Also available at http://math.ucr.edu/home/baez/week158.html
October 16, 2000
This Week's Finds in Mathematical Physics (Week 158)
John Baez
Like lots of mathematicians these days, I'm trying to understand
M-theory. It's a bit difficult, partially because the theory doesn't
really *exist* yet. If it existed, it would explain lots of stuff: on
that everyone agrees. But nobody knows how to formulate M-theory in a
precise way, so you can't open up a paper and stare at "the fundamental
equation of M-theory", or anything like that. There are some conjectures
about what M-theory might be like, but no solid agreement.
One thing that *does* exist is 11-dimensional supergravity. This is
supposed to be some kind of classical limit of M-theory. But the good
thing is, it's a classical field theory with a Lagrangian that you can
write down and ponder to your heart's content. So I'm trying to learn a
bit about this.
Unfortunately, being a mathematician, I like to understand everything
rather carefully, preferably in a conceptual way that doesn't involve
big equations with indices dangling all over the place. This is slowing
me down, because all the descriptions I've seen make 11-dimensional
supergravity look sort of ugly, when in fact it should be really pretty.
The physicists always point out that it's a lot simpler than the
supergravity theories in lower dimensions. On that I agree! But I
don't find it to be quite as simple as I'd like.
Now, mathematicians always whine like this when they are trying to learn
physics that hasn't been pre-processed by some other mathematician. So
just to show that I'm not completely making this stuff up, let me show
you the Lagrangian for 11d supergravity, as taken from the famous string
theory text by Green, Schwarz and Witten (see "week118"):
L = - (1/2k^2) e R
- (1/2) e psibar_M Gamma^{MNP} D_N[omega + omega')/2] psi_P
- (1/48) e F^2
- (sqrt(2)k/384) e
(psibar_M Gamma^{MNPQRS} psi_S + 12 psibar^N Gamma^{PQ} psi^R)
(F + F')_{NPQR}
- (sqrt(2)k/3456) epsilon^{M1 ... M11}
F_{M1 ... M4} F_{M5 ... M8} A_{M9 ... M11}
For comparison, here's the Lagrangian for ordinary gravity:
L = e R
Here e is the volume form and R is the Ricci scalar curvature. Of
course, there is a lot of stuff packed into this "R". General relativity
didn't look so slick when Einstein first made it up! But by now,
mathematicians have gnawed away at it for long enough that there's a
nice theory of differential geometry, where after a few months of work
you learn about "R". And after you've done this work, you realize that
"R" is a very natural concept. I want to get to this point for the
Lagrangian for 11d supergravity, but I'm not there yet.
You'll note that apart from a constant, the Lagrangian for 11d
supergravity starts out basically like the Lagrangian for ordinary
gravity. So *that* part I understand. It's just the other stuff
that's the problem.
Modulo some subtleties discussed below, the whole Lagrangian is built
from just three ingredients, which are the three basic fields in the
theory:
A) a Lorentzian metric g on the 11-dimensional manifold
representing spacetime,
B) a field psi on this manifold which takes values in the real
spin-3/2 representation of SO(10,1),
C) a 3-form field A on this manifold.
Physicists call the metric the "graviton". They call the spin-3/2 field
the "gravitino" or a "Rarita-Schwinger field". And they call the 3-form
a "gauge field", by analogy to the 1-form that appears in electromagnetism.
Above it's written as "A", to remind us of this analogy, but people often
use a "C" instead - for reasons I'll explain later.
Let me say a bit more about these three items. To define a spin-3/2
field on a manifold we need to give the manifold a spin structure.
Locally, we can do this by picking a smoothly varying basis of tangent
vectors. Such a thing is called a "frame field", but it also has other
names: in 4-dimensional spacetime people call it a "tetrad" or
"vierbein", after the German word for "four legs", but in 11-dimensional
spacetime people call it an "elfbein", after the German word for "eleven
legs". Anyway, this frame field determines a spin structure, and also
a metric, if we declare the basis to be orthonormal.
