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Work on quantum gravity has seemed stagnant and stuck for the last couple of years, which is why I've been turning more towards pure math.
Over in string theory they're contemplating a vast "landscape" of possible universes, each with their own laws of physics - one or more of which might be ours. Each one is supposed to correspond to a different "vacuum" or "background" for the marvelous unifying M-theory that we don't completely understand yet. They can't choose the right vacuum except by the good old method of fitting the experimental data. But these days, this time-honored method gets a lot less airplay than the "anthropic principle":
1) Leonard Susskind, The anthropic landscape of string theory, available as hep-th/0302219.
Perhaps this is because it's more grandiose to imagine choosing one theory out of a multitude by discovering that it's among the few that supports intelligent life, than by noticing that it correctly predicts experimental results. Or, perhaps it's because nobody really knows how to get string theory to predict experimental results! Even after you chose a vacuum, you'd need to see how supersymmetry gets broken, and this remain quite obscure.
There's still tons of beautiful math coming out of string theory, mind you: right now I'm just talking about physics.
What about loop quantum gravity? This line of research has always been less ambitious than string theory. Instead of finding the correct theory of everything, its goal has merely been to find any theory that combines gravity and quantum mechanics in a background-free way. But, it has major problems of its own: nobody knows how it can successfully mimic general relativity at large length scales, as it must to be realistic! Old-fashioned perturbative quantum gravity failed on this score because it wasn't renormalizable. Loop quantum gravity may get around this somehow... but it's about time to see exactly how.
Loop quantum gravity follows two main approaches: the so-called "Hamiltonian" or "spin network" approach, which focuses on the geometry of space at a given time, and the so-called "Lagrangian" or "spin foam" approach, which focuses on the geometry of spacetime.
In the last couple of years, the most interesting new work in the Hamiltonian approach has focussed on problems with extra symmetry, like black holes and the big bang. Here's a nontechnical introduction:
2) Abhay Ashtekar, Gravity and the quantum, available as gr-qc/0410054.
and here's some new work that treats the information loss puzzle:
3) Abhay Ashtekar and Martin Bojowald, Black hole evaporation: a paradigm, Class. Quant. Grav. 22 (2005) 3349-3362. Also available as gr-qc/0504029.
However, by focusing on solutions with extra symmetry, one puts off facing the hardest aspects of renormalization, or whatever its equivalent might be in loop quantum gravity.
The other approach - the spin foam approach - got stalled when the most popular model seemed to give spacetimes made mostly of squashed-flat "degenerate 4-simplexes". Various papers have found an effect like this: see "week198" for more details. So, there's definitely a real phenomenon going on here. However, its physical significance remains a bit obscure. The devil is in the details.
In particular, even though the amplitude for a single large 4-simplex in the Barrett-Crane model is dominated by degenerate geometries, certain second derivatives of the amplitude might not - and this may be what really matters. Carlo Rovelli has recently come out with a paper on this:
4) Carlo Rovelli, Graviton propagator from background-independent quantum gravity, available as gr-qc/0508124.
If the idea holds up, I'll be pretty excited. If not, I'll be bummed. But luckily, I've already gone through the withdrawal pains of switching my focus away from quantum gravity. When you do theoretical physics, sometimes you feel the high of discovering hidden truths about the physical universe. Sometimes you feel the agony of suspecting that those "hidden truths" were probably just a bunch of baloney... or, realizing that you may never know. Ultimately nature has the last word.
Math is, at least for me, a less nerve-racking pursuit, since the truths we find can be confirmed simply by discussing them: we don't need to wait for experiment. Math is just as grand as physics, or more so. But it's more wispy and ethereal, since it's about pure pattern in general - not the particular magic patterns that became the world we see. So, the stakes are lower, but the odds are higher.
