### Yang-Mills Yumyums

The gallant kneemo gave me a link to some great slides by Zvi Bern who works on perturbative quantum gravity. Before twistor strings came along he was thinking about the KLT relations between gravity amplitudes and colour free diagrams, such as MHV tree level diagrams for n gluons.

These amplitudes are surprisingly simple, and apparently people don't really understand why! For example, the MHV tree amplitude for 4 gluons in QCD looks like A4 = (k1 + k2)^2 / (k2 + k3)^2.

It does make one wonder about the modelling of Witten's gluon spaces by projective twistor geometry in the context of M theory. Remember that there are three complex moduli of real dimension six, namely M(0,6), M(1,4) and M(2,0). We already know the first one is interesting because it has an orbifold Euler characteristic of minus six. What if we needed to draw little loops on representative Riemann surfaces? There is a nice mathematical way to think about this, but just imagine cutting up the two holed surface from end to end, straight through the two holes and at 90 deg to the correct way to cut a bagel. That cut marks six points on the two holed surface. It turns out that special loops on this surface map to ones on the M(0,6) by taking the six points to the six punctures. Zvi Bern would say that the graviton polarisation tensor is written as a square of gluon polarisation vectors.

Of course one can consider any number of punctures on the sphere to get tree amplitudes for n gluons. All one needs to know is that the compactified real moduli are all tiled by 1-operad Stasheff associahedra, as shown in the work of Devadoss. For example, the three dimensional case of six points is tiled by the 3D 14 vertex polytope.

From both a physical and mathematical point of view, we would like to better understand Yang-Mills theory in 4D. To quote Jaffe and Witten: we would like to prove that for any compact simple gauge group G, a non-trivial quantum Yangâ€“Mills theory exists on R4 and has a mass gap d > 0. One of the ATLAS people recently said:

These amplitudes are surprisingly simple, and apparently people don't really understand why! For example, the MHV tree amplitude for 4 gluons in QCD looks like A4 = (k1 + k2)^2 / (k2 + k3)^2.

It does make one wonder about the modelling of Witten's gluon spaces by projective twistor geometry in the context of M theory. Remember that there are three complex moduli of real dimension six, namely M(0,6), M(1,4) and M(2,0). We already know the first one is interesting because it has an orbifold Euler characteristic of minus six. What if we needed to draw little loops on representative Riemann surfaces? There is a nice mathematical way to think about this, but just imagine cutting up the two holed surface from end to end, straight through the two holes and at 90 deg to the correct way to cut a bagel. That cut marks six points on the two holed surface. It turns out that special loops on this surface map to ones on the M(0,6) by taking the six points to the six punctures. Zvi Bern would say that the graviton polarisation tensor is written as a square of gluon polarisation vectors.

Of course one can consider any number of punctures on the sphere to get tree amplitudes for n gluons. All one needs to know is that the compactified real moduli are all tiled by 1-operad Stasheff associahedra, as shown in the work of Devadoss. For example, the three dimensional case of six points is tiled by the 3D 14 vertex polytope.

From both a physical and mathematical point of view, we would like to better understand Yang-Mills theory in 4D. To quote Jaffe and Witten: we would like to prove that for any compact simple gauge group G, a non-trivial quantum Yangâ€“Mills theory exists on R4 and has a mass gap d > 0. One of the ATLAS people recently said:

*our field must get some serious profit from LHC start-up and first data, and we better teach ourselves right now how to explain Higgs, SUSY and extra dimensions to the public and the media.*Oops. This statement needs a little revision.
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