinite-dimensional projective
space RP∞. Summary of what we've done so
far this year.
To dig deeper, try the Spring notes.
To learn more about the Dijkgraaf-Witten models,
try their original paper, especially starting on page 42:
If you discover any errors in the course notes
please email me, and we'll try to correct them.
We'll keep a list of errors that
haven't been fixed yet.
You can also download
the LaTeX, encapsulated
PostScript and xfig files if for some bizarre reason you want them.
However, I reserve all rights to this work.
© 2005 John Baez and Derek Wise