Speaking of math, I really want to talk about the Streetfest - the conference in honor of Ross Street's 60th birthday. It was a real blast: over sixty talks in two weeks in two cities, Sydney and Canberra. However, I accidentally left my notes from those talks at home before zipping off to Calgary for a summer school on homotopy theory:
5) Topics in Homotopy Theory, graduate summer school at the Pacific Institute of Mathematics run by Kristine Bauer and Laura Scull. Recommended reading material available at http://www.pims.math.ca/science/2005/05homotopy/reading.html
So, I'll say a bit about what I learned at this school.
Dan Dugger spoke about motivic homotopy theory, which was great, because I've been trying to understand stuff from number theory and algebraic geometry like the Weil conjectures, etale cohomology, motives, and Voevodsky's proof of the Milnor conjecture... and thanks to his wonderfully pedagogical lectures, it's all starting to make some sense!
I hope to talk about this someday, but not now.
Alejandro Adem spoke about orbifolds and group cohomology. Purely personally, the most exciting thing here was seeing that orbifolds can also be seen as certain kinds of topological groupoids, or stacks, or topoi... so that various versions of "categorified topology" are actually different faces of the same thing!
I may talk about this someday, too, but not now.
I spoke about higher gauge theory and its relation to Eilenberg-Mac Lane spaces. I may talk about that too someday, but not now.
Dev Sinha spoke about operads, and besides explaining the basics, he said a couple of things that really blew me away. So, I want to talk about this now.
For one, the homology of the little k-cubes operad is a graded version of the Poisson operad! For two, the little 2-cubes operad acts on the space of thickened long knots!
But for this to thrill you like it thrills me, I'd better say a word about operads - and especially little k-cubes operads.
Operads, and especially the little k-cubes operads, were invented by Peter May in the early 1970s to formalize the algebraic structures lurking in "infinite loop spaces". In "week149" I explained what infinite loop spaces are, and how they give generalized cohomology theories, but let's not get bogged down in this motivation now, since operads are actually quite simple.
In its simplest form, an operad is a gizmo that has for each n = 0,1,2,... a set O(n) whose elements are thought of as n-ary operations - operations with n inputs. It's good to draw such operations as black boxes with n input wires and one output:
\ | /
\ | /
\ | /
-----
| f |
-----
|
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For starters these operations are purely abstract things that
don't actually operate on anything. Only when we consider a
"representation" or "action" of an operad do they get incarnated
as actual n-ary operations on some set. The point of operads is
to study their actions.
But, for completeness, let me sketch the definition of an operad. An operad tells us how to compose its operations, like this:
\ / \ | / |
\ / \ | / |
----- ----- -----
| b | | c | | d |
----- ----- -----
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
-----
| a |
-----
|
|
Here we are composing a with b,c, and d to get an operation with 6
inputs called a o (b,c,d).
An operad needs to have a unary operation serving as the identity for composition. It also needs to satisfy an "associative law" that makes a composite of composites like this well-defined:
\ / | \ | / \ /
\ / | \ | / \ /
--- --- --- ---
| | | | | | | |
--- --- --- ---
\ | / /
\ | / /
\ | / /
----- ----- -----
| | | | | |
----- ----- -----
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
-----
| |
-----
|
|
(This picture has a 0-ary operation in it, just to emphasize
that this is allowed.)
That's the complete definition of a "planar operad". In a full-fledged operad we can do more: we can permute the inputs of any operation and get a new operation:
\ / /
/ /
/ \ /
/ /
/ / \
\ | /
-----
| |
-----
|
|
This gives actions of the permutation groups on the sets O(n).
We also demand that these actions be compatible with composition,
in a way that's supposed to be obvious from the pictures. For
example:
\ | / | \ / \\\ / / /
\ | / | \ / \\/ / /
--- --- --- /\\ / /
| a | | b | | c | / \\/ /
--- --- --- / / /
\ / / / / /\\
\ / / / | | \\\
\ / / / | | \\\
/ / --- --- ---
/ \ / = | b | | c | | a |
/ / --- --- ---
/ / \ \ | /
\ | / \ | /
----- -----
| d | | d |
----- -----
| |
| |
and similarly for permuting the inputs of the black boxes on
top.
