### M Theory Lesson 4

Last time we looked at the MHV diagram technique in twistor String theory. Tree amplitudes in N=4 SUSY Yang-Mills might as well be thought of as QCD amplitudes. The real advantage of the MHV technique is that lower loop terms feed into the structure of higher loop ones, so the recursion is highly constructive. Where might this come from? I confess now that we have not been talking about operads without a purpose in mind.

But one thing at a time. The brilliant young Mahndisa has wisely shown enthusiasm for the work of Satyan Devadoss. Devadoss has spent a lot of time focusing on the moduli of punctured spheres M(0,n), or rather the real points of the compactified space. These spaces are tiled by associahedra. Those are the lovely polytopes of Stasheff that we have met a number of times before. So, the simplest case of real moduli for punctured spheres (which look a bit like trees, right?) can be described by a 1-operad.

What I might have neglected to mention before is that Brown has recently studied MZVs and integrals for such moduli, using Motivic Cohomology. In particular, any integral which we would like to associate with physical amplitudes is given in terms of MZVs. M-theory is so much fun, don't you think? Today's homework is to look through the Brown paper and find the Veneziano four point function.

But one thing at a time. The brilliant young Mahndisa has wisely shown enthusiasm for the work of Satyan Devadoss. Devadoss has spent a lot of time focusing on the moduli of punctured spheres M(0,n), or rather the real points of the compactified space. These spaces are tiled by associahedra. Those are the lovely polytopes of Stasheff that we have met a number of times before. So, the simplest case of real moduli for punctured spheres (which look a bit like trees, right?) can be described by a 1-operad.

What I might have neglected to mention before is that Brown has recently studied MZVs and integrals for such moduli, using Motivic Cohomology. In particular, any integral which we would like to associate with physical amplitudes is given in terms of MZVs. M-theory is so much fun, don't you think? Today's homework is to look through the Brown paper and find the Veneziano four point function.

## 3 Comments:

Kea, you know how to give a good workout. It has delayed my meal by an hour, but equation (7.31) on p. 96 is equivalent to the Veneziano four-point function. You may correct me, for I am new at some of this.

I appreciate your mathematical prowess more and more. It makes one pay more attention to M-theory.

Hi Louise

Goodness! Wow. I hope I'm not delaying too many meals. You know, I'm not much of a mathematician myself. Many of these difficult theorems are over my head. But we just need to know how to calculate things, heh?

11 23 06

Hello Kea:

Great post. Now I am up at 5:19AM, but it is worth it because I am learning something. I need to read a bit more before I can comment. Thanks:) And yes, many, many thanks for posting a link to that Devadoss paper a while ago; it was realllllllllly cool!

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