Books on spinors in physics
Martin Hallnaes
like to learn more about spinors and their applications in physics. So my
question is; does anyone know of any good books on the subject, that is to
say books that deal both with the "mathematics of spinors" and their use in
physics?
--
Martin Hallnaes
Lubos Motl
(Roger) Penrose and (Wolfgang) Rindler, "Spinors and spacetime".
Many call it the "Bible" of spinors. But I must say that I could not learn
many important things from that book. When I look back, it surprises me
how can someone write hundreds and hundres pages about four-dimensional
spinors and spin-tensors; of course, they even don't generalize them to
arbitrary dimensions, if I remember well. But you will see how can they
rewrite Einstein equations and many other things in the spinor formalism.
I would have other tips, too, but they are not books on spinors only.
Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
John Baez
Martin Hallnaes <e022...@student.tuwien.ac.at> wrote:
If you're just getting started, you should master spinors in the
context of 3+1-dimensional spacetime, but if you go into string
theory you'll need to understand spinors in any old p+q-dimensional
spacetime (with p plus signs and q minus signs in the metric),
and then you'll have fun learning about the wonderful "Bott
periodicity" phenomenon: many features of spinors repeate
themselves if you increase p or q by 8.
These are my favorite books on the math of spinors:
H. Blaine Lawson, Jr. and Marie-Louise Michelson, Spin Geometry,
Princeton U. Press, Princeton, 1989. (The first few chapters
tell you all the basic facts you need to know about Clifford
algebras, spinors, spin structures, and the Dirac operator -
a very good intro. The rest puts this to use to study questions
in geometry and topology.)
P. Budinich and A. Trautman, The Spinorial Chessboard,
Springer-Verlag, Berlin, 1988. (Out of print, hard to find,
but a great treatment of Clifford algebras and their spinor
representations. Trautman plans to come out with a new edition
in a couple of years, health permitting.)
F. Reese Harvey, Spinors and Calibrations, Academic Press,
Boston, 1990. (More nice charts listing the spinor and pinor
representations in all dimensions and signatures; also, good
information on the relation between spinors, octonions, and
the exceptional groups G2, F4 and E6, all of which show up in
string theory.)
Ian R. Porteous, Clifford Algebras and the Classical Groups,
Cambridge University Press, Cambridge, 1995.
The first three are enough to keep me satisfied most of the
time, but to make contact with the physics literature you really
need to understand the terminology "Dirac spinor", "Weyl spinor",
"Majorana spinor", "Majorana-Weyl spinor", and most mathematicians,
including the authors above, tend to use different terminology
for these various sorts of spinors. For example, mathematicians
often call Weyl spinors "semi-spinors'. So, you need to set up
your own little translation dictionary. This book explains all
these different sorts of spinors, perhaps in more detail than
you need at first:
Pertti Lounesto, Clifford Algebras and Spinors, Cambridge U.
Press, Cambridge, 1997.
(Alas, Lounesto died recently, so he can't argue with me.)
Phyisicists also have an unhealthy fondness for explicit
matrix representations of Clifford algebras, so you'll probably
need to learn their favorite conventions for "gamma matrices"
at some point; here's one place to start:
Antoine Van Proeyen, Tools for supersymmetry,
http://www.arXiv.org/abs/hep-th/9910030
Also, I'll append my own introduction to spinors, just
to get you going. Here I only consider p+1-dimensional and
1+q-dimensional spacetime.
.....................................................................
Also available at http://math.ucr.edu/home/baez/week93.html
October 27, 1996
This Week's Finds in Mathematical Physics - Week 93
John Baez
Lately I've been trying to learn more about string theory. I've always
had grave doubts about string theory, but it seems worth knowing about.
As usual, when I'm trying to learn something I find it helpful to write
about it --- it helps me remember stuff, and it points out gaps in my
understanding. So I'll start trying to explain some string theory in
this and forthcoming Week's Finds.
However: watch out! This isn't going to be a systematic introduction to
the subject. First of all, I don't know enough to do that. Secondly,
it will be very quirky and idiosyncratic, because the aspects of string
theory I'm interested in now aren't necessarily the ones most string
theorists would consider central. I've been taking as my theme of
departure, "What's so great about 10 and 26 dimensions?" When one
reads about string theory, one often hears that it only works in 10
or 26 dimensions --- and the obvious question is, why?
This question leads one down strange roads, and one runs into lots of
surprising coincidences, and spooky things that sound like coindences
but might NOT be coincidences if we understood them better.
For example, when we have a string in 26 dimensions we can think of it
as wiggling around in the 24 directions perpendicular to the
2-dimensional surface the string traces out in spacetime (the "string
worldsheet"). So the number 24 plays an especially important role in
26-dimensional string theory. It turns out that
1^2 + 2^2 + 3^2 + ... + 24^2 = 70^2.
In fact, 24 is the *only* integer n > 1 such that the sum of squares
from 1^2 to n^2 is itself a perfect square. Is this a coincidence?
Probably not, as I'll eventually explain! This is just one of
many eerie facts one meets when trying to understand this stuff.
For starters I just want to explain why dimensions of the form 8k + 2
are special. Notice that if we take k = 0 here we get 2, the
dimension of the string worldsheet. For k = 1 we get 10, the dimension
of spacetime in "supersymmetric string theory". For k = 3 we get 26,
the dimension of spacetime in "purely bosonic string theory". So these
dimensions are important. What about n = k and the dimension 18, I hear
you ask? Well, I don't know what happens there yet... maybe someone can
tell me! All I want to do now is to explain what's good about
8n + 2.
But I need to start by saying a bit about fermions.
Remember that in the Standard Model of particle physics --- the model
that all fancier theories are trying to outdo --- elementary particles
come in 3 basic kinds. There are the basic fermions. In general a
"fermion" is a particle whose angular momentum comes in units of
Planck's constant hbar times 1/2, 3/2, 5/2, and so on. Fermions satisfy
the Pauli exclusion principle --- you can't put two identical fermions
in the same state. That's why we have chemistry: the electrons stack up
in "shells" at different energy levels, instead of all going to the
lowest-energy state, because they are fermions and satisfy the exclusion
principle. In the Standard Model the fermions go like this:
LEPTONS QUARKS
electron electron neutrino down quark up quark
muon muon neutrino strange quark charm quark
tauon tauon neutrino bottom quark top quark
There are three "generations" here, all rather similar to each other.
There are also particles in the Standard Model called "bosons" having
angular momentum in units of hbar times 0,1,2, and so on. Identical
bosons, far from satisfying the exclusion principle, sort of like to all
get into the same state: one sees this in phenomena such as lasers,
where lots of photons occupy the same few states. Most of the bosons
in the Standard Model are called "gauge bosons". These carry the
different forces in the standard model, by which the particles interact:
ELECTROMAGNETIC FORCE WEAK FORCE STRONG FORCE
photon W+ 8 gluons
W-
Z
Finally, there is also a bizarre particle in the Standard Model called the
"Higgs boson". This was first introduced as a rather ad hoc hypothesis:
it's supposed to interact with the forces in such a way as to break the
symmetry that would otherwise be present between the electromagnetic
force and the weak force. It has not yet been observed; finding it would
would represent a great triumph for the Standard Model, while *not*
finding it might point the way to better theories.
Indeed, while the Standard Model has passed many stringent experimental
tests, and successfully predicted the existence of many particles which
were later observed (like the W, the Z, and the charm and top quarks),
it is a most puzzling sort of hodgepodge. Could nature really be this
baroque at its most fundamental level? Few people seem to think so;
most hope for some deeper, simpler theory.
It's easy to want a "deeper, simpler theory", but how to get it? What
are the clues? What can we do? Experimentalists certainly have their
work cut out for them. They can try to find or rule out the Higgs.
They can also try to see if neutrinos, assumed to be massless in the
Standard Model, actually have a small mass --- for while the Standard
Model could easily be patched if this were the case, it would shed
interesting light on one of the biggest mysteries in physics, namely why
the fermions in nature seem not to be symmetric under reflection, or
"parity". Right now, we believe that neutrinos only exist in a
left-handed form, rotating one way but not the other around the axis
they move along. This is intimately related to their apparent
masslessness. In fact, for reasons that would take a while to explain,
the lack of parity symmetry in the Standard Model forces us to assume
all observed fundamental fermions acquire their mass only through
interaction with the Higgs particle! For more on the neutrino mass puzzle,
try:
1) Paul Langacker, Implications of neutrino mass,
http://dept.physics.upenn.edu/neutrino/jhu/jhu.html
And, of course, experimentalists can continue to do what they always do
best: discover the utterly unexpected.
Theorists, on the other hand, have been spending the last couple of
decades poring over the standard model and trying to understand what
it's telling us. It's so full of suggestive patterns and partial
symmetries! First, why are there 3 forces here? Each force goes along
with a group of symmetries called a "gauge group", and electromagnetism
corresponds to U(1), while the weak force corresponds to SU(2) and the
strong force corresponds to SU(3). (Here U(n) is the group of n x n
unitary complex matrices, while SU(n) is the subgroup consisting of
those with determinant equal to 1.) Well, actually the Standard Model
partially unifies the electromagnetic and weak force into the
"electroweak force", and then resorts to the Higgs to explain why these
forces are so different in practice. Various "grand unified theories"
or "GUTs" try to unify the forces further by sticking the group SU(3) x
SU(2) x U(1) into a bigger group --- but then resort to still more
Higgses to break the symmetry between them!
Then, there is the curious parallel between the leptons and quarks in
each generation. Each generation has a lepton with mass, a massless or
almost massless neutrino, and two quarks. The massive lepton has charge
-1, the neutrino has charge 0 as its name suggests, the "down" type
quark has charge -1/3, and the "up" type quark has charge 2/3. Funny
pattern, eh? The Standard Model does not really explain this, although
it would be ruined by "anomalies" --- certain nightmarish problems that
can beset a quantum field theory --- if one idly tried to mess with the
generations by leaving out a quark or the like. It's natural to try
to "unify" the quarks and leptons, and indeed, in grand unified theories
like the SU(5) theory proposed in 1974 of Georgi and Glashow, the quarks
and leptons are treated in a unified way.
Another interesting pattern is the repetition of generations itself.
Why is there more than one? Why are there three, almost the same,
but with the masses increasing dramatically as we go up? The Standard
Model makes no attempt to explain this, although it does suggest that
there had better not be more than 17 quarks --- more, and the strong force
would not be "asymptotically free" (weak at high energies), which would
cause lots of problems for the theory. In fact, experiments strongly
suggest that there are no more than 3 generations. Why?