Voilà!
Now, operads make sense in various contexts. So far we've been talking about operads that have a set O(n) of n-ary operations for each n. These have actions on sets, where each guy in O(n) gets incarnated as a function that eats n elements of some set and spits out an element of that set.
But historically, Peter May started by inventing operads that have a topological space of n-ary operations for each n. These like to act on topological spaces, with the operations getting incarnated as continuous maps.
Most importantly, he invented an operad called the "little k-cubes operad". Here O(n) is the space of ways of putting n nonoverlapping little k-dimensional cubes in a big one. We don't demand that the little cubes are actually cubes: they can be rectangular boxes. We do demand that their walls are nicely lined up with the walls of the big cube:
---------------------
| |
| ----- |
| ----- | | |
|| | | | |
|| | | | | typical
| ----- | | | 3-ary operation in the
| ----- | little 2-cubes operad
| ---------------- |
| | | |
| ---------------- |
| |
---------------------
This is an operation in O(3), where O is the little 2-cubes
operad. Or, at least it would be if I labelled each of the 3
little 2-cubes - we need that extra information.
We compose operations by sticking pictures like this into each of the little k-cubes in another picture like this! I should draw you an example, but I'm too lazy. So, figure it out yourself and check the associative law.
The reason this example is so important is that we get an action of the little k-cubes operad whenever we have a "k-fold loop space".
Starting from a space S equipped with a chosen point *, the k-fold loop space Ωk(S) is the space of all maps from a k-sphere into S that send the north pole to the point *. But this is also the space of all maps from a k-cube into S sending the boundary of the k-cube to the point *.
So, given n such such maps, we can glom them together using an n-ary operation in the little k-cubes operad:
---------------------
|*********************|
|***********-----*****|
|*-----****| |****|
|| |***| |****|
|| |***| |****|
| ----- ***| |****|
|***********-----*****|
|***----------------**|
|**| |*|
|***----------------**|
|*********************|
---------------------
where we map all the shaded stuff to the point *. We get
another map from the k-cube to S sending the boundary to *.
So:
But the really cool part is the converse:
This is too technical to make a good bumper sticker, so if you want people in your neighborhood to get interested in operads, I suggest combining both the above slogans into one:
Like any good slogan, this leaves out some important fine print, but it gets the basic idea across. Modulo some details, being a k-fold loop space amounts to having a bunch of operations: one for each way of stuffing little k-cubes in a big one!
By the way:
Speaking of bumper stickers, I'm in Montreal now, and there's a funky hangout on the Boulevard Saint-Laurent called Cafe π where people play chess - and they sell T-shirts, key rings, baseball caps and coffee mugs decorated with the Greek letter π! The T-shirts are great if you're going for a kind of math-nerd/punk look; I got one to wow the students in my undergraduate courses. I don't usually provide links to commercial websites, but I made an exception for Acme Klein Bottles, and I'll make an exception for Cafe π:
6) Cafe π, http://www.cafepi.ca/
Unfortunately they don't sell bumper stickers.
But where were we? Ah yes - the little k-cubes operad.
The little k-cubes operad sits in the little (k+1)-cubes operad in an obvious way. Indeed, it's a "sub-operad". So, we can take the limit as k goes to ∞ and form the "little ∞-cubes operad". Any infinite loop space gets an action of this... and that's why Peter May invented operads!
You can read more about these ideas in May's book:
7) J. Peter May, The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics 271, Springer, Berlin, 1972.
or for a more gentle treatment, try this expository article:
8) J. Peter May, Infinite loop space theory, Bull. Amer. Math. Soc. 83 (1977), 456-494.
But Dev Sinha told us about some subsequent work by Fred Cohen, who computed the homology and cohomology of the little k-cubes operad.
For this, we need to think about operads in the world of linear algebra. Here we consider operads that have a vector space of n-ary operations for each n, which get incarnated as multilinear maps when they act on some vector space. These are sometimes called "linear operads".
