# August 16, 2006 {#week238}

NASA is trying to built up suspense with this "media advisory":

1) NASA, "NASA Announces Dark Matter Discovery", `http://www.nasa.gov/home/hqnews/2006/aug/HQ_M06128_dark_matter.html`

which says simply:

> Astronomers who used NASA's Chandra X-ray Observatory will host a
> media teleconference at 1 p.m. EDT Monday, Aug. 21, to announce how
> dark and normal matter have been forced apart in an extraordinarily
> energetic collision.

Hmm! What's this about?

Someone nicknamed "riptalon" at Slashdot made a good guess. The media
advisory lists the "briefing participants" as Maxim Markevitch, Doug
Clowe and Sean Carroll. Markevitch and Clowe work with the Chandra X-ray
telescope to study galaxy collisions and dark matter. Last November,
Markevitch gave a talk on this work, which you can see here:

2) Maxim Markevitch, Scott Randall, Douglas Clowe, and Anthony H. Gonzalez, "Insights on physics of gas and dark matter from cluster mergers", available at `http://cxc.harvard.edu/symposium_2005/proceedings/theme_energy.html#abs23`

So, barring any drastic new revelations, we can guess what's up.
Markevitch and company have been studying the "Bullet Cluster", a
bunch of galaxies that has a small bullet-shaped subcluster zipping away
from the center at 4,500 kilometers per second. Here's a picture of it
from the above paper:
$$\href{http://cxc.harvard.edu/symposium_2005/proceedings/theme_energy.html#abs23}{\includegraphics[max width=0.65\linewidth]{../images/bullet_cluster.jpg}}$$
To help you understand this picture a bit: the official name of the
Bullet Cluster is 1E0657-56. The "exposure" for this X-ray photograph
taken by Chanda was apparently 0.5 million seconds --- 140 hours! The
distance scale shown, 0.5 megaparsecs, is about 1.6 million light years.
The cluster itself has a redshift $z = 0.3$, meaning its light has
wavelengths stretched by a factor of 1.3. Under currently popular ideas
on cosmology, this means it's roughly 4 billion light years away.

Anyway, what are we seeing here?

You can see rapidly moving galaxy cluster with a shock wave trailing
behind it. It seems to have hit another cluster at high speed. When this
kind of thing happens, the *gas* in the clusters is what actually
collides --- the individual galaxies are too sparse to hit very often. And
when the gas collides, it gets hot. In this case, it heated up to about
160 million degrees and started emitting X-rays like mad! The picture
shows these X-rays. This may be hottest known galactic cluster.

That's fun. But that's not enough reason to call a press conference.
The cool part is not the crashing of gas against gas. The cool part is
that the dark matter in the clusters was unstopped --- it kept right on
going!

How do people know this? Simple. Folks can see the *gravity* of the dark
matter bending the light from more distant galaxies! It's called
"gravitational lensing". Here are the mass density contours, as seen
by this effect. I guess Clowe took this photo using the Hubble Space
Telescope:
$$\href{http://cxc.harvard.edu/symposium_2005/proceedings/theme_energy.html#abs23}{\includegraphics[max width=0.65\linewidth]{../images/bullet_cluster_lensing.jpg}}$$
So: X-rays show the gas in one place, but gravity shows most of the mass
is somewhere else --- two lumps zipping along unstopped. That's good
evidence that dark matter is for real.

For more try these:

3) M. Markevitch, S. Randall, D. Clowe, A. Gonzalez, and M. Bradac, "Dark matter and the Bullet Cluster", available at `http://www.cosis.net/abstracts/COSPAR2006/02655/COSPAR2006-A-02655.pdf`

4) M. Markevitch, A. H. Gonzalez, D. Clowe, A. Vikhlinin, L. David, W. Forman, C. Jones, S. Murray, and W. Tucker, "Direct constraints on the dark matter self-interaction cross-section from the merging galaxy cluster 1E0657-56", available as [`astro-ph/0309303`](http://arxiv.org/abs/astro-ph/0309303).

5) Maxim Markevitch, "Chandra observation of the most interesting cluster in the Universe", available as [`astro-ph/0511345`](https://arxiv.org/abs/astro-ph/0511345).

6) M. Markevitch, A. H. Gonzalez, L. David, A. Vikhlinin, S. Murray, W. Forman, C. Jones and W. Tucker, "A textbook example of a bow shock in the merging galaxy cluster 1E0657-56", _Astrophys. J._ **567** (2002), L27. Also available as [`astro-ph/0110468`](https://arxiv.org/abs/astro-ph/0110468).

7) Eric Hayashi and Simon D. M. White, "How rare is the Bullet Cluster?", _Mon. Not. Roy. Astron. Soc. Lett._ **370** (2006), L38--L41, available as [`astro-ph/0604443`](https://arxiv.org/abs/astro-ph/0604443).

