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.

Strings 2013: Calabi-Yaus and Moonshine

 

Over at Strings 2013, Shamit Kachru gave a nice talk on Calabi-Yau compactifications and Mathieu Moonshine.  The talk is based on the recent paper arXiv:1306.4981 [hep-th]

Frenkel on Langlands

For those interested in the geometric Langlands and S-duality, Frenkel gave a series of lectures while visiting Columbia University last year.  His lecture style is wonderfully clear and masterful, reminding me of talks by Gukov and Kapustin over at Caltech.  Enjoy.  (Spring 2012 Eilenberg Lectures)

Andrew Wiles and M-theory

Today Andrew Wiles turns 60 (Born: April 11th 1953) and Lubos Motl at TRF has posted a very nice BBC documentary on the solution of Fermat's last theorem.


The video features the chain of conjectures, that when proven, leads to the proof Fermat's last theorem.  The key component of this reasoning involved the proof of the Taniyama-Shimura conjecture, which states that every elliptic curve is really a modular form in disguise.  The problem is doing the proper counting on each side to solidify the correspondence.  The key insight is to count Galois representations, which can be associated to the elliptic curves.  Wiles accomplished this after seven years of solitary work and a last repair of his initial proof.

The Taniyama-Shimura conjecture is, in a sense, a two-dimensional special case of the more general Langlands correspondence.  In the string theory context, Taniyama-Shimura is to the worldsheet, as the more general Langlands is to worldvolumes.  Further work on the Langlands should shed some light on M-theory, and vice versa, as string theory/M-theory so far has shown quantum gravity prefers to work in certain, special dimensions.  To fully understand this, one must go on a journey from number theory to geometry, and back to number theory.  So far, Witten and Kapustin have suggested that Langlands duality and S-duality are related, so it is tempting to conjecture further about the role of U-duality in the general Langlands correspondence.

AMS and Dark Matter

Today the Alpha Magnetic Spectrometer (AMS) Collaboration announced the publication of its first physics result in Physical Review Letters. The AMS Experiment is the most powerful and sensitive particle physics spectrometer ever deployed in space.  In the initial 18 month period of space operations, from May 19, 2011 to December 10, 2012, AMS analyzed 25 billion primary cosmic ray events.  Of these, 6.8 million, were unambiguously identified as electrons and their antimatter counterpart, positrons.  The positron to electron ratio shows no anisotropy indicating the energetic positrons are not coming from a preferred direction in space, but instead a common source, in support of new physics, e.g. dark matter.

AMS has measured the positron fraction (ratio of the positron flux to the combined flux of positrons and electrons) in the energy range 0.5 to 350 GeV  At energies greater than 250 GeV, the spectrum appears to flatten but to study the behavior above 250 GeV requires more statistics – the data reported represents ~10% of the total expected.

String Holes

Gabriele Veneziano has updated his paper on string holes, where he finds that information is not lost for such stringy black holes. He finds, like fuzzball states, the concept of an information free horizon does not make sense.

Giant Graviton Geometry

David Berenstein has written a fresh new paper on the geometry of giant gravitons.  Essentially, he begins by studying N=4 SYM and a Z/2 orbifold, building giant graviton states with their collective coordinates, attaching strings to them and computing the energies of the strings as functions of the collective coordinates.  At the end of the computations it is found that the contribution to the mass of the strings stretched between the giants (at one loop order in the field theory) is a distance squared in the collective coordinate geometry.

The procedure makes contact with the D-brane description, where one can associate the masses to the Higgs mechanism associated to breaking an enhanced gauge symmetry of coincident D-branes down to a diagonal group when the branes are separated.  This Higgsed gauge symmetry is the emergent gauge symmetry, realized in terms of collective coordinates of D-branes.

Motivic Mirror Symmetry

Rolf Schimmrigk's new paper on the arxiv demonstrates that the Omega-motives of exactly solvable mirror pairs of Calabi-Yau threefolds are isomorphic, as expected by mirror symmetry since the L-functions of the Omega-motives of such pairs are predicted to be identical.

String theory suggests a relation between two-dimensional conformal field theory and the geometry of Calabi-Yau varieties.  This comes in the form of relating automorphic forms derived from pure or mixed motives that arise in Calabi-Yau varieties to modular forms that come from Kac-Moody algebras on the worldsheet.  Such relations allow an arithmetic link between the spacetime and worldsheet theory, giving a means to pass from the extra-dimensional geometry to the worldsheet and back.

In the framework of M-theory, it would be of interest to generalize the Calabi-Yau results to G2 manifolds and other higher dimensional compactification spaces.

Quantum Mechanics of Time Travel

 

Seth Lloyd gave a talk on quantum computation with closed time like curves (CTCs), Nov. 4th 2010.  He begins in a light manner with examples of the many worlds vs consistent history perspectives and moves on to considering sending qubits through projective CTCs.  The talk ends with an experiment that is post-selectively equivalent to a projective CTC circuit.

From Black Holes to Motives

Following the work of Rolf Schimmrgk, it is tempting to try to piece together the "big picture", in relating motives to black holes and emergent spacetime.  In the 10D stringy picture, one considers automorphic forms on the worldsheet and constructs the Calabi-Yau varieties induced by those automorphic forms.  One can also consider black holes with Siegal form partition function (modular form) and ask if the form is associated to (motives of) a compactified geometry.  This is the essence of the stringy extension of the geometric Langland's program. 

Hence, the "big question", in quantum gravity, is can one always re-construct spacetime geometries from number theory information?  More examples are needed, but there are tantalizing hints coming from supergravity and topological string theory.