Posts with this logo use 
October 31, 2007
On Noether’s Second (BV, Part VI)
Posted by Urs Schreiber
One aim of
R. Fulp, T. Lada, J. Stasheff
Noether’s Variational Theorem II and the BV formalism
math/0204079
was to
[…] restore […] an emphasis [on] the relevance of Noether’s theorem in […] the BV approach
Namely it is Noether’s second theorem (see page 6 of the above article) for Lagrangian theories which is reincarnated equivalently in the BV statement that
the space of ghosts is canonically isomorphic to that of anti-ghosts.
Meaning that
For every Noether identity there is a symmetry. And vice versa.
In terms of the little toy example (which is not that toy-ish, actually, rather skeletalized, I think), which I talked about last time (see also parts I, II, III, IV), this means that in our little complex
which is induced entirely from a smooth function on a manifold , we have a canonical isomorphism between the first and the last term And this canonical isomorphism is, I think, Noether’s second theorem in this context.
And I’ll claim: this is here nothing but a special case of Cartan’s magic formula (or whatever you call that).
For that to make sense, I’ll first need to say mor precisely how the in is defined in the first place.
October 30, 2007
BV for Dummies (Part V)
Posted by Urs Schreiber
On my way back from Oxford, I am spending a night in a hotel close to Manchester airport to get my plane tomorrow morning. Luckily they have a public terminal here. This allows me to talk a little about BV formalism.
John began his last course on Quantization and Cohomology by focusing a bit of attention on a seemingly boring special case: that of statics instead of dynamics.
Here I’ll do something similar for the BV formalism (Part I, II, III, IV):
I use a 4-term complex of vector spaces to study the simple situation of a compact manifold equipped with a smooth real-valued function which you may think of as a Lagrangian depending only on the fields (not on their derivatives) which are elements of . That complex of vector spaces will extract for us the nature of the critical points of .
If you like, read this in parallel with Jim Stasheff’s hep-th/9712157 from which it follows by truncating the jet space completely down to its 0th component.
On the other hand, if you follow in thoughts the point of view adopted with considerable success by Lyakhovich and Sharapov, who get quite far with thinking of field theory Lagrangians as functions on a finite-dimensional space, you may regard, I guess, the following also as a picture of aspects of the full BV machinery.
In any case, the little pedagogical exercise here is mainly supposed to make explicit the simple nature of the complex we are dealing with, which is essentially
where denotes the Lie algebra of symmetries of our function.
And probably I won’t be able to refrain from making some comments on the higher categorical interpretation of what is going on.
Of course for a general BV situation this may be a higher Lie algebra (a Lie -algebra`) and its action may be weak and all that, but that shall be ignored here for the time being.
Comet Holmes
Posted by John Baez
Have any of you folks seen Comet Holmes? It was just another boring little comet somewhere between Mars and Jupiter when it suddenly got a million times brighter on October 23rd, going from magnitude 17 to magnitude 2.8 in just a few hours!
According to the magazine Sky and Telescope, it’s easy to spot with the naked eye…
Higher Clifford Algebras
Posted by John Baez
Lately Urs has been dreaming of categorified Clifford algebras. But he’s not the only one! We should send one of our spies to this talk tomorrow:
- Chris Douglas, Higher Clifford algebras, Topology Seminar, Chicago University, talk in E203 at 4:30 pm, pre-talk in the same room at 3:00, October 30, 2007.
October 29, 2007
Fundamental Physics: Where We Stand Today
Posted by John Baez
Are most of the entries on this blog too technical for you? Well, try this:
-
John Baez, Fundamental physics: where we stand today, Department of Physics and Astronomy, James Madison University, November 2, 2007.
Since the discovery of the W and Z particles over twenty years ago, few truly novel predictions of fundamental theoretical physics have been confirmed by experiment. On the other hand, observations in astronomy have revealed shocking facts that our theories do not really explain: most of our universe consists of "dark matter" and "dark energy". Where does fundamental physics stand today, and why has theory become divorced from experiment?