An example is the operad for Lie algebras. This one is called "Lie". Lie(n) is the vector space of n-ary operations that one can do whenever one has a Lie algebra. In this example:
a → a
(a,b) → [a,b]
You might think we need a second guy in Lie(2), namely
(a,b) → [b,a]
but the antisymmetry of the Lie bracket says this is linearly dependent on the first one:
[b,a] = -[a,b]
(a,b,c) → [[a,b],c]
(a,b,c) → [b,[a,c]]
You might think we need a third guy in Lie(3), for example
(a,b,c) → [a,[b,c]]
but the Jacobi identity says this is linearly dependent on the first two:
[a,[b,c]] = [[a,b],c] + [b,[a,c]]
You may enjoy trying to show that the dimension of Lie(n) is (n - 1)!, at least for n > 0. There's an incredibly beautiful conceptual proof, and probably lots of obnoxious brute-force proofs.
There's a lot more to say about the Lie operad, but right now I want to talk about the Poisson operad. A "Poisson algebra" is a commutative associative algebra that has a bracket operation {a,b} making it into a Lie algebra, with the property that
{a,bc} = {a,b}c + b{a,c}
So, bracketing with any element is like taking a derivative: it satisfies the product rule.
For this reason, Poisson algebras arise naturally as algebras of observables in classical mechanics - the Poisson bracket of any observable A with an observable H called the "Hamiltonian" tells you the time derivative of A:
dA/dt = {H,A}
This is the beginning of a nice big story.
But, what's got me excited now is how Poisson algebras show up in topology!
To understand this, we need to note that there's a linear operad whose algebras are Poisson algebras. That's not surprising. But, we can get a very similar operad in a rather shocking way, as follows.
Take the little k-cubes operad. This has a space O(n) of n-ary operations for each n. Now take the homology of these spaces O(n), using coefficients in your favorite field, and get vector spaces H(O(n)). By functorial abstract nonsense these form a linear operad. And this is the operad for Poisson algebras!
Alas, we actually have to be a bit more careful. The homology of each space O(n) with coefficients in some field is really a graded vector space over that field. So, taking the homology of the little k-cubes operad gives an operad in the category of graded vector spaces. And, it's the operad whose algebras are graded Poisson algebras with a bracket of degree k-1.
What are those? Well, they're like Poisson algebras, but if a is an element of degree |a| and b is an element of degree |b|, then:
and the usual axioms for a Poisson algebra hold, but sprinkled with minus signs according to the usual yoga of graded vector spaces.
So: whenever we have a k-fold loop space, its homology is a graded Poisson algebra with a bracket of degree k-1.
To get an idea of this works, let me sketch how the product and the bracket work. Suppose we have an space X with an action of the little k-cubes operad:
The equation
{a,bc} = {a,b}c + b{a,c}
then says "moving a around b and c is like moving a around b while c stands by, plus moving a around c while b stands by".
I guess this result can be found here:
9) Frederick Cohen, Homology of Ωn+1Σn+1X and Cn+1X, n > 0, Bull. Amer. Math. Soc. 79 (1973), 1236-1241.
10) Frederick Cohen, Tom Lada and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics 533, Springer, Berlin, 1976.
But, I don't think these old papers talk about graded Poisson operads! Dev Sinha has a paper where he takes these ideas and distills them all into the combinatorics of graphs and trees:
11) Dev Sinha, A pairing between graphs and trees, available as math.QA/0502547.
However, what I really like is how he gets these graphs and trees starting from the homology and cohomology (respectively) of the little k-cubes operad! He first wrote about it here:
12) Dev Sinha, Manifold theoretic compactifications of configuration spaces, available as math.GT/0306385.
Dev Sinha, The homology of the little disks operad, available as math/0610236.
