Finiteness of supergravity theories

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Green, Russo, and Vanhove argue that many more divergences in maximally extended supergravity cancel than some people could think. They're not the first ones who conjecture that the power law divergences could be absent in d=4 N=8 supergravity: Zvi Bern has employed the constraints of unitarity together with the twistor-like template for the amplitudes

• A_{closed} = Extrafactors x A_{open} x A_{open}

to argue that the d=4 N=8 supergravity could be finite in the very same way as d=4 N=4 supersymmetric Yang-Mills theory is finite. Green et al. re-check some old things and calculate new diagrams in maximally extended supergravity using the methods based on string theory and M-theory.



Let me start with a historical comment. In the early 1980s, many people - including Stephen Hawking among the famous ones to the public - would get excited by the maximally extended supergravity because the power of supersymmetry seemed enough to cancel the divergences and give a finite supergravity theory. It was already known that less supersymmetric supergravities can't generate a finite theory in d=4.

The N=8 supergravity itself is phenomenologically unacceptable and one must break SUSY to at most N=1 supersymmetry to get a semi-realistic theory. The coupling of gravity to the other forces may create new potential problems. Moreover, it was widely believed that all theories of gravity, including N=8 supergravity, had to be non-renormalizable in d=4.

Let me say a few words, clarified in a discussion with Martin Roček, about the counterterms needed to cancel the different divergences in d=4 theories of gravity.

One-loop level

At one-loop level, one generates "Riemann^2" divergences in the effective action that need a counterterm. This happens to be no problem in gravity, not even in pure gravity: the "Riemann^2" terms can be written as a combination of the Gauss-Bonnet topological term (the Euler density) that doesn't matter for perturbative physics plus a function of the Ricci tensor and Ricci scalar that vanish on-shell (and that can be removed by a field redefinition anyway). That's a kind of "kinematical accident". The situation is even better for supergravity theories. To see problems, we must go to two loops.




Two-loop level

At the two-loop level, the required counterterms are of the form of "Riemann^3".

With no supersymmetry i.e. for "N=0", Goroff and Sagnotti have calculated in 1985-86 the non-zero two-loop "Riemann^3" counterterm. In dimensional regularization, the required counterterm in the Lagrangian is

• 209 / (2880 epsilon) x 1 / (16 pi^2)^2 x ...
• ... x sqrt(-g) C_{abcd} C^{cdef} C_{ef}^{ab}.

Yes, this essential result showing that pure quantum gravity is already perturbatively inconsistent is cubic in the Weyl tensor. A nice surprise is that with any supersymmetry, even with N=1, two-loop calculations are gonna be harmless. It's because the effective action that you generate can be seen to be supersymmetric but the "Riemann^3" terms admit no supersymmetric completion. So the coefficients of these terms must be zero.

Three-loop level

At the next level, one generates "Riemann^4" terms. Because we have already seen at the two-loop level that the pure gravity is unusable, we only look at supergravities. These three-loop counterterms are believed not to cancel in "N=1" supergravity because no one sees an obvious reason for such a cancellation but no full calculation exists.

For higher supersymmetry, many options were plausible.

One should realize that even for maximal supersymmetry, one obtains a non-renormalizable theory above "d=4". The higher dimension you work with, the worse divergences you obtain. The terms of the kind

• R^{3k+1}

in 11-dimensional gravity arise from the corresponding diagrams in type IIA string theory with "k" loops or less. Note that the powers of "R" can only change by multiples of three because M-theory kind of depends on l_{Planck}^3 only: the M2-brane tension, M5-brane tension, and Newton's constant scale like l_{Planck}^{-3}, l_{Planck}^{-6}, l_{Planck}^9, respectively.

The k-loop diagrams contributing to "Riemann^{3k+1}" are clearly the most important ones. Note that at high values of "k", they will scale like "(2k)!" which is the dominant behavior of the volume of the moduli space of genus-k Riemann surfaces. All these higher-order terms kind of know about string theory.

Incidentally, I just realized a week ago or so that the dependence on the factorial means that if you try to resum the effective action, the minimal term in the resummation will go like

• exp ( -Riemann^{-3/2} )

in the 11-dimensional Planck units which is exactly comparable to the first M2-brane-instanton contributions! It's because the exponent goes like "typical_length^3/l_{Planck}^3". You again see that the 11-dimensional supergravity knows about the membranes i.e. M2-branes as the leading non-local objects contributing to the full physics of M-theory.

Green et al. now argue that in the maximal supergravity, the power-law divergences at "h" loops only occur in spacetime dimensions

• d > 4 + 6/h.

This happens to be exactly the same condition as you obtain from a maximally supersymmetric gauge theory, indicating that supergravity may be morally viewed as the "square" of the gauge theory with half as many supercharges as argued by Bern.