It’s a talk for anyone interested in physics: a few equations at first, when I explain general relativity, but then just words and pictures!
October 27, 2007
Steve Fever
Posted by John Baez
Yesterday when I read my MIT alumni magazine, I was pleased to see a short story by Greg Egan. This magazine is available for free online if you submit to a mildly annoying registration process, so I’ll advertise the story here:
- Greg Egan, Steve Fever.
It’s about an artificial intelligence so stupid it believes what it reads on the internet.
October 26, 2007
Concrete Groups and Axiomatic Theories I
Posted by Guest
Guest post by Todd Trimble
I’d like to take a shot here at explaining some of the ideas on logic that Jim Dolan has been alluding to in his talks in the Geometric Representation Theory seminar, and eventually give an argument for a perhaps surprising idea of his, that “concrete groupoid theory” and “axiomatic theories” are really the same subject, from a kind of Galois theory point of view. This is meant to set the stage for a whole slew of interesting developments, in which we view Jim’s orbi-simplex idea as a geometric description of a general axiomatic theory, which in turn is related to the idea of viewing Tits buildings as “quantized” axiomatic theories, and also perhaps to the theory of classifying toposes and their “Galois theory”. But we’ll get to all that later!
Right now I’d just like to set the scene, and try to flesh out the (somewhat skeletal) description Jim gave of axiomatic theories in precise terms, up to the point where we can at least state the amazing Galois correspondence between groups and theories. In part II, we’ll have a look at proving that this correspondence really works, in part by adapting an interesting argument of Joyal that characterizes analytic functors of species (themselves closely related to the Tale of Groupoidification!).
Geometric Representation Theory (Lecture 6)
Posted by John Baez
Where would a wizard be without his magic wands?
In mathematics, a ‘magic wand’ is any systematic process that you can apply to big chunks of interesting mathematics and get new, more interesting mathematics. Or — more magical still — it’s a mysterious bunch of tricks that feel like they’d be part of a systematic process if only we understood them better.
What are some magic wands? One of the most famous was stolen from physicists: it’s called quantization. Muttering one of several cryptic spells, you can wave this wand over any mathematical concept related to classical mechanics, and hope that — POOF! — it suddenly transforms into an analogous concept related to quantum mechanics. We’ve had huge success with this over the last century, but it’s still poorly understood.
Another magic wand is categorification: replacing any number by a set with that number of elements, replacing any set by a category whose set of isomorphism classes it is, and so on. You could almost say this blog is a shrine to categorification. It too, is still poorly understood. Perhaps when a magic wand’s powers become fully understood, it ceases to count as ‘magic’!
Yet another magic wand is -deformation — closely related to quantization but not the same. It’s a way of modifying mathematical entities that depends on a parameter . Sometimes this parameter has the physical meaning of … but sometimes it’s better to think of it as a power of a prime number! In fact, -deformation was discovered by Gauss long before the quantum was a twinkle in Planck’s eye.
When you have two magic wands at your disposal, you can ask if they commute. First wave one, then the other. First wave the other, then the one. Does the same magic occur? Or at least isomorphic magics?
In lecture 6 of the Geometric Representation Theory seminar, I wave two magic wands — categorification and -deformation — at a humble mathematical entity: the binomial coefficient. It seems they commute. But, puzzles abound!
October 23, 2007
On String- and Chern-Simons n-Transport
Posted by Urs Schreiber
I am making the last preparations for a little journey to Great Britain.
Tomorrow starts the Conference: Lie Algebroids and Lie Groupoids in Differential Geometry in Sheffield. Next Monday then I am invited to speak at the Oxford geometry seminar.
On both occasions I’ll talk about selected topics from
String- and Chern-Simons -Transport
(pdf slides)
in Sheffield with an emphasis on the Baez-Crans type/String-like Lie -algebras and related matters, in Oxford with an emphasis on bundle gerbes.
