Arguments in favor of finiteness are much more sloppy
I have been discussing the maximally supersymmetric supergravity and its conjectured perturbative finiteness many times on this blog. But the immediate reason for a new entry is the following paper by my ex-adviser Tom Banks,
As I discussed in the article about the royal status of 11-dimensional supergravity, 11 dimensions is the maximum number of large dimensions in which one may have a theory with unbroken supersymmetry and a nontrivial particle content at long distances. The theory was discovered in the late 1970s and is known as the 11-dimensional supergravity.
In such a high spacetime dimension, it's of course non-renormalizable as a field theory but we have known since 1995 that it is the right long-distance approximation to a theory that is fully consistent and finite, namely M-theory which is an important cousin of string theory. More precisely, M-theory is another vacuum or limit of string theory that, unlike the 10-dimensional vacua, has 11 spacetime dimensions. The 11-dimensional dynamics may be obtained by sending the string coupling \(g_s\) to infinity either in the \(E_8\times E_8\) heterotic string theory or (which is easier) in type IIA string theory. When we do so, a new, eleventh dimension of spacetime emerges and grows larger.
The elf-dimensional supergravity
This 11-dimensional M-theory may be compactified down to various lower spacetime dimensions. From the viewpoint of the bread-and-butter physics, the most important ones among these compactifications are compactifications on singular 7-dimensional manifolds of \(G_2\) holonomy: it's one of the several major descriptions by which string/M-theory may describe the Universe around us and which has all the desired qualitative properties to agree with all the particles and forces we have observed in our world. These compactifications were recently promoted as the most canonical example of "generic string/M-theoretical compactifications" by Gordon Kane, Bobby Acharya, and their colleagues in their phenomenological papers. Edward Witten has played a key role in the discovery of this new "stringy scenario" a decade ago, too.
However, those compactifications are complicated and only preserve four real supercharges which is equivalent to the \(\NNN=1\) supersymmetry in \(d=4\), the usual extent of supersymmetry that is employed in the realistic model building. From a top-down viewpoint, the more fundamental are the more supersymmetric compactifications, especially the maximal i.e. \(\NNN=8\) supergravity with 32 real supercharges, the same number as the original 11-dimensional theory. This theory is pretty much unique and so is its consistent ultraviolet completion, M-theory on a 7-torus.
The maximal supergravity: symmetries, field content
As we have recalled in the royal article, the 11-dimensional supergravity contains one supermultiplet of fields, the gravitational supermultiplet, which boasts \(2^8=256\) physical polarizations for each (light-like) momentum. One-half of them are fermionic, they're the components of the spin-3/2 gravitino (a maximally constrained tensor product of a spinor and a vector), while the remaining 128 polarizations are bosonic ones. 44 of them are components of the metric tensor; 84 of them are components of the \(C_{\lambda\mu\nu}\) three-form potential that couples to membranes and fivebranes.
What happens if you compactify the theory to 4 dimensions i.e. dimensionally reduce it over 7 dimensions? Some values of the vector-like Lorentz indices will become scalar-like. The spin as understood in \(d=4\) will be smaller than the spin as extracted from \(d=11\). How many scalar fields will we get?
We only get scalar fields from the bosons; the spin-statistics relation always holds in these and other theories. From the metric tensor, we get the components \(g_{\mu\nu}\) where \(\mu,\nu\) go over the 7 dimensions we have compactified on a tiny 7-torus. The number of such components is \(7\times 8/2\times 1 = 28\). These components remember the radii and angles in the 7-torus defining the compactification. The gravitational excitations should be traceless but we will assign the task of keeping the tracelessness to the components in the uncompactified dimensions so there's no elimination of components here.
There are also now-scalar components of the three-form potential, \(C_{\lambda\mu\nu}\). If all the indices are set to some of the numbers in the list of 7 compact dimensions, we get \((7\times 6\times 5)/(3\times 2 \times 1) = 35\). I would have to spend a lot of time to explain the complete justification but we must also count the components of the dual 6-form potential \(\tilde C_{\lambda\mu\nu\pi\rho\sigma}\) that couples to the fivebranes. These six indices may be set to the seven compactified dimensions with one missing exception; so there are seven ways to do so. We get 7 new scalar physical polarizations.
In total, we have \(28+35+7=70\) scalars in the four-dimensional \(\NNN=8\) supergravity. If you were a very good numerologist, you would be able to realize that \(70=133-63\) and because \(133\) and \(63=8^2-1\) are dimensions of the \(E_{7(7)}\equiv E_{7(7)}(\RR)\) and \(SU(8)\) groups, those 70 scalars could be parameterizing the coset or quotient \(E_{7(7)}/SU(8)\) of a noncompact exceptional group and its largest noncompact subgroup.
The arithmetic check above is of course very far from being a full proof that the symmetries are relevant for the theory; but the conclusion is right. The supergravity theory has an exact, internal, noncompact \(E_{7(7)}\) global symmetry. Its \(SU(8)\) subgroup's action on the scalar fields is trivial; so the scalar fields inevitably parameterize the coset. There are various independent ways to see why the noncompact exceptional symmetry exists – a brute force algebraic calculation; arguments that the high degree of SUSY implies that the moduli spaces of scalars have to be cosets; reconciliation of partial stringy symmetries such as T-dualities – but just believe me that the symmetry is there.