I have a vague feeling that this relation between the little k-cubes operad and the Poisson operad is part of a big picture involving braids and quantization. Another hint in this direction is Deligne's Conjecture, now proved in many ways, which says that the operad of singular chains coming from the little 2-disks operad acts on the Hochschild cochain complex of any associative algebra. Since Hochschild cohomology classifies the ways you can deform an associative algebra, this result is related to quantization and Poisson algebras. But, I don't get the big picture! This might help:
13) Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999) 35-72. Also available as math.QA/9904055.
I'd like to ponder this now! But I'm getting tired, and I still need to say how the little 2-cubes operad acts on the space of thickened long knots.
What's a thickened long knot? In k dimensions, it's an embedding of a little k-cube in a big one:
f: [0,1]k → [0,1]k
subject to the condition that the top and bottom of the little cube get mapped to the top and bottom of the big one via the identity map. So, you should imagine a thickened long knot as a fat square rope going from the ceiling to the floor, all tied up in knots.
There are two ways to "compose" thickened long knots.
If you're a knot theorist, the obvious way is to stick one on top of the other - just like the usual composition of tangles. But if you just think of thickened long knots as functions, you can also compose them just by composing functions! This amounts to stuffing one knot inside another... a little hard to visualize, but fun.
Anyway, it turns out that the whole little 2-cubes operad acts on the space of thickened long knots, with the two operations I just mentioned corresponding to this:
---------------------
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| |
| |
| |
| | sticking one thickened long
--------------------- knot on top of another
| |
| |
| |
| |
| |
---------------------
and this:
---------------------
| | |
| | |
| | |
| | |
| | | sticking one thickened long
| | | knot inside another
| | |
| | |
| | |
| | |
| | |
---------------------
This isn't supposed to make obvious sense, but the point is, there
are lots of binary operations interpolating between these two -
one for each binary operation in the little 2-cubes operad!
This gives a new proof that the operation of "sticking one thickened long knot on top of another" is commutative up to homotopy.
And, using these ideas, Ryan Budney has managed to figure out a lot of information about the homotopy type of the space of long knots. Check out these papers:
14) Ryan Budney, Little cubes and long knots, available as math.GT/0309427.
15) Ryan Budney and Frederick Cohen, On the homology of the space of long knots, available as math.GT/0504206.
16) Ryan Budney, Topology of spaces of knots in dimension 3, available as math.GT/0506524.
The paper by Budney and Cohen combines the two ideas I just described - the action of the little 2-cubes operad on thickened long knots and its relation to the Poisson operad. Using these, they show that the rational homology of the space of thickened long knots in 3 dimensions is a free Poisson algebra! They also show that the mod-p homology of this space is a free "restricted Poisson" algebra.
Addendum: Jesse McKeown had a question about the two operations on long thickened knots. Here's what he asked:
Perhaps I'm being too imaginative, but I don't feel very convinced the two operations described towards the end of "week220" are fundamentally different.Here's my reply:VagueSpecifically, in map-composition, can't one stretch all the knottednes of the first composand into an "upper", essentially unknotty portion of the second composand, and similarly squish the knottedness of the second composand into a "lower" section of the big all-encompassing box?
Right! That's exactly what having an action of the little 2-cubes operad says! There's nothing "fundamentally different" between this:--------------------- | | | | | | | | | | sticking one thickened long A: --------------------- knot on top of another | | | | | | | | | | ---------------------and this:--------------------- | | | | | | | | | | | | B: | | | sticking one thickened long | | | knot inside another | | | | | | | | | | | | | | | ---------------------because there is a continuous family of operations interpolating between these - one for each way of sticking two little squares in a big one.But, the process of moving from operation A to operation B is itself nontrivial. If you loop all the way around from A to B to A - moving the two little squares around each other in the big one - you can get a noncontractible loop in the space of long thickened knots!
And, this is what gives the bracket operation on the homology of the space of thickened long knots.
Operads were born to deal with issues like this.
