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February 29, 2008
Charges and Twisted n-Bundles, I
Posted by Urs Schreiber
I want to talk about
twisted -bundles with connection over -dimensional base space
coupled to
(electrically) charged -particles (()-branes);
how they are
sections with covariant derivatives of -bundles with connection
which can be interpreted as
obstructions to lifts through extensions of -groups
or equivalently
magnetic charges
of
magnetically charged ()-particles (()-branes).
A crucial new ingredient compared to my former (I,II) discussion of sections of -bundles is the method from groupoidification: think of an -representation of an -group not as an -functor, but in terms of the corresponding action -groupoid, as described more recently in -associated bundles and sections.
Much of what I say is, in the language of generalized differential cohomology, in the great
D. S. Freed
Dirac Charge Quantization and Generalized Differential Cohomology
(arXiv)
only that what I describe in the language of -parallel transport/-connections (pdf, arXiv, blog, brief slides, detailed slides) is supposed to be a nonabelian generalization of the purely abelian treatment which allows to not just say things like
The Green-Schwarz mechanism says that the Kalb-Ramond 2-bundle coupling to the electric string is twisted by magnetic 5-brane charge.
(discussion around equation (3.5) in Freed’s article)
But also things like
The Green-Schwarz mechanism says that over the 10-dimensional boundary of 11-dimensional supergravity base space the supergravity Chern-Simons 3-bundle obstructing the lift of an -bundle to a -2-bundle trivializes by admitting a global 2-section: the twisted Kalb-Ramond 2-bundle.
(previewed in section 3 of -connections)
where the underlying -bundle as well as its String-2-bundle lift with connections represent nonabelian differential cohomology.
LaTeXified notes on what I will talk about are beginning to evolve as
Sections and covariant derivatives of -algebra connections
(pdf, blog, Bruce Bartlett’s recollection)
and
Twisted -connections
(pdf).
February 28, 2008
Peirce on Mathematics
Posted by David Corfield
By the winter of 1897-8, the jobless philosopher Charles Peirce was financially crippled, continuing his studies despite the cold, unable to heat his house. Regretting this situation, William James, at Harvard, organised for Peirce to give a series of lectures for much needed remuneration. Peirce wanted to talk about formal logic, but James persuaded him that this would severely reduce his audience (“Now be a good boy and think a more popular plan out. I don’t want the audience to dwindle to 3 or 4…”), and that he would be better off discussing “topics of a vitally important character”.
He did adjust the content somewhat, as you can see in the published lectures Reasoning and the Logic of Things, (Kenneth Ketner ed., Harvard University Press, 1992), where you can read some of the correspondence between Peirce and James.
February 27, 2008
(Generalized) Differential Cohomology and Lie Infinity-Connections
Posted by Urs Schreiber
Over the years, Dan Freed, Michael Hopkins and I. M. Singer and others have been developing the theory of Generalized differential cohomology and applied it with great success to various problems appearing in the theory of charged -particles, usually known as NS-branes and D-branes and M-branes.
The idea is:
Given a class in the (generalized) cohomology of a space , regard it as classifying an -bundle-like thing and then find a way to equip that with something like a connection , such that the curvature differential form of that connection reproduces the image of in deRham cohomology (with coefficients in the ring ):
When is ordinary integral cohomology, this reproduces the notion of Cheeger-Simons differential characters, which is a way of talking about equipping line -bundles ((n-1) gerbes) with a connection.
The notion of generalized differential cohomology allows to go beyond that and equip any other kind of cohomology class with a corresponding notion of “connection and curvature”. This has notably been applied to the next interesting generalized cohomology theory after ordinary integral cohomology: K-theory. It turns out that the differential forms appearing in differential K-theory model the RR-fields appearing in string theory.
Here I try to review some basics, provide some links – and then start to relate all this to the theory of parallel -transport and -connections.
New Hire at UCR
Posted by John Baez
Yay!
Julia Bergner has accepted a tenure-track position in the math department here at U. C. Riverside. She’s just finishing her third year as a postdoc at the University of Kansas. She’s done a lot of important work on the ‘homotopy theory of homotopy theories’.
So, life here at UCR just got a lot more interesting for people who like -categories.
February 26, 2008
Impressions on Infinity-Lie Theory
Posted by Urs Schreiber
While talking to people who will hold still and listen, like when I talk to Bruce Bartlett or to Danny Stevenson, I realized that while I talked about integration theory of Lie -algebras here and there, I might not have gotten my point across succinctly.
So, as a kind of meta-response to some aspect of Bruce’s highly appreciated exegesis. I have now prepared a manifesto:
Impressions on -Lie theory
pdf (6 pages)Abstract. I chat about some of the known aspects of the categorified version of Lie theory – the relation between Lie -algebras and Lie -groups – indicate how I am thinking about it, talk about open problems to be solved and ideas for how to solve them.
This starts with very roughly reviewing the basic ideas underlying the work by Getzler, Henriques and Ševera, then exhibits the “shift in perspective” which I keep finding helpful and important, mentions applications like the strict integration of the String Lie 2-algebra using strict path 2-groupoids and ends by quickly sketching how that relates the “fundamental problem of Lie -theory” to the relation between smooth spaces and “DGCAs” which we are having a long discussion about with Todd Trimble and Andrew Stacey, scattered over various threads (notably here, and here).