The metric, in turn, determines the Levi-Civita connection on the
tangent bundle. However, in modern Lagrangians for gravity, people
often treat the frame field and connection as independent variables.
This amounts to dropping the requirement that the connection be
torsion-free (while still requiring that it be metric-preserving). Only
when you work out the equations of motion from the Lagrangian do you get
back the equation saying the connection is torsion-free - and even this
only happens when there are no fields with *spin* around. In these
theories, spin creates torsion! But the torsion doesn't propagate: it
just sits there, determined by other fields. So we are basically just
repackaging the same data when we work with a frame field and connection
instead of a metric.
As a slight variant, instead of working with a frame field and
connection on the tangent bundle, we can work with a frame field and
"spin connection" - a connection on the spin bundle. We need to do this
whenever we have fields with half-integer spin around, as in supergravity.
Okay, so we'll use a frame field and spin connection to describe the
graviton. What about the gravitino? I'm less clear about this, but I
guess the idea is that we think of the spin-3/2 representation of the
Lorentz group SO(10,1) as sitting inside the tensor product of the
spin-1 representation and the spin-1/2 representation. This allows us
to think of the gravitino as a spinor-valued 1-form on spacetime.
That's why people write it as psi_N: the subscript indicates that we've
got some sort of 1-form on our hands. One thing I don't understand is
what, if any, constraints there are on a spinor-valued 1-form to make
it lie in the spin-3/2 representation.
What are spinors like in 11-dimensional spacetime? For this, go back
and reread "week93". You'll see that by Bott periodicity, spinors in
(n+8)-dimensional spacetime are just like spinors in n-dimensional
spacetime, but tensored with R^{16}. So spinors in 11-dimensional
spacetime are a lot like spinors in 3-dimensional spacetime! In 3
dimensions, the double cover of the Lorentz group is just SL(2,R), and
its spinor representation is R^2. Actually these are "real" spinors, or
what physicists call "Majorana" spinors. We could complexify and get
"complex" or "Dirac" spinors - but we won't!
Since the space of Majorana spinors in 3d spacetime is R^2, the space
of Majorana spinors in 11d spacetime is R^2 tensor R^{16} = R^{32}.
The gravitino is a 1-form taking values in this space.
Finally, what about the 3-form that appears in 11d supergravity? Why
is it called a "gauge field"? Well, if you've made it this far, you
probably know that the 1-form in electromagnetism (the "vector potential")
is perfectly suited for integrating along the worldline of a charged
point particle. Classically, the resulting number is just the *action.
In quantum theory, the exponential of the action describes how the
particle's *phase* changes.
If we're dealing with strings instead of point particles, we can pull
the same trick using a 2-form, which is the right sort of thing to
integrate over the 2-dimensional worldsheet of a string. Since people
call the 1-form in electromagnetism A, they naturally took to calling
this 2-form B. People like to study strings propagating in a background
metric that satisfies the vacuum Einstein equations, but they also study
what happens when you throw in a background B field like this, and add a
term to the string action that's proportional to the integral of B over
the string worldsheet. It works out nice when the B field satisfies the
obvious analogues of the vacuum Maxwell equations:
dF = 0, d*F = 0
where the "curvature" or "field strength tensor" F is given by F = dB.
Like Maxwell's equations, these equations are "gauge-invariant", in the
sense that we can change B like this without changing the field strength
tensor:
B -> B + dw,
where w is any 1-form.
Similarly, people believe that M-theory involves 2-dimensional membranes
called "2-branes". A 2-brane traces out a 3-dimensional "world-volume"
in spacetime. The 3-form field in 11d supergravity is perfectly suited
for integrating over this world-volume! So we're really dealing with a
still higher-dimensional analog of electromagnetism. Since we've already
talked about a 1-form A that couples to point particles and a 2-form field
B that couples to strings, it makes sense to call this 3-form C. Lots of
people do that. But I'll stick with Green, Schwarz and Witten, and call
it A. I'll write F for the corresponding field strength (which is 6dA
if we use their nutty normalization).
