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.