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March 31, 2007
Some Notes on Local QFT
Posted by Urs Schreiber
I have just returned from visiting my brother in Berlin. On the train I did some reading and thinking related to fittting algebraic quantum field theory into the general picture™. Here are a couple of random notes.
By the way, we went to the zoo, but Knut was not available. Probably preparing with his manager for the time after his cuteness career. Turns out, though, that there are, for instance, little warthog puppies. Not quite as cute, but lots of fun…
March 29, 2007
This Week’s Finds in Mathematical Physics (Week 248)
Posted by John Baez
In week248 of This Week’s Finds, see movies of coronal mass ejections, auroras, and tornados on the Sun!
Then, continue reading the Tale of Groupoidification — in which we see how spans of groupoids arise naturally in geometry.
March 28, 2007
Relativity on the World-Wide Web
Posted by John Baez
Chris Hillman is back! — with a new, improved guide to online resources on general relativity:
- Chris Hillman, Relativity on the World-Wide Web.
Popular science sites, web tutorials, undergraduate and graduate-level course material online, and a detailed survey of books — everything you need to learn general relativity, no matter where you’re starting!
There are even lots of nice visualization websites, packed with eye candy like this…
March 24, 2007
Philosophising in Brussels
Posted by David Corfield
March 23, 2007
This Week’s Finds in Mathematical Physics (Week 247)
Posted by John Baez
In week247 of This Week’s Finds, read about symmetry — from the appearance of quasicrystals in medieval Islamic tile patterns:
to the news about E8:
to Tale of Groupoidification.
History of Understanding Bundles with Connection using Parallel Transport around Loops
Posted by Urs Schreiber
In the process of finishing a paper, I was today busy collecting some background history literature on the development of the idea that a principal -bundle with connection may equivalently be encoded in its parallel transport around based loops.
March 22, 2007
Branes, Bi-Branes, 2-Vectors, 2-Linear Maps
Posted by Urs Schreiber
The open charged 2-particle looks like and its quantization, , assigns to it a morphism of its 2-vector space of 2-states each of which is a generalized element
When the 2-particle is charged under a line 2-bundle (a line bundle gerbe) the 2-vectors and are Chan-Paton bundles on D-branes, also known as modules for that gerbe.
The space of states is acted on 2-linearly by pull-push through spans which may encode operation like time evolution or gauge transformations like T-duality.
In a chosen 2-basis for , which is an algebra, 2-states appear as modules and 2-linear maps appear as bimodules.
The former fact harmonizes with the term “gerbe module” used for D-branes. In that sense, these bimodules could be addressed as bi-branes.
This is the language now chosen in
Fuchs, Schweigert, Waldorf
Bi-branes: Target Space Geometry for World Sheet topological Defects
Bi-branes: Target Space Geometry for World Sheet topological Defects.
March 21, 2007
Recent Developments in QFT in Leipzig
Posted by Urs Schreiber
The Max Planck Institute for Mathematics in the Sciences in Leipzig hosts a conference
Recent Developments in Quantum Field Theory
July 20 - 22, 2007
March 20, 2007
Generalising Hopf Algebras
Posted by David Corfield
To prepare for the appearance of categorified quantum groups it might be worth taking a look at Gizem Karaali’s On Hopf Algebras and Their Generalizations, in which she describes Hopf algebras and five attempts to generalise them. Much hangs on their representation categories.
Hopfish algebras are briefly touched on. As I noted in the Oct 21 entry of my old blog, according to Alan Weinstein and colleagues,
We call our new objects hopfish algebras, the suffix “oid” and prefixes like “quasi” and “pseudo” having already been appropriated for other uses. Also, our term retains a hint of the Poisson geometry which inspired some of our work.
March 19, 2007
QFT of Charged n-Particle: The Canonical 1-Particle
Posted by Urs Schreiber
A category of histories for the 1-particle, whose Leinster measure reproduces (a discretized approximation of the Euclidean version of) the path integral measure for the charged 1-particle on the real line.
News about E8
Posted by John Baez
The exceptional Lie group E8 is a marvelous 248-dimensional monster, with mysterious connections to the octonions and string theory. Here’s a nice webpage about a new calculation involving :
- American Institute of Mathematics, Mathematicians map E8.