Let's look at that Lagrangian again, and see how much of it we can
understand now:
L = - (1/2k^2) e R
- (1/2) e psibar_M Gamma^{MNP} D_N[(omega + omega')/2] psi_P
- (1/48) e F^2
- (sqrt(2)k/384) e
(psibar_M Gamma^{MNPQRS} psi_S + 12 psibar^N Gamma^{PQ} psi^R)
(F + F')_{NPQR}
- (sqrt(2)k/3456) epsilon^{M1 ... M11}
F_{M1 ... M4} F_{M5 ... M8} A_{M9 ... M11}
The number "k" is just a coupling constant. The quantity "e" is the
volume form cooked up from the frame field. The quantity "R" is the
Ricci scalar cooked up from the spin connection. "psi_N" is the
gravitino field, and physicists write the inner product on spinors as
"psibar_N psi^N", where the "bar" is really an line drawn over the letter
psi. "A" is the 3-form field and "F" is the field strength. There's
also some other weird stuff I haven't explained yet.
Note: the first, middle, and last terms in this Lagrangian only involve
the bosonic fields - not the gravitino. They have the following meanings:
The first term, the "e R" part, is just the Lagrangian for the
gravitational field.
The middle term is, up to a constant, just what I'd call "F ^ *F": the
Lagrangian for the 3-form analog of Maxwell's equations.
The last term is, again up to a constant, just what I'd "F ^ F ^ A".
This is an 11-dimensional analog of the Chern-Simons term F ^ A that
you can add on to the electromagnetic Lagrangian in 3d spacetime.
The other two terms involve the gravitino. This is where I start getting
nervous. We've got this:
- (1/2) e psibar_M Gamma^{MNP} D_N[(omega + omega')/2] psi_P
and this:
- (sqrt(2)k/384) e
(psibar_M Gamma^{MNPQRS} psi_S + 12 psibar^N Gamma^{PQ} psi^R)
(F + F')_{NPQR}
The first one is mainly about how the gravitino propagates in a given
metric - a kind of spin-3/2 analog of the Lagrangian for the Dirac
equation. The second one is mainly about the coupling of the gravitino
to the 3-form field A - it's sort of like the coupling between the electron
and electromagnetic field in QED. But there's some funky stuff going on
here!
The "Gamma" gadgets are antisymmetrized products of gamma matrices,
i.e. Clifford algebra generators. I don't mind that. It's the stuff
involving omega' and F' that confuses me. "omega" is just a name for
the spin connection, so D_v[omega] would mean "covariant differentiation
with respect to the spin connection". But instead of using that, we use
D_v[(omega + omega')/2], where omega' is the "supercovariantization" of
the spin connection. Don't ask me that that means! I know it amounts
to adding some terms that are quadratic in the gravitino field, and I
know it's required to get the whole Lagrangian to be invariant under a
"supersymmetry transformation", which mixes up the gravitino field with
the graviton and 3-form fields. But I don't really understand the
geometrical meaning of what's going on, especially because the
supersymmetry only works "on shell" - i.e., assuming the equations of
motion. Similarly, I guess F' is some sort of "supercovariantization"
of the field strength tensor - but again, it seems fairly mysterious.
Anyway, we can summarize all this by saying we've got gravity, a
gravitino, and a 3-form gauge field interacting in a manner vaguely
reminiscent of how gravity, the electron and the electromagnetic
field interact in the Einstein-Dirac-Maxwell equations - except that
there's a "four-fermion" term where four gravitinos interact directly.
Stepping back a bit, one is tempted to ask: what exactly is so great
about this theory?