If you want to get rid of all power law divergences, you can see that "d > 4" is already making your task impossible. However, "d = 4" is exactly the marginal case. The argument above shows that the logarithmic divergences are probably the worst kind of divergences you kind get in N=8, d=4.

Moreover, if you combine the one-loop kinematical accident with unitarity and supersymmetry, it is more or less healthy to believe that the effect of all these logarithmic divergences can be contained as well as in the "N=4" gauge theory although no complete proof exists.

What does it all mean?

I think it would be fun if the maximal supergravity were UV-finite after all although supergravity is still far from being enough to learn everything we need to learn about particle physics from string theory. It would be interesting to see whether some other, less supersymmetric supergravity theories that naturally arise from string theory could lead to similar cancellations if the additional matter is taken into account.

A new argument supporting the importance of "N=8" supergravity would be a good news for the inevitability of string theory. Why? Because "N=8" supergravity makes the more-or-less manifest existence of 6 or 7 extra dimensions of string theory with all the usual physics of string theory rather obvious.

Let me explain why. Imagine that someone convinces you that the "d=4" gravity should always be completed as in N=8 supergravity or in a similar setup. The N=8 supergravity is not a random theory of gravity. It is a very concrete theory whose 70 real scalar fields live in the coset of groups

• E_{7(7)} (R) / SU(8)

Recall that 133-63=70. In this theory, the mysterious exceptional group "E_{7(7)}", a non-compact version of "E_7", is the global symmetry. Also, there are 56 "U(1)" gauge fields transforming in the fundamental representation of the exceptional group. They're in one-to-one correspondence with different types of charges in four dimensions: Kaluza-Klein momenta (7), wrapped M2-branes (21), wrapped M5-branes (21), and wrapped Kaluza-Klein monopoles (7). Recall that 7+21+21+7=56.

You can easily see that general transformations from this exceptional group don't preserve the quantization of the magnetic fluxes. Only a discrete subgroup of the gauge symmetry can preserve the lattice of possible charges. In other words, the full symmetry at the non-perturbative level is

• E_{7(7)} (Z).

In other words, once you have the "N=8" supergravity, you can automatically derive that the actual maximum symmetry of the theory at the non-perturbative level is the U-duality group of M-theory on a seven-torus. Note that the perturbative part of the physics doesn't care about the flux quantization. The perturbative part of the theory doesn't even force you to have any objects that are electrically or magnetically charged under the "U(1)" symmetries.

However, we have argued using various arguments based on string theory and quantum gravity that a "U(1)" gauge field must not only come together with some charged particles but these particles must in fact be lighter than "g M_{Pl}" where "g" is the gauge coupling. At this scale or above it, the pure SUGRA theory has to break down. The last place above which pure SUGRA must certainly break down is the mass of the lightest black hole microstate.

Assuming that the non-perturbative completion is unique, the maximum symmetry that happens to coincide with the U-duality group is therefore the real symmetry of the non-perturbatively completed supergravity. Therefore, the U-duality is there. The U-duality relates the momentum states with various wrapped M2-branes and M5-branes and the Kaluza-Klein monopoles. In the right limit of the moduli space - the space of vevs of the 70 scalars - you can show that the physics of these BPS objects inevitably includes light strings (wrapped membranes) of type II string theory, and the rest of type II string theory follows because the strings are inevitably interacting and their interactions are again uniquely determined by supersymmetry.

It is quite clear that the rest of physics of maximally supersymmetric vacua of string/M-theory may be derived from non-perturbative consistency of the maximally extended supergravity: that's one of the reasons why the SUGRA community has really merged with the string-theoretical community ten years ago or so.

Note that some critics of science don't distinguish perturbative finiteness from non-perturbative completeness: the latter requires extended objects, among other things, as seen for example in the argument involving "(2k)!" above or the argument about the existence of charged light objects under all "U(1)" groups. Also, the perturbative expansion never converges in normal field theories and the first non-perturbative effects that fix it necessarily occur at the same order as dictated by string/M-theory. Also, the extended objects are found in the SUGRA whether you like it or not: they are found as classical solutions.

I find this belief system about the structure of consistent quantum gravity theories very appealing because with some extra steps in the reasoning, it could lead to a proof that string theory with its extra dimensions (or an equivalent) and all the extra extended objects is inevitable for consistent quantum gravity theories in "d=4" or higher: supersymmetry is necessary for a good UV behavior; that implies extra scalars and noncompact symmetries; these symmetries are broken by flux quantization to a discrete subgroup; extended objects must thus automatically exist in the theory; the only consistent way how they can interact is the way dictated by string/M-theory.

We kind of know that it is true anyway that string theory is necessary but it would be fun to have a semi-rigorous proof. One could expect that if "N=8" SUGRA is perturbatively finite, other limits obtained from other well-defined vacua of string/M-theory - such as the orientifolds relevant for the type IIA intersecting braneworlds - will be perturbatively finite, too. But other people could perhaps see a reason why it's not the case.