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September 30, 2006
Dimensional Analysis and Coordinate Systems
Posted by John Baez
We had a nice conversation on dimensional analysis. Here are some things I learned.
September 28, 2006
Puzzle Pieces Falling Into Place
Posted by Urs Schreiber
There should be a 3-group governing Chern-Simons theory for gauge group . Which one is it?
I would like to present evidence that it should be the strict 3-group #
which is a sub-3-group of the non-strict automorphism 3-group #
Moreover, the canonical lax 2-representation #
for should extend canonically to a lax 3-representation
on endomorphisms of #.
Unless I am mixed up - which is your task to find out - this suggests to relate the correspondence
to higher Schreier theory #.
2-Groups and Algebras
Posted by Urs Schreiber
In another thread #, I am talking with Jim Stasheff and David Roberts about the question how to reconstruct a 2-bundle with connection from its local transition data #.
There are examples where I have some idea at least about certain aspects of the answer.
And there seems to be a pattern:
Example 1 is this: start with transition data on some space with respect to the 2-group coming from the crossed module (characterizing an abelian gerbe #). It is well known that this is equivalent to a -bundle on . happens to be the automorphism group of the algebra of compact operators on . Hence we can find the associated algebra bundle. Regarding each fiber not as a mere algebra, but as the category of modules of that algebra, we do obtain a 2-bundle of sorts. I think one can show that this is the 2-bundle whose local trivializations yields the 3-cocycle we started with #.
Example 2 is the string bundle with string connection by Stolz & Teichner #.
In both cases one can, I think, understand the algebra that the nerve acts on by automorphisms as the 2-vector space on which the 2-group is represented by its canonical 2-representation #.
So, clearly, there is some general mechanism at work which should generalize the above table from and to any strict 2-group. Which mechanism is that?
September 27, 2006
Toleration
Posted by David Corfield
One of the goals of our activity in this joint blog is to further the ends of mathematics and physics through our public conversations. Likewise for philosophy, if not directly through the refinement of -categorical thinking, then indirectly by observation of what it is to partake in an enterprise such as the furthering of mathematics or physics by -categorical means. Naturally, in terms of helping ourselves achieve those ends, we have to consider the question of what we are prepared to tolerate, both in terms of the content and the spirit of any contributions.
Bulk Fields and Induced Bimodules
Posted by Urs Schreiber
As I mentioned recently, Fjelstad, Fuchs, Runkel & Schweigert know how to describe 2-dimensional (rational) conformal field theory in terms of tangle diagrams in modular tensor categories .
There are various hints # that one can understand this formalism from the point of view of 2-transport # with values in .
Notice how this is rather analogous to principal 2-transport #, with values in a 2-group .
Instead of a 2-group, is just a 2-monoid. But the look and feel of both is rather similar: being modular tensor, in particular has left and right duals for all objects.
Many aspects of the diagrams drawn onto the worldsheet in the FRS formalism can be understood # from locally trivializing a 2-transport
which sends pieces of worldsheet to morphisms in .
One aspect of this is however a little troubling: only sees the 2-dimensional parameter space. However, FRS show that bulk field insertions in 2D CFT are represented diagrammatically by insertion points on the worldsheet at which ribbons emanate perpendicular to the worldsheet, into a third dimension.
In fact, this leads to a big story where the entire 2-dimensional field theory is described as the boundary part of a 3-dimensional topological field theory, generalizing the old observation by Witten on the relation between Chern-Simons theory and the Wess-Zumino model. I used to be puzzled about how to capture this 3-dimensional aspect of 2-dimensional CFT in terms of 2-dimensional transport.
But, remember, I also used to be puzzled about how to describe non fake-flat # principal 2-transport. There, the solution is # to pass from a 2-functor
with values in the the 2-group to the pseudo functor (a 3-functor, really!)
with values in the 3-group of automorphisms of .
Given what I said so far, there is really one question one should ask:
What happens if we consider weak 2-functors that send pieces of worldsheet not to a modular tensor category , but to the 3-monoid
of endomorphisms of ?