There are various ways to focus this question a bit. For example: the
Lagrangian for ordinary gravity makes sense in a spacetime of any
dimension. The 11d supergravity Lagrangian, on the other hand, only
makes sense in 11 dimensions. Why is that?
Well, if you ask a physicist, they'll tell you something like this:
Eleven is the maximum spacetime dimension in which one can formulate
a consistent supergravity, as was first recognized by Nahm in his
classification of supersymmetry algebras. The easiest way to see this
is to start in four dimensions and note that one supersymmetry relates
states differing by one half unit of helicity. If we now make the
reasonable assumption that there be no massless particles with spins
greater than two, then we can allow up to a maximum of N = 8
supersymmetries taking us from the helicity -2 through to helicity +2.
Since the minimal supersymmetry generator is a Majorana spinor with
four offshell components, this means a total of 32 spinor components.
Now in a spacetime with D dimensions and signature (1,D-1), the
maximum value of D admitting a 32 component spinor is D = 11.
In case you're wondering, this is from the first paragraph of this book:
1) The World in Eleven Dimensions: Supergravity, Supermembranes and
M-theory, ed. M. J. Duff, Institute of Physics Publishing, Bristol,
1999.
which is a collection of the most important articles on these topics.
It's a fun book to carry around - you can really impress people with the
title. But if you're a mathematician trying to decipher the above
passage, it helps to note a few things.
First, this explanation of why 11d supergravity is good boils down to
saying that it's the biggest, baddest supergravity theory around that
doesn't give particles of spin greater than two when we compactify the
extra dimensions in order to get a 4d theory.
Second, why is it "reasonable" to assume that there aren't massless
particles with spin greater than two? Because it's physics folklore
that quantum field theories with such particles are bad, nasty and evil
- in fact, so evil that nobody even dares explain why! Well, actually
there's a paper by Witten in the above book that contains references to
papers that supposedly explain why particles of spin > 2 are bad. It's
an excellent paper, too:
2) Edward Witten, Search for a realistic Kaluza-Klein theory, Nucl.
Phys. B186 (1981), 412-428.
Maybe someday I'll get up the nerve to read those references.
Third, once we buy into this "spin > 2 bad" idea, the rest of the
argument is largely stuff about spinors and Clifford algebras. This
is easy for mathematicians to learn, at least after a little physics
jargon has been explained. For example, a "Majorana" spinor is just
a real spinor, and "offshell components" refer to the components of a
field that are independent before you impose the equations of motion.
Fourth, if you're a mathematician wondering what "supersymmetry
algebras" are, there are places where you can start learning about
this without needing to know lots of physics:
3) Quantum Fields and Strings: A Course for Mathematicians, 2 volumes,
eds. P. Deligne, P. Etinghof, D. Freed, L. Jeffrey, D. Kazhdan, D. Morrison
and E. Witten, American Mathematical Society, Providence, Rhode Island,
1999.
Unfortunately, this book does not cover supergravity theories.
Fifth, Nahm's classification of supersymmetry algebras looks like the
sort of thing an algebraist should be able to understand, though I
haven't yet understood it. You can find it in Duff's book, or in the
original paper:
4) W. Nahm, Supersymmetries and their representations, Nucl. Phys.
B135 (1978), 149-166.
Next I want to mention some wild guesses and speculations about 11d
supergravity and M-theory. I'm guessing these theories are somehow
a cousin of 3d Chern-Simons theory, related in a way that involves
Bott periodicity. And I'm guessing that there's something deeply
octonionic about this theory. There's probably something wrong about
these guesses, since I can't quite get everything to fall in line.
But there's also probably something right about them.
We've seen two clues already:
First, the 11d spinors are related to 3d spinors via Bott periodicity,
which amounts to tensoring with R^{16} - the space of Majorana spinors
in 8d Euclidean space. Given the relation between octonion, 8d spinors
and Bott periodicity (see "week61" and "week105"), it's also very natural
to think of these Majorana spinors as pairs of octonions.