Charged objects in the theory (its stringy completion)
To have some fun, let me show you one more numerological check that the \(E_{7(7)}\) exceptional symmetry exists. Consider the stringy completion of our \(d=4\) \(\NNN=4\) SUGRA, namely M-theory on a seven-torus, and let's count the number of individual \(U(1)\) gauge symmetries (each of them has a new kind of a charge and the corresponding charged objects, too).
First, remembering Kaluza and Klein 90+ years ago, we know that general relativity compactified on a torus produces \(U(1)\) groups. We get seven of them; the gauge fields are \(A_{\mu}^{(n)}=g_{\mu n}\) where \(\mu\) is the 4-dimensional Lorentz index and \(n\) goes over the seven compact dimensions and labels the seven different \(U(1)\) groups. The charged objects are any particles that are moving in the directions of the compactified seven-torus; recall that quantum mechanics guarantees that the momentum on a compact space is quantized (and, consequently, so is the resulting electric charge).
However, we also have \(A^{(m,n)}_\mu=C_{\mu mn}\). The indices \(m,n\) take 7 possibilities each and due to the antisymmetry, they produce \(7\times 6/2\times 1=21\) different \(U(1)\) fields. The corresponding charged objects are M2-branes with both dimensions wrapped on a two-torus within the spacetime seven-torus.
Similarly, there are five-branes wrapped on five-tori that are charged under similar \(U(1)\) symmetries whose gauge fields are \(\tilde C_{\mu mnopq}\) where \(mnopq\) form a quintuplet of indices chosen from the list of seven compact directions. We get "7 choose 5" which is 21 different \(U(1)\) fields again.
There are some extra charged objects (and their corresponding \(U(1)\) groups) I haven't counted yet. The charged objects are the Kaluza-Klein monopoles which are really 6-branes (we want fully wrapped ones) with 1 additional "systemic" compact spatial dimension. Any of the 7 spacetime toroidal dimensions may play the role of this "systemic" Kaluza-Klein direction so we get 7 independent Kaluza-Klein monopole charges.
In total, we may find \(7+21+21+7=56\) different \(U(1)\) groups. It's only 28 times and not 56 times more electromagnetic fields than in Maxwell's theory because in the 56-based counting, Maxwell's theory has two types of charges, the electric ones and the magnetic monopoles. Now, it is not a coincidence that 56 is the dimension of the fundamental representation of \(E_{7(7)}\) – which is just a continuation of the compact \(E_7\) Lie group. In fact, these charges do transform under the \(E_{7(7)}\) non-compact global symmetry that I have already discussed.
The supergravity itself, i.e. the long-distance limit of M-theory on a seven-torus, doesn't admit any charged objects. The torus is infinitely tiny in this limit so the Kaluza-Klein particles are infinitely heavy and as long as we are at a generic point of the moduli spaces, the other charged objects are related to the Kaluza-Klein particles by the symmetry so they must be equally heavy. And in fact, you could think that in the supergravity, the charged objects and their precise charge quantization conditions don't make any difference at all.
However, in the full stringy theory, namely M-theory on a seven-torus, these charged objects – Kaluza-Klein particles (moving along the new dimensions), wrapped membranes and fivebranes, and wrapped Kaluza-Klein monopoles – are allowed, real, and they make a difference. Because the Kaluza-Klein particles are electromagnetic duals of the wrapped Kaluza-Klein monopoles and the wrapped M2-branes are electromagnetic duals of the wrapped M5-branes – it is not an accident that the list 7,21,21,7 was symmetric – we must remember the Dirac quantization rule for the magnetic charges.
When we impose this rule on all the charges we have, we find out that these charges must belong to a particular lattice. The lattice isn't quite unique but by \(E_{7(7)}\) transformations, you may get all the solutions (all the lattices) from a particular one. It means that the 70-dimensional (real) moduli space labeled by the 70 scalars we started with\[
{\mathcal M} \approx E_{7(7)} / SU(8)
\] is nothing else than the space of all possible lattices which may be chosen as allowed charges of the 56 independent types of electromagnetically charged objects. In fact, the actual moduli space is locally equal to what I just wrote but its exact global structure has some identifications. The moduli space is\[
{\mathcal M} = E_{7(7)}(\ZZ)\backslash E_{7(7)} / SU(8)
\] It's a quotient taken from both sides. The extra discrete group at the beginning of the right hand side is the U-duality group; it's a group of the noncompact symmetry transformations that don't belong to the compact \(SU(8)\) subgroup but that still preserve the lattice of allowed charges.
In other words, the full stringy theory of quantum gravity – which takes things such as the Dirac quantization rules into account – breaks the continuous noncompact \(E_{7(7)}\equiv E_{7(7)}(\RR)\) symmetry down to its \(E_{7(7)}(\ZZ)\) discrete subgroup. It's a good thing: quantum gravity morally prohibits continuous global symmetries so it's good that string theory only keeps a discrete subgroup of it – and this subgroup is actually a group of local symmetries if you look a bit more carefully. (The difference between global and local symmetries is subtle for discrete groups; the existence of monodromies and cosmic strings is what operationally decides about the right adjective.)