September 26, 2006
Fahrenberg and Raussen on Continuous Paths
Posted by Urs Schreiber
Suppose you want to transport something along some path through a space . Before you do so, you need to know what a path in is.
If is a smooth space, we tend to demand a path to be a smooth map
up to reparameterization. ( is the standard interval.)
What exactly is the analog of dividing out by reparameterization of paths in the case that is just a topological space?
Jim Stasheff, being interested in topological notions #, wondered why I kept going on about smooth paths # without ever talking about continuous paths. He was so kind to point me to the work
Ulrich Fahrenberg & Martin Raussen
Reparametrizations of Continuous Paths
Dept. of Mathematical Sciences, Aalborg University
Technical Report R-2006-22
(pdf)
where exactly this issue is investigated.
Our Raison D’être
Posted by David Corfield
Marni Sheppeard reports from the AustMS2006 conference, which, as anyone who knows about Australian mathematics might expect, is holding a category theory session. Dominic Verity is giving one of the talks, in which he considers the raison d’être for higher category theory, and so by extension that of the Café. Of course, we also come here for the coffee.
September 23, 2006
Mathematical Kinds
Posted by David Corfield
I’ve just sent off a paper Mathematical Kinds, or Being Kind to Mathematics to appear in the journal Philosophica. The idea of the paper is to explore the extent to which the language of laws and natural kinds, so much a part of the philosophy of science, is also appropriate to mathematics. To give a fresh example of this phenomenon, let’s consider the classification of finite simple groups.
September 21, 2006
Dimensional Analysis
Posted by John Baez
With your help, I would like to start amassing a collection of wisdom on gnarly issues in physics. Let’s start with dimensional analysis. I thought I had this pretty much figured out, until Kehrli pointed out a couple of things that surprised me:
- Dimensionless constants can depend on our choice of units.
- Dimensionful constants often don’t depend on our choice of units.
Categorification in Uppsala
Posted by David Corfield
Earlier this month the Mathematics Institute at Uppsala University hosted a conference called Categorification in Algebra and Topology, clearly a theme close to our collective heart. As yet there are only a handful of participants’ notes available (Scott Morrison’s are particularly well rendered), although the abstracts refer you to one or two others. What I’d like to have found out is whether there are differences in people’s conception of the scope of categorification.
The Why and Wherefore of History
Posted by David Corfield
Here are some notes for my talk at the Berlin workshop. Fortunately I was upgraded to a 45-minute talk. Even so, I didn’t manage to reach the last part where I discuss David Carr’s ideas.
I would be interested in a discussion here about practitioners’ histories. A couple of examples we might consider are Baez and Lauda’s draft History of n-categorical physics and Ronald Solomon’s A Brief History of the Classification of Finite Simple Groups, BAMS 38(3) 315-382.
September 20, 2006
Differential n-Geometry
Posted by Urs Schreiber
I need it all the time - and yet, I still don’t have it:
a nice arrow-theoretic way to talk about differential -geometry.
I know that greater minds than me have thought about this thoroughly before #. That I still don’t feel like having the satisfactory tools at my disposal probably has two reasons:
a) I am ignorant of what has been done already;
b) I feel the need for something somewhat different than what has been done already.
In order to find out which of the two it is, I want to start going through some gymnastics here in the -Café.
I am thoroughly (maybe hopelessly) motivated by
i) quantum physics
and
ii) the belief that -dimensional QFT lives in .
Maybe this explains point b) above. As far as I am aware, previous work in arrow-theoretic differential geometry was motivated by classical physics and the belief that suffices.
For instance, I believe that we want a notion of differential -forms that take values in -categories, like -functors do.
This belief is a consequence of particular physics applications that I have in mind, which I roughly know how to do already, but which need a more systematic underpinning. In particular, one of my goals is to give a good arrow-theoretic description of an -Dirac operator twisted by an -vector bundle with -connection. Unless I am confused, such a concept is at the heart of -dimensional supersymmetric quantum field theory (at least for and ).
Okay. The gymnastics starts below.