Second, the Chern-Simons-like term F ^ F ^ A in 11d supergravity is akin
to the 3d Chern-Simons Lagrangian F ^ A. But this relation is a bit
odd, since a crucial part of it involves switching from a 1-form gauge
field in the 3d case to a 3-form gauge field in the 11d case. To really
understand this, we first need to understand the geometry of these
generalized "gauge fields". These higher gauge fields are really not
connections on bundles, but connections on "n-gerbes", which are
categorified analogues of bundles. I explained this to some extent in
"week25" and "week151", but the basic idea is that there's an analogy
like this:
1-forms connections on bundles parallel transport of point particles
2-forms connections on gerbes parallel transport of strings
3-forms connections on 2-gerbes parallel transport of 2-branes
4-forms connections on 3-gerbes parallel transport of 3-branes
. . .
. . .
. . .
and so on. Just as connections on bundles naturally give rise to
Chern classes and the Chern-Simons secondary characteristic classes,
the same should be true for these higher analogues of connections.
There is also another clue: as I mentioned in "week118", you can only
write down Lagrangians for supersymmetric membranes in certain dimensions.
There are supposedly 4 basic cases, which correspond to the 4 normed
division algebras:
the 2-brane in dimension 4 - real numbers
the 3-brane in dimension 6 - complex numbers
the 5-brane in dimension 10 - quaternions
the 2-brane in dimension 11 - octonions
Part of the point is that the in these theories there are 1, 2, 4,
or 8 dimensions transverse to the worldvolume of the brane in question.
So 2-branes in 11 dimensions, in particular, are inherently "octonionic".
This seems like a wonderful clue, but so far I don't really understand it.
The evidence is lurking here:
5) T. Kugo and P. Townsend, Supersymmetry and the division algebras,
Nucl. Phys. B221 (1983), 357-380.
6) G. Sierra, An application of the theories of Jordan algebras and
Freudenthal triple systems to particles and strings, Class. Quant.
Grav. 4 (1987) 227.
7) J. M. Evans, Supersymmetric Yang-Mills theories and division algebras,
Nucl. Phys. B298 (1988), 92.
8) M. J. Duff, Supermembranes: the first fifteen weeks, Class. Quant.
Grav. 5 (1988), 189-205.
There are also tantalizing clues scattered through these fascinating
books:
9) Feza Gursey and Chia-Hsiung Tze, On the Role of Division, Jordan, and
Related Algebras in Particle Physics, World Scientific, Singapore, 1996.
10) Jaak Lohmus, Eugene Paal and Leo Sorgsepp, Nonassociative Algebras
in Physics, Hadronic Press, Palm Harbor, Florida, 1994.
However, these books are frustrating to me, because they make some
interesting claims without providing solid evidence.
Anyway, I'll try to keep gnawing away at this bone until I get to the
marrow! Any help would be appreciated.
-----------------------------------------------------------------------
Here is an article by Maxime Bagnoud that answers some of my
questions above....
From: Maxime Bagnoud
Subject: Re: This Week's Finds in Mathematical Physics (Week 158)
Date: 17 Oct 2000 00:00:00 GMT
John Baez wrote:
> One thing that *does* exist is 11-dimensional supergravity.
Unfortunately, only at the classical level, presumably. The quantum theory
doesn't seem to exist, neither. It's non-renormalizable, despite the large
amount of SUSY. We were not sure about this until quite recently, actually
(2 years ago?) You probably know this, but maybe not all the readers of the
"Finds".
> Okay, so we'll use a frame field and spin connection to describe the
> graviton. What about the gravitino? I'm less clear about this, but I
> guess the idea is that we think of the spin-3/2 representation of the
> Lorentz group SO(10,1) as sitting inside the tensor product of the
> spin-1 representation and the spin-1/2 representation. This allows us
> to think of the gravitino as a spinor-valued 1-form on spacetime.