Finally, there is the grand distinction between bosons and fermions.
What does this mean? Here we understand quite a bit from basic
principles. For example, the "spin-statistics theorem" explains why
particles with half-integer spin should satisfy the Pauli exclusion
principle, while those with integer spin should like to hang out
together. This is a very beautiful result with a deep connection to
topology, which I try to explain in
2) John Baez, Spin, statistics, CPT and all that jazz,
http://math.ucr.edu/home/baez/spin.stat.html
But many people have tried to bridge the chasm between bosons and
fermions, unifying them by a principle called "supersymmetry". As in
the other cases mentioned above, when they do this, they then need to
pull tricks to "break" the symmetry to get a theory that fits the
experimental fact that bosons and fermions are very different.
Personally, I'm suspicious of all these symmetries postulated only to be
cleverly broken; this approach was so successful in dealing with the
electroweak force --- modulo the missing Higgs! --- that it seems to
have been accepted as a universal method of having ones cake and eating
it too.
Now, string theory comes in two basic flavors. Purely bosonic
string theory lives in 26 dimensions and doesn't have any fermions in
it. Supersymmetric string theories live in 10 dimensions and have both
bosons and fermions, unified via supersymmetry. To deal with the
fermions in nature, most work in physics has focused on the
supersymmetric case. Just for completeness, I should point out that
there are 5 different supersymmetric string theories: type I, type
IIA, type IIB, E8 x E8 heterotic and SO(32) heterotic. For more on
these, see "week72". We won't be getting into them here. Instead,
I just want to explain how fermions work in different dimensions, and
why nice things happen in dimensions of the form 8k + 2. Most of
what I say is in Section 3 of
3) John H. Schwarz, Introduction to supersymmetry, in Superstrings
and Supergravity, Proc. of the 28th Scottish Universities Summer
School in Physics, ed. A. T. Davies and D. G. Sutherland, University
Printing House, Oxford, 1985.
but mathematicians may also want to supplement this with material
from the book "Spin Geometry" by Lawson and Michelson, cited in
"week82".
To understand fermions in different dimensions we need to understand
Clifford algebras. As far as I know, when Clifford originally invented
these algebras in the late 1800s, he was trying to generalize Hamilton's
quaternion algebra by considering algebras that had lots of different
anticommuting square roots of -1. In other words, he considered
an associative algebra generated by a bunch of guys e_1,...,e_n,
satisfying
e_i^2 = -1
for all i, and
e_i e_j = - e_j e_i
whenever i is not equal to j. I discussed these algebras in "week82"
and I said what they all were --- they all have nice descriptions in terms
of the reals, the complexes, and the quaternions.
These original Clifford algebras are great for studying rotations in
n-dimensional Euclidean space --- please take my word for this for now.
However, here we want to study rotations and Lorentz transformations
in n-dimensional Minkowski spacetime, so we need to work with a slightly
Different kind of Clifford algebra, which was probably invented by Dirac.
In n-dimensional Euclidean space the metric (used for measuring distances)
is
dx_1^2 + dx_2^2 + ... + dx_n^2
while in n-dimensional Minkowski spacetime it is
dx_1^2 + dx_2^2 + ... - dx_n^2
or if you prefer (it's just a matter of convention), you can
take it to be
- dx_1^2 - dx_2^2 - ... + dx_n^2
So it turns out that we need to switch some signs in the definition
of the Clifford algebra when working in Minkowski spacetime.
In general, we can define the Clifford algebra C_{p,q} to be the algebra
generated by a bunch of elements e_i, with p of them being square roots
of -1 and q of them being square roots of 1. As before, we require that
they anticommute:
e_i e_j = - e_j e_i
when i and j are different. Physicists usually call these guys "gamma
matrices". For n-dimensional Minkowski space we can work either
with C_{n-1,1} or C_{1,n-1}, depending on our preference. As Cecile
DeWitt has pointed out, it *does* make a difference which one we use.
With some work, one can check that these algebras go like this:
C_{0,1} R + R C_{1,0} C
C_{1,1} R(2) C_{1,1} R(2)
C_{2,1} C(2) C_{1,2} R(2) + R(2)
C_{3,1} H(2) C_{1,3} R(4)
C_{4,1} H(2) + H(2) C_{1,4} C(4)
C_{5,1} H(4) C_{1,5} H(4)
C_{6,1} C(8) C_{1,6} H(4) + H(4)
C_{7,1} R(16) C_{1,7} H(8)
I've only listed these up to 8-dimensional Minkowski spacetime, and
the cool thing is that after that they sort of repeat --- more precisely,
C_{n+8,1} is just the same as 16 x 16 matrices with entries in C_{n,1},
and C_{1,n+8} is just 16 x 16 matrices with entries in C_{1,n}!
This "period-8" phenomenon, sometimes called Bott periodicity, has
implications for all sorts of branches of math and physics. This is
why fermions in 2 dimensions are a bit like fermions in 10 dimensions
and 18 dimensions and 26 dimensions....
In physics, we describe fermions using "spinors", but there are
different kinds of spinors: Dirac spinors, Weyl spinors, Majorana
spinors, and even Majorana-Weyl spinors. This is a bit technical but
I want to dig into it here, since it explains what's special about
8k + 2 dimensions and especially 10 dimensions.
Before I get technical, though, let me just summarize the point for
those of you who don't want all the gory details. "Dirac spinors"
are what you use to describe spin-1/2 particles that come in both
left-handed and right-handed forms and aren't their own antiparticle
--- like the electron. Weyl spinors have half as many components,
and describe spin-1/2 particles with an intrinsic handedness that
aren't their own antiparticle --- like the neutrino. "Weyl spinors"
are only possible in even dimensions!
Both these sorts of spinors are "complex" --- they have complex-valued
components. But there are also real spinors. These are used for describing
particles that are their own antiparticle, because the operation of
turning a particle into an antiparticle is described mathematically
by complex conjugation. "Majorana spinors" describe spin-1/2 particles
that come in both left-handed and right-handed forms and are their
own antiparticle. Finally, "Majorana-Weyl spinors" are used to describe
spin-1/2 particles with an intrinsic handedness that are their own
antiparticle.
As far as we can tell, none of the particles we've seen are Majorana
or Majorana-Weyl spinors, although if the neutrino has a mass it
might be a Majorana spinor. Majorana and Majorana-Weyl spinors
only exist in certain dimensions. In particular, Majorana-Weyl spinors
are very finicky: they only work in dimensions of the form 8k + 2.
This is part of what makes supersymmetric string theory work in 10
dimensions!
Now let me describe the technical details. I'm doing this mainly
for my own benefit; if I write this up, I'll be able to refer to
it whenever I forget it. For those of you who stick with me, there
will be a little reward: we'll see that a certain kind of supersymmetric
gauge theory, in which there's a symmetry between gauge bosons and
fermions, only works in dimensions 3, 4, 6, and 10. Perhaps
coincidentally --- I don't understand this stuff well enough to know ---
these are also the dimensions when supersymmetric string theory works
classically. (It's the quantum version that only works in dimension 10.)
So: part of the point of these Clifford algebras is that they give
representations of the double cover of the Lorentz group in different
dimensions. In "week61" I explained this double cover business,
and how the group SO(n) of rotations of n-dimensional Euclidean space
has a double cover called Spin(n). Similarly, the Lorentz group
of n-dimensional Minkowski space, written SO(n-1,1), has a double cover
we could call Spin(n-1,1). The spinors we'll discuss are all
representations of this group.
The way Clifford algebras help is that there is a nice way to
embed Spin(n-1,1) in either C_{n-1,1} or C_{1,n-1}, so any
representation of these Clifford algebras gives a representation
of Spin(n-1,1). We have a choice of dealing with real representations or
complex representations. Any complex representation of one of
these Clifford algebras is also a representation of the *complexified*
Clifford algebra. What I mean is this: above I implicitly wanted
C_{p,q} to consist of all *real* linear combinations of products of
the e_i, but we could have worked with *complex* linear combinations
instead. Then we would have "complexified" C_{p,q}. Since the
complex numbers include a square root of minus 1, the complexification
of C_{p,q} only depends on the dimension p + q, not on how many minus
signs we have.
Now, it is easy and fun and important to check that if you complexify R
you get C, and if you complexify C you get C + C, and if you complexify
H you get C(2). Thus from the above table we get this table:
dimension n complexified Clifford algebra
1 C + C
2 C(2)
3 C(2) + C(2)
4 C(4)
5 C(4) + C(4)
6 C(8)
7 C(8) + C(8)
8 C(16)
Notice this table is a lot simpler --- complex Clifford algebras
are "period-2" instead of period-8.
Now the smallest complex representation of the complexified Clifford
algebra in dimension n is what we call a "Dirac spinor". We can figure
out what this is using the above table, since the smallest complex
representation of C(n) or C(n) + C(n) is on the n-dimensional complex
vector space C^n, given by matrix multiplication. Of course, for
C(n) + C(n) there are *two* representations depending on which copy
of C(n) we use, but these give equivalent representations of Spin(n-1,1),
which is what we're really interested in, so we still speak of "the"
Dirac spinors.
So we get:
dimension n Dirac spinors
1 C
2 C^2
3 C^2
4 C^4
5 C^4
6 C^8
7 C^8
8 C^16
The dimension of the Dirac spinors doubles as we go to each new
even dimension.
We can also look for the smallest real representation of C_{n-1,1}
or C_{1,n-1}. This is easy to work out from our tables using
the fact that the algebra R has its smallest real representation
on R, while for C it's on R^2 and for H it's on R^4.
Sometimes this smallest real representation is secretly just the
Dirac spinors *viewed as a real representation* --- we can view C^n
as the real vector space R^{2n}. But sometimes the Dirac spinors
are the *complexification* of the smallest real representation ---
for example, C^{2n} is the complexification of R^n. In this
case folks call the smallest real representation "Majorana spinors".
When we are looking for the smallest real representations, we get
different answers for C_{n-1,1} and C_{1,n-1}. Here is what we get:
n C_{n-1,1} smallest C_{1,n-1} smallest
real rep real rep
1 R + R R Majorana C R^2
2 R(2) R^2 Majorana R(2) R^2 Majorana
3 C(2) R^4 R(2) + R(2) R^2 Majorana
4 H(2) R^8 R(4) R^4 Majorana
5 H(2) + H(2) R^8 C(4) R^8
6 H(4) R^16 H(4) R^16
7 C(8) R^16 H(4) + H(4) R^16
8 R(16) R^16 Majorana H(8) R^32
I've noted when the representations are Majorana spinors. Everything
repeats with period 8 after this, in an obvious way.