As part of a project called the Atlas of Lie Groups and Representations, a team of mathematicians led by Jeffrey Adams have computed the Kazhdan–Lusztig–Vogan polynomials for .
You may have heard some hype about this, because it’s a really big calculation, and the American Institute of Mathematics has coaxed a lot of science reporters to write about it — in part by comparing it to the human genome project.
To see what was really done, try these:
- David Vogan, Narrative of the project to compute Kazhdan–Lusztig polynomials for E8.
- Jeffrey Adams, Technical details.
March 16, 2007
Quantum Logic, Topology and Categories at Oxford
Posted by John Baez
Café regulars tend to enjoy the analogies between computation, logic, topology, and quantum physics, since -categories are a great way to make these analogies precise. There will be a conference touching on these topics in Oxford:
- Algebraic and Topological Methods in Non-Classical Logics III, Saint Anne’s College, University of Oxford, August 5-9, 2007.
It has a number of satellite workshops, including one on Categorical Quantum Logic, August 10-12 at the Oxford University Computing Laboratory, organized by Bob Coecke. I hope to attend that.
Poisson-Sigma Models, Lie Algebroids, Deformations and Higher Analogues in Vienna
Posted by Urs Schreiber
This summer the Erwin-Schrödinger Institute in Vienna hosts a program
Poisson Sigma Models, Lie Algebroids, Deformations and Higher Analogues
Organizers: H. Bursztyn, H. Grosse and T. Strobl
August 1 to September 30, 2007
There seems to be no web page with further details yet.
The list of topics in the title are related to the stuff I was blogging about almost two years ago on the String Coffee Table:
PSM and Algebroids, Part I
PSM and Algebroids, Part II
PSM and Algebroids, Part III
PSM and Algebroids, Part IV
PSM and Algebroids, Part V.
I will spend a period of two weeks in the time Aug 1 to Sept 30 in Vienna, still having to decide which two weeks exactly. If any other -Café-reader will be there, too, we could maybe coordinate our visits. Just drop me an email, if so.
March 15, 2007
Classical vs Quantum Computation (Week 18)
Posted by John Baez
Today we had our last class on Classical vs Quantum Computation for this quarter:
-
Week 18 (Mar. 15) -
2-categories from typed λ-calculi, continued.
The Church-Rosser theorem. Surface diagrams showing
the process of computation. β-reduction as a fold
catastrophe. Thom’s ideas on catastrophe theory.
Challenge: draw the surface diagram for η-reduction.
Which catastrophe does it correspond to?
Blog entry.
Supplementary reading:
- Lucien Dujardin, Catastrophe teacher: an introduction for experimentalists.
- Barnaby P. Hilken, Towards a proof theory of rewriting: the simply-typed 2λ-calculus, Theor. Comp. Sci. 170 (1996), 407–444.
- R. A. G. Seely, Weak adjointness in proof theory in Proc. Durham Conf. on Applications of Sheaves, Springer Lecture Notes in Mathematics 753, Springer, Berlin, 1979, pp. 697–701.
- R. A. G. Seely, Modeling computations: a 2-categorical framework, in Proc. Symposium on Logic in Computer Science 1987, Computer Society of the IEEE, pp. 65–71.
- C. Barry Jay and Neil Ghani, The virtues of eta-expansion, J. Functional Programming 1 (1993), 1–19.
Star-Structures and Daggers
Posted by Urs Schreiber
A question by Bruce Bartlett:
Hi guys,
I’ve got a question about duality for 2-categories. John Baez and Laurel Langford defined what a “monoidal 2-category with duals” was in HDA IV. The basic concept is easy enough to understand. A monoidal 2-category with duals is a 2-category with duals on all levels : duals for objects, morphisms and 2-morphisms.
Thus every 2-morphism has a dual , every morphism has a dual and every object has a dual . That’s the basic picture.
For our purposes here, we can ignore the tensor product and the duals for objects side of things, so don’t worry about that.
Since this is a long post, for the experts I’ll state my question right up. Can anyone help me understand the equation
.