The first of these is, alas, only the abstract of a talk. But it's
worth reading, so I'll quote it in its entirety here:

> 1E0657-56, the "Bullet Cluster", is a merger with a uniquely simple
> geometry. From the long Chandra X-ray observation which revealed a
> classic bow shock in front of a small subcluster, we can derive the
> velocity of the subcluster and its direction of motion. Recent
> accurate weak and strong lensing total mass maps clearly show two
> merging subclusters, including the host of the gas bullet seen in
> X-rays. This cluster provided the first direct, model-independent
> proof of the dark matter existence (as opposed to any modified gravity
> theory) and a direct constraint on the self-interaction cross-section
> of the dark matter particles. I will review these and other related
> results.

The Bullet Cluster is not the only direct evidence for dark matter. In
fact, last year folks claimed to have found a "ghost galaxy" made
mainly of dark matter and cold hydrogen, with very few stars:
$$\href{http://www2.naic.edu/~rminchin/press/}{\includegraphics[max width=0.65\linewidth]{../images/ghost_galaxy.jpg}}$$

8) PPARC, "New evidence for a dark matter galaxy", `http://www.interactions.org/cms/?pid=1023641`

However, Matt Owers informs me that the consensus on this ghost, VIRGOHI
21, is that it's hydrogen stripped off from a galaxy by the "wind" it
felt as it fell into the Virgo Cluster. This effect is called "ram
pressure stripping" --- the gas of a galaxy can be stripped off if the
galaxy is moving rapidly through a cluster, due to interaction with the
gas in the cluster.

Nonetheless, dark matter is seeming more and more real. It thus becomes
ever more interesting to find out what dark matter actually *is*. The
lightest neutralino? Axions? Theoretical physicists are good at
inventing plausible candidates, but finding them is another thing.

Since I'd like to send this off in time to beat NASA, I won't say a
lot more today... just a bit.

Dan Christensen and Igor Khavkine have discovered some fascinating
things by plotting the amplitude of the tetrahedral spin network --- the
basic building block of spacetime in 3d quantum gravity --- as a function
of the cosmological constant.

They get pictures like this:
$$\includegraphics[max width=0.65\linewidth]{../images/tet4-real-med.jpg}$$

9) Dan Christensen and Igor Khavkine, "Plots of $q$-deformed tets",
`http://jdc.math.uwo.ca/spinnet/`

Here the color indicates the real part of the spin network amplitude,
and it's plotted as a function of $q$, which is related to the
cosmological constant by a funky formula I won't bother to write down
here.

You can get some nice books on category theory for free these days:

10) Jiri Adamek, Horst Herrlich and George E. Strecker, _Abstract and Concrete Categories: the Joy of Cats_, available at `http://katmat.math.uni-bremen.de/acc/acc.pdf`

11) Robert Goldblatt, _Topoi: the Categorial Analysis of Logic_, available at `http://cdl.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010`

12) Michael Barr and Charles Wells, _Toposes, Triples and Theories_, available at `http://www.case.edu/artsci/math/wells/pub/ttt.html`

The first two are quite elementary --- don't be scared of the title of
Goldblatt's book; the only complaints I've ever heard about it boil
down to the claim that it's too easy!

You can also download this classic text on synthetic differential
geometry, which is an approach to differential geometry based on
infinitesimals, formalized using topos theory:

13) Anders Kock, _Synthetic Differential Geometry_, available at `http://home.imf.au.dk/kock/`

He asks that we not circulate it in printed form --- electrons are okay,
but not paper.

Next I want to say a *tiny* bit about Koszul duality for Lie algebras,
which plays a big role in the work of Castellani on the M-theory Lie
$3$-algebra, which I discussed in ["Week 237"](#week237).

Let's start with the Maurer-Cartan form. This is a gadget that shows up
in the study of Lie groups. It works like this. Suppose you have a Lie
group $G$ with Lie algebra $\mathrm{Lie}(G)$. Suppose you have a tangent vector at
any point of the group $G$. Then you can translate it to the identity
element of $G$ and get a tangent vector at the identity of $G$. But, this is
nothing but an element of $\mathrm{Lie}(G)$!

So, we have a god-given linear map from tangent vectors on $G$ to the Lie
algebra $\mathrm{Lie}(G)$. This is called a "$\mathrm{Lie}(G)$-valued $1$-form" on $G$, since an
ordinary $1$-form eats tangent vectors and spits out numbers, while this
spits out elements of $\mathrm{Lie}(G)$. This particular god-given $\mathrm{Lie}(G)$-valued
$1$-form on $G$ is called the "Maurer-Cartan form", and denoted $\omega$.