So string theory has a moduli space \({\mathcal M}\) of inequivalent vacua (in the old, quantum field theory era, we would probably talk about inequivalent theories) i.e. of different superselection sectors.
Now you should read two roads from \(\NNN=8\) supergravity to string theory if you haven't read it previously. In that article, I argued that this supergravity theory has two basic bugs from a phenomenological viewpoint. It's too supersymmetric so that it can't produce a realistic spectrum; and it's divergent and inconsistent, at least at the nonperturbative level.
If you work hard to fix either of the two glitches, you are led to the full string/M-theory in both cases. To fix the excessive supersymmetry that forbids realistic quarks and leptons, among other things, you have to find out that the supergravity theory is really a compactification of a master \(d=11\) theory and you have to consider more sophisticated compactifications on the \(G_2\) holonomy manifolds.
To fix the problems with the divergences, you must include all the extra objects and excitations – objects moving or monopole-charged in new Kaluza-Klein dimensions; wrapped branes – which make the theory finite and consistent at very short distances in the same sense in which the W-bosons regulate all the divergences we know from Fermi's non-renormalizable four-fermion theory of the weak nuclear interaction.
There is no doubt that a fully consistent completion of this \(\NNN=8\) supergravity theory has to include black holes, including the charged ones under the \(2\times 28\) \(U(1)\) groups (it's always possible to create charged black holes in electromagnetic fields, even classically or "astrophysically") and quantum mechanics dictates that the charges have to be quantized (Dirac quantization condition). If you think for a while, you will realize that consistency requires the small quantized charged objects such as the Kaluza-Klein stuff and M2-branes and M5-branes and all their interactions and properties are pretty much dictated by consistency, too.
Even if you assume nothing else than consistency, you are led to the full string/M-theory as the only "complete cure" for the problems of all the older theories that approximate string/M-theory.
Perturbative finiteness
The non-perturbative inconsistency of the \(\NNN=8\) SUGRA without the stringy objects is indisputable: it either violates the rules of general relativity that give every region the human right to create charged black holes; or it violates the Dirac quantization rule for such charged objects. Or something else.
However, it has seemed possible to many people including your humble correspondent that the \(\NNN=8\) theory could be perturbatively finite. You could calculate the scattering amplitudes of the gravitons (and their superpartners) in the perturbative expansion, adding one order after another, and you could never experience a divergence. All of them would cancel.
I no longer think it is too likely but let me mention two main arguments – which I consider too sloppy to be given too much weight at this moment. One set of these arguments are papers by Renata Kallosh such as this March 2011 paper. She has argued that the theory is perturbatively finite and has used various methods – including the light cone gauge (which I kind of like) and some "deformed" versions of the noncompact symmetry – to prove her point.
A more general reason why I was very open to the possibility that the theory was finite were the KLT relations: the \(\NNN=8\) supergravity looks like the \(\NNN=4\) gauge theory "squared". Because the gauge theory – coming from open strings – is (not only) perturbatively finite, it seemed plausible to me that the supergravity theory – arising from closed strings (in some moral sense, open strings tensor squared) – may be perturbatively finite as well. Various objects relevant for the supergravity case – including the Riemann surface moduli spaces in string theory – looked like the "squared cousins" of similar objects in the open string theory. So the finiteness of the gauge theory could have been squared and preserved, too.
But the relationship between the amplitudes at the loop level isn't exact so the argument is really sloppy, I think today. So I joined those who find it much more likely that the \(\NNN=8\) supergravity is divergent even perturbatively. Simeon H. was among those who have been telling me it's the likely right answer for years.
Now, in the 2006 article about the finiteness of supergravity theories, I explained that at one-loop, the problems of the quantized Eistein's theory cancel. It's really because the would-be counterterms are of the form \(R^2\), the squared Riemann tensor. There are just three independent ways how to contract the indices (an easy-to-see basis is the Riemann tensor squared, Ricci tensor squared, Ricci scalar squared). One combination is topological (the Euler density) which means that it doesn't influence the local physics (or equations of motion: it is a total derivative) and the other two combinations may be eliminated by a field redefinition that adds a multiple of the Ricci tensor (or Ricci scalar times the metric) to the metric tensor.
However, at two loops, there is a well-known divergence with a complicated coefficient. It must be cancelled by a counterterm. The divergent coefficient of the counterterm arrives along with a new residual finite coefficient that may be adjusted (and must be adjusted to make the theory specific). At higher loops, the vertices combine in infinitely many ways to produce infinitely many different types of counterterms whose values have to be determined. It's not possible using a finite number of experiments. So the quantized general relativity is totally unpredictive; that's the right description why its non-renormalizability is a problem.
If you consider supergravity with some supercharges, they may cancel some leading divergences but they usually just "delay" where the first divergences occur. In combination with other known symmetries, the \(\NNN=8\) supergravity shifts the "leading" possible divergence to seven-loop diagrams. Be sure that these are complicated mathematical objects – one could say that they are sums of millions of complicated 28-dimensional integrals – but many people became very good in dealing with these objects.