> That's why people write it as psi_N: the subscript indicates that we've
> got some sort of 1-form on our hands. One thing I don't understand is
> what, if any, constraints there are on a spinor-valued 1-form to make
> it lie in the spin-3/2 representation.
As you guessed, there is a Clebsch-Gordan relationship like:
1 x 1/2 = 3/2 + 1/2 (where x is tensor product, + is direct sum)
in fact, out of a general spinor-vector, you can form a linear combination of
its components to get a spin 1/2 spinor by multiplying psi_M with a
Gamma^M matrix and summing of course on the vector index. The remaining part
of the representation is irreducible and it's the gravitino. (You can look
for example at Polchinski vol. II, page 23).
I guess that was your question.
> Similarly, people believe that M-theory involves 2-dimensional membranes
> called "2-branes". A 2-brane traces out a 3-dimensional "world-volume"
> in spacetime. The 3-form field in 11d supergravity is perfectly suited
> for integrating over this world-volume! So we're really dealing with a
> still higher-dimensional analog of electromagnetism. Since we've already
> talked about a 1-form A that couples to point particles and a 2-form field
> B that couples to strings, it makes sense to call this 3-form C. Lots of
> people do that. But I'll stick with Green, Schwarz and Witten, and call
> it A. I'll write F for the corresponding field strength (which is 6dA
> if we use their nutty normalization).
>
> Let's look at that Lagrangian again, and see how much of it we can
> understand now:
>
> L = - (1/2k^2) e R
> - (1/2) e psibar_M Gamma^{MNP} D_N[(omega + omega')/2] psi_P
> - (1/48) e F^2
> - (sqrt(2)k/384) e
> (psibar_M Gamma^{MNPQRS} psi_S + 12 psibar^N Gamma^{PQ} psi^R)
> (F + F')_{NPQR}
> - (sqrt(2)k/3456) epsilon^{M1 ... M11}
> F_{M1 ... M4} F_{M5 ... M8} A_{M9 ... M11}
>
> The middle term is, up to a constant, just what I'd call "F ^ *F": the
> Lagrangian for the 3-form analog of Maxwell's equations.
Now, it's time for me to answer one of your old questions! You seem to be
ready to hear the answer (you see, I never forget...).
Why should there be a 5-form in M-theory?
You nicely have replaced F^2 by F/\*F. Cool! Now, we can go further.
A is a 3-form, so F is a 4-form, then *F is a 11-4=7-form, then it should be
the field strength tensor of some 6-form potential, dA_(6)=*F, But a 6-form is
perfectly suited to be integrated over a 6-dimensional world-volume, i.e. a
5-brane! Here comes the M5-brane into the play.
Of course, in 11D SUGRA, the membrane is the fundamental object and the
M5-brane is a solitonic solution, but in a non-perturbative theory, solitonic
solutions can become fundamental at strong coupling and vice-versa. That's
why we expect that the M5-brane will play an important role in M-theory.
The other question was what this had to do with the theory of Smolin?
In the BFSS matrix model, there is only one kind of objects, matrix-valued
1-forms (D0-branes).
These have a nice interpretation in terms of M2-branes (that's how modern-day
physicists write membranes...:->) wrapped on the two light-cone coordinates,
but what is the role of M5-branes in this game is unclear.
While in the matrix model proposed by Smolin in hep-th/0002009, there are more
terms involving also a 4-form, which might be related with a wrapped M5-brane.
This raises the hope that this matrix model might be a better try for a
non-perturbative version of M-theory than the usual BFSS one. But this has to
be investigated in more detail, of course; that's more or less what I'm doing
now.
> Second, why is it "reasonable" to assume that there aren't massless
> particles with spin greater than two? Because it's physics folklore
> that quantum field theories with such particles are bad, nasty and evil
> - in fact, so evil that nobody even dares explain why! Well, actually
> there's a paper by Witten in the above book that contains references to
> papers that supposedly explain why particles of spin > 2 are bad. It's
> an excellent paper, too:
>
> 2) Edward Witten, Search for a realistic Kaluza-Klein theory, Nucl.