Now, we can define exterior derivatives of $\mathrm{Lie}(G)$-valued differential
forms just as we can for ordinary differential forms. So, it's
interesting to calculate $d\omega$ and see what it's like.

The answer is very simple. It's called the Maurer-Cartan equation:
$$d\omega = -\omega\wedge\omega$$
On the right here I'm using the wedge product of $\mathrm{Lie}(G)$-valued
differential forms. This is defined just like the wedge product of
ordinary differential forms, except instead of multiplication of numbers
we use the bracket in our Lie algebra.

I won't prove the Maurer-Cartan equation; the proof is so easy you can
even find it on the Wikipedia:

14) Wikipedia, "Maurer-Cartan form", `http://en.wikipedia.org/wiki/Maurer-Cartan_form`

An interesting thing about this equation is that it shows everything
about the Lie algebra $\mathrm{Lie}(G)$ is packed into the Maurer-Cartan form. The
reason is that everything about the bracket operation is packed into the
definition of $\omega\wedge\omega.$

If you have trouble seeing this, note that we can feed $\omega\wedge\omega$ a pair of
tangent vectors at any point of $G$, and it will spit out an element of
$\mathrm{Lie}(G)$. How will it do this? The two copies of $\omega$ will eat the two
tangent vectors and spit out elements of $\mathrm{Lie}(G)$. Then we take the
bracket of those, and that's the final answer.

Since we can get the bracket of *any* two elements of $\mathrm{Lie}(G)$ using this
trick, $\omega\wedge\omega$ knows everything about the bracket in $\mathrm{Lie}(G)$. You could
even say it's the bracket viewed as a geometrical entity --- a kind of
"field" on the group $G$!

Now, since
$$d\omega = -\omega\wedge\omega$$
and the usual rules for exterior derivatives imply that
$$d^2\omega = 0$$
we must have
$$d(\omega\wedge\omega) = 0$$
If we work this concretely what this says, we must get some identity
involving the bracket in our Lie algebra, since $\omega\wedge\omega$ is just the
bracket in disguise. What identity could this be?

THE JACOBI IDENTITY!

It has to be, since the Jacobi identity says there's a way to take 3
Lie algebra elements, bracket them in a clever way, and get zero:
$$[u,[v,w]] + [v,[w,u]] + [w,[u,v]] = 0$$
while $d(\omega\wedge\omega)$ is a $\mathrm{Lie}(G)$-valued $3$-form that happens to vanish, built
using the bracket.

It also has to be since the equation $d^2 = 0$ is just another way of
saying the Jacobi identity. For example, if you write out the explicit
grungy formula for d of a differential form applied to a list of vector
fields, and then use this to compute $d^2$ of that differential form,
you'll see that to get zero you need the Jacobi identity for the Lie
bracket of vector fields. Here we're just using a special case of that.

The relationship between the Jacobi identity and $d^2 = 0$ is actually
very beautiful and deep. The Jacobi identity says the bracket is a
derivation of itself, which is an infinitesimal way of saying that the
flow generated by a vector field, acting as an operation on vector
fields, preserves the Lie bracket! And this, in turn, follows from the
fact that the Lie bracket is *preserved by diffeomorphisms* --- in other
words, it's a "canonically defined" operation on vector fields.

Similarly, $d^2 = 0$ is related to the fact that d is a natural operation
on differential forms --- in other words, that it commutes with
diffeomorphisms. I'll leave this cryptic; I don't feel like trying to
work out the details now.

Instead, let me say how to translate this fact:

> $d^2\omega = 0$ IS SECRETLY THE JACOBI IDENTITY

into pure algebra. We'll get something called "Kozsul duality". I
always found Koszul duality mysterious, until I realized it's just a
generalization of the above fact.

How can we state the above fact purely algebraically, only using the Lie
algebra $\mathrm{Lie}(G)$, not the group $G$? To get ourselves in the mood, let's
call our Lie algebra simply $L$.

By the way we constructed it, the Maurer-Cartan form is
"left-invariant", meaning it doesn't change when you translate it
using maps like this:
$$
  \begin{aligned}
    Lg\colon G &\to G
  \\x &\mapsto gx
  \end{aligned}
$$
that is, left multiplication by any element $g$ of $G$. So, how can we
describe the left-invariant differential forms on $G$ in a purely
algebraic way? Let's do this for *ordinary* differential forms; to get
$\mathrm{Lie}(G)$-valued ones we can just tensor with $L = \mathrm{Lie}(G)$.

Well, here's how we do it. The left-invariant vector fields on $G$ are
just
$$L$$
so the left-invariant $1$-forms are
$$L^*$$
So, the algebra of all left-invariant diferential forms on $G$ is just the
exterior algebra on $L^*$. And, defining the exterior derivative of such a
form is precisely the same as giving the bracket in the Lie algebra $L$!
And, the equation $d^2 = 0$ is just the Jacobi identity in disguise.