Finally, sometimes there are "Weyl spinors" or "Majorana-Weyl"
spinors. The point is that sometimes the Dirac spinors, or
Majorana spinors, are a *reducible* representation of Spin(1,n-1).
For Dirac spinors this happens in every even dimension, because the
Clifford algebra element Gamma = e_1 ... e_n commutes with everything
in Spin(1,n-1) and Gamma^2 is 1 or -1, so we can break the space of
Dirac spinors into the two eigenspaces of Gamma, which will be smaller
reps of Spin(1,n-1) --- the "Weyl spinors". Physicists usually call this
Gamma thing "gamma_5", and it's an operator that represents parity
transformations. We get "Majorana-Weyl" spinors only when we have
Majorana spinors, n is even, and Gamma^2 = 1, since we are then working
with real numbers and -1 doesn't have a square root. You can work out
Gamma^2 for either and C_{n-1,1} or C_{1,n-1}, and see that we'll
only get Majorana-Weyl spinors when n = 8k + 2.
Whew! Let me summarize some of our results:
n Dirac Majorana Weyl Majorana-Weyl
1 C R
2 C^2 R^2 C R
3 C^2 R^2
4 C^4 R^4 C^2
5 C^4
6 C^8 C^4
7 C^8
8 C^16 R^16 C^8
When there are blanks here, the relevant sort of spinor doesn't
exist. Here I'm not distinguishing Majorana spinors that come from
C_{n-1,1} and those that come from C_{1,n-1}; you can do that with
the previous table. Again, things continue for larger n in an obvious
way.
Now, let's imagine a theory that has a supersymmetry between a gauge
bosons and a fermion. We'll assume there are as many physical degrees of
freedom for the gauge boson as there are for the fermion. Gauge
bosons have n - 2 physical degrees of freedom in n dimensions: for
example, in dimension 4 the photon has 2 degrees of freedom, the left
and right polarized states. So we want to find a kind of spinor that
has n - 2 physical degrees of freedom. But the number of physical
degrees of freedom of a spinor field is half the number of (real)
components of the spinor, since the Dirac equation relates the
components. So we are looking for a kind of spinor that has 2(n - 2)
real components. This occurs in only 4 cases:
n = 3 -> 2(n-2) = 2, and Majorana or Weyl spinors have 2 real components
n = 4 -> 2(n-2) = 4, and Majorana or Weyl spinors have 4 real components
n = 6 -> 2(n-2) = 8, and Weyl spinors have 8 real components
n = 10 -> 2(n-2) = 16, and Majorana-Weyl spinors have 16 real components
Note we count complex components as two real components. And note how
dimension 10 works: the dimension of the spinors grows pretty fast as
n increases, but the Majorana-Weyl condition reduces the dimension by
a factor of 4, so dimension 10 just squeaks by!
John Schwarz explains how nice things happen in the same dimensions
for superstring theory in:
4) John H. Schwarz, Introduction to superstrings, in Superstrings
and Supergravity, Proc. of the 28th Scottish Universities Summer
School in Physics, ed. A. T. Davies and D. G. Sutherland, University
Printing House, Oxford, 1985.
He also makes a tantalizing remark: perhaps these 4 cases correspond
somehow to the reals, complexes, quaternions and octonions. Note:
3 = 1 + 2, 4 = 2 + 2, 6 = 4 + 2 and 10 = 8 + 2. You can never tell
with this stuff... everything is related.
I thank Joshua Burton for helping me overcome my fear of Majorana
spinors, and for correcting a number of embarassing errors in the
first version of this article.
Addendum added July 2001: Long after the above article was written,
Lubos Motl explained where the number 18 shows up in string theory:
>Today we know that the two heterotic string theories are related by
>various dualities. For example, in 17+1 dimension, the lattices Gamma16
>and Gamma8+Gamma8, with an added Lorentzian Gamma_{1,1}, become
>isometric. There is a single even self-dual lattice in 17+1 dimensions,
>Gamma_{17,1}. This is the reason why two heterotic string theories are
>T-dual to each other. The compactification on a circle adds two extra
>U(1)s (from Kaluza-Klein graviphoton and the B-field), and with
>appropriate Wilson lines, a compactification of one heterotic string
>theory on radius R is equivalent to the other on radius 1/R, using
>correct units.
-----------------------------------------------------------------------
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
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumping-off point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
Urs Schreiber
> Addendum added July 2001: Long after the above article was written,
> Lubos Motl explained where the number 18 shows up in string theory:
>
>>Today we know that the two heterotic string theories are related by
>>various dualities. For example, in 17+1 dimension,
I haven't studied this in detail yet, but I seem to understand that the
heterotic string lives in 9+1 spacetime dimensions (because that's the
critical dimension of the right-moving supersymmetric sector) with 26-10 =
16 = 18-2 of the remaining left-moving bosonic fields giving the gauge
group. So in which sense is Lubos Motl here talking about 17+1 dimensions?
Lubos Motl
it is not such a big deal. Spinors in 3 spatial dimensions - for example -
are 2-complex-component representations of the rotational group Spin(3),
which is essentially the same as SO(3), but has two different elements
"rotation by 0" and "rotation by 2.pi". Under the latter, spinors change
their signs.
SO(3) is the orthogonal group whose elements can be written as
exponentials of antisymmetric 3x3 real matrices - called generators of
SO(3). In spin(3) - which is the same as SU(2) - you have 2x2 complex
unitary matrices (M.Mdagger = 1) whose determinant is one (therefore S),
that can be written as exponentials of 2x2 antihermitean matrices.
Spinor is the 2-complex-dimensional "vector", on which SU(2) acts. To see
that SU(2) is the same group as SO(3) - up to the identification of two
elements, note that a natural basis of antisymmetric 3x3 matrices is
0 1 0 0 0 -1 0 0 0
-1 0 0, 0 0 0, 0 0 1
0 0 0 1 0 0 0 -1 0
If you denote these matrices Jx, Jy, Jz, you can see that they satisfy the
same commutation relations as Pauli's matrices multiplied by i/2: just
like the commutator [Jx,Jy]=Jz above, the same is true for the Pauli
matrices, with J replaced by sigma.
If you make a physical rotation in 3-dimensional space by angle PHI around
the z axis, you know how vectors and tensors transform. Spinors, on the
other hand, have 2 components. The first gets multiplied by exp(i.PHI/2)
and the other by exp(-i.PHI/2). If you rotate by 2.pi, the whole spinor
changes its sign - something unphysical for fermions because only
even-order things in fermions have physical meaning.
The 3-spatial dimensional spinors are based on the equivalence
SU(2)/Z2 = SO(3). You can extend them to the full Lorentz group because
SL(2,C)/Z2 = SO(3,1). Spinors can be however generalized to any
dimensions. At any rate, spinors form a representation of the rotational
or Lorentz group, whose dimension is a power of two. The action of the
rotations on spinors is expressed by Gamma matrices, which are mostly
anticommuting matrices of the size 2^k x 2^k.
Your task to learn spinors is equivalent to learning gamma matrices, in a
sense. The basic gamma matrices Gamma_mu satisfy
{Gamma_a, Gamma_b} = delta_{ab} or g_{ab}
where {u,v}=u.v+v.u is the anticommutator. This can be solved by taking
Gamma_a being a tensor product of various Pauli matrices, that satisfy
essentially the same equation. This is why the dimension of spinor
representations is a power of two.
In the simplest nonrelativistic quantum mechanical case, a 2-component
complex spinor (a,b) column describes the amplitude that a spin 1/2
particle such as electron has j_z up or down (+-1/2).
a is the amplitude that j_z = +1/2, b is the amplitude that j_z = -1/2.
Note that the projection of spin can have only two discrete values, +-1/2.
The probability that the projection is up, is |a|^2, that the projection
is down, is |b|^2. Because there are only probabilities, only two
possibilities are consistent with rotational symmetry: there is a way to
rotate a spinor by any angle, and two components of it then determine the
amplitudes that the spin is up/down with respect to a different axis.
As I said above, the rotation around the z-axis changes only the phases of
the two components, and therefore does not change the probabilities that
the spin is up or down, |a|^2, and |b|^2. A different SU(2) rotation would
mix the upper and the lower components.
Sorry, I don't have time to fix the errors in the text above.
David Hillman
John Baez wrote:
> I've only listed these up to 8-dimensional Minkowski spacetime, and
> the cool thing is that after that they sort of repeat --- more precisely,
> C_{n+8,1} is just the same as 16 x 16 matrices with entries in C_{n,1},
> and C_{1,n+8} is just 16 x 16 matrices with entries in C_{1,n}!
> This "period-8" phenomenon, sometimes called Bott periodicity, has
> implications for all sorts of branches of math and physics. This is
> why fermions in 2 dimensions are a bit like fermions in 10 dimensions
> and 18 dimensions and 26 dimensions....
Just curious: does this "period-8" phenomenon bear any relation to the one that
exists in integral quadratic forms? (I happen to be studying them right now.)
Here
the modulus of 8 crops up, related to the signature. In particular, if I
understand
this business correctly, the signature of an invertible (det +/- 1) symmetric
integer
matrix with even entries on the diagonal must be a multiple of 8. (Which I
suppose
means that there probably exists a famous 8 x 8 invertible integer matrix with
even
entries on the diagonal and signature +8. Does anyone know such a matrix?)
John Baez
David Hillman <d...@cablespeed.com> wrote:
>John Baez wrote:
>
>> I've only listed these up to 8-dimensional Minkowski spacetime, and
>> the cool thing is that after that they sort of repeat --- more precisely,
>> C_{n+8,1} is just the same as 16 x 16 matrices with entries in C_{n,1},
>> and C_{1,n+8} is just 16 x 16 matrices with entries in C_{1,n}!
>> This "period-8" phenomenon, sometimes called Bott periodicity, has
>> implications for all sorts of branches of math and physics. This is
>> why fermions in 2 dimensions are a bit like fermions in 10 dimensions
>> and 18 dimensions and 26 dimensions....
>Just curious: does this "period-8" phenomenon bear any relation to the
>one that exists in integral quadratic forms? (I happen to be studying
>them right now.)
It must, since both play important and tightly linked roles in string
theory, but I've never figured out how - mainly because whenever
I glance at a proof that even unimodular lattices can only occur
in dimensions that are multiples of 8, it never seems to have much
to do with Clifford algebras... even though some proofs mention
something called the "spinor genus", which sounds awfully promising.
>In particular, if I understand this business correctly, the signature
of an invertible (det +/- 1) symmetric integer matrix with even entries
>on the diagonal must be a multiple of 8.