> Phys. B186 (1981), 412-428.
I'm not a specialist of this, but higher spins involve the representation
theory of W-algebras, which can hardly be described as easy. Of course,
that's not an argument, but I think that this has prevented many physicists
from pursuing the matter too far.
> Unfortunately, this book does not cover supergravity theories.
As a matter of fact, there are some books on supergravity in 4D, but no books
covering higher-dimensional supergravity theories with a reasonable amount of
explanations.
Of course, people really able to do this properly are a handful on this
planet, and even for them, this would require an enormous amount of work to
get things consistent all the way with a coherent choice of conventions and
check all the horrible formulas. On the other hand, when you hear their talks,
you usually don't get the feeling that they really want you to understand it,
but rather that they try to hide the truth about SUGRA in a well-hidden
"grimoire", maybe somewhere in Wizard's castle.
I hope some other people can shed more light on the subject, for example on
the supercovariantization of the spin connection (which I don't understand
very deeply, neither), maybe Aaron?
In any case, best regards to everyone,
and thanks John for the "This Week's Finds".
Maxime
-----------------------------------------------------------------------
And here is one by Robert Helling:
From: helling@x4u2.desy.de (Robert C. Helling)
Subject: Re: This Week's Finds in Mathematical Physics (Week 158)
Date: 24 Oct 2000 00:00:00 GMT
In article <8skfsg$qnb$1@mortar.ucr.edu>, John Baez wrote,
concerning 11d supergravity:
>I knew that people *thought* it wasn't renormalizable - that's not
>very new - but I didn't know people had become *sure* about it.
Well, it depends a bit on your definition of "non-renormalizable". In a strict
sense, it means that renormalization would require an _infinite_ number
of different counter terms. In order to fix all their coefficients
one would have to do an infinite number of experiments before the
theory becomes predictive. This should be compared to renormalizable
theories that get along with a finite number although their coefficients
have to be adopted a each order of pertubation theory. Better are
superrenormalizable theories that also have a finite number of counter
terms but there coefficients are not changed after some order in
pertubation theory.
The status of supergravity is as follows (in my understanding): Long ago
(what you refer to as thought) people figured out an additional
term in the action that might appears as counter term and that is
invariant under all symmetries of the action (well, in 11d not
all symmetries, the full supermultiplet is not known and is expected
to be infinite but with _fixed_ relative coefficients. So there
is still just one parameter). E.g. in 4D, the situation is simpler because
there a superspace formulation is at hand that allows you to write expressions
that are automatically supersymmetric.
What people didn't know was whether this counter term really arises in
loop integrals. But now, in 11D Deser at al have calculated that a
certain combitation of four Riemann tensors appears as a counterterm
(has a non-zero coefficient) at 2 loop order.
This should be compared to Einstein's theory in 4D: There it was known
that a certain combination of two Weyl tensors does not vanish
by Bianchi identities or is topological. Therefore it is a possible
counterterm. 10 years ago, people did a 3 loop calculation (this is
really hard work!) to show that it actually arises. 4D sugra does not
allow this term and its first possible counter term appears only at the
next loop order. I know somebody personally that spend the last 10 years
doing this calcualtion and hasn't got very far (luckily he still has a
job in physics).
But finding one counter term that was not in the classical action
does not show a theory is non-renormalizable (remember this is a
statement about infinitely many counter terms, so it is about
an infinity of orders of pertubation theory). It might just be
that this one term has been in the classical action just with coefficient
(coupling constant) 0 that is renormalized at higher orders. This
behaviour is highly unlikely but a mathematical possibility.