These slides currently serve for me the purpose of a substitute for our cool-but-non-existing-higher-Wiki and are supposed to be treated as such. They should be comparatively enjoyable to read (on the screen, don’t ever try to to print them) if use is made of the hypertext tools provided by your pdf-reader. (Use the arrow keys to read sequentially, remeber your pdf-reader’s internal back button for convenient hyperlink navigation within the pdf document).
To get going, you might want to surf to section Introduction, subsection Plan and have a look at the menu of links provided there. In sub-subsection Categorfication, local trivialization, differentiation you’ll find an “animated and subtitled” version of the classical transport cube playing the role of a 3-dimensional table of contents.
This classical cube is the one whose first edge is local trivialization, whose second edge is differentiation and whose third edge is categorification. Keeping these three directions in mind should help see the big picture behind the details.
I am looking forward to meeting Bruce Bartlett and Simon Willerton in Sheffield. I had been in Sheffield before only once, about 17 years ago, or so, when I stayed for 2 weeks with a guest family.
October 22, 2007
Geometric Representation Theory (Lecture 5)
Posted by John Baez
Felix Klein had a great idea: a lot of geometry is secretly group theory. Say you’ve got a group of symmetries, and it acts transitively on a set of geometrical figures of some type. This means that
for some subgroup , namely the ‘stabilizer’ subgroup — the subgroup that preserves a figure. So, you get types of figures from subgroups of your symmetry group.
But, there’s a lot more. Say you have some relation between figures of type and figures of type that’s invariant under your symmetry group. For example: ‘a point lies on a line’.
This means you have a subset that’s invariant under the action of . You can think of this as an -shaped matrix of 1’s and 0’s: 1’s where the relation holds, 0’s where it doesn’t. But, such a matrix can be reinterpreted as a linear operator
The invariance condition then means this is an intertwining operator between permutation representations of .
A wonderful fact — though the proof is easy — is that we can get a basis of intertwining operators this way, called ‘Hecke operators’. We get this basis from ‘atomic’ invariant relations, meaning those that can’t be chopped up into a disjunction — a logical ‘or’ — of smaller relations. Another way to think about it: these atomic invariant relations are just -orbits in .
Soon we’ll use all this to take permutation representations of groups and chop them into irreducible representations. So: we’ll turning the insight of Klein around, and use geometry to study group representations!
Today, in the 5th lecture of our Geometric Representation Theory seminar, James Dolan works through some easy examples of Hecke operators.
October 20, 2007
On Lie N-tegration and Rational Homotopy Theory
Posted by Urs Schreiber
In rational homotopy theory one studies spaces “up to finite ambiguity” as Dennis Sullivan put it, namely by considering all forms of homotopy and (co)homology over the rationals (i.e discarding all torsion information).
For an overview see for instance
Kathryn Hess
Rational Homotopy Theory: A Brief Introduction
(2000)
(pdf).
The crucial insight of Dennis Sullivan described in
Dennis Sullivan
Infinitesimal computations in topology
Publications mathématiqeu de l’ I.H.É.S., tome 47 (1977), p. 269-331
(NUMDAM)
was that all rational spaces are obtained from integrating Lie -algebras.
Of course Sullivan didn’t put it that way, nor do many people in rational homotopy theory. Instead they are talking about differential graded commutive algebras, which are freely generated in positive degree, as graded commutative algebras.
Here at the -Café we call (following Jim Stasheff’s suggestion) such dg-algebras “quasi free differential algebras” (qDGCAs) and are fond of the fact that they are dual to codifferential coalgebras, which are the same as -algebras, which are the same as Lie -algebras, which are -fold categorifications of Lie algebras. For a quick reminder on how this works, see Lie -algebra cohomology. For a bestiary of examples, most of them described in both languages, try Zoo of Lie