At the end of the notes I indicate how I am currently trying to address this issue. But I need more time to work this out. Comments are appreciated.
What I learned from Urs
Posted by Urs Schreiber
Guest post by Bruce Bartlett
Dear reader,
I’m sure you’ll agree with me that there is a remarkable person on this blog : Urs. The rate at which he produces new posts and deep ideas is nothing short of a phenomenon. Indeed, he is so fast that perhaps many of you are like me and have been left in the dust long ago!
If so, this post is for you! I was lucky enough to have Urs visit me recently, and after much patience on his part I think I am finally beginning to see the first glimmers of daylight. Let me mention some of the things he explained to me; perhaps it will help some of you to understand what Urs has been going on about.
February 25, 2008
A Question or Two
Posted by David Corfield
Points of a set, , correspond to certain maps from the Boolean algebra of subsets, , to , namely those corresponding to prime ideals of the algebra.
Points of a space, , correspond to certain functors from the topos of locally constant sheaves to Set, via evaluation at a point again. Is there a way to construe this by analogy to the prime ideal story? Is there a ‘spectrum’ around?
How does one characterise the fibre functors to Set which correspond to points? Is there something ‘ideal’ going on?
Cartier’s Mad Day’s Work paper seems to suggest there is such a story going on here. In the same paragraph (p. 404) as the description of the fundamental group as the automorphism group of a fibre functor, he speaks of the Galois group of a field extension in terms of the fields’ spectra.
February 21, 2008
Lurie on Extended TQFT
Posted by Urs Schreiber
Over at the Secret Blogging Seminar Noah Snyder is reporting on talks Jacob Lurie gave on extended TQFT, which is the theory of representations of -categories of cobordisms (in contrast to ordinary TQFT, which is just representations of mere 1-categories of cobordisms):
Noah Snyder
Jacob Lurie on 2-d TQFT
From Noah’s notes, the talks mostly centered on the observation by John Baez and James Dolan
John C. Baez, James Dolan
Higher-dimensional Algebra and Topological Quantum Field Theory
arXiv:q-alg/9503002v2
that this should be essentially about representing the free stable -groupoid and that hence such a representation is fixed by choosing just one object, the image of the point, with suitable dualities on it. See also our recent discussion about that here.
The extended TQFT would then be an assignment of this object to the point, and of the -fold higher “trace” on this object to closed -dimensional manifolds.
In his talk Lurie apparently talked about some new classification results on this. With a little luck, more details will percolate through to us eventually.
Incidentally, while typing this I am on my way back from Edinburgh to Sheffield to meet Bruce Bartlett again. Bruce is writing his PhD thesis, advised by Simon Willerton, on a beautiful description of (higher) Dijkgraaf-Witten theory – a finite group version of Chern-Simons theory – in this extended sense. He finds plenty of fascinating relations between these “higher traces” and finite group representation theory, providing a useful blueprint for and nice insights into what fully extended Chern-Simons theory has to eventually look like.
Logicians Needed Now
Posted by John Baez
Mike Stay and I are writing a paper for a book Bob Coecke is editing: New Structures for Physics. The deadline is coming up soon, and we need your help!
- John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone. (Draft version: rose.pdf.)
We’d really love comments on the ‘logic’ section, because neither of us are professional logicians. I haven’t included the ‘computation’ section in this draft, since it’s embarrassingly far from finished… but Mike knows computation.
February 20, 2008
Kostant on E8
Posted by John Baez
At Riverside we recently heard a fascinating talk:
-
Bertram Kostant, On Some Mathematics in Garrett Lisi’s ‘E8 Theory of Everything’, U. C. Riverside, February 12th.
Abstract: A physicist, Garrett Lisi, has published a highly controversial, but fascinating, paper purporting to go beyond the Standard Model in that it unifies all 4 forces of nature by using as gauge group the exceptional Lie group . My talk, strictly mathematical, will be about an elaboration of the mathematics of which Lisi relies on to construct his theory.
Luckily we had a video camera on hand. So, at the above link you can see streaming or downloadable videos of Kostant’s talk, as well as lecture notes.
February 19, 2008
Categories, Logic and Physics in London, II
Posted by Urs Schreiber
Andreas Döring writes:
Dear all,
we hereby wish to invite you to participate in the second workshop on “Categories, Logic and Foundations of Physics”, which will take place at Imperial College on
Wednesday, 14th May 2008, 11:00 - 18:45, Lecture Theatre 3, Blackett Laboratory.
Our workshop series is aimed at nourishing research in the fields named in the title and at bringing together scientists from the different fields involved. The success of the first workshop already showed that there is substantial interest in these topics and a great potential for collaborations. With the second workshop, we hope to keep and increase the momentum.
Please also have a look at the website.
The videos and slides of the January workshop are online now.
February 18, 2008
Harvard Research Free Online
Posted by John Baez
Last Tuesday, the Harvard Faculty of Arts and Sciences voted overwhelmingly to make their research papers freely available online!
Under the new system, faculty will deposit finished papers in an open-access repository run by the library. The papers will instantly become available for free on the Internet. Authors will still retain their copyright. So, they can still publish an