On another note - in the summer of 2007, Urs Schreiber posted a 3-part article on Batalin-Vilkovisky quantization over at the n-Category Cafe. The third part has links to the other two:
18) Urs Schreiber, Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III), http://golem.ph.utexas.edu/category/2007/08/lyakhonov_and_sharapov_on_qft.html.
and in addition to providing lots of references, it led me back to puzzling about Poisson algebras and the little disks operad. Here's what I wrote, roughly:
Try these:
19) Ezra Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), 265-285. Available at http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104254599
20) Takashi Kimura, Jim Stasheff and Alexander A. Voronov, On operad structures of moduli spaces and string theory, Comm. Math. Phys. 171 (1995), 1-25. Section 3.7: the Batalin-Vilkovisky (BV) algebra. Available at http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104273401
I don't understand this stuff very well!
More precisely:
If you take the space of multivector fields V on a manifold M, and think of V equipped with its wedge product and Schouten bracket, you get the easiest example of a Gerstenhaber algebra.
A Gerstenhaber algebra is an associative supercommutative graded algebra A together with a bracket of degree -1 which makes A into a kind of "graded Poisson algebra with bracket of degree -1". All the usual Poisson algebra axioms hold, but sprinkled with minus signs according to the usual conventions.
If your manifold M is a Poisson manifold, then the space V of multivector fields comes equipped with a differential given by taking the Schouten bracket with the Poisson bivector field Π in V.
Axiomatizing this mess, we get the definition of a Batalin-Vilkovisky algebra: a Gerstenhaber algebra with differential that's compatible with the other structure in a certain way.
There are also lots of Batalin-Vilkovisky algebras that don't come from Poisson manifolds. But just like Poisson manifolds, we can still think of these as describing phase spaces in classical mechanics - in a clever algebraic way. And, that's what BV quantization is all about: figuring out how to treat these Batalin-Vilkovisky algebras as classical phase spaces and quantize them!
All this makes some sense to me. But then it gets weird and mystical...
First, thanks to an old result of Fred Cohen, a Gerstenhaber algebra is the same as an algebra of the operad H(D) - the homology of the little disks operad!
Did I just hear some of you say "Huh?"
Well, let me sketch what that means. The little disks operad is a gadget with a bunch of n-ary operations corresponding to ways of sticking n little disks in a big one. For each n there's a topological space of these n-ary operations. Taking the homology of this topological space, we get a graded vector space. These are the n-ary operations of the operad I'm calling H(D).
While I roughly follow how this works, I don't understand the deep inner meaning. It seems amazing: there's a mystical relation between ways of sticking little 2d disks in bigger ones, and operations you can do on the space of multivector fields on a manifold!
I don't know if the connections to 2d topological and conformal field theory (described in the articles I cite) actually explain this mystical relation, or merely exploit it.
Now, as I said, a Batalin-Vilovisky algebra is a Gerstenhaber algebra with an extra operation. And, Getzler showed that this extra operation corresponds to our ability to twist a little disk 360°. (Until we started twisting like this, we could equally have used little 2-cubes.)
More precisely, Getzler showed that a Batalin--Vilkovisky algebra is the same as an algebra of the homology of the framed little discs operad.
This extra twist of the knife only makes me more curious to know what's really going on here.
Here's a clue that could help. As I explained to Urs a couple days ago, this business of "taking homology" is really some sort of procedure for turning weak ∞-groupoids (i.e. spaces) into stable strict ∞-groupoids (i.e. chain complexes) - followed by taking the homology of the chain complex, which in principle loses even more information, but doesn't in this particular example. That suggests that these Gerstenhaber (and Batalin-Vilkovisky) algebras are really just watered-down chain complex versions of spaces equipped with n-ary operations corresponding to ways of sticking n (framed) little disks into a big disk.
But still: what's really going on? What do classical phase spaces have to do with little 2-dimensional disks???
As far as I'm concerned, the Rosetta Stone on the third page of Getzler's paper only serves to heighten the mystery further!
© 2005 John Baez
baez@math.removethis.ucr.andthis.edu
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