That sounds right: I'm talking about "even unimodular lattices" and
you're talking about "invertible symmetric integer matrices with
even entries on the diagonal", but they're secretly the same thing as
long as your matrix is positive definite... and then the result you
mention reduces to the one I mentioned!
Some definitions might help you to see this: a "lattice" is a
subgroup of R^n that's isomorphic to Z^n. If we give R^n its usual
inner product, an "even" lattice is one such that the inner product
of any two vectors in the lattice is an even integer. A lattice is
"unimodular" if the volume of each cell of the lattice is 1.
To get from an even unimodular lattice to a matrix, pick a basis of
vectors in the lattice and form the matrix of their inner products.
This matrix will then be symmetric, have determinant +-1, and have
even entries down the diagonal. I leave it as an exercise to go
back the other way.
>Which I suppose means that there probably exists a famous 8 x 8 invertible
>integer matrix with even entries on the diagonal and signature +8. Does
>anyone know such a matrix?
Sure:
2 -1 0 0 0 0 0 0
-1 2 -1 0 0 0 0 0
0 -1 2 -1 0 0 0 0
0 0 -1 2 -1 0 0 0
0 0 0 -1 2 -1 0 -1
0 0 0 0 -1 2 -1 0
0 0 0 0 0 -1 2 0
0 0 0 0 -1 0 0 2
It's called E8, and it leads to huge wads of amazing mathematics.
For example, suppose we take 8 dots and connect the ith and jth
dots with an edge if there is a "-1" in the ij entry of the above
matrix. We get this:
o----o---o----o----o----o----o
|
|
o
Now, make a model with one ring for each dot in the above picture,
where the rings link if the corresponding dots have an edge connecting
them. We get this:
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Next imagine this model as living in the 3-sphere. Hollow out all
these rings: actually delete the portion of space that lies inside
them! We now have a 3-manifold M whose boundary dM consists of 8
connected components, each a torus. Of course, a solid torus also
has a torus as its boundary. So attach solid tori to each of these
8 components of dM, but do it via this attaching map:
(x,y) -> (y,-x+2y)
where x and y are the obvious coordinates on the torus, numbers
between 0 and 2pi, and we do the arithmetic mod 2pi. We now have a
new 3-manifold without boundary.
This manifold is called the "Poincare homology sphere". Poincare
invented it as a counterexample to his own conjecture that any
3-manifold with the same homology groups as a 3-sphere must *be*
the 3-sphere. But he didn't invent it this way. Instead, he got
it by taking a regular dodecahedron and identifying its opposite
faces in the simplest possible way, namely by a 1/10th turn.
So, we've gone from E8 to the dodecahedron!
Now let's go back, by a different route.
The fundamental group of the Poincare homology sphere has 120
elements. In fact, we can describe it as follows. The rotational
symmetry group of the dodecahedron has 60 elements. Take the
"double cover" of this 60-element group, namely the 120-element
subgroup of SU(2) consisting of elements that map to rotational
symmetries of the dodecahedron under the double cover
p: SU(2) -> SO(3)
This is the fundamental group of the Poincare homology where.
Now, this 120-element group has finitely many irreducible
representations. One of them just comes from restricting
the 2-dimensional representation of SU(2) to this subgroup:
call that R. There are 8 others: call them R(i) for i = 1,...,8.
Draw a dot for each one, and draw a line from the ith dot to the jth
dot if the tensor product of R and R(i) contains R(j) as a
subrepresentation. We get this picture:
o----o---o----o----o----o----o
|
|
o
Voila! Back to E8.
If you're interested in integral quadratic forms and how that
theorem you mentioned plays a role in string theory, you might like this:
.......................................................................
Also available at http://math.ucr.edu/home/baez/week95.html
This Week's Finds in Mathematical Physics - Week 95
John Baez
[stuff deleted]
It's a bit funny how one of the most curious features of bosonic
string theory in 26 dimensions was anticipated by the number theorist
Edouard Lucas in 1875. I assume this is the same Lucas who is famous
for the Lucas numbers: 1,3,4,7,11,18,..., each one being the sum of
the previous two, after starting off with 1 and 3. They are not
quite as wonderful as the Fibonacci numbers, but in a study of pine
cones it was found that while *most* cones have consecutive Fibonacci
numbers of spirals going around clockwise and counterclockwise, a
small minority of deviant cones use Lucas numbers instead.
Anyway, Lucas must have liked playing around with numbers, because
in one publication he challenged his readers to prove that: "A square
pyramid of cannon balls contains a square number of cannon balls only
when it has 24 cannon balls along its base". In other words, the
only integer solution of
1^2 + 2^2 + ... + n^2 = m^2,
is the solution n = 24, not counting silly solutions like n = 0 and
n = 1.
It seems the Lucas didn't have a proof of this; the first proof is
due to G. N. Watson in 1918, using hyperelliptic functions.
Apparently an elementary proof appears in the following ridiculously
overpriced book:
2) W. S. Anglin, "The Queen of Mathematics: An Introduction to
Number Theory", Kluwer, Dordrecht, 1995. ISBN 0-7923-3287-3.
For more historical details, see the review in
3) Jet Wimp, Eight recent mathematical books, Math. Intelligencer
18 (1996) 72-79,
Unfortunately, I haven't seen these proofs of Lucas' claim, so I don`t
know why it's true. I do know a little about its relation to
string theory, so I'll talk about that.
There are two main flavors of string theory, "bosonic" and
"supersymmetric". The first is, true to its name, just the quantized,
special-relativistic theory of little loops made of some abstract
string stuff that has a certain tension --- the "string tension".
Classically this theory would make sense in any dimension, but
quantum-mechanically, for reasons that I want to explain *someday*
but not now, this theory works best in 26 dimensions. Different
modes of vibration of the string correspond to different particles,
but the theory is called "bosonic" because these particles are all
bosons. That's no good for a realistic theory of physics, because
the real world has lots of fermions, too. (For a bit about
bosons and fermions in particle physics, see "week93".)
For a more realistic theory people use "supersymmetric" string
theory. The idea here is to let the string be a bit more abstract:
it vibrates in "superspace", which has in addition to the usual
coordinates some extra "fermionic" coordinates. I don't want to
get too technical here, but the basic idea is that while the usual
coordinates commute as usual:
x_i x_j = x_j x_i
the fermionic coordinates "anticommute"
y_i y_j = - y_j y_i
while the bosonic coordinates commute with fermionic ones:
x_i y_j = y_j x_i
If you've studied bosons and fermions this will be sort of
familiar; all the differences between them arise from the difference
between commuting and anticommuting variables. For a little glimpse
of this subject try:
4) John Baez, Spin and the harmonic oscillator,
http://math.ucr.edu/home/baez/harmonic.html
As it so happens, supersymmetric string theory --- often abbreviated
to "superstring theory" --- works best in 10 dimensions. There are
five main versions of superstring theory, which I described in
"week74". The type I string theory involves open strings --- little
segments rather than loops. The type IIA and type IIB theories
involve closed strings, that is, loops. But the most popular sort
of superstring theories are the "heterotic strings". A nice
introduction to these, written by one of their discoverers, is:
5) David J. Gross, The heterotic string, in "Workshop on Unified
String Theories", eds. M. Green and D. Gross, World Scientific,
Singapore, 1986, pp. 357-399.
These theories involve closed strings, but the odd thing about
them, which accounts for the name "heterotic", is that vibrations
of the string going around one way are supersymmetric and act as
if they were in 10 dimensions, while the vibrations going around
the other way are bosonic and act as if they were in 26 dimensions!
To get this string with a split personality to make sense, people
cleverly think of the 26 dimensional spacetime for the bosonic part
as a 10-dimensional spacetime times a little 16-dimensional curled-up
space, or "compact manifold". To get the theory to work, it seems that
this compact manifold needs to be flat, which means it has to be a
torus --- a 16-dimensional torus. We can think of any such torus as
16-dimensional Euclidean space "modulo a lattice". Remember, a lattice
in Euclidean space is something that looks sort of like this:
x
x
x x
x x
x x
x x
x x
x x
x x
x x
x
x
Mathematically, it's just a discrete subset L of R^n (n-dimensional
Euclidean space, with its usual coordinates) with the property that
if x and y lie in L, so does jx + ky for all integers j and k. When
we form n-dimensional Euclidean space "modulo a lattice", we decree
two points x and y to be the same if x - y is in L. For example,
all the points labelled x in the figure above count as the same
when we "mod out by the lattice"... so in this case, we get a
2-dimensional torus.
For more on 2-dimensional tori and their relation to complex analysis,
you can read "week13." Here we are going to be macho and plunge right
into talking about lattices and tori in arbitrary dimensions.
To get our 26-dimensional string theory to work out nicely when we
curl up 16-dimensional space to at 16-dimensional torus, it turns
out that we need the lattice L that we're modding out by to have some
nice properties. First of all, it needs to be an "integral" lattice,
meaning that for any vectors x and y in L the dot product x.y must
be an integer. This is no big deal --- there are gadzillions of
integral lattices. In fact, sometimes when people say "lattice" they
really mean "integral lattice". What's more of a big deal is that
L must be "even", that is, for any x in L the inner product x.x is
even. This implies that L is integral, by the identity
(x + y).(x + y) = x.x + 2x.y + y.y
But what's really a big deal is that L must also be "unimodular".
There are different ways to define this concept. Perhaps the easiest
to grok is that the volume of each lattice cell --- e.g., each
parallelogram in the picture above --- is 1. Another way to say it
is this. Take any basis of L, that is, a bunch of vectors in L
such that any vector in L can be uniquely expressed as an integer
linear combination of these vectors. Then make a matrix with the
components of these vectors as rows. Then take its determinant.
That should equal plus or minus 1. Still another way to say it
is this. We can define the "dual" of L, say L*, to be all the
vectors x such that x.y is an integer for all y in L. An integer
lattice is one that's contained in its dual, but L is unimodular if
and only if L = L*. So people also call unimodular lattices
"self-dual". It's a fun little exercise in linear algebra to show
that all these definitions are equivalent.
Why does L have to be an even unimodular lattice? Well, one
can begin to understand this a litle by thinking about what a closed
string vibrating in a torus is like. If you've ever studied the
quantum mechanics of a particle on a torus (e.g. a circle!) you may
know that its momentum is quantized, and must be an element of L*. So
the momentum of the center of mass of the string lies in L*.
On the other hand, the string can also wrap around the torus in
various topologically different ways. Since two points in Euclidean
space correspond to the same point in the torus if they differ by a
vector in L, if we imagine the string as living up in Euclidean space,
and trace our finger all around it, we don't necesarily come back to
the same point in Euclidean space: the same point *plus* any vector
in L will do. So the way the string wraps around the torus is
described by a vector in L. If you've heard of the "winding number",
this is just a generalization of that.
