Tuesday, December 09, 2014

The Father of Motives

With the passing of Alexander Grothendieck, there is much discussion comparing him to Einstein in the mathematics world.  I think it would be more fitting to compare him to Riemann, whose differential geometry made General Relativity possible.  Of course, this is meant in a future tense as it is extremely likely Grothendieck's Motives will play a central role in the nonperturbative completion of M-theory.

Thursday, November 27, 2014

Matrix theory Revisited

Progress has been made on matrix descriptions of the lattices for E10, E11 and K27.  It's a great time to return to Chern-Simons like matrix models with cubic actions.

Saturday, September 20, 2014

Wednesday, July 02, 2014

Unified Theory of Mathematics


What do the Langland's program and Monstrous Moonshine have in common?  The answer lies in M-theory.  Stay tuned.

Friday, April 25, 2014

The Higgs Vacuum is Unstable

 arXiv:1404.4709 [hep-ph]
So far, the experiments at the Large Hadron Collider (LHC) have shown no sign of new physics beyond the Standard Model. Assuming the Standard Model is correct at presently available energies, we can accurately extrapolate the theory to higher energies in order to verify its validity. Here we report the results of new high precision calculations which show that absolute stability of the Higgs vacuum state is now excluded. Combining these new results with the recent observation of primordial gravitational waves by the BICEP Collaboration, we find that the Higgs vacuum state would have quickly decayed during cosmic inflation, leading to a catastrophic collapse of the universe into a black hole. Thus, we are driven to the conclusion that there must be some new physics beyond the Standard Model at energies below the instability scale ΛI109 GeV, which is responsible for the stabilisation of the Higgs vacuum.

Friday, December 13, 2013

Motivic Amplitudes and Generalized Associahedra

With the ongoing amplituhedra hype, it's worthwhile to re-visit motivic amplitudes and associahedra once again.  Motivic amplitudes were created because of the plethora of functional identities amongst generalized polylogarithms that precludes the existence of any particular preferred or canonical functional representation (i.e. formula) for general multi-loop amplitudes.  Such amplitudes are elements of a Hopf algebra, acting as an algebra of functions on a motivic Galois group.  The group structure is contained in the coproduct of the Hopf algebra, enabling one to uncover the hidden Galois symmetries of the amplitudes.

The twistor amplitudes relevant for N=4 SYM can be seen as functions on the 3(n-5) dimensional configuration space Conf_n(CP^3), the space of collections of n points in the projective 3-space CP^3, i.e., projective twistor space.  Goncharov et al. assert that the cluster structure of the space Conf_n(CP^3) underlies the structure of amplitudes in N=4 SYM theory.  It is pleasing to see polygon chordings, associahedra and Stasheff polytope mentioned in the paper.  The chorded polygon approach to twistor string amplitudes was discussed on this blog back in 2011.  It all makes better sense if one recalls Witten's original twistor string paper.

 Recall Witten's diagram for the n=5 MHV amplitude, shown above.  One can view it as spheres (degree one genus zero instantons) connected by a 'twistor field' tube by lines in CP^3 that exchange the twistor field, altogether giving an object topologically equivalent to a sphere. (In the twistor field tube, the helicities are reversed between the two ends because all fields attached to either instanton are considered to be outgoing and because crossing symmetry relates an incoming gluon of one helicity to an outgoing gluon of opposite helicity.  Also, each degree one instanton must have at least two opposite helicities attached to it.)  Witten argued that n-particle MHV amplitudes with two particles of negative helicity and n-2 with positive helicity localize on such degree one genus zero curves.  This is a special case of Witten's more general conjecture that the twistor version of the n particle scattering amplitude is nonzero only if the points are supported on an algebraic curve in twistor space of degree d=q-1+l (where q is number of negative helicity particles and l is the number of loops).

Another way to view it is with a chorded polygon, where the twistor field parts are represented as chords.  As Goncharov et al. note, one can 'mutate' these polygons, and find equivalence classes of diagrams, count up the representative diagrams and it will tell you how many terms will show up in your amplitude.  For higher polygons, where internal triangles appear, one can go back to the sphere picture and see why internal triangles violate Witten's rules for exchanged twistor fields, hence their diagrams don't contribute to the amplitudes.


For the above diagram, imagine three spheres (degree one genus zero instantons) connected by three twistor field tubes that all meet together in a central location.  How can one sensibly assign helicities to the ends of the twistor field tubes?

It would be neat to go beyond Conf_n(CP^3).  To start, it is well known one can identify complex 4-space C^4 with quaternionic 2-space H^2, hence get a map from CP^3 to HP^1.  Then we would be studying Conf_n(HP^1) and transforming lines with SL(2,H)~SO(5,1).  Going a step higher and using octonionic 2-space O^2 gives a map to OP^1 and we can study Conf_n(OP^1) and mapping lines to lines with SL(2,O)~SO(9,1).  We could also study quaternionic 3-space H^3 and map to HP^2 and study Conf_n(HP^2), mapping lines with SL(3,H)~SU*(6).  For the octonions, O^3 gives OP^2 (the Cayley plane) and collineations are given by E6(-26) transformations, where collineations fixing a point in OP^2 are SO(9,1) transformations.  The study of Conf_n(OP^2) would be tantalizing indeed and there are no higher projective spaces over the octonions beyond a plane.  And what exactly would one be scattering in OP^2 anyway?

Tuesday, December 10, 2013


Indeed it was noticed years ago (especially by Marni) that associahedra alone could not capture the full structure of twistor-super Yang-Mills scattering amplitudes.  To capture all the physics, the amplituhedron had to be created.  These amplitudes tell more than just physics, however.  Breaking free from the chains of unitarity and locality takes one deep into the unification of mathematics itself.  The particles dance in projective space, while theorists dream of operads, instantons and infinite Yangian symmetry, followed by a dash of multiple zeta motivic cohomology.

Super Yang-Mills and Division Algebras

It appears Duff et al have completely elucidated the relationship between division algebras and super Yang-Mills theories in D=3,4,6,10 dimensions.  All cases can be recovered from the division algebra pair (O, O), where O is the octonion division algebra, corresponding to D=10, N=1 super Yang-Mills.