Radu Roiban of Penn State University gave a powerful talk about the perturbative finiteness of the N=8 SUGRA in four dimensions. He believes that the S-matrix in this theory is UV finite. They have reconstructed various amplitudes up to three loops from the known cuts and the KLT relations. While the SUGRA Feynman rules contain all possible things such as quartic and arbitrary higher order vertices, you don't need the explicit Feynman diagrams to construct the full three-loop answer.

The answer may be rewritten purely in terms of special Feynman diagrams with cubic vertices. It's because the loop integrals involve polynomials of the momenta in the numerator - because the interactions are derivative interactions - and these polynomials are able to cancel against factors in the denominator, and thus cancel propagators and "shrink" the Feynman diagrams. A visual analysis of possible shapes of the Feynman diagrams - especially those that include triangles and/or bubbles - implies that the divergences of the on-shell four-point function cancel at three loops.

Radu argues that the known N=6 superspace and even the hypothetical N=7 superspace are insufficient to explain some details of this cancellation of UV divergences. Moreover, all known data are consistent with the hypothesis that the degree of UV divergence of the S-matrix amplitudes of N=8 d=4 SUGRA is identical to the degree of divergence in the N=4 d=4 supersymmetric Yang-Mills theory.

The S-matrix of N=4 d=4 supersymmetric Yang-Mills theory is known to be UV finite - all UV divergences cancel. If the naturally sounding relation between the N=8 SUGRA and the N=4 d=4 supersymmetric Yang-Mills theory is correct, it would imply that there are no UV divergences in the N=8 supergravity. Moreover, the infrared divergences only exist in the gauge theory, not in the supergravity theory that is much more weakly coupled in the infrared.

All of us seem to agree that non-perturbatively, the N=8 supergravity clearly can't be a full consistent theory because it's missing all the new particles such as wrapped branes and/or black hole microstates that are essential for the full unitarity etc. Perturbatively, it may be finite to all orders. However, Radu's evidence is not a full proof. People differ in their opinion whether the perturbative finiteness is exact. Some big shots are skeptical and think that the divergences re-appear in diagrams with a sufficient number of loops. I find the available evidence enough to think that the divergence structure of the N=8 supergravity and N=4 d=4 supersymmetric Yang-Mills theory is indeed identical, making the S-matrices of both of these theories UV finite.

Using the KLT rules and the known unitarity conditions, they argue that they can reconstruct the three-loop four-point scattering amplitude and prove that a higher number of types of UV divergences cancel than what you would expect from the theorems based on the known superspaces. That should be viewed as circumstantial evidence for the full finiteness.

Pure spinors in AdS5 x S5

Nathan Berkovits also gave a nice talk about his pure spinor formalism applied to the type IIB background dual to the N=4 d=4 supersymmetric Yang-Mills theory. He reviewed the pure spinor approach to d=10 superparticles and d=10 superstrings, together with its dimensional reduction to d=4, and argued that this can be seen as direct worldsheet evidence of the Maldacena duality using a set of steps that are analogous to the open-closed duality of topological string theory on the resolved vs deformed conifold.

How does it work? In the conifold case, we have a closed string sector in one of the resolutions of the conifold that can be proven to be physically equivalent to the open string sector in the other type of the resolution. Nathan proposes an analogous duality of different theories that are related to N=4 d=4 and AdS5 x S5 of type IIB string theory. You can see that we have an analogy between two dualities here.

AdS5 x S5 with closed strings plays the same role as one of the conifolds above that contains an interesting closed string sector. However, there exists the opposite conifold, and in the AdS5 x S5 duality, the role of this second conifold is played by a theory with an interesting open string sector that can be shown to be equivalent to the N=4 d=4 theory. There are some technical details that are hard to write without maths.

Finally, I have also convinced Nathan to settle a psychological problem we have had with the pure spinor formalism in general and its ability to prove the finiteness of string amplitudes. And he did settle it, as far as I can say. What was that?

In the older version of the pure spinor formalism, you can't construct the b-ghost which makes it hard to define the multiloop scattering amplitudes. First of all, he told me that in a newer, non-minimal formalism, he can already define b(z) explicitly. Nevertheless, both of us wanted to know how a problem was resolved in the older minimal formalism.

In that machinery, the insertion of b(u) - one that is necessary to get the right measure on the moduli space - is replaced by a bilocal operator b~(u,z) where z is different from u, to avoid ill-defined singularities. But the position of z shouldn't matter for the physical amplitudes. But does it? Jacques Distler has convinced me for a while that it did. The z-derivative of b~(u,z) is not BRST exact because it is not even BRST-closed. That would mean that the amplitude depends on the value of z.

However, it does not depend because the z-derivative of b~(u,z) is the stress energy tensor T(z) times something.

So the z-derivative of the full amplitude is equal to the integrand of a correlator of many things that involve one T(z) factor over the moduli space of Riemann surfaces. The insertion of T(z) is equivalent to differentiating the amputated correlator with respect to the moduli space of Riemann surfaces. This derivative thus hopefully integrates to zero assuming that the integrand is well-behaved on the coordinate boundaries of the moduli space. This fact should be true even if the remaining insertions besides T(z) are not BRST-closed so right now I think that it doesn't matter whether the operator CRAP(z) in

- d/dz b~(u,z) = T(z) CRAP(z)

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