So both L and L* are really important here (which has to do with
the fashionable subject of "string duality"), and a bunch
more work shows that they *both* need to be even, which implies
that L is even and unimodular.
Now something cool happens: even unimodular lattices are only
possible in certain dimensions --- namely, dimensions divisible by 8.
So we luck out, since we're in dimension 16.
In dimension 8 there is only *one* even unimodular lattice (up to
isometry), namely the wonderful lattice E8! The easiest way to think
about this lattice is as follows. Say you are packing spheres in n
dimensions in a checkerboard lattice --- in other words, you color
the cubes of an n-dimensional checkerboard alternately red and black,
and you put spheres centered at the center of every red cube, using
the biggest spheres that will fit. There are some little hole left
over where you could put smaller spheres if you wanted. And as you
go up to higher dimensions, these little holes gets bigger! By the
time you get up to dimension 8, there's enough room to put another
sphere OF THE SAME SIZE AS THE REST in each hole! If you do that,
you get the lattice E8. (I explained this and a bunch of other
lattices in "week65", so more info take a look at that.)
In dimension 16 there are only *two* even unimodular lattices. One
is E8 + E8. A vector in this is just a pair of vectors in E8. The
other is called D16+, which we get the same way as we got E8: we
take a checkerboard lattice in 16 dimensions and stick in extra spheres
in all the holes. More mathematically, to get E8 or D16+, we take all
vectors in R^8 or R^16, respectively, whose coordinates are either
*all* integers or *all* half-integers, for which the coordinates add
up to an even integer. (A "half-integer" is an integer plus 1/2.)
So E8 + E8 and D16+ give us the two kinds of heterotic string
theory! They are often called the E8 + E8 and SO(32) heterotic
theories.
In "week63" and "week64" I explained a bit about lattices and Lie
groups, and if you know about that stuff, I can explain why the
second sort of string theory is called "SO(32)". Any compact
Lie group has a maximal torus, which we can think of as some Euclidean
space modulo a lattice. There's a group called E8, described in
"week90", which gives us the E8 lattice this way, and the product
of two copies of this group gives us E8 + E8. On the other hand, we
can also get a lattice this way from the group SO(32) of rotations in
32 dimensions, and after a little finagling this gives us the D16+ lattice
(technically, we need to use the lattice generated by the weights of the
adjoint representation and one of the spinor representations, according
to Gross). In any event, it turns out that these two versions of
heterotic string theory act, at low energies, like gauge field theories
with gauge group E8 x E8 and SO(32), respectively! People seem especially
optimistic that the E8 x E8 theory is relevant to real-world particle
physics; see for example:
6) Edward Witten, Unification in ten dimensions, in "Workshop on
Unified String Theories", eds. M. Green and D. Gross, World Scientific,
Singapore, 1986, pp. 438-456.
Edward Witten, Topological tools in ten dimensional physics, with
an appendix by R. E. Stong, in "Workshop on Unified String Theories",
eds. M. Green and D. Gross, World Scientific, Singapore, 1986, pp.
400-437.
The first paper listed here is about particle physics; I mention
the second here just because E8 fans should enjoy it --- it discusses
the classification of bundles with E8 as gauge group.
Anyway, what does all this have to do with Lucas and his stack
of cannon balls?
Well, in dimension 24, there are *24* even unimodular lattices, which
were classified by Niemeier. A few of these are obvious, like E8 + E8
+ E8 and E8 + D16+, but the coolest one is the "Leech lattice", which is
the only one having no vectors of length 2. This is related to a whole
WORLD of bizarre and perversely fascinating mathematics, like the
"Monster group", the largest sporadic finite simple group --- and also
to string theory. I said a bit about this stuff in "week66", and I will
say more in the future, but for now let me just describe how to get the
Leech lattice.
First of all, let's think about Lorentzian lattices, that is,
lattices in Minkowski spacetime instead of Euclidean space.
The difference is just that now the dot product is defined by
(x_1,...,x_n) . (y_1,...,y_n) = - x_1 y_1 + x_2 y_2 + ... + x_n y_n
with the first coordinate representing time. It turns out that
the only even unimodular Lorentzian lattices occur in dimensions
of the form 8k + 2. There is only *one* in each of those dimensions,
and it is very easy to describe: it consists of all vectors whose
coordinates are either all integers or all half-integers, and whose
coordinates add up to an even number.
Note that the dimensions of this form: 2, 10, 18, 26, etc., are
precisely the dimensions I said were specially important in "week93"
for some *other* string-theoretic reason. Is this a "coincidence"?
Well, all I can say is that I don't understand it.
Anyway, the 10-dimensional even unimodular Lorentzian lattice
is pretty neat and has attracted some attention in string theory:
7) Reinhold W. Gebert and Hermann Nicolai, E10 for beginners,
preprint available as hep-th/9411188
but the 26-dimensional one is even more neat. In particular,
thanks to the cannonball trick of Lucas, the vector
v = (70,0,1,2,3,4,...,24)
is "lightlike". In other words,
v.v = 0
What this implies is that if we let T be the set of all integer
multiples of v, and let S be the set of all vectors x in our lattice
with x.v = 0, then T is contained in S, and S/T is a 24-dimensional
lattice --- the Leech lattice!
Now *that* has all sorts of ramifications that I'm just barely beginning
to understand. For one, it means that if we do bosonic string theory in
26 dimensions on R^26 modulo the 26-dimensional even unimodular lattice,
we get a theory having lots of symmetries related to those of the Leech
lattice. In some sense this is a "maximally symmetric" approach to
26-dimensional bosonic string theory:
8) Gregory Moore, Finite in all directions, preprint available
as hep-th/9305139.
Indeed, the Monster group is lurking around as a symmetry group here!
For a physics-flavored introduction to that aspect, try:
9) Reinhold W. Gebert, Introduction to vertex algebras,
Borcherds algebras, and the Monster Lie algebra, preprint
available as hep-th/9308151
and for a detailed mathematical tour see:
10) Igor Frenkel, James Lepowsky, and Arne Meurman, "Vertex Operator
Algebras and the Monster," Academic Press, 1988.
Also try the very readable review articles by Richard Borcherds, who
came up with a lot of this business:
11) Richard Borcherds, Automorphic forms and Lie algebras.
Richard Borcherds, Sporadic groups and string theory.
These and other papers available at
http://www.pmms.cam.ac.uk/Staff/R.E.Borcherds.html
Well, there is a lot more to say, but I need to go home and pack
for my Thanksgiving travels. Let me conclude by answering a natural
followup question: how many even unimodular lattices are there in
32 dimensions? Well, one can show that there are AT LEAST 80 MILLION!
Lubos Motl
> I haven't studied this in detail yet, but I seem to understand that the
> heterotic string lives in 9+1 spacetime dimensions (because that's the
> critical dimension of the right-moving supersymmetric sector) with 26-10 =
> 16 = 18-2 of the remaining left-moving bosonic fields giving the gauge
> group. So in which sense is Lubos Motl here talking about 17+1 dimensions?
Exactly.
I was not talking about anything such as 17+1-dimensional space (or
spacetime). I was explaining T-duality for heterotic strings, and because
John Baez happened to love the number 18 - ask him for reasons, I don't
know - he used one paragraph from my text for his oversimplified
presentation.
Once again. You described very well that the heterotic string has 26-10=16
extra left-moving bosonic fields (or equivalently 32 fermionic fields).
For consistency (modular invariance), they must be compactified on a
16-dimensional torus that is "even" and "self-dual". Such a torus must
have a unit (16-dimensional) volume and is obtained by modding out R^{16}
by a lattice whose every lattice site has an even length-squared.
There are two such (=even self-dual) 16-dimensional lattices, called
Gamma_{16} and Gamma_{8}+Gamma_{8}, respectively. If you choose them to
define the periodicity of the 16 left-moving bosons, they give rise to two
types of the heterotic string; the corresponding gauge groups are
spin(32)/Z2 and E8 x E8 because the lattices are root lattices of these
two groups, roughly speaking.
However if you consider the heterotic string compactified on a circle
(i.e. you want to keep 8+1=9 dimensions large), you must make one more
left-moving and one more right-moving direction periodic. The relevant
lattice becomes 17+1-dimensional. It must be still even and self-dual, but
in this case, there exists a *unique* lattice of this type in 17+1
dimensions: the lattices Gamma_{16}+Gamma_{1,1} and
Gamma_{8}+Gamma_{8}+Gamma_{1,1} are isometric - because we added two new
directions through which they can be rotated into each other.
This explains that the spin(32)/Z2 heterotic string and the E8 x E8
heterotic string theories are equivalent (T-dual) once you compactify at
least 1 dimension. T-duality, as usually, relates one theory on circle of
radius R with another theory on circle of radius 1/R in some natural
stringy units.
However, I could also compactify e.g. three dimensions and work with a
19+3-dimensional lattice which is also unique. That would not lead to
Baez's favorite number 18 however. ;-) On the other hand, such a
19+3-dimensional lattice would coincide with the 2nd homology of K3
manifolds, including the metric defined by the oriented intersection
numbers. This agreement is the first indication why heterotic strings on a
three-torus (6+1 dimensions remain large) are dual (equivalent) to
M-theory on K3 manifolds (also 6+1 dimensions large). The agreement goes
beyond simple numerical observations such as the 19+3-dimensional space of
charges; it is in fact exact, but unfortunately there is not enough space
here to explain it. ;-)
David Hillman
> In article <3E2EBB42...@cablespeed.com>,
> David Hillman <d...@cablespeed.com> wrote:
>
>>Just curious: does this "period-8" phenomenon bear any relation to the
>>one that exists in integral quadratic forms? (I happen to be studying
>>them right now.)
>
>
> It must, since both play important and tightly linked roles in string
> theory, but I've never figured out how - mainly because whenever
> I glance at a proof that even unimodular lattices can only occur
> in dimensions that are multiples of 8, it never seems to have much
> to do with Clifford algebras... even though some proofs mention
> something called the "spinor genus", which sounds awfully promising.
Thanks for all that fun info (deleted) about E8!!
As for the proof and the spinor genus, all I know so far is this:
1) the spinor genus is the complete invariant for indefinite integral
quadratic forms.
