There’s a conference at the Niels Bohr Institute this week, on Amplitudes in String and Field Theory. Like the conference a few weeks back, this one was funded by the Simons Foundation, as part of Michael Green’s visit here.
The first day featured a two-part talk by Michael Green and Congkao Wen. They are looking at the corrections that string theory adds on top of theories of supergravity. These corrections are difficult to calculate directly from string theory, but one can figure out a lot about them from the kinds of symmetry and duality properties they need to have, using the mathematics of modular forms. While Michael’s talk introduced the topic with a discussion of older work, Congkao talked about their recent progress looking at this from an amplitudes perspective.
Francesca Ferrari’s talk on Tuesday also related to modular forms, while Oliver Schlotterer and Pierre Vanhove talked about a different corner of mathematics, single-valued polylogarithms. These single-valued polylogarithms are of interest to string theorists because they seem to connect two parts of string theory: the open strings that describe Yang-Mills forces and the closed strings that describe gravity. In particular, it looks like you can take a calculation in open string theory and just replace numbers and polylogarithms with their “single-valued counterparts” to get the same calculation in closed string theory. Interestingly, there is more than one way that mathematicians can define “single-valued counterparts”, but only one such definition, the one due to Francis Brown, seems to make this trick work. When I asked Pierre about this he quipped it was because “Francis Brown has good taste…either that, or String Theory has good taste.”
Wednesday saw several talks exploring interesting features of string theory. Nathan Berkovitz discussed his new paper, which makes a certain context of AdS/CFT (a duality between string theory in certain curved spaces and field theory on the boundary of those spaces) manifest particularly nicely. By writing string theory in five-dimensional AdS space in the right way, he can show that if the AdS space is small it will generate the same Feynman diagrams that one would use to do calculations in N=4 super Yang-Mills. In the afternoon, Sameer Murthy showed how localization techniques can be used in gravity theories, including to calculate the entropy of black holes in string theory, while Yvonne Geyer talked about how to combine the string theory-like CHY method for calculating amplitudes with supersymmetry, especially in higher dimensions where the relevant mathematics gets tricky.
Thursday ended up focused on field theory. Carlos Mafra was originally going to speak but he wasn’t feeling well, so instead I gave a talk about the “tardigrade” integrals I’ve been looking at. Zvi Bern talked about his work applying amplitudes techniques to make predictions for LIGO. This subject has advanced a lot in the last few years, and now Zvi and collaborators have finally done a calculation beyond what others had been able to do with older methods. They still have a way to go before they beat the traditional methods overall, but they’re off to a great start. Lance Dixon talked about two-loop five-particle non-planar amplitudes in N=4 super Yang-Mills and N=8 supergravity. These are quite a bit trickier than the planar amplitudes I’ve worked on with him in the past, in particular it’s not yet possible to do this just by guessing the answer without considering Feynman diagrams.
Today was the last day of the conference, and the emphasis was on number theory. David Broadhurst described some interesting contributions from physics to mathematics, in particular emphasizing information that the Weierstrass formulation of elliptic curves omits. Eric D’Hoker discussed how the concept of transcendentality, previously used in field theory, could be applied to string theory. A few of his speculations seemed a bit farfetched (in particular, his setup needs to treat certain rational numbers as if they were transcendental), but after his talk I’m a bit more optimistic that there could be something useful there.
4 gravitons 
Excuse me, but I’d like to know if this was a physics conference or a mathematic conference where the initial isight was physical problems. Before any hostility, It’s a honest doubt and not a criticism. I ever miss the point where this all stills important to the physics itself, although I recognize their importance in mathematics (or mathematical-physics if you prefeer). The empirical evidence needed in physics is the reason why this incredible science is not just “applied mathematics”, of course everything becomes more hard because it.
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There are a lot of ways I could respond to this, but let me settle for a pragmatic one: does it matter?
Is there something these people would be doing differently if they “admitted” they were just doing math? Do you think they’re tricking someone by describing themselves as physicists? I guarantee you the people funding this (in this case, the Simons Foundation) know exactly what they’re funding.
(Also, Zvi Bern’s talk was directly about comparing data to experiment. I know the majority of the talks weren’t, but I wanted to make sure you didn’t miss that one, especially given that a lot of what Zvi is doing wouldn’t have been possible without a lot of this kind of “mathematical” stuff.)
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You comments on single-valued polylogarithms reminded me the connection between gravity and YM mentioned a few years back (I think by Bern and Dixon) that gravity amplitudes are in some sense just the square of the corresponding YM amplitude. However I may be miss-remembering and it is just the integrands that are related this way. Is the single-valued polylogarithms substitution idea just a new way of the looking at the older G=YM^2 relation?
Thank for your blog.
JR
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Indeed, the G=YM^2 relation that Bern and Dixon have worked on just applies to integrands so far. That said, the two ideas are still related, because the best-understood version of this single-valued story for string theory is at tree level, where both it and G=YM^2 can both be understood as consequences of older relations that were derived from string theory.
There is also a version of the single-valued story for one-loop string amplitudes, but unlike at tree level it isn’t yet in a rigorous mathematical form, and it’s still unclear how it might be related to G=YM^2.
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