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November 30, 2006
Puzzle #8
Posted by John Baez
Q: Which 39-year-old female mathematician was rumored in 1999 to be secretly in charge of one of the world’s largest countries?
Warning: since I first posted this puzzle, someone told me the rumor was not only false, but that it was an exaggeration to call this woman a “mathematician”. So: extra credit for more details on that issue!
November 29, 2006
Nicolai on E10 and Supergravity
Posted by Urs Schreiber
We have had several discussions here on how (parts of) the Lie algebra of the gauge group governing 11-dimensional # and 10-dimensional # supergravity can rightly be thought of in terms of semistrict Lie 3-algebras (equivalently: 3-term -algebras).
There are various reasons that make some people expect that these various supergravity theories describe certain facets of some essentially unknown single entity. The working title of this unknown structure is “M-theory”. You’ll see one proposal for a precise statement of this “M-theory hypothesis” in a moment.
In our discussions, I had made a remark on how the various Lie 3-algebras that play a role in supergravity might - or might not - be merged into a single structure here.
John rightly remarked that
This M-theory Lie 3-superalgebra should ultimately be something very beautiful and integrated, not a bunch of pieces tacked together, if M-theory is as Magnificent as it’s supposed to be.
There are various indications that the unifying governing structure behind much of the supergravity zoo are exceptional Kac-Moody algebras, in particular , , and maybe also .
In particular, if one takes 11-dimensional supergravity and compactifies it on a 10-dimensional torus, the resulting 1-dimensional field theory exhibits a gauge symmetry under the gauge group , where denotes something like the group manifold and something like the maximal compact subgroup of .
This, combined with the observation of certain symmetries appearing in the chaotic dynamics of (super)gravity theories near spacelike singularities, has lead a couple of people, most notably H. Nicolai, T. Damour, M. Henneaux, T. Fischbacher and A. Kleinschmidt, to suspect that the classical dynamics encoded in the equations of motion of 11-dimensional supergravity, including its higher order M-theoretic corrections, correspond to geodesic motion on the group manifold of the Kac-Moody group , or rather the coset .
Since is hyperbolic, it is, with current technology, impossible to conceive it in its entirety. Hence all this work is based on a technique, where one uses a certain level truncation of the Kac-Moody algebra to obtain tractable and useful approximations to the full object.
The idea is that expanding geodesic motion on in terms of levels this way, corresponds to expanding a supergravity theory in powers of spatial gradients of its fields close to a spacelike singularity.
For several years now, Hermann Nicolai and collaborators have slowly but steadily checked this hypothesis for low levels.
I had once reviewed some basic aspects of this here.
So far, to the degree of detail that has become accessible, the hypothesis has proven to be correct. And, as the M-theory hypothesis would suggest, not only can 11-dimensional supergravity be found, level by level (up to level 3, so far), in the geodesic motion on , but higher levels seem to correctly reproduce higher order corrections to supergravity which have been derived by other means. Moreover, depending on how one “slices” by means of its subalgebras, one finds that the same geodesic motion also reproduces the other maximal supergravity theories, like 10-dimensional type IIB supergravity and massive 10-dimensional IIA supergravity.
Up to recently, all this work was restricted to the bosonic degrees of freedom of these theories. One of the remarkable aspects of the theory was that it gave rise to the various bosonic fields that accompany the graviton field (the Riemannian metric) in supergravity theories, like the supergravity 3-form, and which ordinarily appear only after one requires supersymmetry.
Still, one would like to check the entire program also against the fermionic fields, like the gravitino. The obvious guess is that these appear on the -side as we pass from the geodesic motion of a spinless particle on to the motion of a spinning particle.
Results on this part of the project are now also appearing. Today has appeared a new preprint, where further progress in this direction is discussed:
Axel Kleinschmidt, Hermann Nicolai
K(E9) from K(E10)
hep-th/0611314.
Among other things, it is discussed how has certain finite-dimensional spinorial representations under which - on the corresponding supergravity side of things - the equation of motion of the gravitino is covariant.
Today Hermann Nicolai visited Hamburg and gave a talk on this stuff:
E. Nicolai
: Prospects and Challenges
(slides).
November 28, 2006
D-Branes from Tin Cans, II
Posted by Urs Schreiber
A brief note on how a 2-section of a transport 2-functor transgressed to the configuration space of the open 2-particle (string) encodes gerbe modules (Chan-Paton bundles) associated to the endpoints of the 2-particle.
November 27, 2006
NIPS 2006
Posted by David Corfield
In a week’s time I shall be in Vancouver attending the NIPS 2006 conference. NIPS stands for Neural Information Processing Systems. I’m looking forward to meeting some of the people whose work I’ve been reading over the past twenty months. Later in the week I shall be speaking up in Whistler at a workshop called ‘Learning when test and training inputs have different distributions’, and hopefully fitting in some skiing.