2) there is a weaker invariant called the genus, which is the complete
invariant for this equivalence relation: A is equivalent to B iff there
is an integral matrix P with determinant +/-1 such that
(P transpose) (A + U) P = B + U
where + is direct sum and U is the 2 x 2 matrix
0 1
1 0
3) there is a slightly weaker invariant than the genus (basically it's
the genus without the dimension) which is the complete invariant for the
Witt equivalence relation, which is generated by the standard relation
(that A is equivalent to B iff there exists an integral matrix P with
determinant +/-1 such that (P transpose) A P = B) and the relation that
A is equivalent to A + U where U is the same 2 x 2 matrix given above.
Three aspects of each of these invariants are:
1) the oddity
2) the p-excess
3) the signature
and this relation holds modulo 8: p-excess + signature = oddity. When
the determinant is not divisible by an odd prime, the p-excess is zero.
When the determinant is not divisible by 2 and the diagonal elements are
all even, the oddity is zero. So it follows that in the case of
determinant +/-1 and all even elements on diagonal, the signature is
zero mod 8.
Why this relation holds, and what the spinor genus has to do with
Clifford algebras, I don't know.
You say that integral quadratic forms are important in string theory.
I'm curious: are indefinite integral quadratic forms important in string
theory? What about Witt equivalence classes of indefinite integral
quadratic forms?
Squark
> ...
> This manifold is called the "Poincare homology sphere". Poincare
> invented it as a counterexample to his own conjecture that any
> 3-manifold with the same homology groups as a 3-sphere must *be*
> the 3-sphere. But he didn't invent it this way. Instead, he got
> it by taking a regular dodecahedron and identifying its opposite
> faces in the simplest possible way, namely by a 1/10th turn.
Hmm... If you consider two adjacent faces of the dodecahedron and
the corresponding opposite faces, the edge at which the two
opposite faces touch is obtained by a 1/2 turn from the edge at
which the "original" faces touch, not by a 1/10 turn. This means
the identification (apparently) doesn't correspond to a
continuous free action by the Z/2Z group, so the result can
hardly be even a topological manifold (which as far as I remember
the Poincare homology sphere _is_). Or can it?
> So, we've gone from E8 to the dodecahedron!
Are their symmetry groups related too? Maybe there's a connection
to the Weyl group of E8, for instance?
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and change the
extension in the obvious way)
Squark
> 1) the spinor genus is the complete invariant for indefinite integral
> quadratic forms.
Okay, now you got me curious. Sorry for the immodest question, but
what in heaven _are_ indefinite intergral quadratic forms?
David Hillman
> David Hillman <d...@cablespeed.com> wrote in message news:<3E32EE22...@cablespeed.com>...
>
>>1) the spinor genus is the complete invariant for indefinite integral
>>quadratic forms.
>
>
> Okay, now you got me curious. Sorry for the immodest question, but
> what in heaven _are_ indefinite intergral quadratic forms?
If A is a symmetric n x n matrix and x = {x1, x2, ..., xn} is a column
vector of variables then
(x transpose) A x
is called the n-ary quadratic form with matrix A. For example if
x = {x,y} and A =
1 1
1 3
then the associated (binary) quadratic form is x^2 + 2xy + 3y^2. So, an
n-ary quadratic form is basically a homogeneous degree 2 polynomial in n
variables. If the entries of A are integers then it's called an integral
quadratic form. (Actually there is a competing definition which
requires that the coefficients of terms in (x transpose) A x be
integers, hence allowing half-integers on the off-diagonal terms of A.)
If y = {y1, ..., yn} then
(x transpose) A y
is called the symmetric bilinear form with matrix A. So the theories of
quadratic forms and symmetric bilinear forms are closely related. If we
consider a change of basis x = P x' where P is an n x n invertible
matrix and B = (P transpose) A P, then B represents the same quadratic
form in the new (primed) basis since then
(x' transpose) B x' = (x transpose) A x.
This is why in quadratic forms people study equivalence relations of the
form A ~ (P transpose) A P.
Just as with metrics, an n-ary quadratic form is called positive
definite if the signature of the matrix is +n, negative definite if the
signature is -n, and indefinite in all other cases.
Arnold Neumaier
> John Baez wrote:
> > ... whenever
> > I glance at a proof that even unimodular lattices can only occur
> > in dimensions that are multiples of 8, it never seems to have much
> > to do with Clifford algebras... even though some proofs mention
> > something called the "spinor genus", which sounds awfully promising.
>
> Thanks for all that fun info (deleted) about E8!!
>
> As for the proof and the spinor genus, all I know so far is this:
>
> 1) the spinor genus is the complete invariant for indefinite integral
> quadratic forms.
A proof of the classification of indefinite integral quadratic forms is in
Milnor & Husemoller, Symmetric bilinear forms, Springer 1973.
Arnold Neumaier
John Baez
> David Hillman <d...@cablespeed.com> wrote in message
>news:<3E32EE22...@cablespeed.com>...
>>1) the spinor genus is the complete invariant for indefinite integral
>>quadratic forms.
> Okay, now you got me curious. Sorry for the immodest question, but
> what in heaven _are_ indefinite integral quadratic forms?
David Hillman gave you one answer; here's another answer
that's equivalent and perhaps more physics-flavored.
Say we start out with Minkowski spacetime or more generally
any spacetime with an indefinite metric:
g = dx_1^2 + .... + dx_p^2 - dy_1^2 - .... - dy_q^2
Take a lattice in this spacetime with the property that for
any two vectors v,w in the lattice, g(v,w) is an integer.
If you take g(v,v) when v is a vector in the lattice, you get
what's called an "integral quadratic form" - a function that
depends on v in a quadratic way, and takes integer values.
It's "indefinite" because sometimes it's positive and sometimes
it's negative.
In article <3E32EE22...@cablespeed.com>,
David Hillman <d...@cablespeed.com> wrote:
>Thanks for all that fun info (deleted) about E8!!
Sure! E8 is one of the things god made when he wanted to see how
crazy he could make mathematics while still keeping it consistent.
>As for the proof and the spinor genus, all I know so far is this:
>
>1) the spinor genus is the complete invariant for indefinite integral
>quadratic forms.
>
>2) there is a weaker invariant called the genus, [....]
I'm just now reading J. H. Conway's book "The Sensual (Quadratic) Form",
which contains a fun explanation of these concepts. You'd probably
love it, because I get the impression it's much more vivid than the
most other books on the subject. However, it doesn't explain what
the spinor genus has to do with spinors. :-(
>You say that integral quadratic forms are important in string theory.
>I'm curious: are indefinite integral quadratic forms important in string
>theory?
Sure! Spacetime has an indefinite metric, after all.
Buried in my last article is a description of how you get
the Leech lattice in 24 dimensions from an indefinite integral
quadratic form in 26 dimensions. A similar trick gets you the
E8 lattice in 8 dimensions from an indefinite integral quadriatc
form in 10 dimensions.
Now, 10 and 26 are the dimensions for superstring theory
and bosonic string theory, respectively. If we take bosonic
string theory on 26-dimensional Minkowski spacetime modulo
the Leech lattice and mod the transformation x -> -x, we get an
utterly amazing string theory. This string theory has the
Monster as a symmetry group! The Monster is the largest
sporadic finite simple group, with
808017424794512875886459904961710757005754368000000000
elements. So, there's some pretty heavy stuff going down
involving string theory and indefinite integral quadratic forms!
>What about Witt equivalence classes of indefinite integral
>quadratic forms?
Dunno.
Here's some stuff about string theory and indefinite integral
quadratic forms. I call them "Lorentzian lattices",
but don't let that scare you - it just means the signature
has all plus signs except for one minus sign.
From "week95":
Lucas must have liked playing around with numbers, because
in one publication he challenged his readers to prove that: "A square
pyramid of cannon balls contains a square number of cannon balls only
when it has 24 cannon balls along its base". In other words, the
only integer solution of
1^2 + 2^2 + ... + n^2 = m^2,
is the solution n = 24, not counting silly solutions like n = 0 and
n = 1.
[....]
First of all, let's think about Lorentzian lattices, that is,
lattices in Minkowski spacetime instead of Euclidean space.
The difference is just that now the dot product is defined by
(x_1,...,x_n) . (y_1,...,y_n) = - x_1 y_1 + x_2 y_2 + ... + x_n y_n
with the first coordinate representing time. It turns out that
the only even unimodular Lorentzian lattices occur in dimensions
of the form 8k + 2. There is only *one* in each of those dimensions,
and it is very easy to describe: it consists of all vectors whose
coordinates are either all integers or all half-integers, and whose
coordinates add up to an even number.
Note that the dimensions of this form: 2, 10, 18, 26, etc., are
precisely the dimensions I said were specially important in "week93"
for some *other* string-theoretic reason [namely, that spinors
work in a specially cool way]. Is this a "coincidence"?
Well, all I can say is that I don't understand it.
Anyway, the 10-dimensional even unimodular Lorentzian lattice
is pretty neat and has attracted some attention in string theory:
7) Reinhold W. Gebert and Hermann Nicolai, E10 for beginners,
preprint available as hep-th/9411188
but the 26-dimensional one is even more neat. In particular,
thanks to the cannonball trick of Lucas, the vector
v = (70,0,1,2,3,4,...,24)
is "lightlike". In other words,
v.v = 0
What this implies is that if we let T be the set of all integer
multiples of v, and let S be the set of all vectors x in our lattice
with x.v = 0, then T is contained in S, and S/T is a 24-dimensional
lattice --- the Leech lattice!
Now *that* has all sorts of ramifications that I'm just barely beginning
to understand. For one, it means that if we do bosonic string theory in
26 dimensions on R^26 modulo the 26-dimensional even unimodular lattice
(and mod the transformation x -> -x), we get a theory having lots of
Squark
> The Monster is the largest
> sporadic finite simple group, with
>
> 808017424794512875886459904961710757005754368000000000
>
> elements.
Wow, that divides by 10^9. Is there a reason for it?
Squark
> Squark wrote:
> > David Hillman <d...@cablespeed.com> wrote in message news:<3E32EE22...@cablespeed.com>...
> >
> >>1) the spinor genus is the complete invariant for indefinite integral
> >>quadratic forms.
> >
> >
> > Okay, now you got me curious. Sorry for the immodest question, but
> > what in heaven _are_ indefinite intergral quadratic forms?
>
> quadratic form. (Actually there is a competing definition which
> requires that the coefficients of terms in (x transpose) A x be
> integers, hence allowing half-integers on the off-diagonal terms of A.)
Whew, it's integral in the sense of integers. Like, Z. I thought
it was gonna be something scary. Okay, now, what does genus have to do
with it, and what the heck do spinors have to do with it ???