The leading candidate 7-loop divergence
Although the supergravity seems to have many symmetries (noncompact symmetries, many supercharges), they don't seem enough to eliminate infinitely many types of candidate divergences. After all, there's just a finite number of such symmetry generators so it would be surprising if they could cure infinitely many diseases.
In this respect, the situation is different from the \(\NNN=4\) gauge theory which celebrated its 35th birthday some months ago. This gauge theory has an infinite-dimensional symmetry that underlies it, the Yangian, and other structures. Although the supergravity theory has a doubled number of supercharges, it seems less constrained than the gauge theory.
At any rate, even when you impose all the symmetries, you see no reason why the theory shouldn't generate a particular leading 7-loop counterterm which may be written in superspace approaches in an extremely simple way: \(\int d^{32}\theta\). It's just the integral over (i.e. total generalized volume of) the full 32-dimensional superspace!
Superspaces that are this huge suck for the other terms so you want to translate this schematic form into components. It turns out that in the language of the component fields, such a counterterm is equivalent to something like \(D^8 R^4\), schematically speaking. So it's the eighth power of the superderivative acting on the fourth power of the Riemann tensor – with various orderings and contractions of indices.
See e.g. this 2010 paper by Elvang, Beisert, Freedman, and others that discusses these counterterms.
Dimensionally speaking, the superderivative \(D\) has the same units as the square root of the curvature tensor (which is bilinear in the ordinary derivatives) so \(D^8\) is dimensionally equivalent to \(R^4\) and \(D^8 R^4\) has the units of \(R^8\). That's OK – thanks to kristan for corrections – because a \(k\)-loop expression should produce terms with units just like \(R^{k+1}\). Recall that the tree-level (0-loop) Einstein-Hilbert action is proportional to \(R\); we canceled the \(R^2\) one-loop corrections; but the \(R^3\) two-loop terms gave us the first lethal divergences in the quantized general relativity. Extrapolate this process to 7 loops and you will get the right units.
The only "hope for perturbative finiteness" could be that this candidate counterterm is produced with a coefficient that is zero. All the contributions to it should cancel. However, there's no known convincing reason why it should be the case; all the known "symmetry weapons" have already been used to cancel other terms.
Now, Tom argues that a problem must arrive at a finite order in the perturbative expansion of the supergravity because it's important for the consistency of black hole physics to break the \(SU(8)\) symmetry, the maximum compact subgroup of the noncompact symmetry. I am confused by his argument because I have believed that the \(SU(8)\) symmetry is exact even in the full string/M-theory.
Tom also writes, much like your humble correspondent, that the classicalization papers are wrong although I haven't studied his reasoning sufficiently carefully to know whether he has the same reasons, less correct reasons, or more correct reasons for the dismissal of these Dvali et al. paper as/than I do (there are surely some similarities in the strategies how to think about these matters).
Tom Banks mentions that "SUGRA has divergences at the 7-loop level" is a "consensus" of the experts. I am a bit surprised by his using this language because while Tom is left-wing, he is not a guy who would promote the "consensus science". After all, he refused to belong to various consensuses himself. ;-) But OK, he is making a sociological point, one that e.g. Renata Kallosh could disagree with.
Shmoits, Shwolins, and related feces
Finally, let me mention that some of the world's most notorious crackpot discussion forums such as "Not Even Wrong" have been running the campaign forcing everyone to believe that the \(\NNN=8\) supergravity had to be finite (they don't understand the difference between perturbatively finite and finite at all so they couldn't discuss this subtlety). Of course, they have never had any arguments – and it seems to disagree with the evidence at this point. It was just decided by some crackpots-in-chief that such a conclusion is more convenient for the dishonest propaganda addressed to the human trash that reads "Not Even Wrong" – that it could be more helpful to invent stories that string theory is not needed.
This approach was not only dishonest but totally irrational at every conceivable level. First, these would-be folks have been irrationally attacking supersymmetry (and, therefore, supergravity as well) almost as intensely as string theory so it's nonsensical to suddenly worship its miraculous (and probably non-existing) properties of a supergravity theory. Second, it's obvious and it's been proven above that supergravity can't be nonperturbatively consistent: it would either prohibit the production of charged black holes (which must be allowed) or it would violate the Dirac quantization rules (making their wave functions multiply-valued etc.).
Third, and it's been also discussed above, the maximal supergravity can't lead to a realistic particle content and the less-supersymmetric versions of it would surely violate the finiteness properties, anyway. Fourth, it's nonsensical to try to put a less complete theory, supergravity, against string theory because the research of supergravity has been incorporated into the research of string/M-theory and everyone realizes that the former is an approximation of the latter. The latter is more complete, more comprehensive, and for many questions, it's totally needed for the consistency.
One must really be a very stupid, brainwashed pile of scum to take any of this dishonest Shmoit-like idiocy seriously.
And that's the memo.
