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.

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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?

Amplituhedron

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.

A 14-dimensional Theory?

With M-theory, F-theory and S-theory in mind, as well as Percacci's SO(3,11) theory, it's fun to look at gradings of exceptional algebras as a starting point for speculation.  The gradings can be lifted to find interesting subgroups of the corresponding exceptional groups.

Some interesting E6 gradings are:

g = E6(6) , 
       II = E6 , 
       II1 = {a1},
       g(0) = so(5,5) + R , 
       g(-1) = M1,2(O')

 g = E6(-26) , 
       II = A2 , 
       II1 = {a1},
       g(0) = so(1,9) + R , 
       g(-1) = M1,2(O)

 g = E6C , 
       II = E6 , 
       II1 = {a1},
       g(0) = so(10)C + C , 
       g(-1) = M1,2(O)C

where we see some nice 10-dimensional space-time signatures arise.

Moving up to 5-gradings, E7 gives an interesting decomposition:

 g = E7(-25), 
        II = C3, 
        II1 = {y1},
        g(0) = so(2,10) + R, 
        dimR g(-1) = 32, 
        dimR g(-2) = 1

where we see a two-time 12-dimensional (2,10) signature arise that reminds us of S-theory constructions with real 32-dimensional Weyl spinor.  S-theory proper is 13-dimensional and one expects to see SO(9,1)xSO(2,1) as well, however.  Alas, E7 also admits the grading:

g = E7(-25), 
        II = E7, 
        II1 = {y2},
        g(0) = so(1,9) + sl(2,R) + R, 
        dimR g(-1) = 32, 
        dimR g(-2) = 10

where the real 32-dimensional Weyl spinor is still present and SO(9,1)xSO(2,1) can be recovered from SO(9,1) and SL(2,R)~Spin(2,1).

Can we go further and see any signs of a higher dimensional theory from gradings of E8?  It appears to be the case:

g = E8(-24), 
        II = F4, 
        II1 = {y4},
        g(0) = so(3,11) + R, 
        dimR g(-1) = 64, 
        dimR g(-2) = 14.

g = E8(8), 
        II = E8, 
        II1 = {y1},
        g(0) = so(7,7) + R, 
        dimR g(-1) = 64, 
        dimR g(-2) = 14

g = E8C, 
        II = E8, 
        II1 = {y1},
        g(0) = so(14,C) + C, 
        dimC g(-1) = 64, 
        dimC g(-2) = 14

There seems to be hints of a 14-dimensional theory lurking in the gradings with a 64-dimensional real or complex spinor.  Viewed from the viewpoint of octonions, E8(8) corresponds to a split-octonion (would be) theory with 14-dimensional (7,7) signature.  E8(-24) arises from the octonions, where the 14-dimensional (3,11) three-time signature is recovered.  The complexified octonion construction gives the E8(C) case with 14-dimensional SO(14,C) symmetry and complex 64-dimensional spinor.

So are the exceptional groups hinting at various higher dimensional theories that go beyond M-theory?  The 5-gradings of E7 and E8 (non-compact and compact) seem to support that hypothesis.

The Monster and the Leech

 

The Monster group is one of those jewels of finite group theory with still mysterious connections to disparate parts of the wide world of mathematics.  The Monster was constructed by Robert Griess as the symmetry group of an algebra structure (Griess algebra) in 196,884 dimensions. His work split the space into three subspaces, and his main task was to show there were symmetries intermingling these subspaces. The dimensions of the subspaces are:         

98,304 + 300 + 98,280 = 196,884

The first number 98,304 = 2¹² × 24 comes from the Golay code in 24 dimensions.

 Geometrically: The 398,034,000 vectors of norm 8 in the Leech fall into 8,292,375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their inverses, and thus describe the vertices of a 24-dimensional orthoplex. Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the Golay code, namely 2¹² × |M₂₄|=4,096×24. 

Remark: The full automorphism group Co_0 of the Leech lattice has order 8,292,375 × 4,096 × 244,823,040. In the octonionic construction of the Leech, Co_0 is generated by F4 transformations.

The second number 300 = 24 + 23 + 22 + … + 3 + 2 + 1 is the dimension of the space of 24-by-24 

 symmetric matrices. 

The third number 96,280 = 196,560 / 2 comes from the Leech Lattice in 24 dimensions, where there are 196,560 vertices closest to a given vertex, forming 98,280 diametrically opposite pairs.

Geometrically: The 196,560 (norm four) vectors that span the Leech lattice are formed by three shells:

1.  3x240=720                      (2lambda, 0, 0)
2.  3x240x16=11,520           (lambda s*, +/-(lambda s*)j, 0)
3.  3x240x16x16=184,320  ((lambda s)j,± lambda k,±( lambda j)k)

So geometrically, the splitting of 196,884 isn't so mysterious.  98,304 comes from symmetry of norm 8 vector crosses that can be taken as the coordinate system of the Leech lattice.  The third number also isn't so mysterious either as it's just half of the norm four vectors that span the Leech lattice.

The j-function is given by:

j(τ)=q −1+744+196,884q+21,493,760q 2+…

where q=exp(i2πτ), and famously 196,883+1=196,884, which is the dimension of the Griess algebra, which can mostly be understood from Leech vectors. Thompson (1979) noted the rest of the coefficients are obtained from the dimensions of Monster’s irreducible representations.

The automorphism group of the Leech 2.Co_1=Co_0 fits inside the monster as 2^24 Co_0, being the centralizer of an involution. Since Co_0 is generated by F4 transformations, the monster has F4 doing work on the 196,884 dimensional space.  In fact, 2¹² × M₂₄ is a maximal 2-local subgroup of Co_0 which can still be expressed in terms of F4 transformations.

F4 performs rotations in BPS black hole charge spaces and its pretty easy to show Co_0 is generated by such U-duality transformations.  Geometrically, the whole Leech lattice itself can be seen as a lattice of BPS black holes by embedding it in the exceptional Jordan algebra.  So the Monster can be partly interpreted as describing Co_0 symmetries of BPS black holes in the Leech lattice that live in a 24+3=27 dimensional charge space, the exceptional Jordan algebra, which has F4 as its automorphism group.

Another approach via Witten, in a pure 3D gravity picture, 196,884 states are Virasoro descendants of the vacuum, where 196883 microstates of the black hole are primaries.  A minimal black hole in this pure three-dimensional gravity transforms as the minimal representation of the monster group.