Best regards,
Squark
------------------------------------------------------------------
Write to me using the following e-mail:
Skvark_N...@excite.exe
(just spell the particle name correctly and use the obvious
extension)
Arnold Neumaier
>
> If A is a symmetric n x n matrix and x = {x1, x2, ..., xn} is a column
> vector of variables then (x transpose) A x
> quadratic form. (Actually there is a competing definition which
> requires that the coefficients of terms in (x transpose) A x be
> integers, hence allowing half-integers on the off-diagonal terms of A.)
The 'correct' definition matching the common use in number theory
(making, e.g., the Leech lattice have an integral quadratic form)
is: A quadratic form Q(x)=x^T A x is integral if it takes integer values
at integral coordiantes. This is the case iff diagonal entries are
integral and off-diagonal entries are half integers.
The right way to look at integral quadratic forms is, however, basis
independent as an integral lattice - see the book
Conway & Sloane, Sphere packings, lattices and groups
This view gives a much better access to their algebraic properties
such as automorphism groups. For example, there is a very interesting
(and essentially unique) unimodular indefinite quadratic form in
26 dimensions... (see again Conway & Sloane).
Arnold Neumaier
John Baez
Squark <fii...@yahoo.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<b22tgd$mgh$1...@glue.ucr.edu>...
>> The Monster is the largest
>> sporadic finite simple group, with
>>
>> 808017424794512875886459904961710757005754368000000000
>>
>> elements.
>Wow, that divides by 10^9. Is there a reason for it?
Yes, god created the Monster this way so Carl Sagan could
say it has BILLIONS AND BILLIONS of elements.
Actually, given our cultural differences, you might not
even get this joke! Sagan had a TV show "Cosmos" in which
he loved to marvel about the BILLIONS AND BILLIONS of stars
in the galaxy, etcetera. At the time people enjoyed copying
him saying this, all the way down to his slightly nasal voice.
Anyway: while I can't really answer your question, I can
say something fun about the number of elements of the Monster.
If we factor it we get
2^46 3^20 5^9 7^6 11^2 13^3 17 19 23 29 31 41 47 59 71.
In the 1970s, the mathematicians Fricke, Ogg and Thompson were
studying the quotient of the hyperbolic plane by various subgroups
of SL(2,R), which acts as isometries of the hyperbolic plane.
Sitting inside SL(2,R) is SL(2,Z), and sitting inside that is
the group Gamma_0(p) consisting of matrices whose lower left
corner is congruent to zero mod p for the prime p. These fellows
were actually considering a somewhat larger group Gamma_0(p)+,
which is the normalizer of Gamma_0(p) inside SL(2,R).
Then they asked the question: if we take the quotient of the
hyperbolic plane by this group Gamma_0(p)+, when does the resulting
Riemann surface have genus zero? And they found that the answer
was: precisely when p is 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,
41, 47, 59 or 71.
Later, Ogg went to a talk on the Monster and noticed that these
primes were precisely the prime factors of the size of the Monster!
He wrote a paper offering a bottle of Jack Daniels whiskey to
anyone who could explain this fact. This was the beginning of a
subject called "Monstrous Moonshine"... the mysterious relation
between the Monster, the group SL(2,R), and Riemann surfaces.
It turns out that lying behind Monstrous Moonshine is a certain
string theory having the Monster as symmetries. So, the Monster
may someday turn out to be important in physics. However, much
remains mysterious about all of this, if not downright monstrous.
Patrick Schaaf
>> 808017424794512875886459904961710757005754368000000000
>Wow, that divides by 10^9. Is there a reason for it?
That's divided by 2^46, 3^20, 5^9, 7^6, 11^2, 13^3, 17, 19, ...
9 of the 2's conspire with the 5's to give the appearance of
10^9. Looks like a normal factorization, doesn't it?
best regards
Patrick (who knows how to use dc(1))
Michael Weiss
: It turns out that lying behind Monstrous Moonshine is a certain
: string theory having the Monster as symmetries. So, the Monster
: may someday turn out to be important in physics. However, much
: remains mysterious about all of this, if not downright monstrous.
There's an on-line webseminar about Monstrous Moonshine by Borcherds:
http://hplbwww2.hpl.hp.com/brims/websems/colloq98/borcherds/sem.html
Borcherds, incidentally, credits Conway with the phrase "Monstrous
Moonshine".
John Baez
Here's some more about the dodecahedron and E8:
In article <939044f.03012...@posting.google.com>,
Squark <fii...@yahoo.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<b0oha4$6gr$1...@glue.ucr.edu>...
>> This manifold is called the "Poincare homology sphere". Poincare
>> invented it as a counterexample to his own conjecture that any
>> 3-manifold with the same homology groups as a 3-sphere must *be*
>> the 3-sphere. But he didn't invent it this way. Instead, he got
>> it by taking a regular dodecahedron and identifying its opposite
>> faces in the simplest possible way, namely by a 1/10th turn.
>Hmm... If you consider two adjacent faces of the dodecahedron and
>the corresponding opposite faces, the edge at which the two
>opposite faces touch is obtained by a 1/2 turn from the edge at
>which the "original" faces touch, not by a 1/10 turn.
Really? I don't have the energy to visualize this right now.
>This means
>the identification (apparently) doesn't correspond to a
>continuous free action by the Z/2Z group, so the result can
>hardly be even a topological manifold (which as far as I remember
>the Poincare homology sphere _is_).
It's a smooth manifold.
> Or can it?
Hmm... Greg Kuperberg said that a 1/10th turn works, and I'm
inclined to believe him, since he's smart. I don't think there's
an action of Z/2 floating around here, but I don't see why you
think there *needs* to be.
Here's a more "highbrow" way of getting the Poincare homology
sphere which makes it more obviously a smooth manifold. Start
out with the rotational symmetry group of the dodecahedron
(or icosahedron, if you prefer). This is a 60-element subgroup
of SO(3). Now look at its inverse image under the double cover
SU(2) -> SO(3)
This is a 120-element subgroup of SU(2), say G. Now form the
quotient space SU(2)/G. This is the Poincare homology sphere, S!
Since SU(2) is just the ordinary 3-sphere, which is simply connected,
we are seeing that the ordinary 3-sphere is the universal cover of
the Poincare homology sphere.
In fact, you can pick a fundamental domain for the action of
G on SU(2) which looks like a dodecahedron. This gives a "tiling"
of SU(2) by 120 dodecahedra. When we form SU(2)/G, we get a space
that looks like a dodecahedron with its faces identified in some
manner.
Maybe you can try to see if the opposite faces are identified
by means of a 1/10 turn.
In fact, the above "tiling" of the 3-sphere by 120 dodecahedra is really
just the "120-cell" - one of the six regular polytopes in 4 dimensions.
>> So, we've gone from E8 to the dodecahedron!
>Are their symmetry groups related too?
Good point! I think so.
>Maybe there's a connection to the Weyl group of E8, for instance?
Interesting idea. Maybe I should explain a bit about what
you mean here.
The E8 lattice is the densest lattice packing of spheres in 8
dimensions. In this lattice each sphere touches 240 others,
and the density of the lattice is pi^4/384, or about .2537.
If we lived in 8 dimensions, I bet a lot of crystals would have
this lattice as their crystal structure! Squark is asking for
relations between the rotation/reflection symmetry group of this
lattice - the "Weyl group of E8" - and some symmetry group of the
dodecahedron.
They are quite different groups, since while the rotation group of
the dodecahedron has a mere 60 elements, and its double cover G has 120
elements, the Weyl group of E8 has a whopping
128 x 27 x 5 x 8! = 696,729,600
elements. But, here's an avenue to explore....
The group SU(2) has a nice description as the group of *unit
quaternions*, that is, things of the form
(a,b,c,d) = a + bI + cJ + dK
where a,b,c,d are real numbers with a^2 + b^2 + c^2 + d^2 = 1, and
I,J, and K satisfy
IJ = -JI = K, JK = -KJ = I, KI = -IK = J, I^2 = J^2 = K^2 = -1
It's natural to ask what the group G - the double cover of the rotation
group of the dodecahedron - looks like explicitly in terms of unit
quaternions. Conway and Sloane have given a nice description. Let's
write (a,b,c,d) for a + bI + cJ + dK, write Phi for the golden ratio:
Phi = (1 + sqrt(5))/2 = 1.61803398874989484820458683437....
and phi for the inverse of the golden ratio:
phi = Phi^{-1} = Phi - 1 = 0.61803398874989484820458683437....
Then the elements of G are of the form
(+-1, 0, 0, 0),
(+-1/2, +-1/2, +-1/2, +-1/2),
(0, +-1/2, +-phi/2, +-Phi/2)
and everything else that can be gotten by *even* permutations of the
coordinates. (Check that there are 120 and that they are closed under
multiplication!)
Charming, but what does it have to do with E8? Well, note that if we
take all *finite sums* of elements of G we get a subring of the
quaternions. Conway and Sloane call this the "icosians."
Any icosian is of the form
a + bI + cJ + dK
where a,b,c, and d live in the "golden field" Q(Phi) - this is the field
of numbers of the form
x + sqrt(5) y
where x and y are rational. Thus we can think of an icosian as an
8-tuple of rational numbers. We don't get all 8-tuples, however, but
only those lying in a given lattice.
In fact, we can put a norm on the icosians as follows. First of all,
there is usual quaternionic norm
||a + bI + cJ + dK||^2 = a^2 + b^2 + c^2 + d^2
But for an icosian this is always of the form x + sqrt(5)y for some
rational x and y. It turns out we can define a new norm on the icosians
by setting
|a + bI + cJ + dK|^2 = x + y.
With respect to this new norm, the icosians form a lattice that fits
isometrically in 8-dimensional Euclidean space. This is none other
than E8!
Now, since the group G acts on itself by left and right multiplication,
and conjugation too, it acts on the icosians in three ways, so it acts
on the E8 lattice in three ways!
BUT, I haven't checked that any of these actions of G on the icosians
preserves the "new" norm. In fact it seems unlikely now that I think
about it. But, I'm too lazy to do the calculation.
Anyway, if we can find an action of G (or any group) on the icosians
that preserves the new norm, we'll get a homomorphism from G (or
any group) to the Weyl group of E8.
John Baez
dodecahedra face-to-face to get a 4-dimensional Platonic
solid, or glue the faces of a single dodecahedron
to each other to get the Poincare homology sphere....
In article <b2qer3$lea$1...@glue.ucr.edu>, John Baez <ba...@galaxy.ucr.edu> wrote:
>Squark <fii...@yahoo.com> wrote:
>>ba...@galaxy.ucr.edu (John Baez) wrote:
>>> This manifold is called the "Poincare homology sphere". Poincare
>>> got it by taking a regular dodecahedron and identifying its opposite
>>> faces in the simplest possible way, namely by a 1/10th turn.