I have been discussing the maximally supersymmetric supergravity and its conjectured perturbative finiteness many times on this blog. But the immediate reason for a new entry is the following paper by my ex-adviser Tom Banks,
Arguments Against a Finite \(\NNN=8\) SupergravityIn this text, I want to review the \(\NNN=8\) supergravity in \(d=4\), its field content and "other objects content", the arguments that have been raised for its perturbative finiteness, the explicit 7-loop counterterm that is allowed by all the symmetries, and the argument why its coefficient is probably nonzero.
As I discussed in the article about the royal status of 11-dimensional supergravity, 11 dimensions is the maximum number of large dimensions in which one may have a theory with unbroken supersymmetry and a nontrivial particle content at long distances. The theory was discovered in the late 1970s and is known as the 11-dimensional supergravity.
In such a high spacetime dimension, it's of course non-renormalizable as a field theory but we have known since 1995 that it is the right long-distance approximation to a theory that is fully consistent and finite, namely M-theory which is an important cousin of string theory. More precisely, M-theory is another vacuum or limit of string theory that, unlike the 10-dimensional vacua, has 11 spacetime dimensions. The 11-dimensional dynamics may be obtained by sending the string coupling \(g_s\) to infinity either in the \(E_8\times E_8\) heterotic string theory or (which is easier) in type IIA string theory. When we do so, a new, eleventh dimension of spacetime emerges and grows larger.
The elf-dimensional supergravity
This 11-dimensional M-theory may be compactified down to various lower spacetime dimensions. From the viewpoint of the bread-and-butter physics, the most important ones among these compactifications are compactifications on singular 7-dimensional manifolds of \(G_2\) holonomy: it's one of the several major descriptions by which string/M-theory may describe the Universe around us and which has all the desired qualitative properties to agree with all the particles and forces we have observed in our world. These compactifications were recently promoted as the most canonical example of "generic string/M-theoretical compactifications" by Gordon Kane, Bobby Acharya, and their colleagues in their phenomenological papers. Edward Witten has played a key role in the discovery of this new "stringy scenario" a decade ago, too.
However, those compactifications are complicated and only preserve four real supercharges which is equivalent to the \(\NNN=1\) supersymmetry in \(d=4\), the usual extent of supersymmetry that is employed in the realistic model building. From a top-down viewpoint, the more fundamental are the more supersymmetric compactifications, especially the maximal i.e. \(\NNN=8\) supergravity with 32 real supercharges, the same number as the original 11-dimensional theory. This theory is pretty much unique and so is its consistent ultraviolet completion, M-theory on a 7-torus.
The maximal supergravity: symmetries, field content
As we have recalled in the royal article, the 11-dimensional supergravity contains one supermultiplet of fields, the gravitational supermultiplet, which boasts \(2^8=256\) physical polarizations for each (light-like) momentum. One-half of them are fermionic, they're the components of the spin-3/2 gravitino (a maximally constrained tensor product of a spinor and a vector), while the remaining 128 polarizations are bosonic ones. 44 of them are components of the metric tensor; 84 of them are components of the \(C_{\lambda\mu\nu}\) three-form potential that couples to membranes and fivebranes.
What happens if you compactify the theory to 4 dimensions i.e. dimensionally reduce it over 7 dimensions? Some values of the vector-like Lorentz indices will become scalar-like. The spin as understood in \(d=4\) will be smaller than the spin as extracted from \(d=11\). How many scalar fields will we get?
We only get scalar fields from the bosons; the spin-statistics relation always holds in these and other theories. From the metric tensor, we get the components \(g_{\mu\nu}\) where \(\mu,\nu\) go over the 7 dimensions we have compactified on a tiny 7-torus. The number of such components is \(7\times 8/2\times 1 = 28\). These components remember the radii and angles in the 7-torus defining the compactification. The gravitational excitations should be traceless but we will assign the task of keeping the tracelessness to the components in the uncompactified dimensions so there's no elimination of components here.
There are also now-scalar components of the three-form potential, \(C_{\lambda\mu\nu}\). If all the indices are set to some of the numbers in the list of 7 compact dimensions, we get \((7\times 6\times 5)/(3\times 2 \times 1) = 35\). I would have to spend a lot of time to explain the complete justification but we must also count the components of the dual 6-form potential \(\tilde C_{\lambda\mu\nu\pi\rho\sigma}\) that couples to the fivebranes. These six indices may be set to the seven compactified dimensions with one missing exception; so there are seven ways to do so. We get 7 new scalar physical polarizations.
In total, we have \(28+35+7=70\) scalars in the four-dimensional \(\NNN=8\) supergravity. If you were a very good numerologist, you would be able to realize that \(70=133-63\) and because \(133\) and \(63=8^2-1\) are dimensions of the \(E_{7(7)}\equiv E_{7(7)}(\RR)\) and \(SU(8)\) groups, those 70 scalars could be parameterizing the coset or quotient \(E_{7(7)}/SU(8)\) of a noncompact exceptional group and its largest noncompact subgroup.
The arithmetic check above is of course very far from being a full proof that the symmetries are relevant for the theory; but the conclusion is right. The supergravity theory has an exact, internal, noncompact \(E_{7(7)}\) global symmetry. Its \(SU(8)\) subgroup's action on the scalar fields is trivial; so the scalar fields inevitably parameterize the coset. There are various independent ways to see why the noncompact exceptional symmetry exists – a brute force algebraic calculation; arguments that the high degree of SUSY implies that the moduli spaces of scalars have to be cosets; reconciliation of partial stringy symmetries such as T-dualities – but just believe me that the symmetry is there.