In a way you could say all of our use of experience to make predictions encounters the problem addressed by the workshop. If we include time as as one of the input variables, then our experience or ‘training sample’ has been gathered in the past, and we hope to apply it to situations in the future. Or from our experience gathered here, we expect certain things to happen there. How is it, though, that sometimes you know time, space, or some other variable, don’t matter, whereas other times you know they do?
November 25, 2006
Puzzle #7
Posted by John Baez
November 24, 2006
2-Monoid of Observables on String-G
Posted by Urs Schreiber
The baby version of the Freed-Hopkins-Teleman result, as explained by Simon Willerton, suggests that we should be thinking of the modular tensor category
that govers -Chern Simons theory and Wess-Zumino-Witten theory rather in terms of the representation category
of a central extension of the action groupoid
of the adjoint action of on itself.
This monoidal category should arise as the 2-monoid of observables # that acts on the 2-space of states over a point as we consider the 3-particle propagating on a target space that resembles .
In turn, this 2-monoid of observables should arise # as the endomorphisms of the trivial transport on target space
Here I would like to show that when we model target space as
where is the strict string 2-group #, coming from the crossed module # , then sections on configuration space of the boundary of the 3-particle form a module category for
As discussed elsewhere (currently at the end of these notes), this should imply that states over a point are a module for
November 23, 2006
The Baby Version of Freed-Hopkins-Teleman
Posted by Urs Schreiber
Recently I had discussed # one aspect of the paper
Simon Willerton
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
math.QA/0503266 .
There are many nice insights in that work. One of them is a rather shockingly simple explanation of the nature of the celebrated Freed-Hopkins-Teleman result # - obtained by finding its analog for finite groups.
Here I will briefly say what Freed-Hopkins-Teleman have shown for Lie groups, and how Simon Willerton finds the analog of that for finite groups.
The 1-Dimensional 3-Vector Space
Posted by Urs Schreiber
I feel a certain need for 3-vector spaces, for 3-reps of 3-groups on 3-vector spaces. And things like that. But 1-dimensional 3-vector spaces would do.
Here I shall talk about how, for any braided abelian monoidal category , the 3-category
plays the role of the 3-category of canonical 1-dimensional 3-vector spaces.
Moreover, I would like to point out how morphisms between almost-trivial line-3-bundles with connection give rise to the 3-category of twisted bimodules that I talked about recently #.
This 3-category is a beautiful gadget. For and any commutative ring,
is discussed in the last part of
R. Gordon, A.J. Power and R. Street,
Coherence for tricategories,
Memoirs of the American Math. Society 117 (1995) Number 558.
John Baez describes this guy in TWF 209.
I first got interested in it here, but for a dumb reason it took me until last night to realize that this is the 3-category of canonical 1-dimensional 3-vector spaces that I was looking for all along.
For reading on, you have to leave the room and go to this file:
A Third Model of the String Lie 2-Algebra
Posted by John Baez
One of the main themes of this blog is categorification: taking mathematical structures that are sets with extra structure, and replacing equations by isomorphisms to make them into categories. A wonderful fact is that any Lie algebra has a god-given one-parameter family of categorifications . We already have two ways to construct this gadget. Now this paper gives a third:
-
Friederich Wagemann, On Lie algebra crossed modules,
Communications in Algebra 34 (2006), 1699-1722.
Abstract: This article constructs a crossed module corresponding to the generator of the third cohomology group with trivial coefficients of a complex simple Lie algebra. This generator reads as , constructed from the Lie bracket and the Killing form . The construction is inspired by the corresponding construction for the Lie algebra of formal vector fields in one formal variable on , and its subalgebra , where the generator is usually called Godbillon-Vey class.
November 21, 2006
Classical vs Quantum Computation (Week 7)
Posted by John Baez
Here are this week’s notes on Classical versus Quantum Computation:
- Week 7 (Nov. 21) - The untyped lambda-calculus, continued. “Building a computer” inside the free cartesian closed category on an object with . Operations on booleans. The "if-then-else" construction. Addition and multiplication of Church numerals. Defining functions recursively: the astounding Fixed Point Theorem.
Basic Question on Homs in 2-Cat
Posted by Urs Schreiber
I have
John W. Gray
Formal Category Theory: Adjointness for 2-Categories
Springer, 1974
in front of me, but I haven’t absorbed it yet. I am looking for information about the following question:
In the world of strict 2-categories, strict 2-functors, pseudonatural transformations and modifications of these, consider three 2-categories
How is
related to
?
Here denotes the 2-category of 2-functors from to , pseudonatural transformations and modifications.
I am interested in this question, because it seems - unless I am hallucinating - to play a role in the construction of extended 2-dimensional quantum field theories #.
I see that one answer to this question is provided by item iii) of theorem I.4.14 of Gray’s text. But I need to better understand what this theorem tells me in practice.