To be a bit more formal about this, let's think of $L$ as a graded vector
space where everything is of degree zero. Then $L^*$ is the same sort of
thing, but we should *add one to the degree* to think of guys in here as
$1$-forms. Let's use $S$ for the operation of "suspending" a graded
vector space --- that is, adding one to the degree. Then the exterior
algebra on $L^*$ is the "free graded-commutative algebra on $SL^*$".

So far, just new jargon. But this lets us state the observation of the
penultimate paragraph in a very sophisticated-sounding way. Take a
vector space $L$ and think of it as a graded vector space where everything
is of degree zero. Then:

> Making the free graded-commutative algebra on $SL^*$ into a
> *differential* graded-commutative algebra is the same as making $L$ into
> a Lie algebra.

This is a basic example of "Koszul duality". Why do we call it
"duality"? Because it's still true if we switch the words
"commutative" and "Lie" in the above sentence!

> Making the free graded Lie algebra on $SL^*$ into a *differential*
> graded Lie algebra is the same as making $L$ into a commutative algebra.

That's sort of mind-blowing. Now the equation $d^2 = 0$ secretly encodes
the *commutative law*.

So, we say the concepts "Lie algebra" and "commutative algebra" are
Koszul dual. Interestingly, the concept "associative algebra" is its
own dual:

> Making the free graded associative algebra on $SL^*$ into a
> *differential* graded associative algebra is the same as making $L$ into
> an associative algebra.

This is the beginning of a big story, and I'll try to say more later.
If you get impatient, try the book on operads mentioned in
["Week 191"](#week191), or else these:

15) Victor Ginzburg and Mikhail Kapranov, "Koszul duality for quadratic operads", _Duke Math. J._ **76** (1994), 203--272. Also "Erratum", _Duke Math. J._ **80** (1995), 293.

16) Benoit Fresse, "Koszul duality of operads and homology of partition posets", _Homotopy theory and its applications (Evanston, 2002)_, Contemp. Math. **346** (2004), 115--215. Also available at `http://math.univ-lille1.fr/~fresse/PartitionHomology.html`

The point is that Lie, commutative and associative algebras are all
defined by "quadratic operads", and one can define for any such operad
$\mathcal{O}$ a "dual" operad $\mathcal{O}^*$ such that:

> Making the free graded $\mathcal{O}$-algebra on $SL^*$ into a *differential* graded
> $\mathcal{O}$-algebra is the same as making $L$ into an $\mathcal{O}^*$-algebra.

And, we have $\mathcal{O}^{**} = \mathcal{O}$, hence the term "duality".

This has always seemed incredibly cool and mysterious to me. There are
other meanings of the term "Koszul duality", and if really understood
them I might better understand what's going on here. But, I'm feeling
happy now because I see this special case:

> Making the free graded-commutative algebra on $SL^*$ into a
> *differential* graded-commutative algebra is the same as making $L$ into
> a Lie algebra.

is really just saying that the exterior derivative of left-invariant
differential forms on a Lie group encodes the bracket in the Lie
algebra. That's something I have a feeling for. And, it's related to
the Maurer-Cartan equation... though notice, I never completely spelled
out how.

------------------------------------------------------------------------

**Addenda:** Let me say some more about how $d^2 = 0$ is related to the
fact that $d$ is a canonically defined operation on differential forms.
Being "canonically defined" means that $d$ commutes with the action of
diffeomorphisms. Saying that $d$ commutes with "small" diffeomorphisms ---
those connected by a path to the identity --- is the same as saying
$$d L_v = L_v d$$
where $v$ is any vector field and $L_v$ is the corresponding "Lie
derivative" operation on differential forms. But, Weil's formula says
that
$$L_v = i_v d + d i_v$$
where $i_v$ is the "interior product with $v$", which sends $p$-forms to
$(p-1)$-forms. If we plug Weil's formula into the equation we're
pondering, we get
$$d (i_v d + d i_v) = (i_v d + d i_v) d$$
which simplifies to give
$$d^2 i_v = i_v d^2$$
So, as soon as we know $d^2 = 0$, we know $d$ commutes with small
diffeomorphisms. Alas, I don't see how to reverse the argument.

Similarly, as soon as we know the Jacobi identity, we know the Lie
bracket operation on vector fields is preserved by small
diffeomorphisms, by the argument outlined in the body of this Week. This
argument is reversable.

So, maybe it's an exaggeration to say that $d^2 = 0$ and the Jacobi
identity say that $d$ and the Lie bracket are preserved by diffeomorphisms
--- but at least they *imply* these operations are preserved by *small*
diffeomorphisms.