Charged objects in the theory (its stringy completion)
To have some fun, let me show you one more numerological check that the \(E_{7(7)}\) exceptional symmetry exists. Consider the stringy completion of our \(d=4\) \(\NNN=4\) SUGRA, namely M-theory on a seven-torus, and let's count the number of individual \(U(1)\) gauge symmetries (each of them has a new kind of a charge and the corresponding charged objects, too).
First, remembering Kaluza and Klein 90+ years ago, we know that general relativity compactified on a torus produces \(U(1)\) groups. We get seven of them; the gauge fields are \(A_{\mu}^{(n)}=g_{\mu n}\) where \(\mu\) is the 4-dimensional Lorentz index and \(n\) goes over the seven compact dimensions and labels the seven different \(U(1)\) groups. The charged objects are any particles that are moving in the directions of the compactified seven-torus; recall that quantum mechanics guarantees that the momentum on a compact space is quantized (and, consequently, so is the resulting electric charge).
However, we also have \(A^{(m,n)}_\mu=C_{\mu mn}\). The indices \(m,n\) take 7 possibilities each and due to the antisymmetry, they produce \(7\times 6/2\times 1=21\) different \(U(1)\) fields. The corresponding charged objects are M2-branes with both dimensions wrapped on a two-torus within the spacetime seven-torus.
Similarly, there are five-branes wrapped on five-tori that are charged under similar \(U(1)\) symmetries whose gauge fields are \(\tilde C_{\mu mnopq}\) where \(mnopq\) form a quintuplet of indices chosen from the list of seven compact directions. We get "7 choose 5" which is 21 different \(U(1)\) fields again.
There are some extra charged objects (and their corresponding \(U(1)\) groups) I haven't counted yet. The charged objects are the Kaluza-Klein monopoles which are really 6-branes (we want fully wrapped ones) with 1 additional "systemic" compact spatial dimension. Any of the 7 spacetime toroidal dimensions may play the role of this "systemic" Kaluza-Klein direction so we get 7 independent Kaluza-Klein monopole charges.
In total, we may find \(7+21+21+7=56\) different \(U(1)\) groups. It's only 28 times and not 56 times more electromagnetic fields than in Maxwell's theory because in the 56-based counting, Maxwell's theory has two types of charges, the electric ones and the magnetic monopoles. Now, it is not a coincidence that 56 is the dimension of the fundamental representation of \(E_{7(7)}\) – which is just a continuation of the compact \(E_7\) Lie group. In fact, these charges do transform under the \(E_{7(7)}\) non-compact global symmetry that I have already discussed.
The supergravity itself, i.e. the long-distance limit of M-theory on a seven-torus, doesn't admit any charged objects. The torus is infinitely tiny in this limit so the Kaluza-Klein particles are infinitely heavy and as long as we are at a generic point of the moduli spaces, the other charged objects are related to the Kaluza-Klein particles by the symmetry so they must be equally heavy. And in fact, you could think that in the supergravity, the charged objects and their precise charge quantization conditions don't make any difference at all.
However, in the full stringy theory, namely M-theory on a seven-torus, these charged objects – Kaluza-Klein particles (moving along the new dimensions), wrapped membranes and fivebranes, and wrapped Kaluza-Klein monopoles – are allowed, real, and they make a difference. Because the Kaluza-Klein particles are electromagnetic duals of the wrapped Kaluza-Klein monopoles and the wrapped M2-branes are electromagnetic duals of the wrapped M5-branes – it is not an accident that the list 7,21,21,7 was symmetric – we must remember the Dirac quantization rule for the magnetic charges.
When we impose this rule on all the charges we have, we find out that these charges must belong to a particular lattice. The lattice isn't quite unique but by \(E_{7(7)}\) transformations, you may get all the solutions (all the lattices) from a particular one. It means that the 70-dimensional (real) moduli space labeled by the 70 scalars we started with\[
{\mathcal M} \approx E_{7(7)} / SU(8)
\] is nothing else than the space of all possible lattices which may be chosen as allowed charges of the 56 independent types of electromagnetically charged objects. In fact, the actual moduli space is locally equal to what I just wrote but its exact global structure has some identifications. The moduli space is\[
{\mathcal M} = E_{7(7)}(\ZZ)\backslash E_{7(7)} / SU(8)
\] It's a quotient taken from both sides. The extra discrete group at the beginning of the right hand side is the U-duality group; it's a group of the noncompact symmetry transformations that don't belong to the compact \(SU(8)\) subgroup but that still preserve the lattice of allowed charges.
In other words, the full stringy theory of quantum gravity – which takes things such as the Dirac quantization rules into account – breaks the continuous noncompact \(E_{7(7)}\equiv E_{7(7)}(\RR)\) symmetry down to its \(E_{7(7)}(\ZZ)\) discrete subgroup. It's a good thing: quantum gravity morally prohibits continuous global symmetries so it's good that string theory only keeps a discrete subgroup of it – and this subgroup is actually a group of local symmetries if you look a bit more carefully. (The difference between global and local symmetries is subtle for discrete groups; the existence of monodromies and cosmic strings is what operationally decides about the right adjective.)