>>Hmm... If you consider two adjacent faces of the dodecahedron and
>>the corresponding opposite faces, the edge at which the two
>>opposite faces touch is obtained by a 1/2 turn from the edge at
>>which the "original" faces touch, not by a 1/10 turn.
>Really? I don't have the energy to visualize this right now.
I think I can visualize it better now.
>>This means the identification (apparently) doesn't correspond to a
>>continuous free action by the Z/2Z group, so the result can
>>hardly be even a topological manifold (which as far as I remember
>>the Poincare homology sphere _is_).
Yes, the Poincare homology sphere is a smooth manifold -
and no, there's really no problem here.
First, note that if you stick 4 identical regular dodecahedra together
so they share a vertex, there will be a little "wiggle room".
To get rid of this wiggle room you need to curl them up into the
fourth dimension a bit. If you keep gluing more dodecahedra
together this way, you eventually get a 4d Platonic solid built
from 120 dodecahedra, 4 touching at each vertex, 3 touching at
each edge, with each dodecahedron touching 12 others.
If you have trouble visualizing this, be thankful that there's
a picture on the web:
http://www.weimholt.com/andrew/120.html
Click on "Layer by Layer Progression" to see first how 12
dodecahedra are glued to a single one - note the wiggle room!
Then see how the rest of the layers are attached. One obtains
a "foldout model" of a 4d Platonic solid called the 120-cell.
Unfortunately, this website does not show you how to fold this
model into the fourth dimension.
Anyway, as I explained in my previous post, the Poincare
homology sphere is the quotient of 120-cell by the action
of a 120-element group. The result is a single dodecahedron
with its faces glued to each other!
More precisely, each face is glued to the opposite face
with a 1/10 turn, and this means that *4 faces meet at each
vertex*. I think this business of 4 faces meeting at each
vertex was what had you worried. But in fact, it's good!
It's just what you'd expect from taking a quotient of the
120-cell, where 4 dodecahedra meet at each vertex.
On a separate note:
>BUT, I haven't checked that any of these actions of G on the icosians
>preserves the "new" norm. In fact it seems unlikely now that I think
>about it. But, I'm too lazy to do the calculation.
>
>Anyway, if we can find an action of G (or any group) on the icosians
>that preserves the new norm, we'll get a homomorphism from G (or
>any group) to the Weyl group of E8.
I checked one example last night and it seemed that in this
example, the action of G on the icosians via conjugation preserved
the new norm. If true, this would give a homomorphism from
G (and in fact the symmetry group of the dodecahedron) to the
Weyl group of E8. I suppose when I'm feeling in a computational
mood I should check this.
Hmmm, it could be obvious from the McKay correspondence.
I'll do anything to avoid doing a bunch of multiplication....
Lagrange said mathematics is the avoidance of unnecessary
thought, but sometimes it's necessary to think a lot to see
how to avoid unnecessary thought!
Squark
> Here's some more about the dodecahedron and E8:
>
> In article <939044f.03012...@posting.google.com>,
> Squark <fii...@yahoo.com> wrote:
>
> >Hmm... If you consider two adjacent faces of the dodecahedron and
> >the corresponding opposite faces, the edge at which the two
> >opposite faces touch is obtained by a 1/2 turn from the edge at
> >which the "original" faces touch, not by a 1/10 turn.
>
> Really? I don't have the energy to visualize this right now.
Well, it's kinda obvious: the operation of taking the opposite point
sends tangent vectors into minus themselves (if the tangent vectors
are looked upon as embedded in the larger 3D space), not into
themselves after a 1/10 turn (here I'm thinking of the tangent
vector pointing from the center of one face to that of the adjacent
one, for instance).
> Hmm... Greg Kuperberg said that a 1/10th turn works, and I'm
> inclined to believe him, since he's smart. I don't think there's
> an action of Z/2 floating around here, but I don't see why you
> think there *needs* to be.
Well, okay, maybe not the action of Z/2 but probably the action of
some group, no? Anyhow the identification just doesn't seem
"continuous".
> ...Now form the
> quotient space SU(2)/G. This is the Poincare homology sphere, S!
>
> Since SU(2) is just the ordinary 3-sphere, which is simply connected,
> we are seeing that the ordinary 3-sphere is the universal cover of
> the Poincare homology sphere.
Hold on!! So, the Poincare homology sphere is 3-dimensional not
2-dimensional. Are you saying I should have in fact taken the interior
of the dodecahedron back there? I was thinking of it as surface, a
tiling of the sphere if you wish, the whole time. Well, that
definitely puts things into persepective. Points on the edges of the
dodecahedron may be glued in sets larger than 2, so what. When we glue
the torus from a square the vertices get all glued together also.
> In fact, you can pick a fundamental domain for the action of
> G on SU(2) which looks like a dodecahedron. This gives a "tiling"
> of SU(2) by 120 dodecahedra. When we form SU(2)/G, we get a space
> that looks like a dodecahedron with its faces identified in some
> manner.
>
> Maybe you can try to see if the opposite faces are identified
> by means of a 1/10 turn.
I definitely will!
> Anyway, if we can find an action of G (or any group) on the icosians
> that preserves the new norm, we'll get a homomorphism from G (or
> any group) to the Weyl group of E8.
Sounds kinky enough to work. Got to think about it some time.
John Baez
Squark <fii...@yahoo.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote in message
>news:<b2qer3$lea$1...@glue.ucr.edu>...
>> Here's some more about the dodecahedron and E8:
>> In article <939044f.03012...@posting.google.com>,
>> Squark <fii...@yahoo.com> wrote:
>> >Hmm... If you consider two adjacent faces of the dodecahedron and
>> >the corresponding opposite faces, the edge at which the two
>> >opposite faces touch is obtained by a 1/2 turn from the edge at
>> >which the "original" faces touch, not by a 1/10 turn.
>> Really? I don't have the energy to visualize this right now.
>Well, it's kinda obvious: the operation of taking the opposite point
>sends tangent vectors into minus themselves (if the tangent vectors
>are looked upon as embedded in the larger 3D space), not into
>themselves after a 1/10 turn (here I'm thinking of the tangent
>vector pointing from the center of one face to that of the adjacent
>one, for instance).
Okay, now I see what you mean.
But:
>> ...Now form the
>> quotient space SU(2)/G. This is the Poincare homology sphere, S!
>> Since SU(2) is just the ordinary 3-sphere, which is simply connected,
>> we are seeing that the ordinary 3-sphere is the universal cover of
>> the Poincare homology sphere.
>Hold on!! So, the Poincare homology sphere is 3-dimensional not
>2-dimensional.
Yeah, sure! Here's what I said about it earlier in this thread:
..........................................................................
/\ /\ /\ /\ /\ /\ /\
/ \ / \ / \ / \ / \ / \ / \
/ \ \ \ \ \ \ \
/ / \ / \ / \ / \ / \ / \ \
\ \ / \ / \ / \ / \ / \ / /
\ \ \ \ \ /\ \ \ /
\ / \ / \ / \ / \ \ \ / \ /
\/ \/ \/ \/ / \/ \ \/ \/
/ \
\ /
\ /
\ /
\/
Imagine this model as living in the 3-sphere. Hollow out all
these rings: actually delete the portion of space that lies inside
them! We now have a 3-manifold M whose boundary dM consists of 8
connected components, each a torus. Of course, a solid torus also
has a torus as its boundary. So attach solid tori to each of these
8 components of dM, but do it via this attaching map:
(x,y) -> (y,-x+2y)
where x and y are the obvious coordinates on the torus, numbers
between 0 and 2pi, and we do the arithmetic mod 2pi. We now have a
new 3-manifold without boundary.
This manifold is called the "Poincare homology sphere". Poincare
invented it as a counterexample to his own conjecture that any
3-manifold with the same homology groups as a 3-sphere must *be*
the 3-sphere.
..........................................................................
It turn out that there are lots of counterexamples to Poincare's
original conjecture: they're called "homology 3-spheres".
Poincare then went on to guess that any *simply connected*
manifold with the same homology groups as the 3-sphere is the
3-sphere; this is the famous and still unsolved "Poincare conjecture".
If there were a counterexample to the Poincare conjecture, it
would have to be a homology 3-sphere. That's one reason people
like homology 3-spheres. Also, these days, people like to
use topological quantum field theory to get invariants of homology
3-spheres. Buzzwords: Casson invariant, Floer theory.
>Are you saying I should have in fact taken the interior
>of the dodecahedron back there?
Yes, most definitely. We are taking the solid dodecahedron
and identifying opposite faces with a little twist. I think
I'm beginning to visualize it better now - see my last post
in this thread.
Also try this webpage by Tony Smith:
http://www.innerx.net/personal/tsmith/PDS3.html#PDS3
and this one:
http://www.math.unl.edu/~mbritten/ldt/poincare.html
which is a nice introduction to the Poincare conjecture.
PS - This website points out that there are contractible
3-manifolds that are not homeomorphic to R^3. So, you
have to be careful. :-)
David Hillman
John Baez wrote:
> In article <3E2EBB42...@cablespeed.com>,
> David Hillman <d...@cablespeed.com> wrote:
>
>
>>John Baez wrote:
>>
>>
>>>I've only listed these up to 8-dimensional Minkowski spacetime, and
>>>the cool thing is that after that they sort of repeat --- more precisely,
>>>C_{n+8,1} is just the same as 16 x 16 matrices with entries in C_{n,1},
>>>and C_{1,n+8} is just 16 x 16 matrices with entries in C_{1,n}!
>>>This "period-8" phenomenon, sometimes called Bott periodicity, has
>>>implications for all sorts of branches of math and physics. This is
>>>why fermions in 2 dimensions are a bit like fermions in 10 dimensions
>>>and 18 dimensions and 26 dimensions....
>>
>
>>Just curious: does this "period-8" phenomenon bear any relation to the
>>one that exists in integral quadratic forms? (I happen to be studying
>>them right now.)
>
>
> It must, since both play important and tightly linked roles in string
> theory, but I've never figured out how - mainly because whenever
> I glance at a proof that even unimodular lattices can only occur
> in dimensions that are multiples of 8, it never seems to have much
> to do with Clifford algebras... even though some proofs mention
> something called the "spinor genus", which sounds awfully promising.
Just saw today in a library a book by Cassel called "Rational Quadratic
Forms" which makes it clear that the spinor genus does have something to
do with the spinors you guys know and love. As I recall it mentioned
spin space, double covers, Clifford algebras. Looked like a quite
readable book.