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This Week's Finds in Mathematical Physics (Week 238)

  1. Aug 18, 2006 #1
    Also available as http://math.ucr.edu/home/baez/week238.html

    August 16, 2006
    This Week's Finds in Mathematical Physics (Week 238)
    John Baez

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

    1) NASA, NASA Announces Dark Matter Discovery,

    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 it 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

    So, barring any drastic new revelations, we can guess what's up.
    Markevitch and company have been studying the "Bullet Cluster", a
    a bunch of galaxies that has a small bullet-shaped subcluster zipping
    away from the center at 4,500 kilometers per second.

    It seems that one of the rapidly moving galaxies in this subcluster
    has hit a bystander galaxy - I'm not sure, but a high-speed collision
    of galaxies occurred. When this kind of thing happens, the *gas* in
    the galaxies is what actually collides; stars are too thinly spread 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! In fact, this may be hottest known galactic cluster.

    That's fun. But that's not 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 galaxies 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 galaxies further away!
    So: X-rays show the gas here, but gravity shows most of the mass is
    somewhere else. That's good evidence that dark matter is for real.

    For more try these:

    3) Maxim Markevitch, Chandra observation of the most interesting
    cluster in the Universe, available at astro-ph/0511345.

    4) 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.

    5) 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

    So, dark matter is seeming more and more real. In fact, last year
    folks found evidence for "ghost galaxies" made mainly of dark matter
    and cold hydrogen, with very few stars:

    6) PPARC, New evidence for a dark matter galaxy,

    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:

    7) Dan Christensen and Igor Khavkin, Plots of q-deformed tets,

    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

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

    8) Jiri Adamek, Horst Herrlich and George E. Strecker,
    Abstract and Concrete Categories: the Joy of Cats, available at

    9) Robert Goldblatt, Topoi: the Categorial Analysis of Logic,
    available at

    10) Michael Barr and Charles Wells, Toposes, Triples and Theories,
    available at http://www.case.edu/artsci/math/wells/pub/ttt.html [Broken]

    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:

    11) Anders Kock, Synthetic Differential Geometry, available at

    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 "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 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 Lie(G)!

    So, we have a god-given linear map from tangent vectors on G to the
    Lie algebra Lie(G). This is called a "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 Lie(G). This particular god-given
    Lie(G)-valued 1-form on G is called the "Maurer-Cartan form", and
    denoted omega.

    Now, we can define exterior derivatives of 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 ^ omega

    On the right here I'm using the wedge product of 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:

    12) Wikipedia, Maurer-Cartan form,

    An interesting thing about this equation is that it shows
    everything about the Lie algebra 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 ^ omega.

    If you have trouble seeing this, note that we can feed omega ^ omega
    a pair of tangent vectors at any point of G, and it will spit out
    an element of Lie(G). How will it do this? The two copies of omega
    will eat the two tangent vectors and spit out elements of 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 Lie(G) using
    this trick, omega ^ omega knows everything about the bracket in
    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 ^ omega

    and the usual rules for exterior derivatives imply that

    d(d omega) = 0

    we must have

    d(omega ^ omega) = 0

    If we work this concretely what this says, we must get some identity
    involving the bracket in our Lie algebra, since omega ^ omega is just
    the bracket in disguise. What identity could this be?


    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 ^ omega) is a 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 on vector fields, preserves
    their 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:


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

    How can we state the above fact purely algebraically, only
    using the Lie algebra 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:

    L_g: G -> G
    x |-> gx

    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 Lie(G)-valued ones we can just tensor with L = Lie(G).

    Well, here's how we do it. The left-invariant vector fields on G
    are just


    so the left-invariant 1-forms are


    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

    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 "week191",
    or else these:

    13) 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.

    14) 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

    The point is that Lie, commutative and associative algebras are all
    defined by "quadatic operads", and one can define for any such operad
    O a "dual" operad O* such that:

    Making the free graded O-algebra on SL* into a *differential*
    graded O-algebra is the same as making L into an O*-algebra.

    And, we have O** = 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

    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at


    For a table of contents of all the issues of This Week's Finds, try


    A simple jumping-off point to the old issues is available at


    If you just want the latest issue, go to

    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Aug 18, 2006 #2
    John Baez wrote:
    > So, dark matter is seeming more and more real. In fact, last year
    > folks found evidence for "ghost galaxies" made mainly of dark matter
    > and cold hydrogen, with very few stars:
    > 6) PPARC, New evidence for a dark matter galaxy,
    > http://www.interactions.org/cms/?pid=1023641
    > 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.

    It is a very curious thing that theoretical physicists almost
    exclusively assume that the dark matter is composed of particle-mass
    objects, most of which have never been seen except on paper.

    Actual observations, on the other hand, have repeatedly detected large
    populations of stellar-mass dark matter objects with mass function
    peaks at 0.15 solar masses and 0.5 solar masses. See Calchi Novati et
    al's astro-ph/0607358 at www.arxiv.org for a very recent review of the
    empirical evidence and its interpretation.

    For what it's worth, well before any microlensing experiments began, it
    was definitively predicted that the microlensing teams would find mass
    peaks at 0.15 solar masses and 0.58 solar masses (Astrophysical Journal
    322, 34-36,1987).

    It could be argued that solving the dark matter enigma may be the
    critical goal of 21st century physics. Given the lessons of the last
    400 years, we should probably be ready for some mighty big surprises.
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