So string theory has a moduli space \({\mathcal M}\) of inequivalent vacua (in the old, quantum field theory era, we would probably talk about inequivalent theories) i.e. of different superselection sectors.
Now you should read two roads from \(\NNN=8\) supergravity to string theory if you haven't read it previously. In that article, I argued that this supergravity theory has two basic bugs from a phenomenological viewpoint. It's too supersymmetric so that it can't produce a realistic spectrum; and it's divergent and inconsistent, at least at the nonperturbative level.
If you work hard to fix either of the two glitches, you are led to the full string/M-theory in both cases. To fix the excessive supersymmetry that forbids realistic quarks and leptons, among other things, you have to find out that the supergravity theory is really a compactification of a master \(d=11\) theory and you have to consider more sophisticated compactifications on the \(G_2\) holonomy manifolds.
To fix the problems with the divergences, you must include all the extra objects and excitations – objects moving or monopole-charged in new Kaluza-Klein dimensions; wrapped branes – which make the theory finite and consistent at very short distances in the same sense in which the W-bosons regulate all the divergences we know from Fermi's non-renormalizable four-fermion theory of the weak nuclear interaction.
There is no doubt that a fully consistent completion of this \(\NNN=8\) supergravity theory has to include black holes, including the charged ones under the \(2\times 28\) \(U(1)\) groups (it's always possible to create charged black holes in electromagnetic fields, even classically or "astrophysically") and quantum mechanics dictates that the charges have to be quantized (Dirac quantization condition). If you think for a while, you will realize that consistency requires the small quantized charged objects such as the Kaluza-Klein stuff and M2-branes and M5-branes and all their interactions and properties are pretty much dictated by consistency, too.
Even if you assume nothing else than consistency, you are led to the full string/M-theory as the only "complete cure" for the problems of all the older theories that approximate string/M-theory.
Perturbative finiteness
The non-perturbative inconsistency of the \(\NNN=8\) SUGRA without the stringy objects is indisputable: it either violates the rules of general relativity that give every region the human right to create charged black holes; or it violates the Dirac quantization rule for such charged objects. Or something else.
However, it has seemed possible to many people including your humble correspondent that the \(\NNN=8\) theory could be perturbatively finite. You could calculate the scattering amplitudes of the gravitons (and their superpartners) in the perturbative expansion, adding one order after another, and you could never experience a divergence. All of them would cancel.
I no longer think it is too likely but let me mention two main arguments – which I consider too sloppy to be given too much weight at this moment. One set of these arguments are papers by Renata Kallosh such as this March 2011 paper. She has argued that the theory is perturbatively finite and has used various methods – including the light cone gauge (which I kind of like) and some "deformed" versions of the noncompact symmetry – to prove her point.
A more general reason why I was very open to the possibility that the theory was finite were the KLT relations: the \(\NNN=8\) supergravity looks like the \(\NNN=4\) gauge theory "squared". Because the gauge theory – coming from open strings – is (not only) perturbatively finite, it seemed plausible to me that the supergravity theory – arising from closed strings (in some moral sense, open strings tensor squared) – may be perturbatively finite as well. Various objects relevant for the supergravity case – including the Riemann surface moduli spaces in string theory – looked like the "squared cousins" of similar objects in the open string theory. So the finiteness of the gauge theory could have been squared and preserved, too.
But the relationship between the amplitudes at the loop level isn't exact so the argument is really sloppy, I think today. So I joined those who find it much more likely that the \(\NNN=8\) supergravity is divergent even perturbatively. Simeon H. was among those who have been telling me it's the likely right answer for years.
Now, in the 2006 article about the finiteness of supergravity theories, I explained that at one-loop, the problems of the quantized Eistein's theory cancel. It's really because the would-be counterterms are of the form \(R^2\), the squared Riemann tensor. There are just three independent ways how to contract the indices (an easy-to-see basis is the Riemann tensor squared, Ricci tensor squared, Ricci scalar squared). One combination is topological (the Euler density) which means that it doesn't influence the local physics (or equations of motion: it is a total derivative) and the other two combinations may be eliminated by a field redefinition that adds a multiple of the Ricci tensor (or Ricci scalar times the metric) to the metric tensor.
However, at two loops, there is a well-known divergence with a complicated coefficient. It must be cancelled by a counterterm. The divergent coefficient of the counterterm arrives along with a new residual finite coefficient that may be adjusted (and must be adjusted to make the theory specific). At higher loops, the vertices combine in infinitely many ways to produce infinitely many different types of counterterms whose values have to be determined. It's not possible using a finite number of experiments. So the quantized general relativity is totally unpredictive; that's the right description why its non-renormalizability is a problem.
If you consider supergravity with some supercharges, they may cancel some leading divergences but they usually just "delay" where the first divergences occur. In combination with other known symmetries, the \(\NNN=8\) supergravity shifts the "leading" possible divergence to seven-loop diagrams. Be sure that these are complicated mathematical objects – one could say that they are sums of millions of complicated 28-dimensional integrals – but many people became very good in dealing with these objects.