Actually showing a theory to be non-renormalizable is as hard
as showing a theory is renormalizable (not too long ago a
Nobel prize was awarded for such a proof ;-))
Now for your point: "Is renormalizability a must?". I think it
is very old fashioned to give an affirmative answer to this
question. A more modern answer would probably be: It's fine
for a theory to be non-renormalizable as long as it is only
an effective theory. Fermi psi^4 theory is not renormalizable
and is a nice theory of weak interactions as long as one
stays away from the EW breaking scale.
The appearance of the infinity of counter terms just shows
that there is some understanding of the high energy degrees
of freedom missing. And there will be a more fundamental
theory lurking around that reduces to this effective
theory for small energies.
So for a string theorist, non-renormalizability for sugra
is just fine: It's just the low energy effective theory
of string or M theory. It does not contain all degrees
of freedom, just the light ones. One way of thinking about
this is that string theory is just a fancy way of regulating
sugra. It supplies finite coefficients for the infinity
of possible counter terms. For example, in 10D sugra has
a one loop counterterm of the form R^4. This is just
an infinity in sugra. But in string theory, this has to be
a finite number, and in fact it is. It is
zeta(3) = sum_n n^(-3).
The same thing is expected for 11D sugra and M-Theory. But
as long as nobody really knows what M-Theory really is this
does not help very much.
Let me add a personal remark: In hep-th/9905183 we have
tried to do exactly this thing for M(atrix)-Theory, but
as it turned out, there are problems remaining.
>>> Unfortunately, this book does not cover supergravity theories.
>>As a matter of fact, there are some books on supergravity in 4D, but no books
>>covering higher-dimensional supergravity theories with a reasonable amount of
>>explanations.
>I've noticed! It's scandalous!
>>Of course, people really able to do this properly are a handful on this
>>planet, and even for them, this would require an enormous amount of work to
>>get things consistent all the way with a coherent choice of conventions and
>>check all the horrible formulas.
I know that at least three of the sugra hot shots of the eighties
independently started such projects and there are sugra_book.tex files
of various stages on their hard disks. They al gave up or made it a really
long term project since they figured out that it would cost them years to
basically redo all calculations in a coherent formalism.
This is just a horrible mess. Dealing with fermions just increases the pain.
Doing a calculation twice you never get the same signs. I have already spend
days figuring out what + h.c. in the stony brook textbook on 4D sugra meant
(actually, it should have read - h.c. since what was computed was a anti-
hermitian quantity). They never stated what their conventions for hermititan
conjugation are. Does it also reverse the order of differential operators?
What about index positions (remember, for anticommuting variables
psi^a phi_a = - psi_a phi^a) and all these kinds of things?
In addition, the old guys that have done many of the calcualtions use very
strange (aka "convenient") conventions, like
psi^2 = 1/2 psi^a psi_a
or they raise and lower SL(2,C) not with the epsilon tensor, but with i times
the epsilon tensor (relate this to h.c.!) This is just a mess and you always
get the feeling that you are wasting your time with such things but in the end
your calculations are not even reliable!
This was all 4D, but the horror starts in higher dimensions. There gamma matrix
algebra becomes interesting. Again there are N+1 conventions if N people
work on something and you have to have hunderets of Fierz identities at hand.
I know a grad student that spend months working them out on a computer and
thought it would be a good service to the community to write a paper like
"Gamma identities and Fierzing in diverse dimensions". This would probably
be like the PhysRep by Slansky and Lie algebra stuff. But his advisor told
him not to do that "This is your capital. Put it in your drawer and lock it.
Be sure, erverybody in the field has such a drawer!"
And this is why there will never be such a text. But I heard people say that
working out for yourself that 11d sugra is indeed supersymmetric is a good
exercise. I have never done it.
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Albert Einstein Institute Potsdam
Max Planck Institute For Gravitational Physics
and
2nd Institute for Theoretical Physics
DESY / University of Hamburg
Email helling@x4u2.desy.de Fon +49 40 8998 4706
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