The leading candidate 7-loop divergence
Although the supergravity seems to have many symmetries (noncompact symmetries, many supercharges), they don't seem enough to eliminate infinitely many types of candidate divergences. After all, there's just a finite number of such symmetry generators so it would be surprising if they could cure infinitely many diseases.
In this respect, the situation is different from the \(\NNN=4\) gauge theory which celebrated its 35th birthday some months ago. This gauge theory has an infinite-dimensional symmetry that underlies it, the Yangian, and other structures. Although the supergravity theory has a doubled number of supercharges, it seems less constrained than the gauge theory.
At any rate, even when you impose all the symmetries, you see no reason why the theory shouldn't generate a particular leading 7-loop counterterm which may be written in superspace approaches in an extremely simple way: \(\int d^{32}\theta\). It's just the integral over (i.e. total generalized volume of) the full 32-dimensional superspace!
Superspaces that are this huge suck for the other terms so you want to translate this schematic form into components. It turns out that in the language of the component fields, such a counterterm is equivalent to something like \(D^8 R^4\), schematically speaking. So it's the eighth power of the superderivative acting on the fourth power of the Riemann tensor – with various orderings and contractions of indices.
See e.g. this 2010 paper by Elvang, Beisert, Freedman, and others that discusses these counterterms.
Dimensionally speaking, the superderivative \(D\) has the same units as the square root of the curvature tensor (which is bilinear in the ordinary derivatives) so \(D^8\) is dimensionally equivalent to \(R^4\) and \(D^8 R^4\) has the units of \(R^8\). That's OK – thanks to kristan for corrections – because a \(k\)-loop expression should produce terms with units just like \(R^{k+1}\). Recall that the tree-level (0-loop) Einstein-Hilbert action is proportional to \(R\); we canceled the \(R^2\) one-loop corrections; but the \(R^3\) two-loop terms gave us the first lethal divergences in the quantized general relativity. Extrapolate this process to 7 loops and you will get the right units.
The only "hope for perturbative finiteness" could be that this candidate counterterm is produced with a coefficient that is zero. All the contributions to it should cancel. However, there's no known convincing reason why it should be the case; all the known "symmetry weapons" have already been used to cancel other terms.
Now, Tom argues that a problem must arrive at a finite order in the perturbative expansion of the supergravity because it's important for the consistency of black hole physics to break the \(SU(8)\) symmetry, the maximum compact subgroup of the noncompact symmetry. I am confused by his argument because I have believed that the \(SU(8)\) symmetry is exact even in the full string/M-theory.
Tom also writes, much like your humble correspondent, that the classicalization papers are wrong although I haven't studied his reasoning sufficiently carefully to know whether he has the same reasons, less correct reasons, or more correct reasons for the dismissal of these Dvali et al. paper as/than I do (there are surely some similarities in the strategies how to think about these matters).
Tom Banks mentions that "SUGRA has divergences at the 7-loop level" is a "consensus" of the experts. I am a bit surprised by his using this language because while Tom is left-wing, he is not a guy who would promote the "consensus science". After all, he refused to belong to various consensuses himself. ;-) But OK, he is making a sociological point, one that e.g. Renata Kallosh could disagree with.
Shmoits, Shwolins, and related feces
Finally, let me mention that some of the world's most notorious crackpot discussion forums such as "Not Even Wrong" have been running the campaign forcing everyone to believe that the \(\NNN=8\) supergravity had to be finite (they don't understand the difference between perturbatively finite and finite at all so they couldn't discuss this subtlety). Of course, they have never had any arguments – and it seems to disagree with the evidence at this point. It was just decided by some crackpots-in-chief that such a conclusion is more convenient for the dishonest propaganda addressed to the human trash that reads "Not Even Wrong" – that it could be more helpful to invent stories that string theory is not needed.
This approach was not only dishonest but totally irrational at every conceivable level. First, these would-be folks have been irrationally attacking supersymmetry (and, therefore, supergravity as well) almost as intensely as string theory so it's nonsensical to suddenly worship its miraculous (and probably non-existing) properties of a supergravity theory. Second, it's obvious and it's been proven above that supergravity can't be nonperturbatively consistent: it would either prohibit the production of charged black holes (which must be allowed) or it would violate the Dirac quantization rules (making their wave functions multiply-valued etc.).
Third, and it's been also discussed above, the maximal supergravity can't lead to a realistic particle content and the less-supersymmetric versions of it would surely violate the finiteness properties, anyway. Fourth, it's nonsensical to try to put a less complete theory, supergravity, against string theory because the research of supergravity has been incorporated into the research of string/M-theory and everyone realizes that the former is an approximation of the latter. The latter is more complete, more comprehensive, and for many questions, it's totally needed for the consistency.
One must really be a very stupid, brainwashed pile of scum to take any of this dishonest Shmoit-like idiocy seriously.
And that's the memo.
Why \(\NNN=8\) supergravity is probably divergent at 7 loops
![Why \(\NNN=8\) supergravity is probably divergent at 7 loops]()
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May 29, 2012
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